Lyapunov Analysis and Applications

Stefen Hui  Stanislaw H. Żak

June 19, 1999

Contents

Preface 1

1 Mathematical Background 3 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Matrices and Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 The Kronecker Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.6 Linear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.7 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.8 First Order Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . 24

1.8.1 The Bellman-Gronwall Lemma . . . . . . . . . . . . . . . . . . . . . 25 1.9 Linearizing Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 26 1.10 Solving the State Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.10.1 Solution of Uncontrolled System . . . . . . . . . . . . . . . . . . . . . 28 1.10.2 Solution of Controlled System . . . . . . . . . . . . . . . . . . . . . . 30

1.11 Matrix Calculus Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2 Stability 33 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 Basic Definitions of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3 Lyapunov Stability of Linear Systems . . . . . . . . . . . . . . . . . . . . . . 39 2.4 The Hurwitz and Routh Stability Criteria . . . . . . . . . . . . . . . . . . . 46 2.5 The Discrete-Time Lyapunov Equation . . . . . . . . . . . . . . . . . . . . . 51 2.6 Stability of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.7 Lyapunov Like Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.8 Nonlinear Discrete-Time System Stability . . . . . . . . . . . . . . . . . . . . 68

2.8.1 Notation and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.8.2 LaSalle’s Invariance Principle . . . . . . . . . . . . . . . . . . . . . . 69

2.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3 Dynamical System Control 73 3.1 What Is a System? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.2 Open-Loop Versus Closed-Loop . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.3 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.4 Centrifugal Governor Stability Analysis . . . . . . . . . . . . . . . . . . . . . 83 3.5 Robust Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.6 Practical Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4 Sliding Modes 105 4.1 Simple Variable Structure Systems . . . . . . . . . . . . . . . . . . . . . . . 105 4.2 Sliding Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.3 A Simple Sliding Mode Controller . . . . . . . . . . . . . . . . . . . . . . . . 114 4.4 Sliding in Multi-Input Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.5 Sliding Mode and System Zeros . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.6 Properties of Equivalent Control . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.7 Switching Surface Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.8 Bounded Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.8.1 Closed-Loop System . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.8.2 Estimating Stability Regions . . . . . . . . . . . . . . . . . . . . . . . 132

4.9 State Estimation of Uncertain Systems . . . . . . . . . . . . . . . . . . . . . 144 4.9.1 Discontinuous Estimators . . . . . . . . . . . . . . . . . . . . . . . . 144 4.9.2 Boundary Layer Estimators . . . . . . . . . . . . . . . . . . . . . . . 146

4.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5 Sliding Modes in Solving Optimization Problems 149 5.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.2 Background Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.3 Neural Network Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.3.1 Reaching Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.3.2 Convergence Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

5.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6 Neural Networks 169

6.1 Radial-Basis Function Networks . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.1.1 Interpolation Using RBF Networks . . . . . . . . . . . . . . . . . . . 171 6.1.2 Identification of a Single-Input Single-State System . . . . . . . . . . 174 6.1.3 Learning Algorithm for the RBF Identifier . . . . . . . . . . . . . . . 175 6.1.4 Growing RBF Network . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.1.5 Identification of Multivariable Systems . . . . . . . . . . . . . . . . . 179

6.2 Hopfield Neural Network Model . . . . . . . . . . . . . . . . . . . . . . . . . 184 6.3 Analog-to-Digital Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.4 Hopfield Network Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . 196

6.4.1 Hopfield Network Model Analysis . . . . . . . . . . . . . . . . . . . . 196 6.4.2 Single Neuron Stability Analysis . . . . . . . . . . . . . . . . . . . . . 199 6.4.3 Stability Analysis of the Network . . . . . . . . . . . . . . . . . . . . 202

6.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

7 Fuzzy Systems 209 7.1 Motivation and Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . 209 7.2 Standard Additive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 7.3 Fuzzy Logic Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 7.4 Stabilization Using Fuzzy Models . . . . . . . . . . . . . . . . . . . . . . . . 232

7.4.1 Fuzzy Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 7.4.2 Constructing the Design Model . . . . . . . . . . . . . . . . . . . . . 234 7.4.3 Stabilizability of Fuzzy Models . . . . . . . . . . . . . . . . . . . . . 240 7.4.4 A Lyapunov Based Stabilizer . . . . . . . . . . . . . . . . . . . . . . 245

7.5 Stability of Discrete Fuzzy Models . . . . . . . . . . . . . . . . . . . . . . . . 251 7.6 Fuzzy Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

7.6.1 The Comparison Method for Linear Systems . . . . . . . . . . . . . . 258 7.6.2 Stability Analysis of the Closed-Loop System . . . . . . . . . . . . . 259

7.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

A Models 265 A.1 Ground Vehicle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 A.2 Stick Balancer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 iv CONTENTS Bibliography 287

Preface

This book is suitable for adavnced undergraduate engineering or applied mathematics students or practicing engineers who are interested in applications of the Lyapunov’s direct method method and related techniques to the design of control systems and other applications. There are thorough treatises on the stability analysis using Lyapunov’s direct method—see, for example, Yoshizawa [129], Hahn [41], or Rouche, Habets and Laloy [97]. These books, however, are technically demanding. Our book, on the other hand, contains elementary exposition of the Lyapunov’s direct method followed by a number of applications.

2 CONTENTS

Chapter 1

Mathematical Background

The aim of this chapter is to review some of the basic mathematical techniques that will be applied in later chapters.

1.1 Notation

We start with Cantor’s definition of a set: “A set is any collection of definite, distinguishable objects of our thought, to be conceived as a whole” [73, page 1]. The objects, in the definition, are the elements or members of the set. Usually, we will use capital letters to denote sets and lower case letters to denote elements of the set. Thus, if X is a set, then we write x ∈ X to mean that x is an element of X. When an object x is not an element of a set X, we write x ∉ X. Often, we will use the curly bracket notation for sets. For example, {x : x < 7} denotes the set of all x less than 7. The colon, :, following x is read “such that”. We will be working with the following sets:

N = {1, 2, 3, …}, the set of all positive integers, Z = {0, 1, −1, 2, −2, …}, the set of all integers, ℝ, the set of all real numbers, ℂ, the set of all complex numbers.

1.2 Vectors

We define an n-dimensional column vector to be an n-tuple of numbers

[ x_{1} ; x_{2} ; ⋮ ; x_{n} ] ,  (1.1)

where n is a positive integer. Thus a vector is an ordered set of scalars. We will use lowercase bold letters, for example x, to denote such n-tuples. We call the numbers x_{1}, x_{2}, …, x_{n} the coordinates or components of the vector x. The number of components of a vector is its dimension. When referring to a particular component of a vector, we use its component, for example x_{i}, or the subscripted name of the vector, for example (x)_{i} to denote x_{i}. The set of all n-tuples, that is n-vectors, with real coefficients will be denoted ℝ^{n} and called the real n-space. The transpose of the vector x, denoted x^{T}, is an n-dimensional row vector

x^{T} = [ x_{1} ; x_{2} ; ⋮ ; x_{n} ]^{T} = [ x_{1} x_{2} ⋯ x_{n} ] .  (1.2)

The sum of two n-vectors x and y is an n-vector, denoted x + y, whose i-th component is x_{i} + y_{i}. The following rules are satisfied:

1. (x + y) + z = x + (y + z), that is, vector addition is associative.

2. x + y = y + x, which means that vector addition is commutative.

3. Let 0 = [ 0 0 ⋯ 0 ]^{T} denote the zero vector. Then,

0 + x = x + 0 = x.

4. Let x = [ x_{1} x_{2} ⋯ x_{n} ]^{T} and let −x = [ −x_{1} −x_{2} ⋯ −x_{n} ]^{T}. Then,

x + (−x) = 0.

The product of an n-vector x by a scalar c is an n-th vector, denoted cx, whose i-th component is cx_{i}.

For two vectors x and y from ℝ^{n}, we define their scalar product, also called the inner product, to be

⟨x, y⟩ = x^{T} y = x_{1}y_{1} + x_{2}y_{2} + ⋯ + x_{n}y_{n}.  (1.3)

The scalar product is commutative,

⟨x, y⟩ = ⟨y, x⟩,

and distributivity over vector addition holds,

⟨x, y + z⟩ = ⟨x, y⟩ + ⟨x, z⟩.

Two vectors x and y are said to be orthogonal if ⟨x, y⟩ = 0.

A vector norm, denoted ∥ · ∥, satisfies the following properties:

VN 1. ∥x∥ > 0 for any nonzero vector.

VN 2. For any scalar c, ∥cx∥ = |c|∥x∥.

VN 3. For any two vectors x and y, ∥x + y∥ ≤ ∥x∥ + ∥y∥—the triangle inequality.

The Euclidean norm, or the 2-norm, of a vector x ∈ ℝ^{n}, denoted ∥x∥_{2}, is defined as

∥x∥_{2} = \sqrt{⟨x, x⟩} = \sqrt{x_{1}^{2} + x_{2}^{2} + ⋯ + x_{n}^{2}}.  (1.4)

Note that ∥x∥_{2}^{2} = ⟨x, x⟩ = x^{T} x. The Euclidean norm is a special class of norms called the Hölder or p-norms defined by

∥x∥_{p} = (|x_{1}|^{p} + ⋯ + |x_{n}|^{p})^{1/p} , p ≥ 1.  (1.5)

Other often used norms that are special cases of the above class of norms are the 1-norm and the infinity norm defined respectively as

∥x∥_{1} = |x_{1}| + ⋯ + |x_{n}|  (1.6)

and

∥x∥_{∞} = max_{i} |x_{i}|.  (1.7)

Most of the time, we will be using the Euclidean norm for vectors. So for the sake of brevity, we will write ∥x∥ to mean ∥x∥_{2}.

[figure: y; x; z = cy; 0; x − z]

Figure 1.1: Computing the projection of x along y.

Theorem 1.1 (Pythagoras) If x and y are orthogonal, then

∥x + y∥^{2} = ∥x∥^{2} + ∥y∥^{2}.

Proof To prove the theorem, we use the above properties of the scalar product to obtain

∥x + y∥^{2} = (x + y)^{T} (x + y)
= ∥x∥^{2} + 2⟨x, y⟩ + ∥y∥^{2}
= ∥x∥^{2} + ∥y∥^{2},

since by assumption ⟨x, y⟩ = 0. The proof of the theorem is complete.

□

We now introduce the notion of projection. Let x, y be two vectors, where y ≠ 0. Let z = cy, c ∈ ℝ, be the vector such that the vector x − z is orthogonal to y—see Figure 1.1 for an illustration of the above construction. Thus,

(x − z)^{T} y = 0.  (1.8)

Substituting z = cy into (1.8) yields

(x − z)^{T} y = (x − cy)^{T} y = 0.  (1.9)

Hence,

c = \frac{⟨x, y⟩}{∥y∥^{2}}.  (1.10)

The number c given by (1.10) is called the component of x along y. The projection of x along y is the vector

z = cy = \frac{⟨x, y⟩}{∥y∥^{2}}y.  (1.11)

Theorem 1.2 (Cauchy-Schwarz Inequality) Let x, y ∈ ℝ^{n}. Then,

|⟨x, y⟩| ≤ ∥x∥∥y∥.  (1.12)

Proof If y = 0, then both sides of (1.12) are equal to 0, and the assertion holds. Suppose that y ≠ 0, and let c denote the component of x along y. Hence, c is given by (1.10). We then write

x = x − z + z = x − cy + cy.  (1.13)

For an illustration of (1.13), consult Figure 1.1. Applying the Pythagoras Theorem to (1.13), we obtain

∥x∥^{2} = ∥x − cy∥^{2} + ∥cy∥^{2} = ∥x − cy∥^{2} + c^{2}∥y∥^{2}.  (1.14)

Hence,

c^{2}∥y∥^{2} ≤ ∥x∥^{2}.  (1.15)

On the other hand, by (1.10),

c^{2}∥y∥^{2} = \frac{⟨x, y⟩^{2}}{∥y∥^{4}} ∥y∥^{2} = \frac{⟨x, y⟩^{2}}{∥y∥^{2}} .  (1.16)

Combining (1.15) and (1.16) yields

\frac{⟨x, y⟩^{2}}{∥y∥^{2}} ≤ ∥x∥^{2}.  (1.17)

Multiplying both sides of (1.17) by ∥y∥^{2} and taking the square root, concludes the proof. □

Theorem 1.3 (Triangle inequality) Let x, y be two vectors in ℝ^{n}. Then,

∥x + y∥ ≤ ∥x∥ + ∥y∥.  (1.18)

8 CHAPTER 1. MATHEMATICAL BACKGROUND Proof Both sides of (1.18) are nonnegative. Hence, to prove that (1.18) holds it suffices to show that the squares of the both sides satisfy the desired inequality,

(x + y)^{T} (x + y) ≤ (∥x∥ + ∥y∥)^{2} .  (1.19)

Expanding (x + y)^{T} (x + y) and applying the Schwarz inequality yields

(x + y)^{T} (x + y) = ∥x∥^{2} + 2⟨x, y⟩ + ∥y∥^{2}
≤ |x∥^{2} + 2∥x∥∥y∥ + ∥y∥^{2}
= (∥x∥ + ∥y∥)^{2} ,  (1.20)

which completes the proof of the theorem.

□

Theorem 1.4 Let x, y be two vectors in ℝ^{n}. Then, |∥x∥ − ∥y∥| ≤ ∥x − y∥.

Proof We have

∥x∥ = ∥ (x − y) + y∥ ≤ ∥x − y∥ + ∥y∥.

Hence,

∥x∥ − ∥y∥ ≤ ∥x − y∥.  (1.21)

The above implies that

∥x − y∥ = ∥y − x∥ ≥ ∥y∥ − ∥x∥ = − (∥x∥ − ∥y∥) .  (1.22)

Combining (1.21) and (1.22) yields

|∥x∥ − ∥y∥| ≤ ∥x − y∥

which completes the proof.

□

1.3 Matrices and Determinants

Let m, n be two integers such that m, n ≥ 1. An array of numbers

[ \begin{matrix}a_{11} & a_{12} & ⋯ & a_{1n} \\ a_{21} & a_{22} & ⋯ & a_{2n} \\ ⋮ & ⋮ & ... & ⋮ \\ a_{m1} & a_{m2} & ⋯ & a_{mn} \end{matrix} ]

is called a matrix. Thus a matrix is a set of scalars subject to two orderings. We say that it is an m by n matrix, or m × n matrix. This matrix has m rows and n columns. If m = n, then we say that it is a square matrix. An m × n matrix A whose elements are real numbers will be denoted as A ∈ ℝ^{m×n}. If the elements of the matrix A are complex numbers, then we will write A ∈ ℂ^{m×n}. We define the zero matrix, denoted O, to be the matrix such that a_{ij} = 0 for all i, j. We define the n × n identity matrix, denoted I_{n}, as the matrix having diagonal components all equal to 1 and all other components all equal to 0.

Given an m×n matrix A. The n×m matrix B such that b_{ji} = a_{ij} is called the transpose of A and is denoted A^{T}. Thus, the transpose of a matrix is obtained by changing rows into columns and vice versa. A matrix A which is equal to its transpose, that is, A = A^{T}, is called symmetric.

We now define the sum of two matrices. Given two matrices A and B of the same size. We define A + B to be the matrix whose elements are a_{ij} + b_{ij}, that is, we add matrices of the same size componentwise.

Let c be a number and A be a matrix. We define cA to be the matrix whose components are ca_{ij}. Thus, to obtain cA, we multiply each component of A by c.

We now define the product of two matrices. Let A be an m × n matrix and B be an n × s matrix. Let a_{i}^{T} denote the i-th row of A, and b_{j} the j-th column of B, Then, the product of AB is an m × s matrix C defined as

C = AB = [ \begin{matrix}a_{1}^{T} b_{1} & a_{1}^{T} b_{2} & ⋯ & a_{1}^{T} b_{s} \\ a_{2}^{T} b_{1} & a_{2}^{T} b_{2} & ⋯ & a_{2}^{T} b_{s} \\ ⋮ & ⋮ & ... & ⋮ \\ a_{m}^{T}b_{1} & a_{m}^{T}b_{2} & ⋯ & a_{m}^{T}b_{s} \end{matrix} ] .  (1.23)

It is not always true that AB = BA, that is, matrix multiplication is not necessarily commutative.

Let A, B, and C be matrices such that A, B can be multiplied, B, C can be multiplied, and B, C can be added. Then,

A(B + C) = AB + AC.  (1.24)

The above property is called the distributive law. If x is a number, then

A(xB) = x(AB).  (1.25)

Furthermore, matrix multiplication is associative since

(AB)C = A(BC).  (1.26)

Suppose now that A, B can be multiplied. Then,

(AB)^{T} = B^{T} A^{T} .  (1.27)

Thus the transpose of a product is the product of transposed matrices in reverse order. Let A be an n × n matrix. An inverse of A, if it exists, is an n × n matrix B such that

AB = BA = I_{n}.  (1.28)

If an inverse exists, then it is unique. A square matrix that is invertible is said to be nonsingular. We denote the inverse of A by A^{−1}. Hence, we have

A^{−1}A = AA^{−1} = I_{n}.

The transpose of an inverse is the inverse of the transpose,

(A^{−1})^{T} = (A^{T} )^{−1} ,  (1.29)

and we denote it by A^{−T}.

We now present the Sherman-Morrison-Woodbury formula for inverting a special kind of matrices.

Lemma 1.1 If A, D, and D^{−1} + CA^{−1}B are invertible, then

(A + BDC)^{−1} = A^{−1} − A^{−1}B (D^{−1} + CA^{−1}B)^{−1} CA^{−1}.

Proof We prove the Sherman-Morrison-Woodbury formula by verification. We have

(A + BDC) (A^{−1} − A^{−1}B (D^{−1} + CA^{−1}B)^{−1} CA^{−1})
= (A + BDC) A^{−1}
− (A + BDC) A^{−1}B (D^{−1} + CA^{−1}B)^{−1} CA^{−1}
= I + BDCA^{−1}
− (B + BDCA^{−1}B) (D^{−1} + CA^{−1}B)^{−1} CA^{−1}
= I + BDCA^{−1}
−BD (D^{−1} + CA^{−1}B) (D^{−1} + CA^{−1}B)^{−1} CA^{−1}
= I + BDCA^{−1} − BDCA^{−1}
= I,

as desired.

□

In the case when D = I, the Sherman-Morrison-Woodbury formula takes the form

(A + BC)^{−1} = A^{−1} − A^{−1}B (I + CA^{−1}B)^{−1} CA^{−1}.  (1.30)

Formula (1.30) is sometimes referred to as the matrix inversion formula.

A set of nonzero vectors {q_{1}, q_{2}, …, q_{n}} is said to be orthogonal if

⟨q_{i}, q_{j}⟩ = 0, for i ≠ j.

If each vector in an orthogonal set is of unity length, that is, ∥q_{i}∥ = 1 for all i = 1, 2 …, n, then the set of vectors is called orthonormal. A square matrix with orthonormal columns will be called a unitary matrix. Sometimes in the literature, a unitary matrix is called an orthogonal matrix. If Q ∈ ℝ^{n×n} is a unitary matrix, then

Q^{T} Q = QQ^{T} = I_{n}.  (1.31)

For a square, n × n, matrix A, its trace, denoted traceA, is the sum of its diagonal elements,

traceA = ∑_{i=1}^{n} a_{ii}.  (1.32)

If B is also an n × n matrix, then

trace(AB) = trace(BA).  (1.33)

The determinant of a square matrix A, denoted detA, can be evaluated using the Laplace expansion as follows. Let c_{ij} denote the cofactor corresponding to the element a_{ij} of an n×n matrix A. The cofactor c_{ij} is (−1)^{i+j} times the determinant of the (n−1)×(n−1) submatrix of A that is obtained by deleting the i-th row and j-th column of A. Then, for any fixed i, 1 ≤ i ≤ n, we have

det A = ∑_{j=1}^{n} a_{ij}c_{ij}.  (1.34)

This is the expansion of the determinant along the i-th row. In a similar fashion, we can expand the determinant along a column. The determinant has the following properties:

D 1. det A = det A^{T}

D 2. det (AB) = det A det B = det (BA)

D 3. det O = 0

D 4. det I_{n} = 1.

If A is n × n and v is a nonzero n × 1 vector such that for some scalar λ,

Av = λv,  (1.35)

then v is an eigenvector corresponding to the eigenvalue λ. In order that λ be an eigenvalue of A, it is necessary and sufficient for the matrix λI_{n}−A to be singular, that is, det (λI_{n} − A) = 0. This leads to an n-th order polynomial equation in λ of the form

det (λI_{n} − A) = λ^{n} + a_{n−1}λ^{n−1} + ⋯ + a_{1}λ + a_{0} = 0.  (1.36)

This equation, called the characteristic polynomial equation, must have n, possibly nondistinct, complex roots which are the eigenvalues of A. The spectral radius of A, denoted ρ(A) is defined as

ρ(A) = max_{1≤i≤n} |λ_{i}(A)|,

where λ_{i}(A) denotes the i-th eigenvalue of A.

Any m × n matrix A can be represented as

A = USV^{T} ,  (1.37)

where U is an m × m unitary matrix, V is an n × n unitary matrix, and S is an m × n diagonal matrix of the form

S = diag(σ_{1}, σ_{2}, …, σ_{p}), where p = min(m, n) and σ_{i} ≥ 0, i = 1, 2, …, p.

The nonnegative numbers {σ_{i}} are called the singular values of A, and (1.37) is called the singular value decomposition (SVD) of A. Following the convention, we will write

σ_{1} ≥ σ_{2} ≥ ⋯ ≥ σ_{p},

so that σ_{1} = σ_{1}(A) denotes the largest singular value of A. The singular values of A are the square roots of the eigenvalues of A^{T} A if m ≥ n or of AA^{T} if m < n. If A is symmetric, its singular values are the absolute values of its eigenvalues.

1.4 Quadratic Forms

A real quadratic form is the product x^{T} Qx, where Q is a specified n × n matrix with real coefficients and x is an n × 1 real, column, vector. Without loss of generality Q can be taken to be symmetric, that is, Q = Q^{T}. Indeed, if Q is not symmetric, we can always replace it with a symmetric matrix without changing the quadratic form values. Indeed, since z = x^{T} Qx is a scalar, hence z^{T} = z, and we have

(x^{T} Qx)^{T} = x^{T} Q^{T} x = x^{T} Qx.

Therefore, we can write

x^{T} Qx = x^{T} (\frac{1}{2}Q + \frac{1}{2}Q^{T} ) x = x^{T} (\frac{Q + Q^{T}}{2} ) x,

where \frac{1}{2} (Q + Q^{T} ) is a symmetric matrix. Thus, the quadratic form values are unchanged if Q is replaced by the symmetric matrix \frac{1}{2} (Q + Q^{T} ). From now on, in our study of quadratic forms, we assume that Q is symmetric.

A quadratic form x^{T} Qx, or equivalently, the matrix Q is called positive semi-definite if x^{T} Qx ≥ 0 for all x. It is positive definite if x^{T} Qx > 0 for all x ≠ 0. If Q is positive semi-definite, then we write Q ≥ 0, if it is positive definite, we write Q > 0. We write Q_{1} ≥ Q_{2} to mean that Q_{1} − Q_{2} ≥ 0. We say that Q is negative semi-definite if x^{T} Qx ≤ 0 for all x, it is negative definite if x^{T} Qx < 0 for all x ≠ 0. Note that Q ≤ 0 if and only if −Q ≥ 0.

We now present tests for definiteness properties of symmetric matrices. We state these tests in terms of principal minors of a given real, symmetric, n × n matrix Q with elements q_{ij}. We will also give tests for definiteness in terms of eigenvalues of Q. The principal minors of Q are det Q itself and the determinants of submatrices of Q obtained by removing successively an i-th row and i-th column. Specifically, for p = 1, 2, …, n, the principal minors are

Δ_{p} ( \begin{matrix}i_{1}, & i_{2}, & …, & i_{p} \\ i_{1}, & i_{2}, & …, & i_{p} \end{matrix} ) = det [ \begin{matrix}q_{i_{1}i_{1}} & q_{i_{1}i_{2}} & ⋯ & q_{i_{1}i_{p}} \\ q_{i_{2}i_{1}} & q_{i_{2}i_{2}} & ⋯ & q_{i_{2}i_{p}} \\ ⋮ & ⋮ & ... & ⋮ \\ q_{i_{p}i_{1}} & q_{i_{p}i_{2}} & ⋯ & q_{i_{p}i_{p}} \end{matrix} ] .

Since Q = Q^{T}, its eigenvalues are all real.

Theorem 1.5 A symmetric matrix is positive definite (or positive semi-definite) if and only if its eigenvalues are all positive (nonnegative).

If a given matrix Q ∈ ℝ^{n×n} is symmetric, then its eigenvalues are all real and there exists a unitary matrix U such that

U^{T} QU = [ \begin{matrix}λ_{1} & 0 & ⋯ & 0 \\ 0 & λ_{2} & ⋯ & 0 \\ ⋮ & & ... & ⋮ \\ 0 & 0 & ⋯ & λ_{n} \end{matrix} ] = Λ.  (1.38)

Using the above, we can easily show that a symmetric positive semi-definite matrix Q has a positive semi-definite symmetric square root, denoted Q^{1/2} or \sqrt{Q}, that satisfies

Q^{1/2}Q^{1/2} = Q.  (1.39)

Indeed, it follows from Theorem 1.5 that if Q = Q^{T} ≥ 0 then its eigenvalues are all nonnegative. Hence,

Λ^{1/2} = [ \begin{matrix}λ_{1}^{1/2} & 0 & ⋯ & 0 \\ 0 & λ_{2}^{1/2} & ⋯ & 0 \\ ⋮ & & ... & ⋮ \\ 0 & 0 & ⋯ & λ_{n}^{1/2} \end{matrix} ]  (1.40)

is well defined. Using the fact that U^{T} U = I_{n}, we can define

Q^{1/2} = UΛ^{1/2}U^{T} ,  (1.41)

which has the desired properties.

If Q = Q^{T} is positive semi-definite, and not positive definite, then some of its eigenvalues are equal to zero. Suppose that Q has r nonzero eigenvalues. Then, there is a unitary matrix, say U, such that

U^{T} QU = [ \begin{matrix}λ_{1} & 0 & ⋯ & 0 & 0 & ⋯ & 0 \\ 0 & λ_{2} & ⋯ & 0 & 0 & ⋯ & 0 \\ ⋮ & & ... & ⋮ & & ... & ⋮ \\ 0 & 0 & ⋯ & λ_{r} & 0 & ⋯ & 0 \\ 0 & 0 & ⋯ & 0 & 0 & ⋯ & 0 \\ ⋮ & & & & & ... & ⋮ \\ 0 & 0 & ⋯ & 0 & 0 & ⋯ & 0 \end{matrix} ] .  (1.42)

Let V ∈ ℝ^{n×r} be a matrix that consists of the first r columns of the matrix U, and let

C = [ \begin{matrix}λ_{1}^{1/2} & 0 & ⋯ & 0 \\ 0 & λ_{2}^{1/2} & ⋯ & 0 \\ ⋮ & & ... & ⋮ \\ 0 & 0 & ⋯ & λ_{r}^{1/2} \end{matrix} ] V^{T} .  (1.43)

We obtain a full rank factorization of Q,

Q = C^{T} C.  (1.44)

We can generalize the above, using (1.37), to obtain a full rank factorization of a non-square matrix. Specifically, given an m × n matrix A of rank r, then we can represent it as

A = BC,

where B is an m × r matrix and C is a r × n matrix such that

rank(B) = rank(C) = rank(A) = r.

1.5 The Kronecker Product

Let A be an m × n and B be a p × q matrices. The Kronecker product of A and B is denoted A ⊗ B and is a mp × nq matrix defined by

A ⊗ B = [ \begin{matrix}a_{11}B & a_{12}B & ⋯ & a_{1n}B \\ a_{21}B & a_{22}B & ⋯ & a_{2n}B \\ ⋮ & ⋮ & ... & ⋮ \\ a_{m1}B & a_{m2}B & ⋯ & a_{mn}B \end{matrix} ] .  (1.45)

Thus, the matrix A⊗B consists of mn blocks. Using the definition of the Kronecker product, we can verify the following two properties:

(A ⊗ B) (C ⊗ D) = AC ⊗ BD,  (1.46)
(A ⊗ B)^{T} = A^{T} ⊗ B^{T} .  (1.47)

Let

X = [ x_{1}, x_{2}, …, x_{m} ]

be a n × m matrix, where x_{i}, i = 1, 2, …, m are the columns of X. Then, the stacking operator is defined as

s(X) = [ x_{1}^{T} , x_{2}^{T} , …, x_{m}^{T} ]^{T}
= [ x_{11} x_{21} ⋯ x_{n1} x_{12} x_{22} ⋯ x_{1m} ⋯ x_{1n} ⋯ x_{nm} ]^{T} (1.48)

is the column nm-vector formed from the columns of X taken in order. Let now A be an n × n, and C and X be n × m matrices. Then, the matrix equation

AX = C  (1.49)

can be written as

(I_{m} ⊗ A) s(X) = s(C).  (1.50)

Let B be an m × m matrix, then the matrix equation

XB = C  (1.51)

can be written as

(B^{T} ⊗ I_{n}) s(X) = s(C).  (1.52)

Using the above two facts, we can verify that the matrix equation

AX + XB = C,  (1.53)

where A is n × n, B is m × m, and C is n × m, can be written as

(I_{m} ⊗ A + B^{T} ⊗ I_{n}) s(X) = s(C).  (1.54)

Let λ_{i}, w_{i} be the eigenvalues and eigenvectors, respectively of A, and μ_{j} and v_{j} the eigenvalues and eigenvectors of m × m matrix B. Then,

(A ⊗ B) (w_{i} ⊗ v_{j}) = Aw_{i} ⊗ Bv_{j}
= λ_{i}w_{i} ⊗ μ_{j}v_{j}
= λ_{i}μ_{j} (w_{i} ⊗ v_{j}) .  (1.55)

Thus, the eigenvalues of A ⊗ B are λ_{i}μ_{j}, and their respective eigenvectors are w_{i} ⊗ v_{j} for i = 1, 2, …, n, j = 1, 2, …, m.

We use the above result to study the matrix equation given by (1.53). We represent (1.53) as

D s(X) = s(C),  (1.56)

where

D = I_{m} ⊗ A + B^{T} ⊗ I_{n}.  (1.57)

The solution to (1.56) is unique if, and only if, the mn × mn matrix D is nonsingular. To find the condition for this to hold, consider the following matrix

(I_{m} + εB^{T} ) ⊗ (I_{n} + εA) = I_{m} ⊗ I_{n} + εD + ε^{2}B^{T} ⊗ A  (1.58)

whose eigenvalues are

(1 + εμ_{j}) (1 + ελ_{i}) = 1 + ε (μ_{j} + λ_{i}) + ε^{2}μ_{j}λ_{i}  (1.59)

because

λ_{i} (I_{n} + A) = 1 + λ_{i} (A) .

Comparing terms in ε, we conclude that the eigenvalues of D are λ_{i} + μ_{j}, i = 1, 2, …, n, j = 1, 2, …, m. Hence, D is nonsingular if and only if

λ_{i} + μ_{j} ≠ 0.  (1.60)

The above is the necessary and sufficient condition for the solution X of the matrix equation AX + XB = C to be unique.

Another useful identity is

s(ABC) = (C^{T} ⊗ A) s(B)  (1.61)

For further information on the subject of the Kronecker product, we refer the reader to Brewer [14].

1.6 Linear Maps

A function

F : ℝ^{n} → ℝ^{m}

is said to be linear if it satisfies the following two properties:

LM 1. For any x, y in ℝ^{n}, we have

F (x + y) = F (x) + F (y),

LM 2. If c is a number, then

F (c x) = cF (x).

Linear functions are sometimes referred to as linear maps or linear operators.

Linear functions from ℝ^{n} into ℝ^{m} can be identified with the m × n real matrices by

F : ℝ^{n} → ℝ^{m} is linear if and only if F (x) = Ax,

where A ∈ ℝ^{m×n}.

If F : ℝ^{n} → ℝ^{m} is linear, then there exists a constant L ≥ 0 such that

∥F (x)∥ ≤ L∥x∥  (1.62)

for all x ∈ ℝ^{n}. The smallest such constant L for which this is true is denoted by ∥F ∥ and is called the norm of the linear map F. If F is the linear map induced by the matrix A, then we define ∥A∥ = ∥F ∥. For any matrix A and vector norm ∥ · ∥, we can compute the matrix norm using the formula

∥A∥ = sup_{x≠0} \frac{∥Ax∥}{∥x∥}.  (1.63)

The matrix norm, that is, the norm of the linear map F given by (1.63) is equivalent to

∥A∥ = max_{∥x∥=1} ∥Ax∥  (1.64)

A matrix norm produced using (1.63) or (1.64) is called the induced norm. Note that

∥Ax − Ay∥ = ∥A(x − y)∥ ≤ ∥A∥∥x − y∥,

that is, the linear map, F, represented by the matrix A is uniformly continuous.

We will now comment on the notation used in (1.63) and (1.64). First, the same symbol is used for the matrix norm as for the norm of a vector. Second, the norms appearing on the right side of (1.63) and (1.64) are the norms of the vectors x ∈ ℝ^{n}, and Ax ∈ ℝ^{m}. Thus, if ℝ^{n} is equipped with the vector norm ∥ · ∥_{s} while ℝ^{m} is equipped with the vector norm ∥ · ∥_{t}, then (1.64) should be interpreted as

∥A∥ = max_{∥x∥_{s}=1} ∥Ax∥_{t}.

1.6. LINEAR MAPS 19 Vector norms and a matrix norm are said to be compatible or consistent if they satisfy,

∥Ax∥_{t} ≤ ∥A∥∥x∥_{s}.  (1.65)

We will now show that (1.64) is well defined. For this, we need the following lemma. Lemma 1.2 A vector norm ∥ · ∥ : ℝ^{p} → ℝ is a uniformly continuous function. Proof By Theorem 1.4, on page 8,

|∥x∥ − ∥y∥| ≤ ∥x − y∥.

Hence, for any ε > 0, we can take δ = ε such that for any x, y ∈ ℝ^{p}, if ∥x − y∥ < δ then |∥x − y∥| < ε, as was to be shown.

□

By the continuity of the vector norm and the Theorem of Weierstrass, a vector x_{0} can be found such that ∥x_{0}∥ = 1 and ∥A∥ = ∥Ax_{0}∥ = max_{∥x∥=1} ∥Ax∥. Thus, the expression (1.64) is well defined.

The matrix norm satisfies the following conditions: MN 1. ∥A∥ > 0 if A ≠ O and ∥O∥ = 0. MN 2. ∥cA∥ = |c|∥A∥, where c is a constant. MN 3. ∥A + B∥ ≤ ∥A∥ + ∥B∥. MN 4. ∥AB∥ ≤ ∥A∥∥B∥.

The most frequently used matrix norms are the p-norms

∥A∥_{p} = max_{∥x∥_{p}=1} ∥Ax∥_{p}.

The induced matrix norms corresponding to the one-, two-, and infinity-norms for vectors are:

• ∥A∥_{1} = max_{j} ∥a_{j}∥_{1}, that is, ∥A∥_{1} is the maximum of absolute column sum;

• ∥A∥_{2} = σ_{1}(A), which is the largest singular value of A;

• ∥A∥_{∞} = max_{i} ∥a_{i}^{T} ∥_{1}, which is the maximum of absolute row sum.

A matrix norm that is not induced by a vector norm is the Frobenius norm, defined for an m × n matrix A as

∥A∥_{F} = (∑_{i=1}^{m} ∑_{j=1}^{n} a_{ij}^{2})^{1/2} .  (1.66)

The Frobenius norm satisfies the relation

∥A∥_{F}^{2} = trace (A^{T} A) .  (1.67)

The Frobenius norm is compatible with the Euclidean vector norm, that is,

∥Ax∥_{2} ≤ ∥A∥_{F} ∥x∥_{2}.  (1.68)

1.7 Vector Spaces

A given set of vectors of the same dimension {x_{1}, x_{2}, …, x_{n}} is linearly independent if the zero vector, 0, can be written as a non-trivial linear combination of the given vectors,

α_{1}x_{1} + α_{2}x_{2} + ⋯ + α_{n}x_{n} = 0, with α_{i} ≠ 0 for at least one i.

In other words, the set of vectors {x_{1}, x_{2}, …, x_{n}} is linearly independent if

α_{1}x_{1} + α_{2}x_{2} + ⋯ + α_{n}x_{n} = 0, implies α_{1} = α_{2} = ⋯ = α_{n} = 0.

A set S containing vectors of dimension n is a subspace of ℝ^{n} if for any scalars α and β,

x, y ∈ S implies αx + βy ∈ S.  (1.69)

It follows from the above that a subspace must contain the zero vector, 0. The set containing only the zero vector is a subspace. This set, {0}, is called a trivial subspace.

A matrix-vector product

Ax = [ a_{1} a_{2} ⋯ a_{n} ] [ x_{1} ; x_{2} ; ⋮ ; x_{n} ] = x_{1}a_{1} + x_{2}a_{2} + ⋯ + x_{n}a_{n}  (1.70)

is a linear combination of the columns of the matrix. Thus, linear dependence of the columns of A is equivalent to the condition

Az = 0 for some vector z ≠ 0.  (1.71)

Linear independence of the columns of A is equivalent to the condition

Az = 0 implies z = 0.  (1.72)

The set of all vectors that are linear combinations of the columns of A is called the range, or image, of A and will be denoted range(A). Thus,

range(A) = {v : v = Ax for some x}.  (1.73)

For a given m × n matrix A and an m-vector b, we have b ∈ range(A) if and only if there exists an n-vector x such that Ax = b. We say that a system of linear equations represented by Ax = b is compatible if b ∈ range(A).

The range of A is a subspace of ℝ^{m}. Indeed, for any b_{1}, b_{2} ∈ range(A), we have b_{1} = Ax_{1} and b_{2} = Ax_{2} for some x_{1} and x_{2}. Using the fact that A represents a linear map, for any scalars α_{1} and α_{2},

α_{1}b_{1} + α_{2}b_{2} = α_{1}Ax_{1} + α_{2}Ax_{2} = A (α_{1}x_{1} + α_{2}x_{2}) .  (1.74)

Hence, if b_{1}, b_{2} ∈ range(A), then α_{1}b_{1} + α_{2}b_{2} ∈ range(A), which means that the range of A is a subspace of ℝ^{m}.

Similarly, we define the range of A^{T} to be the set

range(A^{T}) = {y : y = A^{T} z for some z} .  (1.75)

For a non-trivial subspace S there is a unique smallest positive integer called the dimension of the subspace S and will be denoted dim S. The dimension of the subspace is equal to the smallest number of linearly independent vectors of the subspace such that every vector of the subspace can be expressed as a linear combination of these vectors. Any such set of vectors is called a basis of the subspace. The vectors in a basis are not unique.

The number of linearly independent columns of a matrix A is called the rank of A and will be denoted rank(A). The rank of A is also equal to the number of linearly independent rows of A. For the product of a m × n matrix A and a n × p matrix B, we have Sylvester’s inequalities:

rank(A) + rank(B) − n ≤ rank(AB) ≤ min{rank(A), rank(B)}.  (1.76)

For an m × n matrix A, the set of all n-vectors orthogonal to the rows of A is called the kernel, or null space, of A and will be denoted ker(A). Hence,

ker(A) = {x : Ax = 0} .  (1.77)

The kernel of A is a subspace of ℝ^{n}. Indeed, if x_{1}, x_{2} ∈ ker(A), then Ax_{1} = 0 and Ax_{2} = 0. Hence, for any scalars α_{1} and α_{2}, we have

A (α_{1}x_{1} + α_{2}x_{2}) = α_{1}Ax_{1} + α_{2}Ax_{2} = 0  (1.78)

because A represents a linear map. Thus, if x_{1}, x_{2} ∈ ker(A), then α_{1}x_{1} + α_{2}x_{2} ∈ ker(A), which means that ker(A) is a subspace of ℝ^{n}. Similarly, we define the kernel of A^{T} to be the set

ker (A^{T} ) = {w : A^{T} w = 0} .  (1.79)

The intersection of range(A) and ker (A^{T} ) contains only the zero vector, that is,

range(A) ∩ ker (A^{T} ) = {0}.  (1.80)

For a given subspace S of ℝ^{n}, the orthogonal complement S^{⊥} of S is defined by

S^{⊥} = {x ∈ ℝ^{n} : x^{T} y = 0 for all y ∈ S} .  (1.81)

The vectors in ker (A^{T} ) are orthogonal to the vectors in range(A), and vice versa. We say that range(A) and ker (A^{T} ) are orthogonal complements of one another and write

range(A)^{⊥} = ker (A^{T} ) ,  (1.82)

and

ker (A^{T} )^{⊥} = range(A).  (1.83)

Similarly,

range (A^{T} )^{⊥} = ker (A) ,  (1.84)

and

ker (A)^{⊥} = range (A^{T} ) .  (1.85)

If A is an m × n matrix with linearly independent columns, which implies that m ≥ n, then

rank(A) = n = dim range(A),

and

dim ker (A^{T} ) = m − n.

We now define a normed space. The concept of a norm in a vector space is a generalization of the length of a vector in ℝ^{3}.

Definition 1.1 A vector space with a norm is called a normed space.

By the normed vector space ℝ^{n}, we usually mean the Euclidean space, which is ℝ^{n} with the norm ∥x∥_{2} = \sqrt{x_{1}^{2} + ⋯ + x_{n}^{2}}. The vector space ℝ^{n} equipped with any of the p-norm forms a normed vector space. The absolute value is a norm in ℝ.

Let Ω be a compact subset of ℝ^{n} and let C(Ω) denote the set of continuous functions f : ℝ^{n} → ℝ on Ω. Thus, we view functions as vectors in the function space C(Ω). Then, the function ∥ · ∥ : C(Ω) → ℝ defined as

∥f∥ = max_{x∈Ω} |f(x)|,  (1.86)

satisfies the norm axioms. The function space C(Ω), with scalars from ℝ and with the norm (1.86) is a linear normed space. In particular, the space of all real-valued functions, f(x), continuous on some interval [a, b] of the real line with the norm defined by

∥f∥ = max_{x∈[a,b]} |f(x)|,  (1.87)

is an example of a linear normed space. This space is denoted C([a, b]).

We will also be interested in sequences of elements of C([a, b]). Let f_{1}, f_{2}, … be a sequence of elements of C([a, b]). Denote this sequence by {f_{k}}. In the following, f_{k}(x), means the value of the function f_{k} at x, while f_{k} means the function itself. We say that the sequence {f_{k}} converges (pointwise) on the interval [a, b] if there exists a function f such that for every x ∈ [a, b], the sequence {f_{k}(x)} converges to f(x). In other words, the sequence {f_{k}} converges (pointwise) if there exists a function f such that for a given x ∈ [a, b] and a given ε > 0, we can find N = N(x, ε) such that

|f(x) − f_{k}(x)| < ε for k > N.  (1.88)

The sequence {f_{k}} converges uniformly, on the interval [a, b], if there is a function f such that for any ε > 0 there exists N = N(ε), depending on ε but not on x, such that (1.88) holds for every x ∈ [a, b]. The distinction between convergence and uniform convergence is that in the later case, the same N can be used for any value of x to show the convergence of the sequence {f_{k}(x)}. One can show that the limit of an uniformly convergent sequence of continuous functions is continuous. It can be verified that the norm (1.87) defines the uniform convergence since {f_{k}} converges uniformly to f if and only if max_{x∈[a,b]} |f(x) − f_{k}(x)| → 0. This the reason the norm (1.87) is sometimes called the norm of uniform convergence. It can be shown that there is no norm on C([a, b]) that defines pointwise convergence to an element in C([a, b])—see Debnath and Mikusiński [23, page 12].

1.8 First Order Differential Equation

Let ℝ denote the set of real numbers and let D ⊆ ℝ^{2} be a domain, that is, an open connected nonempty subset of ℝ^{2}. Let f be a real valued function defined and continuous on D. Let ẋ = dx/dt denote the derivative of x with respect to t. We call

ẋ = f(t, x)  (1.89)

an ordinary differential equation of the first order. By a solution of the differential equation (1.89) on an open interval I = {t ∈ ℝ : t_{1} < t < t_{2}}, we mean a real valued continuously differentiable function ϕ defined on I such that (t, ϕ(t)) ∈ D for all t ∈ I and

ϕ̇(t) = f(t, ϕ(t)) for all t ∈ I.

Definition 1.2 Given (t_{0}, x_{0}) ∈ D, the initial value problem for (1.89) is

ẋ = f(t, x), x(t_{0}) = x_{0}.  (1.90)

A function ϕ is a solution of (1.90) if ϕ is a solution of the differential equation (1.89) on some interval I containing t_{0} and x(t_{0}) = x_{0}.

We can represent the initial value problem (1.90) equivalently by an integral equation,

ϕ(t) = x_{0} + ∫_{t_{0}}^{t} f(s, ϕ(s))ds.  (1.91)

To prove this equivalence, let ϕ be a solution of the initial value problem (1.90). Then x(t_{0}) = x_{0} and

ϕ̇(t) = f(t, ϕ(t)) for all t ∈ I.

Integrating the above from t_{0} to t, we obtain

∫_{t_{0}}^{t} ϕ̇(s)ds = ∫_{t_{0}}^{t} f(s, ϕ(s))ds.

Hence,

ϕ(t) − x_{0} = ∫_{t_{0}}^{t} f(s, ϕ(s))ds.

Therefore, ϕ is a solution of the integral equation (1.91).

Conversly, let ϕ be a solution of the integral equation (1.91). Then ϕ(t_{0}) = x_{0}, and differentiating both sides of (1.91) with respect to t gives ϕ̇(t) = f(t, ϕ(t)). Therefore, ϕ is also a solution of the initial value problem (1.90).

The general ordinary differential linear equation of the first order has the form

ẋ(t) = a(t)x(t) + b(t),  t ∈ (t_{1}, t_{2}),  (1.92)

where a(t) and b(t) are assumed to be continuous on the interval (t_{1}, t_{2}). We associate with equation (1.92) the initial condition

x(t_{0}) = x_{0}, x_{0} ∈ ℝ.

To find the solution to (1.92), we write it in the form

ẋ(t) − a(t)x(t) = b(t).  (1.93)

We then multiply (1.93) by the “integrating factor”

e^{− ∫_{t_{0}}^{t} a(s)ds}  (1.94)

to get

ẋ(t)e^{− ∫_{t_{0}}^{t} a(s)ds} − a(t)x(t)e^{− ∫_{t_{0}}^{t} a(s)ds} = e^{− ∫_{t_{0}}^{t} a(s)ds}b(t).  (1.95)

The above equation can be represented in an equivalent form as

\frac{d}{dt} \{x(t)e^{− ∫_{t_{0}}^{t} a(s)ds}\} = e^{− ∫_{t_{0}}^{t} a(s)ds}b(t).  (1.96)

Integrating both sides of (1.96) yields

x(t)e^{− ∫_{t_{0}}^{t} a(s)ds} − x_{0} = ∫_{t_{0}}^{t} e^{− ∫_{t_{0}}^{τ} a(s)ds}b(τ)dτ.  (1.97)

Hence,

x(t) = e^{∫_{t_{0}}^{t} a(s)ds} [x_{0} + ∫_{t_{0}}^{t} e^{− ∫_{t_{0}}^{τ} a(s)ds}b(τ)dτ ] ,  (1.98)

or in an equivalent form

x(t) = e^{∫_{t_{0}}^{t} a(s)ds}x_{0} + ∫_{t_{0}}^{t} e^{∫_{τ}^{t} a(s)ds}b(τ)dτ  (1.99)

1.8.1 The Bellman-Gronwall Lemma

Lemma 1.3 Let v(t) and w(t) be continuous real functions of t. Let w(t) ≥ 0, and c be a real constant. If

v(t) ≤ c + ∫_{0}^{t} w(s)v(s)ds,  (1.100)

then

v(t) ≤ ce^{∫_{0}^{t} w(s)ds}.  (1.101)

Proof Denote the right-hand side of (1.100) by x(t), that is, let

x(t) = c + ∫_{0}^{t} w(s)v(s)ds,  (1.102)

and let

z(t) = x(t) − v(t).  (1.103)

Note that z(t) ≥ 0, and x(t) is differentiable. Differentiating both sides of (1.102) and using (1.103) yields

ẋ(t) = w(t)v(t)
= w(t)x(t) − w(t)z(t).  (1.104)

We use (1.99) to represent (1.104) in the equivalent integral form as

x(t) = e^{∫_{0}^{t} w(s)ds}x_{0} − ∫_{0}^{t} e^{∫_{τ}^{t} w(s)ds}w(τ)z(τ)dτ.  (1.105)

Since z(t) ≥ 0 and w(t) ≥ 0, we have

x(t) ≤ e^{∫_{0}^{t} w(s)ds}x_{0}.  (1.106)

From the definition of x(t) given by (1.102), we have x(0) = c. Since

v(t) = x(t) − z(t) ≤ x(t),  (1.107)

because z(t) ≥ 0, hence,

v(t) ≤ ce^{∫_{0}^{t} w(s)ds},

and the proof is completed.

□

The generalized Bellman-Gronwall lemma allows us to convert the investigation of integral equations into integral inequalities.

1.9 Linearizing Differential Equations

We now consider a dynamic system modeled by a set of nonlinear differential equations

\frac{dx_{1}}{dt} = f_{1}(x_{1}, x_{2}, …, x_{n}, u_{1}, u_{2}, …, u_{m})
\frac{dx_{2}}{dt} = f_{2}(x_{1}, x_{2}, …, x_{n}, u_{1}, u_{2}, …, u_{m})
⋮
\frac{dx_{n}}{dt} = f_{n}(x_{1}, x_{2}, …, x_{n}, u_{1}, u_{2}, …, u_{m}).

We assume that the functions f_{i}, i = 1, 2, …, n are continuously differentiable. The above set of differential equations can be represented in vector form as

ẋ = f(x, u).

Let u_{0} = [ u_{10} u_{20} ⋯ u_{m0} ]^{T} be a constant input that forces the system to settle into a constant, equilibrium, state x_{0} = [ x_{10} x_{20} ⋯ x_{n0} ]^{T}, that is,

f(x_{0}, u_{0}) = 0.

We now perturb the equilibrium state by allowing

x = x_{0} + Δx, u = u_{0} + Δu.

Taylor’s expansion yields

\frac{d}{dt}x = \frac{d}{dt}x_{0} + \frac{d}{dt}Δx
= f(x_{0} + Δx, u_{0} + Δu)
= f(x_{0}, u_{0}) + \frac{∂f}{∂x}(x_{0}, u_{0}) Δx + \frac{∂f}{∂u}(x_{0}, u_{0}) Δu + higher order terms,

where

\frac{∂f}{∂x}(x_{0}, u_{0}) = [ \begin{matrix}\frac{∂f_{1}}{∂x_{1}} & ⋯ & \frac{∂f_{1}}{∂x_{n}} \\ ⋮ & & ⋮ \\ \frac{∂f_{n}}{∂x_{1}} & ⋯ & \frac{∂f_{n}}{∂x_{n}} \end{matrix} ]|_{{x=x_{0} \atop u=u_{0}}}and \frac{∂f}{∂u}(x_{0}, u_{0}) = [ \begin{matrix}\frac{∂f_{1}}{∂u_{1}} & ⋯ & \frac{∂f_{1}}{∂u_{m}} \\ ⋮ & & ⋮ \\ \frac{∂f_{n}}{∂u_{1}} & ⋯ & \frac{∂f_{n}}{∂u_{m}} \end{matrix} ]|_{{x=x_{0} \atop u=u_{0}}}

are the Jacobian matrices of f with respect x and u respectively, evaluated at the equilibrium point. Note that

\frac{d}{dt}x = \frac{d}{dt}x_{0} + \frac{d}{dt}Δx = \frac{d}{dt}Δx

because x_{0} is constant. Furthermore f(x_{0}, u_{0}) = 0. Let

A = \frac{∂f}{∂x}(x_{0}, u_{0}) and B = \frac{∂f}{∂u}(x_{0}, u_{0}).

By neglecting higher order terms we arrive at the linear approximation

\frac{d}{dt}Δx = AΔx + BΔu.

Similarly, if the outputs of the nonlinear system model are of the form

y_{1} = h_{1}(x_{1}, x_{2}, …, x_{n}, u_{1}, u_{2}, …, u_{m})
y_{2} = h_{2}(x_{1}, x_{2}, …, x_{n}, u_{1}, u_{2}, …, u_{m})
⋮
y_{p} = h_{p}(x_{1}, x_{2}, …, x_{n}, u_{1}, u_{2}, …, u_{m}),

or in vector notation

y = h(x, u),

then Taylor’s series expansion can again be used to yield the linear approximation of the above output equations. Indeed, let

y = y_{0} + Δy,

then we obtain

Δy = CΔx + DΔu,

where

C = \frac{∂h}{∂x}(x_{0}, u_{0}) = [ \begin{matrix}\frac{∂h_{1}}{∂x_{1}} & ⋯ & \frac{∂h_{1}}{∂x_{n}} \\ ⋮ & & ⋮ \\ \frac{∂h_{p}}{∂x_{1}} & ⋯ & \frac{∂h_{p}}{∂x_{n}} \end{matrix} ]|_{{x=x_{0} \atop u=u_{0}}} ∈ ℝ^{p×n},

and

D = \frac{∂h}{∂u}(x_{0}, u_{0}) = [ \begin{matrix}\frac{∂h_{1}}{∂u_{1}} & ⋯ & \frac{∂h_{1}}{∂u_{m}} \\ ⋮ & & ⋮ \\ \frac{∂h_{p}}{∂u_{1}} & ⋯ & \frac{∂h_{p}}{∂u_{m}} \end{matrix} ]|_{{x=x_{0} \atop u=u_{0}}} ∈ ℝ^{p×m}

are the Jacobian matrices of h with respect x and u respectively, evaluated at the equilibrium point [x_{0}^{T} u_{0}^{T}]^{T}.

1.10 Solving the State Equations

1.10.1 Solution of Uncontrolled System

Given a dynamical system’s time-invariant model of the form

ẋ(t) = Ax(t)  (1.108)

subject to an initial condition

x(0) = x_{0},  (1.109)

where A ∈ ℝ^{n×n}. The solution to (1.108) subject to (1.109) is

x(t) = e^{At}x_{0}

The above is a generalization of the situation when A is a scalar. The exponential matrix e^{At} is defined as

e^{At} = I_{n} + tA + \frac{t^{2}}{2!}A^{2} + \frac{t^{3}}{3!}A^{3} + ⋯

Observe that

\frac{d}{dt}e^{At} = Ae^{At}
= e^{At}A.

Suppose now that the initial condition x(0) = x_{0} is replaced by a more general condition

x(t_{0}) = x_{0},

where t_{0} is a given initial time instant. Then, the solution to ẋ(t) = Ax(t) subject to x(t_{0}) = x_{0} is

x(t) = e^{A^{(t−t_{0})}}x_{0}

The matrix e^{A^{(t−t_{0})}} is often written as

e^{A^{(t−t_{0})}} = Φ (t, t_{0})

and is called the state transition matrix, since it relates the state at any instant of time t_{0} to the state at any other time t. There are many methods for calculating the state transition matrix. For an overview of some of them, we refer the reader to the classic paper on the subject by Moler and Van Loan [81]. In the case when the matrix A is of low dimensions, we can use the formula that results from applying the Laplace transform to the time-invariant system ẋ = Ax, x(0) = x_{0}. The Laplace transform of the above matrix differential equation is

sX(s) − x_{0} = AX(s).

Hence

X(s) = (sI_{n} − A)^{−1} x_{0},

and the inverse transform yields

e^{At} = L^{−1} {(sI_{n} − A)^{−1}} ,  (1.110)

where L^{−1} denotes the inverse Laplace transform operator.

1.10.2 Solution of Controlled System

Consider a model of a controlled dynamical system,

ẋ(t) = Ax(t) + Bu(t)  (1.111)

subject to a given initial condition x(0) = x_{0}, where B is an n × m matrix, and m < n as is usually in practice. Premultiplying both sides of (1.111) by the integrating factor, e^{−At}, yields

e^{−At}ẋ(t) = e^{−At}Ax(t) + e^{−At}Bu(t).

We then rearrange the above equation as

e^{−At}ẋ(t) − e^{−At}Ax(t) = e^{−At}Bu(t).  (1.112)

Noting that

\frac{d}{dt} (e^{−At}x(t)) = −Ae^{−At}x(t) + e^{−At}ẋ(t),

we can represent (1.112) in the form

\frac{d}{dt} (e^{−At}x(t)) = e^{−At}Bu(t).

Integrating both sides of the resulting equation gives

e^{−At}x(t) − x(0) = ∫_{0}^{t} e^{−Aτ}Bu(τ)dτ.

Hence

x(t) = e^{At}x(0) + ∫_{0}^{t} e^{A(t−τ)}Bu(τ)dτ.

If the initial condition is x(t_{0}) = x_{0}, then integration from t_{0} to t yields

x(t) = e^{A(t−t_{0})}x(t_{0}) + ∫_{t_{0}}^{t} e^{A(t−τ)}Bu(τ)dτ  (1.113)

As in the case of the uncontrolled system, we can also use the Laplace transform technique to obtain the solution to (1.111) subject to the initial condition x(0) = x_{0}. Indeed, taking the Laplace transform of the above equation and performing some manipulations yields

X(s) = (sI_{n} − A)^{−1} x(0) + (sI_{n} − A)^{−1} BU(s).

Hence

x(t) = L^{−1} {(sI_{n} − A)^{−1} x(0)} + L^{−1} {(sI_{n} − A)^{−1} BU(s)} .

If x_{0} = x(t_{0}), then

x(t) = L^{−1} {e^{−t_{0}s} (sI_{n} − A)^{−1} x(t_{0})} + L^{−1} {e^{−t_{0}s} (sI_{n} − A)^{−1} BU(s)} .

1.11 Matrix Calculus Formulas

Below useful formulas from matrix calculus are listed that involve computing the gradient (with respect to x). Recall that in our convention the gradient is a column vector. We have:

∇ (x^{T} y) = ∇ (y^{T} x) = y  (1.114)
∇ (x^{T} x) = 2x  (1.115)
∇ (x^{T} Ay) = Ay  (1.116)
∇ (y^{T} Ax) = A^{T} y  (1.117)
∇ (x^{T} Ax) = (A + A^{T} ) x  (1.118)

Let f : ℝ^{n} → ℝ^{n} and g : ℝ^{n} → ℝ^{n} be two maps, and let \frac{∂f}{∂x} and \frac{∂g}{∂x} be the Jacobian matrices of f and g, respectively. Then,

∇ (f^{T} (x)g(x)) = ∇ (g^{T} (x)f(x)) = (\frac{∂f}{∂x} )^{T} g + (\frac{∂g}{∂x})^{T} f  (1.119)

For an n × n symmetric matrix Q,

∇ (f^{T} (x)Qf(x)) = 2 (\frac{∂f}{∂x} )^{T} Qf  (1.120)

1.12 Notes

For more information on linear algebra, and in particular matrix theory, the reader is referred Gelfand [36] and Gantmacher [35]. Analyses of numerical methods for matrix computations can be found in Faddeeva [26], Wilkinson [124], Householder [49], Golub and Van Loan [38], as well as in Gill, Murray, and Wright [37]. Lang [65] and Seeley [104] are recommended for a review of calculus of several variables. A straightforward approach to mathematical analysis is presented in Binmore [12]. Principles of real analysis can be found in Natanson [85], Bartle [7], and Rudin [98]. Some facts from basic functional analysis that we touched upon in this chapter, are discussed, for example, by Maddox [73], and Debnath and Mikusiński [23]. Lucid expositions of differential equations are in Driver [25], Miller and Michel [79], and in Boyce and DiPrima [13].

The English mathematician, G. H. Hardy (1877–1947) argues in [42, page 113] that in truly great theorems and their proofs, “there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions.” He further states that “A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.”

Stability

2.1 Introduction

In general, it may be difficult to obtain an explicit solution of nonlinear differential equations modeling a dynamical system. However, often we do not need to know the explicit solution of the modeling equations. Instead, we are interested in the behavior of the solution as time tends to infinity. Furthermore, in many models of physical systems a “small” change in the initial conditions results in a “small” change in the solution. Specifically, if a system is perturbed from its rest, or equilibrium position, then it starts to move. We can roughly say that the equilibrium position is stable if the system does not go far from this position for small initial perturbations. This concept may be illustrated, as in Willems [126], Brogan [15], or in Barnett [6], by means of a ball resting in equilibrium on a sheet of metal bent into various shapes with cross-sections depicted in Figure 2.1. This figure illustrates four possible scenarios. If we neglect frictional forces, then a small perturbation from the equilibrium leads to:

(1) oscillatory motion about equilibrium irrespective of the magnitude of the initial perturbation. In this case the equilibrium position is globally stable,

(2) the new position is also an equilibrium and the ball will remain in it. Here the equilibrium position is stable,

(3) the ball moving away without returning to the equilibrium. This equilibrium position is unstable,

(4) oscillatory motion about the equilibrium, unless the initial perturbation is so large that

Figure 2.1: Illustration of stable and unstable equilibrium positions.

the ball is forced to oscillate about the new equilibrium position. This equilibrium position is locally stable.

If friction is taken into account, then the oscillatory motion steadily decreases until the ball returns to the equilibrium positions as in items (1) and (4) above. We refer to such equilibrium positions as being asymptotically stable. In our discussion of stability concepts, we implicitly assume, as in item (4), that there is some “small” region for allowed initial perturbations for which subsequent motions converge to the equilibrium position. If however, as in item (1), any initial perturbation yields a motion that converges to the equilibrium position, then we have global asymptotic stability, which is also referred to as asymptotic stability in the large.

The above informal discussion should serve as a preparation for an in-depth analysis of stability of dynamical systems. We begin our analysis by formally defining the equilibrium state. We then define stability, asymptotic stability, and other relevant notions.

2.2 Basic Definitions of Stability

We consider a class of dynamical systems modeled by the equation:

ẋ(t) = f(t, x(t)), x(t_{0}) = x_{0},

where x(t) ∈ ℝ^{n} is the state vector and f : ℝ × ℝ^{n} → ℝ is a vector-valued function with the components

f_{i}(t, x_{1}, x_{2}, …, x_{n}) : ℝ × ℝ^{n} → ℝ, i = 1, 2, …, n.

We assume that f_{i}’s are continuous and have continuous first partial derivatives. Under these assumptions, the solution to ẋ(t) = f(t, x(t)), x(t_{0}) = x_{0}, exists and is unique. We denote this solution by x(t) = x(t; t_{0}, x_{0}). If the f_{i}’s do not depend explicitly on t, then the system represented by ẋ(t) = f(t, x(t)) = f(x(t)) is called autonomous, otherwise it is called non-autonomous. An equilibrium point (at time t^{∗} ∈ ℝ) is a constant vector, say x_{e}, such that

f(t, x_{e}) = 0 for all t ≥ t^{∗}.

Other terms for equilibrium point are stationary point, critical point, rest position, and equilibrium state. Observe that an equilibrium point x_{e} is a constant solution to

ẋ(t) = f(t, x(t)), x(t_{0}) = x_{e}.

If x_{e} is an equilibrium point, we can arrange for this equilibrium point to be transferred to the origin of ℝ^{n} by introducing the new variable

x̃ = x − x_{e}.

We assume that this has been done for an equilibrium point under consideration. Therefore, we have

f(t, 0) = 0 for all t.

There may be several equilibrium points for a given system. We assume that there is no other equilibrium point in a neighborhood of the origin of ℝ^{n}. Thus, the origin is an isolated equilibrium point.

We now introduce basic definitions of stability in the sense of A. M. Lyapunov^{1} (1857-1918). In the following considerations we use the standard Euclidean norm of a vector. Thus, if x ∈ ℝ^{n} then

∥x∥ = ∥x∥_{2} = (x^{T} x)^{\frac{1}{2}} .

Definition 2.1 An equilibrium state x_{e} is said to be stable if for any given t_{0} and any positive scalar ε, there exists a positive scalar δ = δ(t_{0}, ε) such that if ∥x(t_{0}) − x_{e}∥ < δ, then ∥x(t) − x_{e}∥ < ε for all t ≥ t_{0}.

1Sometimes transliterated in the literature as Liapunov or Liapunoff.

Definition 2.2 An equilibrium state x_{e} is said to be attractive if for any given t_{0} there is a positive scalar δ_{1} = δ_{1}(t_{0}) such that if ∥x(t_{0}) − x_{e}∥ < δ_{1} then,

lim_{t→∞} x(t; t_{0}, x_{0}) = x_{e}.

We can restate Definition 2.2 as follows. An equilibrium state x_{e} is attractive if for any t_{0} there is a positive scalar δ_{1} = δ_{1}(t_{0}) such that if ∥x(t_{0}) − x_{e}∥ < δ_{1}, then for any given ε_{1} > 0 there exists a T = T (ε_{1}, t_{0}, x_{0}) such that ∥x(t; t_{0}, x_{0}) − x_{e}∥ < ε_{1} for all t > t_{0} + T.

Definition 2.3 An equilibrium state x_{e} is asymptotically stable if it is stable and attractive.

Definition 2.4 An equilibrium state x_{e} is said to be uniformly stable if the positive scalar δ = δ(t_{0}, ε) introduced in Definition 2.1 can be taken independent of the initial time t_{0}.

Definition 2.5 An equilibrium point x_{e} is said to be uniformly attractive, if the positive scalars δ_{1} = δ_{1}(t_{0}) and T = T (ε_{1}, t_{0}, x_{0}) introduced in Definition 2.2 are independent of t_{0}.

The last two definitions allow us to introduce the notion of uniform asymptotic stability.

Definition 2.6 An equilibrium point x_{e} is said to be uniformly asymptotically stable if it is uniformly stable and uniformly attractive.

Notice that the solution x(t) = x_{e} must first be stable in order to qualify to be asymptotically stable. This is to prevent a system trajectory from straying arbitrarily far from the equilibrium state before converging toward it. Note also that the concepts of Lyapunov stability and asymptotic stability are local concepts. They apply to the system’s behavior in the small or in a small neighborhood about the equilibrium point (see Hsu and Meyer [50]).

Definition 2.7 An equilibrium point x_{e} is said to be unstable if there is an ε > 0 so that for any δ > 0 there exists x(t_{0}) such that if ∥x(t_{0}) − x_{e}∥ < δ, then ∥x(t_{1}) − x_{e}∥ ≥ ε for some t_{1} > t_{0}.

A two dimensional geometric interpretation of the above definitions is shown in Figure 2.2. We assume that x_{e} = 0 is the equilibrium state. If the equilibrium state is stable, as in Figure 2.2(a), then given an outer circle of radius ε, there exists an inner circle of radius δ such that trajectories starting within the δ-circle never leave the ε-circle. If the equilibrium state is asymptotically stable, then trajectories in addition tend to the equilibrium state as t tends to ∞. This situation is depicted in Figure 2.2(b). Finally, the equilibrium state is unstable, as in Figure 2.2(c), then there exists an ε-circle such that for every δ-circle there exists a trajectory starting within it that leaves the ε-circle at some later time. The corresponding geometrical interpretation of the above described scenarios in one dimension is given in Figure 2.3.

Figure 2.2: Illustration of stability concepts in two dimensions; (a) stable equilibrium state in the sense of Lyapunov, (b) asymptotically stable equilibrium state, (c) unstable equilibrium state.

38 CHAPTER 2. STABILITY

(a) stable equilibrium state in the sense of Lyapunov (b) asymptotically stable equilibrium state

(c) unstable equilibrium state

Figure 2.3: Illustration of stability concepts in one dimension.

2.3 Lyapunov Stability of Linear Systems

We know that a linear time-invariant dynamical system modeled by the equation

ẋ = Ax, x(t_{0}) = x_{0}  (2.1)

is asymptotically stable if and only if, the solution to equation (2.1) decays to zero as t → ∞ for any initial state x_{0}. We can view the vector x(t) ∈ ℝ^{n} as defining the coordinates of a point moving in an n-dimensional state space ℝ^{n}. In an asymptotically stable system modeled by equation (2.1), this point converges to the origin of ℝ^{n}. We can argue that if a trajectory is converging to the origin of the state space, then it should be possible to find a family of nested surfaces described by V (x_{1}, x_{2}, …, x_{n}) = c, c ≥ 0, such that monotonically decreasing values of c give surfaces progressively shrinking in on the origin with the limiting surface V (x_{1}, x_{2}, …, x_{n}) = 0 being the origin x = 0. Furthermore, along every trajectory of (2.1), the parameter c should steadily decrease. We now introduce some notation and definitions that we use in the ensuing discussion.

Let V (x) = V (x_{1}, x_{2}, …, x_{n}) : ℝ^{n} → ℝ be a real-valued function and let S ⊆ ℝ^{n} be a compact region containing the origin x = 0 in its interior.

Definition 2.8 We say that the function V = V (x) is positive semi-definite (p.s.d.) in S 40 CHAPTER 2. STABILITY if

1. V is continuously differentiable, that is, V ∈ C^{1},

2. V (0) = 0,

3. V (x) ≥ 0 for all x ∈ S. Definition 2.9 We say that the function V = V (x) is positive definite (p.d.) in S if

1. V ∈ C^{1},

2. V (0) = 0,

3. V (x) > 0 for all x ∈ S \ {0}.

The symbol S \ {0} in item 3 above denotes the set {x : x ∈ S, x ≠ 0}.

Negative semi-definite (n.s.d.) and negative definite (n.d.) functions are defined in a similar fashion by reversing inequality signs in item 3 in both definitions above.

There are many functions that describe a nested family of surfaces. In the stability analysis of linear systems one class is best suited. This is the positive definite quadratic form, a label attached to a nested family of ellipses and their extensions to higher dimensional spaces—ellipsoids for n = 3, and hyper-ellipsoids for n ≥ 4.

Let

V (x) = x^{T} P x,

where P is a real symmetric n × n matrix. The time derivative of V (x(t)) evaluated on a solution of ẋ = Ax is

\frac{dV (x(t))}{dt} = V̇
= ẋ^{T} P x + x^{T} P ẋ.

Substituting ẋ = Ax into the above equation yields

V̇ = x^{T} A^{T} P x + x^{T} P Ax
= x^{T} (A^{T} P + P A)x.

Let

Q = −(A^{T} P + P A).

Then,

V̇ = −x^{T} Qx.

We are now ready to state and prove the following theorem concerning the stability of the system ẋ = Ax, or equivalently, the matrix A ∈ ℝ^{n}.

Theorem 2.1 (A. M. Lyapunov, 1892) The system ẋ = Ax, x(t_{0}) = x_{0}, is asymptotically stable if and only if for any given real symmetric positive definite (r.s.p.d.) matrix Q the solution P to the continuous Lyapunov matrix equation,

A^{T} P + P A = −Q

is also (real symmetric) positive definite.

Proof (⇒) We prove necessity, for asymptotic stability, by contradiction. We have −x^{T} Qx < 0 for all x ≠ 0. We assume that A is an asymptotically stable matrix and that for some nonzero x_{0} we have x_{0}^{T} P x_{0} ≤ 0. From the discussion before the Theorem, it follows that

\frac{d}{dt}(x^{T} (t)P x(t)) = −x^{T} (t)Qx(t) < 0.

The above means that V (x(t)) = x^{T} (t)P x(t) decreases on the trajectories of the system ẋ = Ax as t → ∞. On the other hand, since A is asymptotically stable, we have

lim_{t→∞} x(t) = 0.

Hence

lim_{t→∞} x^{T} (t)P x(t) = lim_{x(t)→0} x^{T} (t)P x(t)
= 0.

Thus, we arrived at a contradiction. This completes the proof of the first part of the Theorem that states that asymptotic stability of A implies that for any r.s.p.d matrix Q the solution P to the equation A^{T} P + P A = −Q is r.s.p.d..

(⇐) We prove sufficiency, also by contradiction. We assume that

x^{T} P x > 0 and − x^{T} Qx < 0 for all x ≠ 0,

and that the matrix A is not asymptotically stable. This means that the solution to ẋ(t) = Ax(t), x_{0} ≠ 0, which has the form x(t) = e^{A(t−t_{0})}x_{0}, is bounded away from the origin x = 0. Thus, there is a constant b > 0 such that

x^{T} (t)Qx(t) > b > 0 for all t > t_{0}.

Integrating \frac{d}{dt}(x^{T} (t)P x(t)) = −x^{T} (t)Qx(t), we obtain

x^{T} (t)P x(t) = x^{T} (t_{0})P x(t_{0}) + ∫_{t_{0}}^{t}(−x^{T} (s)Qx(s))ds.

Taking into account that −x^{T} (t)Qx(t) < −b < 0 for all t > t_{0}, we get for t > t_{0},

x^{T} (t)P x(t) = x^{T} (t_{0})P x(t_{0}) + ∫_{t_{0}}^{t}(−x^{T} (s)Qx(s))ds
< x^{T} (t_{0})P x(t_{0}) − b(t − t_{0}).

The above implies that for a sufficiently large t we have x^{T} (t)P x(t) < 0, which contradicts the assumption that x^{T} (t)P x(t) > 0. This means that A must be asymptotically stable, and the proof is complete.

□

Example 2.1 Let

A = [ \begin{matrix}0 & 1 \\ −1 & −2 \end{matrix} ] ,

and Q = I_{2}. We solve the Lyapunov equation A^{T} P + P A = −Q for P. Assume P to have the form,

P = [ \begin{matrix}p_{1} & p_{2} \\ p_{2} & p_{3} \end{matrix} ] = P^{T} .

Then, we have

A^{T} P + P A = [ \begin{matrix}0 & −1 \\ 1 & −2 \end{matrix} ] [ \begin{matrix}p_{1} & p_{2} \\ p_{2} & p_{3} \end{matrix} ] + [ \begin{matrix}p_{1} & p_{2} \\ p_{2} & p_{3} \end{matrix} ] [ \begin{matrix}0 & 1 \\ −1 & −2 \end{matrix} ]
= [ \begin{matrix}−2p_{2} & −p_{3} + p_{1} − 2p_{2} \\ −p_{3} + p_{1} − 2p_{2} & 2p_{2} − 4p_{3} \end{matrix} ]
= [ \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ] .

Solving the corresponding linear equations yields

P = [ \begin{matrix}3/2 & 1/2 \\ 1/2 & 1/2 \end{matrix} ] .

The solution matrix P is positive definite. Hence, the matrix A must have its eigenvalues in the open left-half plane. Indeed, the eigenvalues of A are located at −1.

Definition 2.10 A positive definite function V (x) whose time derivative evaluated on the solutions of the system ẋ = Ax is negative definite (or negative semi-definite) is called a Lyapunov function for this system.

Lyapunov’s theorem gives us a necessary and sufficient condition for asymptotic stability of linear time-invariant dynamical systems. To check if a given matrix A is asymptotically stable or not, it is enough to take an arbitrary r.s.p.d. matrix Q, solve the equation A^{T} P + P A = −Q for real symmetric P and check if the obtained P is positive definite, or not. If P > 0, then A is asymptotically stable. If P is not positive definite, then A cannot be asymptotically stable. Since Q can be an arbitrary r.s.p.d. matrix, Q = I_{n}, the n × n identity matrix, is a good choice. It is important to note that there would be no use, as the following example shows, in first choosing a matrix P and then calculating Q from the equation A^{T} P + P A = −Q.

Example 2.2 Let

A = [ \begin{matrix}−1 & 3 \\ 0 & −1 \end{matrix} ] ,

which is asymptotically stable. Let

P = [ \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ] = I_{2} > 0.

Then,

Q = −(A^{T} P + P A) = [ \begin{matrix}2 & −3 \\ −3 & 2 \end{matrix} ]

which is indefinite. Thus, unless Q turns out to be definite, nothing can be inferred about asymptotic stability from the Lyapunov theorem if we start with P and then calculate Q.

We now give a geometric interpretation of the Lyapunov theorem. First, note that the function

V̇ = \frac{dV (x(t))}{dt}

can be expressed, using the chain rule, as

V̇ = ∇^{T} V ẋ = ∥∇V ∥ ∥ẋ∥ cos θ,

Figure 2.4: Geometric interpretation of the Lyapunov theorem.

where θ is the angle between the vectors ∇V and ẋ. If ∥ẋ∥ = 1, then ∇^{T} V ẋ can be viewed as the rate of increase of V at x in the direction ẋ. Thus, V̇ < 0 if and only if, 90 deg < θ < 270 deg. In such a case the representative point moves in the direction of decreasing V. We illustrate the above discussion in Figure 2.4.

We now prove a variant of Theorem 2.1, which is very useful in numerical verification of stability.

Theorem 2.2 A matrix A is asymptotically stable if and only if for any Q = C^{T} C, C ∈ ℝ^{p×n}, such that the pair (A, C) is observable, the solution P to the Lyapunov matrix equation A^{T} P + P A = −C^{T} C is also positive definite.

Proof We first prove the necessity condition (⇒) for asymptotic stability. If A is asymptotically stable, that is, ℜλ_{i}(A) < 0, then we will show that

P = ∫_{0}^{∞} e^{A^{T} t}C^{T} Ce^{At}dt  (2.2)

is the symmetric positive definite solution to the Lyapunov equation. Indeed, the integrand of (2.2) is a sum of terms of the form

t^{k}e^{λ_{i}t},

where λ_{i} is the i-th eigenvalue of A. Since ℜλ_{i}(A) < 0, the integral in (2.2) exists because the terms in P do not diverge when t = ∞. Indeed, let −α_{i} = ℜλ_{i}, where α_{i} > 0, then

lim_{t→∞} t^{k}e^{−α_{i}t} = lim_{t→∞} \frac{t^{k}}{e^{α_{i}t}} ,

We differentiate the numerator and denominator of the above k times and use L’Hospital’s rule to get

lim_{t→∞} \frac{t^{k}}{e^{α_{i}t}} = lim_{t→∞} \frac{kt^{k−1}}{α_{i}e^{α_{i}t}}
= lim_{t→∞} \frac{k(k − 1)t^{k−1}}{α_{i}^{2}e^{α_{i}t}}
⋮
= lim_{t→∞} \frac{k!}{α_{i}^{k}e^{t}}
= 0.  (2.3)

Thus, by (2.3), the formula for P given by (2.2) is well defined. Next, note that P = P^{T}. The matrix P is also positive definite since the pair (A, C) is observable. Substituting the expression for P into the Lyapunov equation and performing manipulations yields

A^{T} P + P A = ∫_{0}^{∞} A^{T} e^{A^{T} t}C^{T} Ce^{At}dt + ∫_{0}^{∞} e^{A^{T} t}C^{T} Ce^{At}Adt
= ∫_{0}^{∞} \frac{d}{dt} (e^{A^{T} t}C^{T} Ce^{At}) dt
= e^{A^{T} t}C^{T} Ce^{At}|_{0}^{∞} .
= −C^{T} C.

The sufficiency part (⇐) can be proven using the arguments we used while proving the sufficiency part of the Lyapunov Theorem and noting that the observability of the pair (A, C) implies that x^{T} (t)C^{T} Cx(t) is not identically zero along any nonzero solution of ẋ(t) = Ax(t).

□

We now show that if A is asymptotically stable, then the Lyapunov equation

A^{T} P + P A = −Q

has a unique solution for every Q. Uniqueness can be proven in many ways. We use the Kronecker product in our proof. Using the results of Section 1.5, we represent the Lyapunov equation, A^{T} P + P A = −Q, as

(I_{n} ⊗ A^{T} + A^{T} ⊗ I_{n}) s(P) = −s(Q).

The condition for a solution s(P) of the above equation to be unique is

λ_{i}(A^{T}) + λ_{j}(A) ≠ 0, i, j = 1, 2, …, n.

Since λ_{i}(A^{T}) = λ_{i}(A), and A is asymptotically stable by assumption, the condition λ_{i}(A^{T})+ λ_{j}(A) ≠ 0 is met, and hence the Lyapunov equation has a unique solution.

2.4 The Hurwitz and Routh Stability Criteria

In this section, we use the Lyapunov theorem to derive two tests for determining if the zeros of an n-th degree polynomial are located in the open left-half complex plane. Consider the n-th degree polynomial

p(s) = a_{n}s^{n} + a_{n−1}s^{n−1} + ⋯ + a_{1}s + a_{0}  (2.4)

with real coefficients. Associated with the polynomial p(s) is the n × n real Hurwitz matrix

H = [ \begin{matrix}a_{n−1} & a_{n−3} & a_{n−5} & a_{n−7} & ⋯ \\ a_{n} & a_{n−2} & a_{n−4} & a_{n−6} & ⋯ \\ 0 & a_{n−1} & a_{n−3} & a_{n−5} & ⋯ \\ 0 & a_{n} & a_{n−2} & a_{n−4} & ⋯ \\ ⋮ & & & ⋮ & ... \end{matrix} ] .  (2.5)

In our further discussion we assume, without loss of generality, that a_{n} = 1. The leading principal minors of the Hurwitz matrix are

H_{1} = a_{n−1}, H_{2} = det [ \begin{matrix}a_{n−1} & a_{n−3} \\ 1 & a_{n−2} \end{matrix} ] , H_{3} = det [ \begin{matrix}a_{n−1} & a_{n−3} & a_{n−5} \\ 1 & a_{n−2} & a_{n−4} \\ 0 & a_{n−1} & a_{n−3} \end{matrix} ] , …, H_{n} = det H.

(2.6)

We next define

b_{1} = H_{1}, b_{2} = \frac{H_{2}}{H_{1}} , b_{3} = \frac{H_{3}}{H_{2}H_{1}} ,

and

b_{k} = \frac{H_{k−3}H_{k}}{H_{k−2}H_{k−1}} for k = 4, 5, …, n.

Thus, for example, b_{4} = \frac{H_{1}H_{4}}{H_{2}H_{3}} . Let

B = [ \begin{matrix}0 & 1 & 0 & 0 & ⋯ & 0 & 0 & 0 \\ −b_{n} & 0 & 1 & 0 & ⋯ & 0 & 0 & 0 \\ 0 & −b_{n−1} & 0 & 1 & ⋯ & 0 & 0 & 0 \\ ⋮ & & & & & & & ⋮ \\ 0 & 0 & 0 & 0 & ⋯ & −b_{3} & 0 & 1 \\ 0 & 0 & 0 & 0 & ⋯ & 0 & −b_{2} & −b_{1} \end{matrix} ] ∈ ℝ^{n×n}.

For example, for n = 4, we have

B = [ \begin{matrix}0 & 1 & 0 & 0 \\ −b_{4} & 0 & 1 & 0 \\ 0 & −b_{3} & 0 & 1 \\ 0 & 0 & −b_{2} & −b_{1} \end{matrix} ] .

Let

A = [ \begin{matrix}0 & 1 & 0 & ⋯ & 0 & 0 \\ 0 & 0 & 1 & ⋯ & 0 & 0 \\ ⋮ & & & & & ⋮ \\ 0 & 0 & 0 & ⋯ & 0 & 1 \\ −a_{0} & −a_{1} & −a_{2} & ⋯ & −a_{n−2} & −a_{n−1} \end{matrix} ] ∈ ℝ^{n×n}.

Lemma 2.1 The matrices B and A are similar, that is, there is a nonsingular matrix T ∈ ℝ^{n×n} such that

B = T^{−1}AT .  (2.7)

Proof The similarity matrix is lower triangular and has the form

T = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & ⋯ & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & ⋯ & 0 \\ t_{31} & 0 & 1 & 0 & 0 & 0 & ⋯ & 0 \\ 0 & t_{42} & 0 & 1 & 0 & 0 & ⋯ & 0 \\ t_{51} & 0 & t_{53} & 0 & 1 & 0 & ⋯ & 0 \\ 0 & t_{62} & 0 & t_{64} & 0 & 1 & ⋯ & 0 \\ ⋮ & & & & & & & ⋮ \end{matrix} ] .  (2.8)

To determine the nonfixed coefficients t_{ij} of the matrix T, we represent (2.7) as

T B = AT ,  (2.9)

and compare both sides of (2.9) to determine the coefficients t_{ij}. We obtain, for example, t_{31} = −b_{n}, t_{42} = −b_{n} − b_{n−1}, etc.

□

Let P be an n × n diagonal matrix defined as

P = diag [ b_{1}b_{2} ⋯ b_{n}, b_{1}b_{2} ⋯ b_{n−1}, ⋯, b_{1}b_{2}, b_{1} ] ,  (2.10)

and Q be an n × n diagonal matrix defined as

Q = diag [ 0 0 ⋯ 0 2b_{1}^{2} ] .

We represent the matrix Q as Q = c^{T} c, where

c = [ 0 0 ⋯ 0 \sqrt{2}b_{1} ] .

Note that the pair (B, c) is observable. The matrices B, P, and Q satisfy the Lyapunov matrix equation,

B^{T} P + P B = −Q.  (2.11)

By Theorem 2.2, on page 44, applied to (2.11), the matrix B is asymptotically stable if and only if the symmetric matrix P is positive definite. By Theorem ??, the matrix P is positive definite if and only if its leading principal minors are positive. Using the above results we can prove the following theorem:

Theorem 2.3 (Hurwitz) The real polynomial p(s) of degree n with positive coefficients a_{i}, has its zeros in the open left-half plane if and only if the leading principal minors of the Hurwitz matrix are positive.

Proof The polynomial p(s) has its zeros in the open left plane if and only if the matrix B is asymptotically stable. This in turn is true if and only if the diagonal elements of P, given by (2.10), are positive. We have

b_{1} = H_{1}, b_{1}b_{2} = H_{2},

and for k = 3, …, n,

b_{1}b_{2} ⋯ b_{k} = \frac{H_{k}}{H_{k−2}} .

Thus, the diagonal elements of P are all positive, if and only if the leading principal minors of the Hurwitz matrix are positive, which completes the proof.

□

We will now prove the Routh necessary and sufficient condition for the real polynomial (2.4) to have its zeros in the open left-half plane. We first use the polynomial coefficients to form the following (n + 1)-row Routh array:

s^{n} row a_{n} a_{n−2} a_{n−4} a_{n−6} ⋯
s^{n−1} row a_{n−1} a_{n−3} a_{n−5} a_{n−7} ⋯
s^{n−2} row r_{21} r_{22} r_{23} r_{24} ⋯
s^{n−3} row r_{31} r_{32} r_{33} r_{34} ⋯
⋮ ⋮ ⋮
s^{0}  row r_{n1}

(2.12)

The elements that comprise the s^{n} and s^{n−1} rows are appropriately ordered coefficients of p(s). The elements of the s^{n−2} row are formed from its preceding two rows. They are:

r_{21} = −\frac{1}{a_{n−1}} det [ \begin{matrix}a_{n} & a_{n−2} \\ a_{n−1} & a_{n−3} \end{matrix} ] = \frac{a_{n−1}a_{n−2} − a_{n}a_{n−3}}{a_{n−1}}
r_{22} = −\frac{1}{a_{n−1}} det [ \begin{matrix}a_{n} & a_{n−4} \\ a_{n−1} & a_{n−5} \end{matrix} ] = \frac{a_{n−1}a_{n−4} − a_{n}a_{n−5}}{a_{n−1}}
r_{23} = −\frac{1}{a_{n−1}} det [ \begin{matrix}a_{n} & a_{n−6} \\ a_{n−1} & a_{n−7} \end{matrix} ] = \frac{a_{n−1}a_{n−6} − a_{n}a_{n−7}}{a_{n−1}}
⋮  ⋮

The elements of the s^{n−3} row of the Routh array are generated using its preceding two rows. They are:

r_{31} = −\frac{1}{r_{21}} det [ \begin{matrix}a_{n−1} & a_{n−3} \\ r_{21} & r_{22} \end{matrix} ] = \frac{r_{21}a_{n−3} − r_{22}a_{n−1}}{r_{21}}
r_{32} = −\frac{1}{r_{21}} det [ \begin{matrix}a_{n−1} & a_{n−5} \\ r_{21} & r_{23} \end{matrix} ] = \frac{r_{21}a_{n−5} − r_{23}a_{n−1}}{r_{21}}
r_{33} = −\frac{1}{r_{21}} det [ \begin{matrix}a_{n−1} & a_{n−7} \\ r_{21} & r_{24} \end{matrix} ] = \frac{r_{21}a_{n−7} − r_{24}a_{n−1}}{r_{21}}
⋮  ⋮

Subsequent rows of the array are formed from their preceding rows in an analogous fashion. Undefined positions can be filled with zeros. The last two rows of the array will have one term each. We will now formulate the stability test that in some texts (for example [96] or [127]) is also called the Routh-Hurwitz criterion.

Theorem 2.4 (Routh) The zeros of the polynomial p(s) are all in the open left-half plane if and only if the elements of the first column of the Routh-Hurwitz array are all positive.

Proof We perform elementary row operations on the Hurwitz matrix that are equivalent to premultiplying it by lower triangular matrices with ones on the diagonal. For example,

[ \begin{matrix}1 & 0 & 0 & 0 & ⋯ \\ −a_{n}/a_{n−1} & 1 & 0 & 0 & ⋯ \\ 0 & 0 & 1 & 0 & ⋯ \\ ⋮ & & & ... & ⋮ \\ 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}a_{n−1} & a_{n−3} & a_{n−5} & a_{n−7} \\ a_{n} & a_{n−2} & a_{n−4} & a_{n−6} \\ 0 & a_{n−1} & a_{n−3} & a_{n−5} \\ 0 & a_{n} & a_{n−2} & a_{n−4} \\ ⋮ & & ⋮ & ... \end{matrix} ] = [ \begin{matrix}a_{n−1} & a_{n−3} & a_{n−5} & a_{n−7} \\ 0 & r_{21} & r_{22} & r_{23} \\ 0 & a_{n−1} & a_{n−3} & a_{n−5} \\ 0 & a_{n} & a_{n−2} & a_{n−4} \\ ⋮ & & ⋮ & ... \end{matrix} ] .

Let U be the product of the matrices corresponding to the above described elementary operations on H. Then,

U = [ \begin{matrix}1 & 0 & ⋯ & 0 \\ u_{12} & 1 & ⋯ & 0 \\ ⋮ & & ... & ⋮ \\ u_{n1} & u_{n2} & ⋯ & 1 \end{matrix} ] ,

where, for example, u_{12} = −a_{n}/a_{n−1}. Premultiplying the Hurwitz matrix (2.5) by the matrix U given above yields

UH = [ \begin{matrix}a_{n−1} & a_{n−3} & a_{n−5} & ⋯ & ⋯ \\ 0 & r_{21} & r_{22} & ⋯ & r_{2,n−1} \\ 0 & 0 & r_{31} & ⋯ & r_{3,n−2} \\ ⋮ & & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & ⋯ & r_{n1} \end{matrix} ] .  (2.13)

Since the matrix U is lower triangular with the ones on the diagonal, the leading principal minors of H and UH are the same, where

H_{1} = a_{n−1}, H_{2} = a_{n−1}r_{21}, …, H_{i} = a_{n−1}r_{21} ⋯ r_{i1}, i = 1, 2, …, n.

Hence,

a_{n−1} = H_{1}, and r_{i1} = \frac{H_{i}}{H_{i−1}} for i = 2, …, n.

By the Hurwitz Theorem, the zeros of the polynomial p(s) are in the open left-half plane if the coefficients r_{i1}, i = 1, 2 …, n, which are the elements of the first column of the Routh array, are all positive. The proof is complete.

□

Note that the Routh theorem could be formulated as follows: The zeros of the polynomial p(s) are all in the open left-half plane if and only if the elements of the first column of the Routh array all have the same sign and are nonzero.

If the polynomial coefficients are negative, we can multiply both sides of the equation p(s) = 0 by −1 thus making the coefficients positive and then we can proceed with the Hurwitz or Routh test to determine the stability of instability of a given polynomial.

A necessary, but not sufficient, condition for the polynomial p(s) to have all its zeros in the open left-half plane is that all its coefficients are nonzero and have the same sign.

It is possible to extend the above Routh stability criterion so that it can be used to find the number of polynomial zeros with positive real parts. The number of zeros of p(s) with positive real parts is equal to the number of sign changes in the first column of the Routh array. For a proof of this statement using the law of inertia of quadratic forms, see, for example, Kaczorek [55, Section 8.4.3].

2.5 The Discrete-Time Lyapunov Equation

In the following, we present a background material that we will use in constructing a controller for a discrete time chaotic system. We begin with the Lyapunov approach to the stability analysis of of the discrete-time linear time-invariant systems modeled by

x(k + 1) = Ax(k), x(k_{0}) = x_{0}.  (2.14)

By analogy with the continuous-time case, consider a quadratic positive definite form,

V (x(k)) = x^{T} (k)P x(k).

We evaluate the first difference of V, defined ΔV = V (x(k+1))−V (x(k)), on the trajectories of (2.14) to obtain

ΔV (x(k)) = V (x(k + 1)) − V (x(k))
= x^{T} (k + 1)P x(k + 1) − x^{T} (k)P x(k)
= x^{T} (k)A^{T} P Ax(k) − x^{T} (k)P x(k)
= x^{T} (k) (A^{T} P A − P ) x(k),  (2.15)

where we replaced x(k + 1) in the above with the right-hand side of (2.14), that is, with Ax(k). For the solutions of (2.14) to decrease the “energy function” V it is necessary and sufficient that ΔV (x(k)) < 0, or equivalently, that for some positive definite Q,

ΔV (x(k)) = −x^{T} (k)Qx(k).  (2.16)

Combining (2.14) and (2.16), we obtain the so called discrete-time Lyapunov equation,

A^{T} P A − P = −Q  (2.17)

We will now show that the matrix A has its eigenvalues in the open unit disk if and only if for any positive definite Q the solution P to (2.17) is positive definite. We can use the Kronecker product, and in particular relation (1.61) to represent the discrete-time Lyapunov equation as

(I_{n^{2}} − A^{T} ⊗ A^{T} ) s(P) = s(Q),  (2.18)

where ⊗ denotes the Kronecker product and s(·) is the stacking operator. It is easy to see that the above equation has a unique solution if and only if

1 − λ_{i}λ_{j} ≠ 0, for all i, j = 1, 2, …, n,

where {λ_{i}} is the set of eigenvalues of A. Note that |λ_{i}| < 1 is a sufficient condition for (2.18) and hence (2.17) to have a unique solution.

2.6 Stability of Nonlinear Systems

In this section, we will be concerned with stability analysis of nonlinear time-varying dynamical system models described by

ẋ(t) = f(t, x(t)), x(t_{0}) = x_{0},

where x(t) ∈ ℝ^{n} is the state vector, and f is a vector-valued function.

To proceed, we need the following lemma. Denote by B_{q}(0) an open ball with the center at 0 and radius q. Let S ⊂ ℝ^{n} be a compact set that contains the origin x = 0 in its interior.

Lemma 2.2 For a continuously differentiable function V : ℝ^{n} → ℝ, where V (0) = 0, we have V (x) > 0 for all x ∈ S \ {0}, if and only if there is a continuous, strictly increasing, function ψ : ℝ → ℝ such that ψ(0) = 0, ψ(r) > 0 for r > 0, and V (x) ≥ ψ(r), where r = ∥x∥, and r ≤ R = max{q : B_{q}(0) ⊆ S}.

Proof Sufficiency ( ⇐) is clear.

To prove necessity ( ⇒) define the function κ : ℝ → ℝ,

κ(y) = min_{∥x∥=y^{x∈S}} V (x)  (2.19)

Note that the function κ is well defined since the set

{x : ∥x∥ = y} ∩ S  (2.20)

is compact and V is continuous. Therefore, by the Weierstrass theorem on page ??, the function V achieves its minimum over the set (2.20).

We will now show that κ is continuous on B_{R}(0). Then, we will construct an appropriate function ψ satisfying the hypotheses of the lemma. To show that κ is continuous on B_{R}(0), we will use the fact that V is uniformly continuous in S. To this end let x^{(1)} ∈ S be such that ∥x^{(1)}∥ = y and κ(y) = V (x^{(1)}). Next, let Δy be an increment of y and let x^{(2)} ∈ S be a vector such that y + Δy = ∥x^{(2)}∥. For example, we can take

x^{(2)} = x^{(1)} + \frac{x^{(1)}}{∥x^{(1)}∥}Δy.

Observe that

V (x^{(2)}) ≥ κ(y + Δy).

Choose now a vector x^{(3)} such that

∥x^{(3)}∥ = y + Δy, and V (x^{(3)}) = κ(y + Δy).

Note that for any vector, say x^{(4)}, such that ∥x^{(4)}∥ = y we have V (x^{(4)}) ≥ κ(y)—see Figure 2.5. We can take

x^{(4)} = x^{(3)} − \frac{x^{(3)}}{∥x^{(3)}∥}Δy.

By the uniform continuity of V, for any ϵ > 0 there exists a δ > 0 such that if

∥x^{(1)} − x^{(2)}∥ < δ and ∥x^{(3)} − x^{(4)}∥ < δ,

then

|V (x^{(1)}) − V (x^{(2)})| < ϵ and |V (x^{(3)}) − V (x^{(4)})| < ϵ.

From |V (x^{(1)}) − V (x^{(2)})| < ϵ, we obtain

−ϵ < V (x^{(1)}) − V (x^{(2)}) < ϵ.

Thus

V (x^{(1)}) > V (x^{(2)}) − ϵ.

From |V (x^{(3)}) − V (x^{(4)})| < ϵ we get

V (x^{(3)}) > V (x^{(4)}) − ϵ.

54 CHAPTER 2. STABILITY Figure 2.5: A two-dimensional illustration of the proof of the continuity of the function κ given by (2.19). From the definition of the function κ it follows that

V (x^{(2)}) ≥ V (x^{(3)})

and

V (x^{(4)}) ≥ V (x^{(1)}) .

Since V (x^{(1)}) > V (x^{(2)}) − ϵ, we obtain

V (x^{(1)}) > V (x^{(3)}) − ϵ

and therefore

−ϵ < V (x^{(1)}) − V (x^{(3)}) .  (2.21)

Since V (x^{(3)}) > V (x^{(4)}) − ϵ,

V (x^{(3)}) > V (x^{(1)}) − ϵ,

and thus

V (x^{(1)}) − V (x^{(3)}) < ϵ.  (2.22)

Combining (2.21) and (2.22), we get |V (x^{(1)}) − V (x^{(3)})| < ϵ,

Figure 2.6: Constructing the function ψ given by (2.23).

or equivalently,

|κ(y) − κ(y + Δy)| < ϵ

if

Δy < δ,

which proves the continuity of κ.

We are now ready to construct a function ψ that satisfies the hypotheses of the lemma. It has the form

ψ(r) = r min_{r≤y≤R} \frac{κ(y)}{y}  (2.23)

Note that ψ is continuous, ψ(0) = 0, and it is positive and strictly increasing—see Figure 2.6 for a graphical illustration of the way ψ is constructed from κ. The proof of the lemma is now complete.

□

We can characterize a positive definite function V in another way as in the following lemma.

Lemma 2.3 A function V : ℝ^{n} → ℝ is positive definite, with respect to x = 0, if and only if there exists a continuous nondecreasing function α : ℝ → ℝ, such that α(0) = 0, α(r) > 0 for r > 0, and V (x) ≥ α(r), where r = ∥x∥, and r ≤ R = max{q : B_{q}(0) ⊆ S}.

A possible choice for α is

α(r) = min_{r≤∥x∥≤R} V (x).

Note that the function α is nondecreasing because as r increases, the minimum is taken over a smaller region. Furthermore, α(r) > 0 whenever r > 0 since V (x) > 0 for x ∈ S \ {0}. The function α is continuous since V is continuous.

So far we only considered positive definite, or negative definite functions, that are timeinvariant. We now extend our definition of positive definiteness to time-varying functions of the form W (t, x). We first define a time-varying positive semi-definite function.

Definition 2.11 A real-valued function W : ℝ × ℝ^{n} → ℝ is positive semi-definite, with respect to x = 0, in a region S ⊆ ℝ^{n} if for all t and all x in S

(i) W is continuous and has continuous partial derivatives with respect to all of its arguments, that is, W ∈ C^{1},

(ii) W (t, 0) = 0,

(iii) W (t, x) ≥ 0.

Definition 2.12 A real-valued function W : ℝ × ℝ^{n} → ℝ is positive definite, with respect to x = 0, in a region S ⊆ ℝ^{n} if for all t and all x in S

(i) W ∈ C^{1},

(ii) W (t, 0) = 0,

(iii) there exists a positive definite time-invariant function V such that W (t, x) ≥ V (x).

We next introduce a notion of a decrescent function.

Definition 2.13 A time-varying function W : ℝ × ℝ^{n} → ℝ is decrescent in a region S if a positive definite function Ṽ exists such that for all x in S and all t

|W (t, x)| ≤ Ṽ (x),

that is, W tends to zero uniformly with respect to t as ∥x∥ → 0.

Consider now a dynamical system modeled by ẋ(t) = f(t, x(t)) subject to the initial condition x(t_{0}) = x_{0}. Suppose that the system has an isolated equilibrium point at x_{e}, that is, f(t, x_{e}) = 0 for all t. Let W (t, x) be a time-dependent real-valued function whose time derivative along the solution x(t) = x(t; t_{0}, x_{0}) of the given differential equation, is

\frac{dW (t, x)}{dt} = Ẇ (t, x) = \frac{∂W}{∂t} + (∇W)^{T} ẋ.

Then, we have the following theorem concerning the stability of the equilibrium point in the sense of Lyapunov.

Theorem 2.5 If in some neighborhood S of the equilibrium state x_{e}, there is a positive definite function W, with respect to x_{e}, such that its time derivative along any solution of ẋ(t) = f(t, x(t)) is negative semi-definite in S, then the equilibrium state x_{e} is stable. If W is also decrescent, then the equilibrium state is uniformly stable.

Proof Without loss of generality, we assume that x_{e} = 0. Since W is positive definite, by Lemma 2.2 there is a strictly increasing function ψ_{1}(r), ψ(0) = 0, such that

W (t, x) ≥ ψ_{1}(∥x∥),

where ∥x∥ ≤ R, and R is the radius of the largest ball contained in S. Let

ψ_{2}(t_{0}, r) = max_{∥x∥≤r} W (t_{0}, x).

Then

ψ_{1}(r) ≤ ψ_{2}(t_{0}, r).

Thus, for any ϵ > 0 there is a positive δ that depends on ϵ and t_{0}, such that

ψ_{2}(t_{0}, δ) < min{ψ_{1}(ϵ), ψ_{1}(R)}.

Figure 2.7 illustrates how to determine a δ given an ϵ. By assumption \frac{dW (t,x)}{dt} ≤ 0 along any solution of the system. Hence W (t, x) is a non-increasing function and therefore

W (t, x(t; t_{0}, x_{0})) ≤ W (t_{0}, x_{0}) for all t ≥ t_{0}.

If ∥x_{0}∥ < δ, then for t ≥ t_{0} we have

ψ_{1}(ϵ) > ψ_{2}(t_{0}, δ) ≥ W (t_{0}, x_{0}) ≥ W (t, x(t; t_{0}, x_{0})) ≥ ψ_{1}(∥x(t; t_{0}, x_{0})∥).

Therefore,

ψ_{1}(ϵ) > ψ_{1}(∥x(t; t_{0}, x_{0})∥).

Figure 2.7: Determining a δ given an ϵ.

Since ψ_{1} is strictly increasing, we have to have

∥x(t; t_{0}, x_{0})∥ < ϵ

for all t ≥ t_{0}. This means that the null solution is (locally) stable, which completes the proof of the first part of the theorem.

If W is also decrescent, then there is a nondecreasing function ψ̃_{2} : ℝ → ℝ, ψ̃_{2}(0) = 0, and

W (t, x) ≤ ψ̃_{2}(∥x∥)

for all x such that ∥x∥ ≤ R and all t. Indeed, since W (t, x) is positive definite and decrescent, there exists a positive definite V (x) such that W (t, x) ≤ V (x). Then, we can take ψ̃_{2} defined as

ψ̃_{2}(r) = max_{∥x∥≤r} V (x).

Let

ψ̃_{2}(δ) < min{ψ_{1}(ϵ), ψ_{1}(R)}.

Note that δ depends now only on ϵ, and not on t_{0} as before. To show the uniform stability of the null solution, we repeat the argument above with ψ̃_{2} replacing ψ_{2}. This completes the proof of the theorem.

□

The next theorem gives us a sufficiency condition for an equilibrium point to be uniformly asymptotically stable.

Theorem 2.6 If in some neighborhood S of the given equilibrium point x_{e} there is a positive definite, with respect to x_{e}, and decrescent function W such that its time derivative along any solution of the system is negative definite in S, then the equilibrium state is uniformly asymptotically stable.

Proof As before, without loss of generality, we assume that x_{e} = 0. The previous theorem ensures uniform stability of the equilibrium state x_{e}. Thus, only the uniform convergence remains to be shown. From the assumptions of the theorem it follows that there exist strictly increasing functions ψ_{1} and ψ_{3}, and a non-decreasing function ψ_{2} such that

ψ_{1}(0) = ψ_{2}(0) = ψ_{3}(0),
ψ_{1}(∥x∥) ≤ W (t, x) ≤ ψ_{2}(∥x∥),
ψ_{3}(∥x∥) ≤ −Ẇ (t, x),

for all t and all x such that ∥x∥ ≤ R. Given an ϵ_{1}. We define positive constants δ_{1}, δ_{2}, and T as follows:

ψ_{2}(δ_{1}) < ψ_{1}(R),  (2.24)
ψ_{2}(δ_{2}) < min{ψ_{1}(ϵ_{1}), ψ_{2}(δ_{1})},  (2.25)
T = ψ_{1}(R)/ψ_{3}(δ_{2}).  (2.26)

Note that δ_{1} is independent of t_{0}, and T depends on ϵ_{1} and δ_{1}. We shall prove the theorem in two stages. First, we shall show that for some t_{2} ∈ [t_{0}, t_{0} + T],

∥x(t_{2}; t_{0}, x_{0})∥ < δ_{2}

whenever ∥x_{0}∥ < δ_{1}. Then, we shall show that once ∥x(t_{2}; t_{0}, x_{0})∥ < δ_{2}, we have

∥x(t; t_{0}, x_{0})∥ < ϵ_{1} for all t ≥ t_{2}.

We begin with proving the claim that for some t_{2} ∈ [t_{0}, t_{0} + T],

∥x(t_{2}; t_{0}, x_{0})∥ < δ_{2}

whenever ∥x_{0}∥ < δ_{1}. We prove this claim by contradiction. We thus assume that for all t ∈ [t_{0}, t_{0} + T],

∥x(t; t_{0}, x_{0})∥ ≥ δ_{2}  (2.27)

if ∥x_{0}∥ < δ_{1}. Using (2.27), (2.26), and (2.24), we obtain

0 < ψ_{1}(δ_{2}) ≤ W [t_{0} + T, x(t_{0} + T; t_{0}, x_{0})]
= W (t_{0}, x_{0}) + ∫_{t_{0}}^{t_{0}+T} Ẇ [t, x(t; t_{0}, x_{0})]dt
≤ W (t_{0}, x_{0}) − ∫_{t_{0}}^{t_{0}+T} ψ_{3}(∥x(t; t_{0}, x_{0})∥)dt
≤ W (t_{0}, x_{0}) − T ψ_{3}(δ_{2})
≤ ψ_{2}(∥x_{0}∥) − T ψ_{3}(δ_{2})
< ψ_{2}(δ_{1}) − ψ_{1}(R)
< 0,

which is a contradiction, and thus we proved the first part of the theorem. We now proceed with the second part. From the first part we have that if ∥x_{0}∥ < δ_{1}, then for some t_{2} ∈ [t_{0}, t_{0} + T] we have ∥x(t_{2}; t_{0}, x_{0})∥ < δ_{2}. For t ≥ t_{2} we have

ψ_{1}(∥x(t; t_{0}, x_{0})∥) ≤ W (t, x(t; t_{0}, x_{0})) ≤ W (t_{2}, x(t_{2}; t_{0}, x_{0})) < ψ_{2}(δ_{2}) < ψ_{1}(ϵ_{1}).

Hence

ψ_{1}(∥x(t; t_{0}, x_{0})∥) < ψ_{1}(ϵ_{1}),

and since ψ_{1} is strictly increasing,

∥x(t; t_{0}, x_{0})∥ < ϵ_{1}

for all t ≥ t_{2} which proves the uniform convergence. The proof of the theorem is thus complete.

□

Example 2.3 Given a dynamical system model

ẋ(t) = [ \begin{matrix}−0.75 & 0.3536 \\ 0.3536 & −0.5 \end{matrix} ] x(t), x(0) = x_{0}.

The solution P = P^{T} to the Lyapunov matrix equation A^{T} P +P A = −Q, where Q = 2I_{2}, is

P = [ \begin{matrix}2 & \sqrt{2} \\ \sqrt{2} & 3 \end{matrix} ] .

The eigenvalues of P are

λ_{1} = λ_{min}(P) = 1, λ_{2} = λ_{max}(P) = 4.

2.6. STABILITY OF NONLINEAR SYSTEMS 61 In general, for a symmetric matrix P,

λ_{min}(P)∥x∥^{2} ≤ x^{T} P x ≤ λ_{max}(P)∥x∥^{2}.

Therefore, we can take

ψ_{1}(∥x∥) = λ_{min}(P)∥x∥^{2}, ψ_{2}(∥x∥) = λ_{max}(P)∥x∥^{2}, and ψ_{3}(∥x∥) = λ_{min}(Q)∥x∥^{2}. In our example, λ_{min}(Q) = 2.

• Given ε = 4. We will find δ > 0 such that if ∥x_{0}∥ ≤ δ, then ∥x(t)∥ ≤ ε for t ≥ 0.

We obtain δ = δ(ε) by solving the equation,

ψ_{1}(ε) = ψ_{2}(δ),

to get

δ = \sqrt{\frac{λ_{min}(P)}{λ_{max}(P)}}ε = 2.

• Given x_{0} = [ 8 6 ]^{T} and ε_{1} = 2. We will find T such that ∥x(t)∥ ≤ ε_{1} for all t ≥ T.

We have ∥x_{0}∥_{2} = 10. Let δ_{1} = 10. We then compute R, δ_{2}, and T from the equations:

ψ_{2}(δ_{1}) = ψ_{1}(R)
ψ_{2}(δ_{2}) = min (ψ_{1}(ε_{1}), ψ_{2}(δ_{1}))
T = \frac{ψ_{1}(R)}{ψ_{3}(δ_{2})}.

Using the first equation, we find

R = \sqrt{\frac{λ_{max}(P)}{λ_{min}(P)}}δ_{1} = 20.

We then solve for δ_{2} ≤ δ_{1} to get

δ_{2} = \sqrt{\frac{λ_{min}(P)}{λ_{max}(P)}}ε_{1} = 1.

Hence,

T = \frac{λ_{min}(P)R^{2}}{λ_{min}(Q)δ_{2}^{2}} = \frac{λ_{min}(P)\frac{λ_{max}(P)}{λ_{min}(P)}δ_{1}^{2}}{λ_{min}(Q)\frac{λ_{min}(P)}{λ_{max}(P)}ε_{1}^{2}} = \frac{λ_{max}^{2}(P)δ_{1}^{2}}{λ_{min}(Q)λ_{min}(P)ε_{1}^{2}} = 200.

The above theorems were concerned with local stability. We now proceed to prove sufficiency conditions for global stability. For this we need the following definitions. We first define a radially unbounded or coercive function.

Definition 2.14 A time-varying function W : ℝ × ℝ^{n} → ℝ is radially unbounded, or coercive, if as ∥x∥ → ∞, the function W tends to infinity uniformly with respect to t, that is, for any given positive real number M, there is a positive real number N so that W (t, x) > M for all t and all x whenever ∥x∥ > N.

The next two definitions are concerned with Lagrange stable equilibrium states.

Definition 2.15 The equilibrium state x_{e} is called Lagrange stable, or bounded, if for every real number d > 0 there exists a positive real number b = b(t_{0}, d) such that if

∥x_{0} − x_{e}∥ < d,

then

∥x(t; t_{0}, x_{0}) − x_{e}∥ < b

for all t ≥ t_{0}.

Definition 2.16 The equilibrium state x_{e} is uniformly bounded if the number b in Definition 2.15 can be taken independent of the initial time t_{0}.

We now define uniformly asymptotically stable in the large equilibrium state.

Definition 2.17 If the equilibrium state x_{e} is uniformly stable, uniformly bounded, and all trajectories are uniformly convergent to x_{e}, then this equilibrium state is called uniformly asymptotically stable in the large, or globally uniformly asymptotically stable.

The following theorem gives a sufficiency condition for an equilibrium state to be globally uniformly asymptotically stable.

Theorem 2.7 If there is a function W that is positive definite for all x ≠ x_{e}, W (t, x_{e}) = 0, decrescent, radially unbounded, and if its time derivative along any solution of the system is negative definite for all x, then the equilibrium state x_{e} is globally uniformly asymptotically stable.

Figure 2.8: Determining a constant b > 0 given d > 0.

Proof Assume, for simplicity, that x_{e} = 0. By Theorem 2.5, the equilibrium state x_{e} is uniformly stable. Similarly as in Theorem 2.6, we construct the functions ψ_{1}, ψ_{2}, and ψ_{3}. Since, by assumption, W is radially unbounded, ψ_{1}(∥x∥) → ∞ as ∥x∥ → ∞. Therefore, for any constant d > 0, there is a constant b > 0 such that ψ_{1}(b) > ψ_{2}(d). See Figure 2.8 for an illustration of the procedure that can be used to find the constant b. If ∥x_{0}∥ < d, then ∥x(t; t_{0}, x_{0})∥ < b for all t ≥ t_{0}. Indeed, since

ψ_{1}(b) > ψ_{2}(d) ≥ W (t_{0}, x_{0}) ≥ W (t, x(t; t_{0}, x_{0})) ≥ ψ_{1}(∥x(t; t_{0}, x_{0})∥),

then

ψ_{1}(b) > ψ_{1}(∥x(t; t_{0}, x_{0})∥).

The above implies that

∥x(t; t_{0}, x_{0})∥ < b

since ψ_{1} is strictly increasing. Hence all trajectories of the system are uniformly bounded. To prove their convergence to the equilibrium state, we define δ_{2}, T, and b, given d and ϵ_{1}, as

ψ_{2}(d) < ψ(b),
ψ_{2}(δ_{2}) < ψ_{1}(ϵ_{1}),
T = ψ_{1}(b)/ψ_{3}(δ_{2}).

If ∥x_{0}∥ < d, then for all t > t_{0} + T, we have

∥x(t; t_{0}, x_{0})∥ < ϵ_{1},

which can be shown using the same arguments as in the proof of Theorem 2.6 with b playing the role of R, and d playing the role of δ_{1}. This proves that all trajectories of the system converge uniformly to the equilibrium state. Hence, the equilibrium state is globally uniformly asymptotically stable.

□

We now give a theorem that can be used as a test for checking the instability of nonlinear dynamical systems. This Instability Test is illustrated in Figure 2.9.

Theorem 2.8 Let x_{e} be an isolated equilibrium state of the system ẋ = f(t, x), that is, f(t, x_{e}) = 0. If there exists a function W : ℝ × ℝ^{n} → ℝ such that in the region S = {x : ∥x − x_{e}∥ ≤ R}

(i) W ∈ C^{1},

(ii) W (t, x_{e}) = 0,

(iii) Ẇ (t, x) > 0 for all x ∈ S \ {x_{e}},

(iv) in every neighborhood of x_{e} in S, for any t_{0} there is at least one point x at which W (t_{0}, x) > 0,

then the equilibrium state x_{e} is unstable.

Proof Without loss of generality, assume that x_{e} = 0. By assumption (iv), for any t_{0} and any δ, 0 < δ < R there exists x_{δ}, 0 < ∥x_{δ}∥ < δ such that W (t_{0}, x_{δ}) > 0. The system trajectory x(t) = x(t; t_{0}, x_{δ}) cannot approach the equilibrium state x_{e} = 0 as t → ∞ because W (t, 0) = 0 but W (t_{0}, x_{δ}) > 0 and Ẇ (t, x) > 0 for x ≠ x_{e}. Since Ẇ is continuous, because W ∈ C^{1}, and the trajectory is bounded away from the equilibrium state, there is a constant m > 0 such that for all t ≥ t_{0},

Ẇ (t, x(t)) ≥ m > 0,

as long as

∥x(t)∥ ≤ R, x ≠ 0.

Hence,

W (t, x) = W (t_{0}, x_{δ}) + ∫_{t_{0}}^{t} Ẇ (s, x(s))ds ≥ W (t_{0}, x_{δ}) + m(t − t_{0}).  (2.28)

Figure 2.9: An illustration of the Instability Test.

The function W is continuous and therefore bounded on the compact set S. However, the right hand side of (2.28) is unbounded. Therefore the trajectory x(t; t_{0}, x_{δ}) cannot remain infinitely long in S. Thus, for some t_{1} > t_{0} the trajectory reaches the boundary of the region ∥x∥ ≤ R. Hence, given any ϵ, 0 < ϵ < R, then for any δ > 0, no matter how small, there is at least one trajectory such that ∥x_{δ}∥ < δ but ∥x(t; t_{0}, x_{δ})∥ ≥ ϵ for some t_{1} > t_{0}. Therefore, the equilibrium state is unstable.

□

Corollary 2.1 If there exists a function W : ℝ × ℝ^{n} → ℝ such that in the region S = {x : ∥x − x_{e}∥ ≤ R}

(i) W ∈ C^{1},

(ii) W (t, x_{e}) = 0,

(iii) Ẇ (t, x) < 0 for all x ∈ S \ {x_{e}},

(iv) in every neighborhood of x_{e} in S, for any t_{0} there is at least one point x at which W (t_{0}, x) < 0,

then the equilibrium state x_{e} is unstable.

2.7 Lyapunov Like Analysis

Lemma 2.4 (Barbalat) Given a differentiable function f = f(t) : ℝ → ℝ such that ḟ is uniformly continuous. If lim_{t→∞} f(t) exists, then

lim_{t→∞} ḟ(t) = 0.

Proof We prove the lemma by contraposition. We assume that ḟ(t) is bounded away from 0 as t → ∞. Therefore, we can construct an infinite sequence of t_{i}’s such that t_{i} → ∞ as i → ∞, and for each t_{i} we have |ḟ(t_{i})| ≥ ε for some ε > 0. By assumption ḟ is uniformly continuous. Hence, given ε/2 > 0 there is δ_{ε/2} > 0 so that for any t and t_{i} such that |t − t_{i}| < δ_{ε/2}, we have

| |ḟ(t)| − |ḟ(t_{i})| | ≤ |ḟ(t) − ḟ(t_{i})| < \frac{ε}{2}.  (2.29)

Suppose that we have

ḟ(t_{i}) ≥ ε.

Then, it follows from (2.29) that for any t such that |t − t_{i}| < δ_{ε/2},

\frac{ε}{2} > | |ḟ(t_{i})| − |ḟ(t)| | ≥ |ε − |ḟ(t)|| > ε − ḟ(t).

Hence,

ḟ(t) > \frac{ε}{2}.

Using the above, we obtain

∫_{t_{i}}^{t_{i}+δ_{ε/2}} ḟ(t)dt > ∫_{t_{i}}^{t_{i}+δ_{ε/2}} \frac{ε}{2}dt
= \frac{ε}{2}δ_{ε/2}.  (2.30)

However,

∫_{t_{i}}^{t_{i}+δ_{ε/2}} ḟ(t)dt = f(t_{i} + δ_{ε/2}) − f(t_{i}).  (2.31)

Combining (2.30) and(2.31) yields

f(t_{i} + δ_{ε/2}) − f(t_{i}) > \frac{ε}{2}δ_{ε/2} for every i,

which contradicts the assumption that f(t) has a finite limit as t → ∞. The proof of the lemma is complete.

□

A simple sufficient condition for a differentiable function to be uniformly continuous is boundedness of its derivative. Thus if a function f is twice differentiable, then its derivative, ḟ, is uniformly continuous if its second derivative, f̈, is bounded.

The following lemma follows immediately from Barbalat’s lemma.

Lemma 2.5 (“Lyapunov-Like Lemma”) Given a real-valued function W (t, x) such that

1. W (t, x) is bounded below;

2. Ẇ (t, x) is negative semi-definite;

3. Ẇ (t, x) is uniformly continuous in time,

then

Ẇ (t, x) → 0 as t → ∞.

Example 2.4 Investigate stability of the following dynamical system model:

[ ẋ_{1} ; ẋ_{2} ] = [ \begin{matrix}−2x_{1} + x_{2}u \\ −x_{1}u \end{matrix} ] ,  (2.32)

where the input u is bounded.

Consider the Lyapunov function candidate

V = ∥x∥^{2} = x_{1}^{2} + x_{2}^{2}.

Note that V is bounded below. The time derivative of V evaluated on the trajectories of the system model (2.32) is

V̇ = 2x_{1}ẋ_{1} + 2x_{2}ẋ_{2} = 2x_{1} (−2x_{1} + x_{2}u) − 2x_{2}x_{1}u = −2x_{1}^{2} ≤ 0.

Thus, the system modeled by (2.32) is uniformly stable. Furthermore, using arguments similar to the ones in the proof of Theorem 2.7, we can show the system’s trajectories are uniformly bounded, and hence x_{1} and x_{2} are bounded. We now use Lemma 2.5 to show that in addition, x_{1} → 0 as t → ∞. To use this Lemma, we have to show that V̇ is uniformly continuous in t. For this it is enough to show that the derivative of V̇ is bounded. The derivative of V̇ is

V̈ = −4x_{1}ẋ_{1} = −4x_{1} (−2x_{1} + x_{2}u) ,

which is bounded because u is bounded by assumption and we showed that x_{1} and x_{2} are also bounded. Hence, V̇ is uniformly continuous. The function V satisfies all the conditions of Lemma 2.5 and therefore x_{1} → 0 as t → ∞.

2.8 Nonlinear Discrete-Time System Stability

2.8.1 Notation and Definitions

In this section, we analyze stability of dynamical systems modeled by the difference equation

x(k + 1) = F (x(k)),  (2.33)

where k ∈ Z_{+}, Z_{+} denotes the set of nonnegative integers, x : Z_{+} → ℝ^{n}, that is, x(k) is the state of the system model (2.33) at time k, and F : ℝ^{n} → ℝ^{n}. We assume that the vector-valued function F is continuous on ℝ^{n}, that is, if x^{j} → y then F (x^{j}) → F (y). If x is a vector, then F (x) is a vector and is the value of the function F at x.

If x is a function, the symbol F x denotes the composition of functions

(F x) (k) = F (x(k)).

Thus the equation x(k + 1) = F (x(k)) can also be written as x(k + 1) = F x(k). The solution to the initial value problem

x(k + 1) = F x(k), x(0) = x^{0}

is

x(k) = x^{k} = F^{k} (x^{0}) ,

where F^{k} is the kth iterate of F, that is, F^{0} = I_{n}, and F^{k} = F F^{k−1}.

A state x_{e} ∈ ℝ^{n} is a fixed point or an equilibrium state of the system x(k + 1) = F x(k) if F (x_{e}) = x_{e}. Thus the solution of the system that starts at x_{e} stays at x_{e} for all k ≥ 0.

A subset A ⊂ ℝ^{n} is called invariant if F (A) ⊂ A, that is, if x ∈ A then F (x) ∈ A. It is clear that an equilibrium solution is invariant.

The solution through x^{0} is bounded if there is a constant M such that ∥F^{k} (x^{0}) ∥ ≤ M for all k ≥ 0.

Consider again a particular solution F^{k} (x^{0}). A point y ∈ ℝ^{n} is a limit point or a point of accumulation of F^{k} (x^{0}) if there is a subsequence k_{i} such that F^{k_{i}} (x^{0}) → y. The set of all limit points of F^{k} (x^{0}) is called the limit set and denoted Ω (x^{0}).

A sequence {x^{k}} is said to approach a set A if

ρ (x^{k}, A) = inf {∥x^{k} − y∥ : y ∈ A} → 0.

Lemma 2.6 Let F be continuous. If F^{k} (x^{0}) is bounded for all k ≥ 0, then Ω (x^{0}) is non-empty, compact invariant set. Furthermore, F^{k} (x^{0}) → Ω (x^{0}) as k → ∞.

Proof Since {x^{k}} is bounded, by the Bolzano-Weierstrass theorem, stated on page ??, there is at least one limit point of {x^{k}} and so Ω (x^{0}) is non-empty and bounded. The set Ω (x^{0}) contains only its limit points and hence it is closed. Since it is also bounded, it is compact.

We now show that Ω (x^{0}) is invariant. For this let y ∈ Ω (x^{0}). Hence, there is a subsequence k_{i} such that x^{k_{i}} → y. Since F is continuous, x^{k_{i}+1} = F (x^{k_{i}}) → F (y) as i → ∞. Therefore F (y) ∈ Ω (x^{0}) which implies that Ω (x^{0}) is invariant. Note that we do not have to have y = F (y).

Finally, we show that F^{k} (x^{0}) → Ω (x^{0}) as k → ∞. We prove the statement by contradiction. First observe that ρ (x^{k}, Ω (x^{0})) is bounded since both {x^{k}} and Ω (x^{0}) are. Assume that ρ (x^{k}, Ω (x^{0})) does not converge to zero. Then, there is a subsequence k_{i} such that x^{k_{i}} → y and ρ (x^{k_{i}}, Ω (x^{0})) → a > 0. But since y ∈ Ω (x^{0}),

ρ (x^{k_{i}}, Ω (x^{0})) ≤ ∥x^{k_{i}} − y∥ → 0,

and thus ρ (x^{k_{i}}, Ω (x^{0})) → 0 which is a contradiction. The proof of the lemma is now complete.

□

2.8.2 LaSalle’s Invariance Principle

Let V : ℝ^{n} → ℝ. Define the Lyapunov difference of V relative to x(k + 1) = F x(k), or F, as

ΔV (x) = V (F (x)) − V (x).

If x(k) is a solution of x(k + 1) = F x(k), then the Lyapunov difference of V relative to F is

ΔV (x(k)) = V (x(k + 1)) − V (x(k)).

If ΔV (x(k)) ≤ 0, then V is non-increasing along solutions of the system.

We say that V is a Lyapunov function for x(k + 1) = F x(k) on a set G ⊆ ℝ^{n} if

1. V is continuous on ℝ^{n}, and

2. ΔV (x(k)) ≤ 0 for all x(k) ∈ G.

Theorem 2.9 (LaSalle’s Invariance Principle) If

1. V is a Lyapunov function for x(k + 1) = F x(k) in G, and

2. the solution F^{k} (x^{0}) is bounded and in G,

then, there is a number c such that F^{k} (x^{0}) → M∩V^{−1}(c), where M is the largest invariant set contained in the set E = {x ∈ ℝ^{n} : ΔV = 0} ∩ Ḡ, where Ḡ denotes the closure of G.

Proof Since x^{k} = F^{k} (x^{0}) is bounded and in G, by Lemma 2.6 the limit set Ω (x^{0}) is non-empty and in G. Furthermore, x^{k} tends to Ω (x^{0}). Now {V (x^{k})} is non-increasing and bounded below, so V (x^{k}) → c for some c. If y ∈ Ω (x^{0}), then there is a subsequence k_{i} such that x^{k_{i}} → y, and hence V (x^{k_{i}}) → V (y), that is, V (y) = c. Thus V (Ω (x^{0})) = c, and hence Ω (x^{0}) ⊂ V^{−1}(c). Since V (Ω (x^{0})) = c and Ω (x^{0}) is invariant, we have ΔV (Ω (x^{0})) = 0. In summary

x^{k} → Ω (x^{0}) ⊂ {x ∈ ℝ^{n} : ΔV = 0} ∩ Ḡ ∩ V^{−1}(c).

Since Ω (x^{0}) is invariant we have Ω (x^{0}) ⊂ M, and the proof of the theorem is complete.

□

Using the above theorem we can study stability properties of difference equations.

The general approach of the LaSalle Invariance Principle can be used to prove a result concerning the instability of isolated equilibrium states. An equilibrium point x_{e} is isolated if there is an ε > 0 such that there is no other equilibrium point, other than x_{e}, in the ball B_{ε} (x_{e}) = {x : ∥x − x_{e}∥ < ε}.

Theorem 2.10 Let x_{e} be an isolated equilibrium state of x(k + 1) = F x(k). Let ΔV be positive definite with respect to x_{e}, and let V take positive values arbitrarily close to x_{e}, and V (x_{e}) = 0, then x_{e} is unstable.

Proof We prove the theorem by contradiction. We thus assume that x_{e} is stable. Let ε > 0 be so small that ΔV (x) > 0 for all x ∈ B_{ε} (x_{e}) \ {x_{e}}. Since by assumption x_{e} is stable, there is δ > 0 so that if x^{0} ∈ B_{δ} (x_{e}) then x^{k} = F^{k} (x^{0}) ∈ B_{ε} (x_{e}) for all k ≥ 0. By assumption there is a point x^{0} such that V (x^{0}) > 0. Since x^{k} is bounded and remains in B_{ε} (x_{e}), by Lemma 2.6 the sequence {x^{k}} tends to {x : ΔV (x) = 0} ∩ B_{ε} (x_{e}) = x_{e}. Another way to arrive at the above observation is as follows. Note that −V is a Lyapunov function because Δ(−V) < 0. Since x^{k} is bounded we can apply the Invariance Principle to conclude that x^{k} tends to {x : Δ(−V (x)) = 0}. But {x : Δ(−V (x)) = 0} = {x_{e}}. Thus, indeed x^{k} → x_{e}. Using the above, we conclude that V (x^{k}) → V (x_{e}) = 0. However, ΔV (x^{k}) > 0 and so

V (x^{k}) ≥ V (x^{k−1}) ≥ ⋯ V (x^{0}) > 0.

Hence, V (x^{k}) > 0. The above contradiction completes the proof.

□

2.9 Notes

A classical text on stability of dynamical systems is by Hahn [41]. The accounts of stability analysis using the direct method of Lyapunov are by Yoshizawa [129] and Rouche et al. [97]. Stability of discrete dynamical systems is presented by Hahn in [40] and LaSalle [66]. Extensive coverage of stability issues can also be found in Willems [126], Slotine and Li [106], Vidyasagar [118], and Khalil [59].

There is an interesting story behind the origins of the Routh test. The problem of investigating the stability of a liner dynamical system in terms of the location of the roots of its characteristic equation was posed by Maxwell [75]. On page 271 of his 1868 paper, Maxwell writes, “This condition is mathematically equivalent to the condition that all the possible roots, and all the possible parts of the impossible roots, of a certain equation shall be negative.” By impossible roots, Maxwell means complex roots and possible parts mean real parts in today’s terminology. He continues, “I have not been able completely to determine these conditions for equations of a higher degree than the third; but I hope that the subject will obtain the attention of mathematicians.” A complete general solution to the problem was published by Routh in 1877. Maxwell and Routh were classmates at Cambridge University. In 1854, Routh was placed first in their final math examination ahead of Maxwell. A story says, Smith [109, page 55], that apparently Maxwell was so sure of coming first on this exam that he did not even bother to rise early to hear the exam results read out in the Senate House. Maxwell sent his servant instead. When the servant came back, Maxwell asked, “Well, tell me who’s second!”. Obviously, he did not expect to receive the reply, “You are, sir”. Years later Routh received the prestigous Adams Prize, in 1877, for solving the problem posed by Maxwell in 1868. According to an account by Truxal [115, page 527], Routh presented his solution with the opening remark, “It has recently come to my attention that my good friend James Clerk Maxwell has had difficulty with a rather trivial problem....”

72 CHAPTER 2. STABILITY

Dynamical System Control

3.1 What Is a System?

A system can be viewed as a collection of interacting components. The term system can be used to describe a physical, biological, or economical unit. An electric motor, an airplane, as well as a biological unit such as the human arm are examples of systems. A system is characterized by two properties. They are:

1. interrelations between the components that are contained within the system, and

2. the system boundaries that separate the components within the system from the components outside.

The system boundaries can be real or imagined. They are elastic in the sense that we may choose, at any stage of the system analysis, to consider only a part of the original system as a system on its own. We call it a subsystem of the original system. On the other hand, we may decide to expand the boundaries of the original system to include new components. The interactions between the system components are governed by physical laws. Thus, we can characterize the system by its physical quantities. In dealing with systems, we will be interested in the effects of external quantities upon the behavior of the system quantities. We refer to the external quantities acting on the system as the inputs to the system. The condition or the state of the system are described by the state variables. The state variables provide the information that together with the knowledge of the system inputs enable us to determine the future state of the system. A dynamical system is one whose state changes with time. In practice it is often not possible or too expensive to measure or determine the values of all of the state variables. Instead, only their subset or combination can be

[figure: Input; System; Output]

Figure 3.1: Schematic representation of a system.

measured. The system quantities whose behavior can be measured or observed are referred to as the system outputs. In Figure 3.1, we schematically represent system’s boundaries using a box. The input signal is processed by the system to provide an output signal variable.

In engineering applications, when dealing with dynamical systems, we are interested in specifying the system inputs that force the system states or outputs to behave with time in some prespecified manner. That is, we are interested in controlling the system states or outputs. This is accomplished by means of a controller whose task is to produce the required system’s inputs. An interconnection of the system and a controller is called a control system. Constructing a controller is a part of the control problem. The essential elements of the control problem, as described, for example, by Owens [87, page 181], are:

1. a specified objective for the system,

2. a model of a dynamical system to be controlled,

3. a set of admissible controllers, and

4. a means of measuring the performance of any given control strategy to evaluate its effectiveness.

We now examine these elements one by one. The objective of a control system is to complete some specified task. This can be expressed as a combination of constraints on the output or state variables and limits on the time available to complete the control objective. For example, the objective of a controller might be to force the output, of a single output system, to settle within a certain percentage of the given value after prespecified time.

The first step in the controller design procedure is the construction of a truth model of the dynamics of the process to be controlled. The truth model is a simulation model that includes all the relevant characteristics of the process. The truth model is too complicated for use in the controller design. Thus, we need to develop a simplified model that can be used to design a controller. Such a simplified model is labeled by Friedland [33] as the design model. The design model should capture the essential feature of the process. A common model of a dynamical system is the finite set of ordinary differential equations of the form:

ẋ(t) = f(t, x(t), u(t)), x(t_{0}) = x_{0}
y(t) = h(t, x(t), u(t)),

where the state x ∈ ℝ^{n}, the input u ∈ ℝ^{m}, the output y ∈ ℝ^{p}, f and h are vector valued functions with f : ℝ × ℝ^{n} × ℝ^{m} → ℝ^{n} and h : ℝ × ℝ^{n} × ℝ^{m} → ℝ^{p}. Another common model of a dynamical system is the finite set of difference equations,

x(k + 1) = f(k, x(k), u(k)), x(k_{0}) = x_{0}
y(k) = h(k, x(k), u(k)),

where x(k) = x(kh), u(k) = u(kh), h is the sampling interval, and k ≥ 0 is an integer.

In our subsequent discussion, we regard the dynamical system and its model as equivalent. During the modeling process, properties of the system, physical constraints, and technical requirements should be taken into account. By itself the pair (f(t, x, u), h(t, x, u)) has little engineering significance until constraints on the state x, the control u, and the output y are also given. Control input signals are obtained from physical devices capable of providing only a limited amount of energy. We term the class of controllers that can be considered for the given control design problem as the set of admissible controllers.

We can evaluate the performance of any given control law by visual inspection of the transient characteristics of the controlled system after the design is complete. But this method of assessment of the effectiveness of the given control strategy is highly inaccurate. We therefore try to quantify the means of evaluating performance in the form of a performance index or cost functional at the beginning of the design process. We construct a numerical index of performance whose value reflects the quality of any admissible controller in accomplishing the system objective. This performance index is assumed to have the property that its numerical value decreases as the quality of the controller increases. This is the reason why we sometimes refer to the performance index as the penalty functional. The “best” controller then is the one that produces the smallest value of the performance index. We call an admissible controller that simultaneously ensures the completion of the system objective and the minimization of the performance index an optimal controller for the system.

[figure: Controller; Reference input; Desired output; Controller output; Plant input; Plant; Actual output]

Figure 3.2: Schematic representation of an open-loop control system.

3.2 Open-Loop Versus Closed-Loop

We distinguish between two types of control systems. They are:

• open-loop control systems, and

• closed-loop control systems. An open-loop control system usually contains:

1. A process to be controlled, labeled plant,

2. The controlling variable of the plant, called the plant input, or just input for short,

3. The controlled variable of the plant, called the plant output, or just output for short,

4. A reference input, which dictates the desired value of the output,

5. A controller that acts upon the reference input in order to form the system input which

would force the behavior of the output in accordance with the reference signal. Note that the plant and controller themselves can be considered as systems on their own. A schematic representation of an open-loop system is depicted in Figure 3.2. In an open-loop control system the output has no influence on the input or reference signal. The controller operates without taking into account the output. Thus, the plant input is formed with no influence of the output. A household appliance such as an iron is a simple example of an open-loop control system. In this example, we consider the iron itself as the plant or process to be controlled. Here we set the temperature control mechanism for the desired temperature. This setting, for example very hot, is the reference input. We can view the potentiometer,

[figure: Desired output; Reference input; +; −; Actuating signal; Controller; Controller output; Plant input; Plant; Actual output; Sensor; Measured output]

Figure 3.3: Schematic representation of a closed-loop control system.

inside the iron, as a controller, or a control actuator. It regulates the temperature of the iron by allowing the necessary amount of resistance to produce the desired temperature. The plant’s output is the temperature of the heated iron. Not every control system is an openloop system. Another type of a control system is a closed-loop system. We can convert an open-loop system into a closed-loop system by adding, to an open-loop system, the following components:

6. The feedback loop where the output signal is measured with a sensor and then the measured signal is fed back to the summing junction,

7. The summing junction, where the measured output signal is subtracted from the reference, command, input signal in order to generate an error signal, also labeled as an actuating signal.

A schematic of a closed-loop control system is shown in Figure 3.3. In a closed-loop system the error signal causes an appropriate action of the controller, which in turn “instructs” the plant to behave in a certain way in order to approach the desired output, as specified by the reference input signal. Thus, in the closed-loop system, output information is fed back to the controller, and the controller then appropriately modifies the plant output behavior. A controller, also called a compensator, can be placed either in the forward loop, or in the feedback loop.

If the goal of the controller is to maintain the output at a constant value, then we have a regulator control system. A “centrifugal speed regulator”, or the “flyball governor”, commonly known as Watt’s governor, is an example of a regulator in a closed-loop system. A schematic of Watt’s governor is shown in Figure 3.4. The task of centrifugal speed regulator 78 CHAPTER 3. DYNAMICAL SYSTEM CONTROL

Figure 3.4: A simplified schematic of the flyball governor. is to control setting of a throttle valve automatically to maintain the desired engine speed. The nominal speed is set by adjusting the valve in the throttle. As the engine speeds up the weights are thrown outwards, the throttle partially closes and the engine slows down. As the engine slows down below its nominal speed, the throttle is opened up and the engine gains its speed. The operation of an engine equipped with a centrifugal speed regulator can be represented using the block diagram of Figure 3.3. The reference input to the closed-loop system is the desired engine speed. The desired engine speed is obtained by appropriately setting the valve. The system output is the actual engine speed. The flyball governor is our controller, while the engine represents the plant. The control signal generated by the controller is the setting of the throttle valve.

We mention at this juncture that Watt’s application of the centrifugal speed governor in 1788, to regulate the speed of the steam engine, marks the starting point for the development of automatic control, and in particular feedback control, as a science [72]. Before 1788, steam engines were controlled manually using the throttle valve. The flyball governor spread widely and quickly. It has been estimated that by 1868 there were about 75,000 flyball governors working in England alone [72]. The flyball governor is credited for the hatching of the Industrial Revolution. We devise a mathematical model of this revolutionary invention in the following section, and in Section ?? we perform its stability analysis.

If the goal of the controller is to force the output to follow a desired trajectory, then we have a servomechanism or tracking control system. To design a controller, we need first to analyze the system quantitatively. The analysis requires a mathematical or linguistic description of the interrelations between the system quantities themselves as well as the interrelations between system quantities and system inputs. The mathematical description of these interrelations is called mathematical modeling. We devote the next section to mathematical modeling. Constructing models using linguistic description of the process to be controlled will be discussed in Subsection 7.4.1.

3.3 Mathematical Modeling

The goal of mathematical modeling is to provide a mathematical description of the interrelations between the system quantities as well as the relations between these quantities and external inputs. In general, the more accurate the mathematical model is, the more complex are the modeling equations. This means that during the mathematical modeling process we may be forced to finding a middle course between accuracy of the model and its complexity. Our goal is to generate the most accurate possible description of the given system while still

Figure 3.5: Schematic diagram of mathematical modeling process.

being able to carry out the model analysis and computations in a reasonable amount of time. At first, we tend to make oversimplifying assumptions in order to obtain a very simple model. However, we should follow the Principle of Maximum Simplicity which states that ”everything should be as simple as possible—but no simpler”. After extracting some information about the system behavior based on this first, possibly crude model, it maybe desirable to construct a more accurate model. Thus, we try to create a mathematical description of a given physical process which incorporates realistic assumptions and physical constraints. We then extract information from this model and compare it with the experimental data. We perform this comparative study to determine the value of the obtained model. If the model appears to be inadequate for our needs, then it must be adjusted. The process of mathematical modeling is depicted in the schematic diagram in Figure 3.5. In the process of mathematical modeling, one can use different types of equations and their combinations like algebraic, ordinary and partial differential equations, difference, integral, or functional equations. In this book, only difference and ordinary differential equations will be used to model dynamical systems.

We now derive a mathematical model the centrifugal governor. In our modeling, we follow Pontryagin [91, pages 213–220]. For an alternative approach to modeling the governor see Maxwell [75] and Fuller [34]. Some, like for example MacFarlane [72, page 250], claim

Figure 3.6: Forces acting on the weights in the governor.

that Maxwell founded the theory of automatic control systems with his 1868 paper “On governors”. However, this landmark paper is written in a terse and cryptic style. Fuller [34] does a wonderful job elucidating, correcting, and expanding Maxwell’s paper. Mayr [76, Chapter X] gives a fascinating historical account of the application of the centrifugal governor to the speed regulation of the steam engine and tracks the origins of the flyball governor.

A schematic of the governor is depicted in Figure 3.4, while forces acting on the governor are shown in Figure 3.6. For simplicity of notation, we assume that the length of the arms, on which the weights are fastened, are of unity length. If the weights rotate with angular velocity ω_{gov}, then the arms form an angle φ with the vertical position. Let v_{weight} denote the linear velocity of the weights, and r be the radius of the circle along the weights rotate. Then, v_{weight} = ω_{gov}r. Note that r = sin φ. The resulting centrifugal force acting on each weight is

m\frac{v_{weight}^{2}}{r} = mω_{gov}^{2}r

= mω_{gov}^{2} sin φ.

Simultaneously, a gravitational force mg acts on each weight. In addition, a frictional force in the hinge joints acts on each weight. We assume that the frictional force is proportional to φ̇ with the friction coefficient b. Summing forces along the axis perpendicular to an arm yields an equation of motion of the weight

mφ̈L = mω_{gov}^{2} sin φ cos φ − mg sin φ − bφ̇
= \frac{1}{2}mω_{gov}^{2} sin 2φ − mg sin φ − bφ̇.  (3.1)

Note that mφ̈L = mφ̈ since by assumption L = 1. The governor is connected to the engine via a gear with the gear ratio N such that

ω_{gov} = Nω,  (3.2)

where ω is the angular velocity of the flywheel. Substituting (3.2) into (3.1) gives

mφ̈ = \frac{1}{2}mN^{2}ω^{2} sin 2φ − mg sin φ − bφ̇.

Let I be the moment of inertia of the flywheel, τ_{1} the torque delivered by the engine, and τ_{l} the load torque. Then, the equation modeling the engine is

Iω̇ = τ_{1} − τ_{l}.  (3.3)

The collar of the governor is connected with the valve so that at the nominal value of φ = φ_{0} the engine delivers the nominal torque τ_{d} that results in the desired angular velocity ω = ω_{d} of the flywheel. We model the interaction of the collar and the valve using the equation

τ_{1} = τ_{d} + κ(cos φ − cos φ_{0}),  (3.4)

where κ > 0 is a proportionality constant that depends on the governor. Substituting (3.4) into (3.3) yields

Iω̇ = τ_{1} − τ_{l}
= τ_{d} + κ(cos φ − cos φ_{0}) − τ_{l}
= κ cos φ − τ,

where τ = τ_{l} − τ_{d} + κ cos φ_{0}. We thus have two equations that model the governor. They are:

mφ̈ = \frac{1}{2}mN^{2}ω^{2} sin 2φ − mg sin φ − bφ̇,
Iω̇ = κ cos φ − τ.

3.4. CENTRIFUGAL GOVERNOR STABILITY ANALYSIS 83 We use the state variables

x_{1} = φ, x_{2} = φ̇, and x_{3} = ω

to represent the above model in state space format,

\begin{matrix}ẋ_{1} & = & x_{2} \\ ẋ_{2} & = & \frac{1}{2}N^{2}x_{3}^{2} sin 2x_{1} − g sin x_{1} − \frac{b}{m}x_{2} \\ ẋ_{3} & = & \frac{κ}{I} cos x_{1} − \frac{τ}{I} . \end{matrix} \}  (3.5)

3.4 Centrifugal Governor Stability Analysis

We first linearize the model of Watt’s governor that we derived in Appendix ??. Recall that the equations modeling the governor have the form:

ẋ_{1} = x_{2}
ẋ_{2} = \frac{1}{2}N^{2}x_{3}^{2} sin 2x_{1} − g sin x_{1} − \frac{b}{m}x_{2}
ẋ_{3} = \frac{κ}{I} cos x_{1} − \frac{τ}{I} .

We assume that the equilibrium state has the form

x_{e} = [ x_{1e} 0 x_{3e} ]^{T} .

Equating the right-hand sides of the engine-governor equations to zero yields

x_{2e} = 0
N^{2} x_{3e}^{2} = \frac{g}{cos x_{1e}}
cos x_{1e} = \frac{τ}{κ}.

The Jacobian matrix of the engine-governor model evaluated at the equilibrium state is

\frac{∂f}{∂x}|_{x=x_{e}} = [ \begin{matrix}0 & 1 & 0 \\ N^{2}x_{3e}^{2} cos 2x_{1e} − g cos x_{1e} & −\frac{b}{m} & N^{2}x_{3e} sin 2x_{1e} \\ −\frac{κ}{I} sin x_{1e} & 0 & 0 \end{matrix} ] .

We use the trigonometric identities cos 2α = cos^{2} α − sin^{2} α, sin 2α = 2 sin α cos α, and the previously obtained relation N^{2} x_{3e}^{2} = \frac{g}{cos x_{1e}} to eliminate N^{2} x_{3e}^{2} from the Jacobian matrix. Performing needed manipulations, we obtain

\frac{∂f}{∂x}|_{x=x_{e}} = [ \begin{matrix}0 & 1 & 0 \\ −\frac{g sin^{2} x_{1e}}{cos x_{1e}} & −\frac{b}{m} & \frac{2g sin x_{1e}}{x_{3e}} \\ −\frac{κ}{I} sin x_{1e} & 0 & 0 \end{matrix} ] .

Hence, the linearized model of the engine-governor system can be represented as

[ \begin{matrix}Δẋ_{1} \\ Δẋ_{2} \\ Δẋ_{3} \end{matrix} ] = [ \begin{matrix}0 & 1 & 0 \\ −\frac{g sin^{2} x_{1e}}{cos x_{1e}} & −\frac{b}{m} & \frac{2g sin x_{1e}}{x_{3e}} \\ −\frac{κ}{I} sin x_{1e} & 0 & 0 \end{matrix} ] [ \begin{matrix}Δx_{1} \\ Δx_{2} \\ Δx_{3} \end{matrix} ] .  (3.6)

We will now use the Routh stability criterion to analyze stability of the linearized model of the flyball governor. The governor’s linearized model is given by (3.6). The model’s characteristic equation is

det [ \begin{matrix}s & −1 & 0 \\ \frac{g sin^{2} x_{1e}}{cos x_{1e}} & s + \frac{b}{m} & −\frac{2g sin x_{1e}}{x_{3e}} \\ \frac{κ}{I} sin x_{1e} & 0 & s \end{matrix} ] = s^{3} + \frac{b}{m}s^{2} + \frac{g sin^{2} x_{1e}}{cos x_{1e}} s + \frac{2gκ sin^{2} x_{1e}}{Ix_{3e}}
= s^{3} + a_{2}s^{2} + a_{1}s + a_{0}.  (3.7)

The coefficients of the above third order polynomial are all positive. Therefore, the necessary condition for stability is satisfied. We next construct the Routh array for (3.7) to get,

\begin{matrix}s^{3} & 1 & a_{1} \\ s^{2} & a_{2} & a_{0} \\ s^{1} & \frac{1}{a_{2}} (a_{2}a_{1} − a_{0}) \\ s^{0} & a_{0} \end{matrix}  (3.8)

Since the characteristic polynomial coefficients are all positive, the linearized model is asymptotically if and only if

a_{2}a_{1} − a_{0} = \frac{b}{m} \frac{g sin^{2} x_{1e}}{cos x_{1e}} − \frac{2gκ sin^{2} x_{1e}}{Ix_{3e}} > 0.  (3.9)

Let

ν = \frac{x_{3e}}{2κ cos x_{1e}} .

Using the above notation, we express (3.9) as

\frac{bI}{m} ν > 1.  (3.10)

It follows from condition (3.10) that 1. an increase of the mass m of the weights of the governor has a deteriorating effect on

the governor’s stability;

2. a decrease of the friction coefficient b has also a harmful effect on stability;

3. a decrease of the moment of inertia I of the flywheel harms stability;

4. a decrease of the coefficient ν harms stability.

The above deductions are attributed by Pontryagin [91, page 219] to the Russian engineer Vyshnegradskiy. Vyshnegradskiy used these observations to propose, in 1876, design rules for governors to regulate engines without engine speeds fluctuating.

3.5 Robust Controller Design

In this section, we consider the problem of constructing stabilizing state-feedback controllers for nonlinear dynamical systems whose mathematical models contain uncertainties. The design of the controllers is based on the Lyapunov theory presented in the previous sections. The class of dynamical systems considered here is modeled by

ẋ(t) = f(t, x(t)) + G(t, x(t)) (u(t) + ξ(t, x(t))) ,  x(t_{0}) = x_{0},  (3.11)

where x(t) ∈ ℝ^{n}, u(t) ∈ ℝ^{m}, and ξ(t, x(t)) ∈ ℝ^{m} represents the lumped uncertainties. We assume that the uncertain element ξ is bounded by a known continuous non-negative function ρ, that is,

∥ξ(t, x(t))∥ ≤ ρ(t, x(t)).

We also assume that the equilibrium state x = 0 of the uncontrolled nominal system

ẋ(t) = f(t, x(t))  (3.12)

is globally uniformly asymptotically stable, and that we were able to construct a Lyapunov function W : ℝ × ℝ^{n} → ℝ corresponding to this equilibrium state. The Lyapunov function W is assumed to satisfy the following conditions: there exist three continuous and strictly increasing functions ψ_{i}, i = 1, 2, 3, such that

\begin{matrix}ψ_{i}(0) & = & 0, i = 1, 2, 3, \\ lim_{r→∞} ψ_{1}(r) & = & ∞, \\ ψ_{1}(∥x∥) & ≤ & W (t, x) ≤ ψ_{2}(∥x∥), \\ ψ_{3} (∥x∥) & ≤ & −Ẇ (t, x) \end{matrix} \}  (3.13)

for all t and all x. We will show that if we apply the control strategy

u(t, x) = \{ \begin{matrix}−\frac{μ(t,x)}{∥μ(t,x)∥}ρ(t, x) & if & (t, x) ∉ Ω \\ ∈ \{p ∈ ℝ^{m} : ∥p∥ ≤ ρ(t, x)\} & if & (t, x) ∈ Ω, \end{matrix}  (3.14)

where μ(t, x) = G^{T} (t, x)∇xW (t, x), and Ω = {(t, x) : μ = 0}, then the equilibrium state x = 0 is a globally uniformly asymptotically stable solution of the closed-loop system. For this, consider the Lyapunov function W for the uncontrolled nominal system. Then, the corresponding time derivative of W evaluated on the solutions of the closed-loop system is

Ẇ (t, x) = \frac{∂W}{∂t} + ∇x^{T}W (f(t, x) + G(t, x) (u + ξ(t, x)))
≤ −ψ_{3}(∥x∥) + ∇x^{T}W G(t, x)u + ∇x^{T}W G(t, x)ξ(t, x)
≤ −ψ_{3}(∥x∥) + μ^{T} (t, x)u + ∥μ(t, x)∥ρ(t, x).  (3.15)

For (t, x) ∉ Ω,

Ẇ (t, x) ≤ −ψ_{3}(∥x∥) − μ^{T} (t, x)\frac{μ(t, x)}{∥μ(t, x)∥}ρ(t, x) + ∥μ(t, x)∥ρ(t, x)
= −ψ_{3}(∥x∥) − \frac{∥μ(t, x)∥^{2}}{∥μ(t, x)∥} ρ(t, x) + ∥μ(t, x)∥ρ(t, x)
= −ψ_{3}(∥x∥) − ∥μ(t, x)∥ρ(t, x) + ∥μ(t, x)∥ρ(t, x)
= −ψ_{3}(∥x∥)
< 0.  (3.16)

For (t, x) ∈ Ω, we have μ(t, x) = 0, and using (3.15), we obtain

Ẇ (t, x) ≤ −ψ_{3}(∥x∥)
< 0.  (3.17)

Thus, W is also a Lyapunov function for the closed-loop system, which means that the uncertain system (3.11) driven by the control law (3.14) is indeed globally uniformly asymptotically stable.

Example 3.1 For the uncertain system model

ẋ = Ax + ξ_{1}(x) + b (u + ξ_{2}(x)) ,

where

A = [ \begin{matrix}−2 & 1 \\ 0 & −3 \end{matrix} ] , b = [ \begin{matrix}0 \\ 1 \end{matrix} ] , ∥ξ_{1}(x)∥ ≤ 1.75∥x∥, and |ξ_{2}| ≤ 10∥x∥ + 0.5|u|,

design stabilizing control strategy of the form (3.14).

The uncontrolled nominal system ẋ = Ax is asymptotically stable. Let

Q = [ \begin{matrix}4 & −1 \\ −1 & 6 \end{matrix} ] .

The eigenvalues of Q are λ_{min}(Q) = 3.5858 and λ_{max}(Q) = 6.4142. Solving the Lyapunov matrix equation A^{T} P + P A = −Q yields

P = I_{2}.

The proposed controller has the form

u = −ρ(x)\frac{μ(x)}{|μ(x)|} = −ρ(x)\frac{b^{T} P x}{|b^{T} P x|} = −ρ(x)\frac{b^{T} x}{|b^{T} x|} = −ρ(x)sign (b^{T} x) .  (3.18)

Our goal is to determine a bound ρ^{∗}(x) > 0 such that for any ρ(x) ≥ ρ^{∗}(x) the uncertain closed-loop system is globally uniformly asymptotically stable. To begin, consider

V (x) = x^{T} P x = x^{T} x

as the Lyapunov function candidate for the closed-loop system. Evaluating the time derivative of V on the trajectories of the closed-loop system yields

\frac{d}{dt}V (x) = 2x^{T} ẋ
= 2x^{T} (Ax + ξ_{1}(x) + b (u + ξ_{2}(x)))
= −x^{T} Qx + 2x^{T} ξ_{1}(x) + 2x^{T} bu(x) + 2x^{T} bξ_{2}(x).  (3.19)

Substituting into the above the expression for the controller given by (3.18), taking the norms, and using bounds on the uncertain elements, we obtain

\frac{d}{dt}V (x) ≤ −λ_{min}(Q)∥x∥^{2} + 2∥x∥∥ξ_{1}(x)∥ − 2ρ(x)\begin{matrix}|x^{T} b| & 2 \\ | & x & T & b & | \end{matrix} + 2 |x^{T} b| |ξ_{2}(x)|
≤ (−3.5858 + 3.5)∥x∥^{2} − 2ρ(x) |x^{T} b| + 2 (10∥x∥ + 0.5ρ(x)) |x^{T} b| .(3.20)

In the above, we used the fact that

|u| = ρ(x).

Collecting the terms in (3.20) yields

\frac{d}{dt}V (x) ≤ (−3.5858 + 3.5)∥x∥^{2} + (20∥x∥ − ρ(x)) |x^{T} b| .  (3.21)

Figure 3.7: Model-reference control system block diagram with input scaling.

Therefore, \frac{dV}{dt} < 0 for any

ρ(x) ≥ ρ^{∗}(x) = 20∥x∥.  (3.22)

In summary, the uncertain closed-loop system will be globally uniformly asymptotically stable if the controller gain ρ(x) is chosen to satisfy (3.22).

We now use the Lyapunov approach to design a model-reference tracking controller. A block diagram of the model-reference control system is shown in Figure 3.7. Model-reference control can be used for plants with nonlinearities and time-varying parameters. One of the assumptions that is made while designing a model-reference controller is that a plant model is in a controller companion form. Here, we propose a general method for designing model-reference controllers for a class of multi-input systems that does not require a plant model to be in a specific form. The crux of our method is in an appropriate selection of the model-reference system. Consider a plant model of the form

ẋ = Ax + Bu,  (3.23)

where A ∈ ℝ^{n×n} and B ∈ ℝ^{n×m}. The model-reference signal generator is described by

ẋ_{d} = A_{d}x_{d} + B_{d}r  (3.24)

where A_{d} ∈ ℝ^{n×n}, B_{d} ∈ ℝ^{n×m}, and r ∈ ℝ^{m} is a command signal that is chosen so that x_{d} represents a desired state trajectory of the plant. The matrix A_{d} of the reference system is chosen from the set

{A_{d} : A_{d} = A − BK_{d}, K_{d} ∈ ℝ^{m×n}} ,  (3.25)

and B_{d} = B. Thus, the reference system is state-feedback equivalent with the plant model (3.23). We assume that the pair (A, B) is controllable. Therefore, we can select A_{d} to have its eigenvalues in prescribed locations in the open-left hand plane. To proceed further, we define the tracking error e as

e(t) = x(t) − x_{d}(t).  (3.26)

Oup goal is to construct a state-feedback control law so that

lim_{t→∞} ∥e(t)∥ = 0.  (3.27)

We now form the differential equation that describes the error dynamics. For this we evaluate the time derivative of the error vector to get

ė(t) = ẋ(t) − ẋ_{d}(t)
= Ax(t) + Bu − A_{d}x_{d}(t) − Br(t).  (3.28)

Recall that A_{d} = A − BK_{d}, hence A = A_{d} + BK_{d}. Substituting this expression for A into the error equation (3.28), we obtain

ė(t) = A_{d}(x(t) − x_{d}(t)) + B(u + K_{d}x(t) − r(t))
= A_{d}e(t) + B(u + K_{d}x(t) − r(t)).  (3.29)

We will use a Lyapunov function approach in our further analysis that will result in a control law forcing the error e(t) to go asymptotically to zero. As a Lyapunov function candidate for (3.29), we take

V (e) = e^{T} P e,  (3.30)

where P solves the Lyapunov equation

A_{d}^{T} P + P A_{d} = −Q  (3.31)

for some Q = Q^{T} > 0. Since by construction A_{d} is asymptotically stable, the solution to the Lyapunov equation (3.31) is positive definite, that is, P = P^{T} > 0. We next evaluate the time derivative of V (e(t)) on the trajectories of (3.29). We obtain

V̇ (e) = 2e^{T} P ė
= 2e^{T} P (A_{d}e + B(u + K_{d}x − r))
= e^{T} (A_{d}^{T} P + P A_{d})e + 2e^{T} P B(u + K_{d}x − r).  (3.32)

Using (3.31) and taking norms of the above yields

V̇ (e) ≤ −e^{T} Qe + 2e^{T} P Bu + 2 ‖e^{T} P B‖ ∥K_{d}x∥ + 2 ‖e^{T} P B‖ ∥r∥.  (3.33)

If e is such that B^{T} P e = 0, then

V̇ (e) ≤ −e^{T} Qe
< 0.  (3.34)

We now analyze the case when B^{T} P e ≠ 0. Let the proposed control law be

u = −ρ(x, r)\begin{matrix}B & T & P & e \\ ‖ & B & T & P & e & ‖ \end{matrix},  (3.35)

where ρ(x, r) = ∥K_{d}x∥ + ∥r∥. Substituting (3.35) into (3.33), we obtain

V̇ (e) ≤ −e^{T} Qe − 2 ‖e^{T} P B‖ (ρ(x, r) − ∥K_{d}x∥ − ∥r∥)
= −e^{T} Qe
< 0.

Since for all e ∈ ℝ^{n}, we have V (e) > 0 and V̇ (e) < 0, the equilibrium point e = 0 of the error equation (3.29) is globally uniformly asymptotically stable, which means that x(t) → x_{d}(t).

Example 3.2 We now apply the above tracking controller to the vehicle model derived in the Subsection A.1, where δ_{r} = 0. That is, we wish to construct a tracking controller for a front wheel steered vehicle. The first step is to select the poles of the model reference system. We choose the model reference system to have the following set of asymptotically stable poles:

{−4 − 2j, −4 + 2j, −4.5, −5.0}.  (3.36)

The pair (A, b) is controllable, hence we can find k_{d} such that the eigenvalues of the matrix A_{d} = A − bk_{d} are at the above selected locations. The feedback gain is

k_{d} = [ 0.0329 2.2854 0.2429 −0.1653 ] ,  (3.37)

and the resulting model reference plant matrix is

A_{d} = [ \begin{matrix}−6.6638 & −107.1281 & −29.6350 & 7.7479 \\ 0 & 0 & 1 & 0 \\ −0.9205 & −65.8195 & −10.8362 & 4.7603 \\ −1 & −18.3 & 0 & 0 \end{matrix} ] .  (3.38)

The next step is to find P that solves the Lyapunov equation, A_{d}^{T} P + P A_{d} = −Q. We take Q = I_{4}. Solving the Lyapunov equation, we get

P = [ \begin{matrix}0.1145 & 0.7530 & −0.1809 & −0.0966 \\ 0.7530 & 34.6174 & −0.2883 & −3.3437 \\ −0.1809 & −0.2883 & 0.5144 & 0.0522 \\ −0.0966 & −3.3437 & 0.0522 & 0.6216 \end{matrix} ] .  (3.39)

Now that we have P, we are ready to construct the controller. We have

u = −ρ(x, r)\begin{matrix}b & T & P & e \\ | & b & T & P & e & | \end{matrix}
= −(|k_{d}x| + |r|)sign(b^{T} P (x − x_{d}))

= − (|[ \begin{matrix}0.0329 \\ 2.2854 \\ 0.2429 \\ −0.1653 \end{matrix} ]^{T} x| + |r|) sign ([ \begin{matrix}0.1571 \\ 26.9931 \\ 6.3324 \\ −3.0250 \end{matrix} ]^{T} (x − x_{d})) .  (3.40)

Our goal is to have the reference input signal u_{r} to correspond to a desired vehicle path. This requires scaling the reference input. The scaling factor is

η = (lim_{s→0} c(sI_{4} − A_{d})^{−1}b)^{−1} ,  (3.41)

where c = [ 0 0 0 1 ]. Note that η is well defined because the matrix A_{d} is asymptotically stable. In our example, η = −0.1653. Thus, we have r = ηu_{r}. The reference input is u_{r} = 4h(t − 1 sec), where h(t) is the unit step function.

The performance of the controller, for zero initial conditions, is shown in Figures 3.8 and 3.9. In Figure 3.8, the closed-loop system output is shown in the upper subplot, while the desired output is shown in the bottom subplot. The vehicle tracks the desired path with no noticeable steady-state error. The control action is shown in Figure 3.9.

92 CHAPTER 3. DYNAMICAL SYSTEM CONTROL Figure 3.8: Plots of the actual output of the closed-loop system and the desired output.

Figure 3.9: Plot of the controller action versus time.

The proposed tracking controller is discontinuous in the state, which results in its high activity referred to as chattering. Chattering can be reduced by using a continuous version of the above controller. The continuous controller is obtained by introducing a so called boundary layer that smooths out the discontinuous controller. We discuss the problem of constructing a boundary layer controller in the following section.

3.6 Practical Stability

A drawback to the discontinuous controllers, discussed in the previous section, is that rapid switching tends to excite high frequency modes of the plant and this may degrade the performance of the controller. These problems can be alleviated by adding a so-called boundary layer to the discontinuous controller resulting in a continuous control strategy. However, there is a price tag for making a discontinuous controller a continuous one. The closedloop system driven by a boundary layer controller may only be uniformly ultimately bounded rather than asymptotically stable. Specifically, we can only prove that a neighborhood of the equilibrium point of interest, rather than the equilibrium point itself, is asymptotically stable. We refer to this kind of stability as practical stability. The notions we are concerned with in this section are spelled out in the following three definitions.

Definition 3.1 The closed ball B̄_{q}(0) = {x : ∥x∥ ≤ q} is uniformly stable if for any ε > q, there is a δ = δ(ε) > 0 so that if ∥x(t_{0})∥ ≤ δ, then ∥x(t)∥ ≤ ε for t ≥ t_{0}.

For the benefit of the reader we now rephrase the definition of uniform boundedness from Section 2.6.

Definition 3.2 The solutions of ẋ = f(t, x) are uniformly bounded if for any given d > 0, there exists b = b(d) < ∞ so that for each solution x(t) = x(t; t_{0}, x_{0}) with ∥x_{0}∥ ≤ d, we have ∥x(t)∥ ≤ b for all t ≥ t_{0}.

We next define the uniform ultimate boundedness of a system, or for short, practical stability.

Definition 3.3 The solutions of ẋ = f(t, x) are uniformly ultimately bounded, or practically stable, with respect to the ball B̄_{q}(0) if for every d > 0, there is T (q, d) such that for each solution x(t) = x(t; t_{0}, x_{0}) with ∥x_{0}∥ ≤ d, we have x(t) ∈ B̄_{q}(0) for t ≥ t_{0} + T (q, d).

We now consider a dynamical system model (3.11),

ẋ(t) = f(t, x(t)) + G(t, x(t)) (u(t) + ξ(t, x(t))) , x(t_{0}) = x_{0},

where x(t) ∈ ℝ^{n}, u(t) ∈ ℝ^{m}, and ξ(t, x(t)) ∈ ℝ^{m} represents the lumped uncertainties. We assume that the uncertain element ξ is bounded by a known continuous non-negative function ρ, that is, ∥ξ(t, x(t))∥ ≤ ρ(t, x(t)). As in Section 3.5, we assume that the equilibrium state x = 0 of the uncontrolled nominal system ẋ(t) = f(t, x(t)) is globally uniformly asymptotically stable, and that we were able to construct a Lyapunov function W : ℝ×ℝ^{n} → ℝ satisfying the conditions (3.13) for all t and all x.

We showed in the previous section, that if we apply the discontinuous control strategy

u(t, x) = \{ \begin{matrix}−\frac{μ(t,x)}{∥μ(t,x)∥}ρ(t, x) & if & (t, x) ∉ Ω \\ ∈ \{p ∈ ℝ^{m} : ∥p∥ ≤ ρ(t, x)\} & if & (t, x) ∈ Ω, \end{matrix}

where μ(t, x) = G^{T} (t, x)∇xW (t, x), and Ω = {(t, x) : μρ(t, x) = 0}, then the equilibrium state x = 0 is a globally uniformly asymptotically stable solution of the closed-loop system. However, the above controller is discontinuous in x. Following Corless and Leitmann [21], we modify it to make it continuous by introducing a boundary layer. The proposed boundary layer controller is

u(t, x) = \{ \begin{matrix}−\frac{μ(t,x)ρ(t,x)}{∥μ(t,x)ρ(t,x)∥}ρ(t, x) & if & ∥μ(t, x)ρ(t, x)∥ > ν \\ ∥p(t, x)∥ ≤ ρ(t, x) & if & ∥μ(t, x)ρ(t, x)∥ ≤ ν, \end{matrix}  (3.42)

where ν is the thickness of the boundary layer about the naturally defined switching surface {(t, x) : ∥μ(t, x)ρ(t, x)∥ = 0}, and p is a continuous function such that

p = −\frac{μ(t, x)ρ(t, x)}{∥μ(t, x)ρ(t, x)∥}ρ(t, x),

when ∥μ(t, x)ρ(t, x)∥ = ν. A particular example of the above type of a controller is

u(t, x) = \{ \begin{matrix}−\frac{μ(t,x)ρ(t,x)}{∥μ(t,x)ρ(t,x)∥}ρ(t, x) & if & ∥μ(t, x)ρ(t, x)∥ > ν \\ −\frac{μ(t,x)}{ν}ρ^{2}(t, x) & if & ∥μ(t, x)ρ(t, x)∥ ≤ ν. \end{matrix}  (3.43)

We shall now show that the closed-loop system driven by the boundary layer controller (3.42) is practically stable with respect to a neighborhood of the zero state. This neighborhood can be made arbitrarily small by letting ν → 0. Before we state the results, we first compute the time derivative of a Lyapunov function candidate on the trajectories of the closed-loop system. Consider the Lyapunov function W for the uncontrolled nominal system. Then, the corresponding time derivative of W evaluated on the solutions of the closed-loop system is

Ẇ (t, x) = \frac{∂W}{∂t} + ∇x^{T}W (f(t, x) + G(t, x) (u + ξ(t, x)))
≤ −ψ_{3}(∥x∥) + ∇x^{T}W G(t, x)u + ∥μ(t, x)ρ(t, x)∥.

We now consider two cases. The first case is when ∥μ(t, x)ρ(t, x)∥ > ν. Taking into account the controller structure for this case yields

Ẇ (t, x) ≤ −ψ_{3}(∥x∥) − ∇x^{T}W G(t, x)\frac{μ(t, x)ρ(t, x)}{∥μ(t, x)ρ(t, x)∥}ρ(t, x) + ∥μ(t, x)ρ(t, x)∥
= −ψ_{3}(∥x∥) − \frac{∥μ(t, x)ρ(t, x)∥^{2}}{∥μ(t, x)ρ(t, x)∥} + ∥μ(t, x)ρ(t, x)∥
= −ψ_{3}(∥x∥).

If, on the other hand, ∥μ(t, x)ρ(t, x)∥ ≤ ν, then using (3.42), we obtain

Ẇ (t, x) ≤ −ψ_{3}(∥x∥) + ∇x^{T}W G(t, x)u + ∥μ(t, x)ρ(t, x)∥
≤ −ψ_{3}(∥x∥) + ∥∇x^{T}W G(t, x)∥ ∥u∥ + ∥μ(t, x)ρ(t, x)∥
≤ −ψ_{3}(∥x∥) + 2∥μ(t, x)ρ(t, x)∥
≤ −ψ_{3}(∥x∥) + 2ν.

We conclude that for all t and all x, we have

Ẇ (t, x) ≤ −ψ_{3}(∥x∥) + 2ν.

To proceed further we need an assumption on the function ψ_{3}. If ψ_{3} is bounded and

lim_{r→∞} ψ_{3}(r) = l < ∞  (3.44)

then we assume

2ν < l.  (3.45)

Proposition 3.1 The closed-loop system driven by the boundary layer controller is uniformly bounded.

Proof Let R be such that

ψ_{3}(R) = 2ν.

Figure 3.10: Finding R for a given ν.

Note that R is well defined by virtue of (3.45) and the fact that ψ_{3} is strictly increasing. Since ψ_{3} is strictly increasing, we can write

R = ψ_{3}^{−1}(2ν).

Figure 3.10 illustrates the above construction.

Given d > 0, where ∥x_{0}∥ ≤ d. Let

d̃= max{d, R}.

We claim that

b = b(d) = ψ_{1}^{−1}(ψ_{2}(d̃)),

that is, the system, or equivalently its null solution, is uniformly bounded. In other words, if ∥x_{0}∥ ≤ d then ∥x(t; t_{0}, x_{0})∥ ≤ b for t ≥ t_{0}. First note that ∥x_{0}∥ ≤ d̃and that ψ_{1}(b) = ψ_{2}(d̃). Hence d̃≤ b—see Figure 3.11 for an illustration of this construction. We have

Ẇ (x(t)) ≤ −ψ_{3}(∥x(t)∥) + 2ν
= −ψ_{3}(∥x(t)∥) + ψ_{3}(R).  (3.46)

For ∥x_{0}∥ > R and ∥x(t)∥ > R, Ẇ < 0 and therefore

ψ_{1}(b) = ψ_{2}(d̃)

Figure 3.11: Finding b for a given d.

≥ W (t_{0}; t_{0}, x_{0})
≥ W (t; t_{0}, x_{0})
≥ ψ_{1}(∥x(t)∥),

which implies that ∥x(t; t_{0}, x_{0})∥ ≤ b for t ≥ t_{0}.

If, on the other hand, ∥x_{0}∥ ≤ R, then since Ẇ (x(t)) ≤ −ψ_{3}(∥x(t)∥) + 2ν, the solution may reach the sphere ∥x∥ = R at some time t_{2} ≥ t_{0}. However, by (3.46), Ẇ < 0 outside ∥x∥ = R. Therefore, the solution cannot leave the ball ∥x∥ ≤ b. Thus the null solution is uniformly bounded.

□

We illustrate Proposition 3.1 with Figure 3.12. The initial condition in Figure 3.12 satisfies ∥x_{0}∥ < d, and d < R. The trajectory is confined to stay within the ball of radius b = ψ_{1}^{−1}(ψ_{2}(R)).

We now show, using arguments similar to those of Corless and Leitmann [21], that the closed-loop system is uniformly ultimately bounded with respect to the ball of radius b̃ > ψ_{1}^{−1}(ψ_{2}(R)).

Theorem 3.1 If x(t) = x(t; t_{0}, x_{0}) is a solution for [t_{0}, ∞) with ∥x_{0}∥ ≤ d, then for a given

b̃ > ψ_{1}^{−1}(ψ_{2}(R)),  (3.47)

98 CHAPTER 3. DYNAMICAL SYSTEM CONTROL Figure 3.12: A two-dimensional illustration of an uniformly ultimately bounded solution. ∥x(t; t_{0}, x_{0})∥ ≤ b̃ for all t ≥ t_{0} + T (b̃, d), where

T (b̃, d) = \{ \begin{matrix}0 & if & d ≤ R̃ \\ \frac{ψ_{2}(d)−ψ_{1}(R̃)}{ψ_{3}(R̃)−2ν} & if & d > R̃, \end{matrix}

and

ψ_{2} \binom{}{R̃} = ψ_{1}(b̃).  (3.48)

Proof From assumption (3.47),

ψ_{1}(b̃) > ψ_{2}(R),

and by (3.48),

ψ_{1}(b̃) = ψ_{2}(R̃).

Hence,

R̃ > R,  (3.49)

by (3.13), where we assumed that ψ_{2} is strictly increasing. We consider two cases. The first case is when d ≤ R̃. Then by uniform boundedness, for all t ≥ t_{0},

∥x(t)∥ < b,

for any b such that ψ_{1}(b) > ψ_{2}(R̃). We let b̃ = b(d). Hence in this case T (b̃, d) = 0.

We now consider the case when d > R̃. We show, by contradiction, that for some t_{1} ∈ [t_{0}, t_{0} + T (b̃, d)]

∥x(t_{1})∥ ≤ R̃.

From this we shall conclude the result by referring to the previous case. So suppose that

∥x(t)∥ > R̃

for all t ∈ [t_{0}, t_{0} + T (b̃, d)]. Then,

ψ_{1} (∥x (t_{0} + T (b̃, d); t_{0}, x_{0}) ∥) ≤ W (t_{0} + T (b̃, d), x (t_{0} + T (b̃, d); t_{0}, x_{0}))
= W (t_{0}, x_{0}) + ∫_{t_{0}}^{t_{0}+T (b̃,d)} Ẇ dt
= W (t_{0}, x_{0}) − ∫_{t_{0}}^{t_{0}+T (b̃,d)} (−Ẇ ) dt
≤ ψ_{2}(d) − ∫_{t_{0}}^{t_{0}+T (b̃,d)} (ψ_{3}(R̃) − 2ν) dt.

Note that ψ_{3}(R̃) − 2ν > 0 because by (3.49), R̃ > R. We proceed with the evaluation to get

ψ_{1} (∥x (t_{0} + T (b̃, d); t_{0}, x_{0}) ∥) ≤ ψ_{2}(d) − (ψ_{3}(R̃) − 2ν) T (b̃, d)
= ψ_{2}(d) − (ψ_{3}(R̃) − 2ν) \frac{ψ_{2}(d) − ψ_{1}(R̃)}{ψ_{3}(R̃) − 2ν}
= ψ_{1} \binom{}{R̃} .

Thus, since ψ_{1} is strictly increasing

∥x (t_{0} + T (b̃, d); t_{0}, x_{0}) ∥ ≤ R̃

which contradicts our assumption that for all t ∈ [t_{0}, t_{0} + T (b̃, d)] it should have been ∥x(t)∥ > R̃. Hence, there exists t_{1} ∈ [t_{0}, t_{0} + T (b̃, d)] such that ∥x(t_{1})∥ ≤ R̃. Then using the same argument as in the previous case we can show that

∥x(t)∥ < b̃

for all t ≥ t_{1} and the proof is complete.

□

We have the following consequence of the above theorem.

100 CHAPTER 3. DYNAMICAL SYSTEM CONTROL

Figure 3.13: Constructing δ given ε for an uniformly stable ball.

Corollary 3.1 The ball

{x : ∥x∥ ≤ ψ_{1}^{−1}(ψ_{2}(R))}

is uniformly stable. Proof Let

ε > ψ_{1}^{−1}(ψ_{2}(R)),

and consider δ = δ(ε) obtained from

ψ_{1}(ε) = ψ_{2}(δ).

Obviously δ ≥ R—see Figure 3.13. Hence by uniform boundedness if ∥x_{0}∥ ≤ δ, then ∥x(t)∥ ≤ ε for all t ≥ t_{0} which means that the ball is uniformly stable.

□

Example 3.3 Consider a dynamical system model

ẋ = Ax + b(u + h(t, x)),

where

A = [ \begin{matrix}0 & 1 \\ −1 & −1 \end{matrix} ] , b = [ \begin{matrix}0 \\ 1 \end{matrix} ] , and |h(t, x)| ≤ 2.

Let P be the solution of the matrix Lyapunov equation A^{T} P + P A = −2I_{2}, and let

u = \{ \begin{matrix}−2 sign(b^{T} P x) & if & |b^{T} P x| ≥ ν \\ −2\frac{b^{T}P x}{ν} & if & |b^{T} P x| < ν, \end{matrix}

where ν = 1. Suppose that the initial condition, x(0), is such that ∥x(0)∥_{2} ≤ 10. Find the radius, b, of a ball such that ∥x(t)∥_{2} ≤ b for all t ≥ 0.

Solving the Lyapunov equation A^{T} P + P A = −2I_{2} yields

P = [ \begin{matrix}3 & 1 \\ 1 & 2 \end{matrix} ] .

We next evaluate \frac{d}{dt} (x^{T} P x) on the trajectories of the closed-loop system. For the case when |b^{T} P x| ≥ ν, we get

\frac{d}{dt} (x^{T} P x) ≤ −2∥x∥^{2}.

For the case when |b^{T} P x| < ν, we obtain

\frac{d}{dt} (x^{T} P x) = −2∥x∥^{2} + 2x^{T} P b(u + h)
≤ −2∥x∥^{2} − 4\frac{|x^{T} P b|^{2}}{ν} + 4 |x^{T} P b|
≤ −2∥x∥^{2} + 4ν
= −2∥x∥^{2} + 4.

Hence, \frac{d}{dt} (x^{T} P x) < 0 outside of the ball

B̄_{R}(0) = \{x : ∥x∥ ≤ R = \sqrt{2}\}.

To find the radius, b, of a ball, such that ∥x(t)∥_{2} ≤ b for all t ≥ 0 and ∥x(0)∥_{2} ≤ 10 = d, we first observe that

λ_{min}(P)∥x∥^{2} = ψ_{1}(∥x∥) ≤ x^{T} P x ≤ ψ_{2}(∥x∥) = λ_{max}(P)∥x∥^{2}.

Let d̃= max{d, R} = 10. Then, the radius b is computed by solving the equation

ψ_{1}(b) = ψ_{2}(d̃),

that is,

λ_{min}(P)b^{2} = λ_{max}(P)d̃^{2}.

We obtain

b = d̃\sqrt{\frac{λ_{max}(P)}{λ_{min}(P)}} = 10\sqrt{\frac{(5 + \sqrt{5})/2}{(5 − \sqrt{5})/2}} = 16.18.

Figure 3.14: Plot of the linear saturation function.

Example 3.4 In this example, we investigate the performance of a boundary layer tracking controller on the tracking problem considered in Example 3.2. Specifically, we replace the discontinuous controller given by (3.40) with a boundary layer continuous controller, that is obtained by replacing the sign nonlinearity with a linear saturation nonlinearity. A linear saturation nonlinearity, denoted by sat_{ν}, can be described by

sat_{ν}(x) = \frac{1}{2ν} (|x + ν| − |x − ν|) ,  (3.50)

where the parameter ν controls the width of the boundary layer. In other words,

sat_{ν}(x) = \{ \begin{matrix}1 & if & x ≥ ν \\ \frac{1}{ν}x & if & −ν < x < ν \\ −1 & if & x ≤ ν. \end{matrix}  (3.51)

A plot of (3.50) for ν = 0.5 is shown in Figure 3.14. A boundary layer version of the discontinuous tracking controller (3.40) is

u = −ρ(x, r)sat_{ν} (b^{T} P e)

Figure 3.15: Plot of the boundary layer controller action versus time.

= −(|k_{d}x| + |r|)sat_{ν}(b^{T} P (x − x_{d}))

= − (|[ \begin{matrix}0.0329 \\ 2.2854 \\ 0.2429 \\ −0.1653 \end{matrix} ]^{T} x| + |r|) sat_{ν} ([ \begin{matrix}0.1571 \\ 26.9931 \\ 6.3324 \\ −3.0250 \end{matrix} ]^{T} (x − x_{d})) .  (3.52)

The plot of the actual output, Y, of the closed-loop system driven by (3.52) for ν = 0.5 and zero initial conditions was indistinguishable from that shown in Figure 3.8 when the discontinuous controller was used. The boundary layer controller action versus time is shown in Figure 3.15. Chattering is basically eliminated in this case.

3.7 Notes

A landmark paper on mathematical description of linear dynamical systems from control point of view is by Kalman [56]. For more examples of dynamical systems models from mechanical and electrical engineering, the reader is referred to Truxal [115], Banks [5], Raven [94], Mohler [80], and D’Azzo and Houpis [22]. Wertz [123] has many excellent real-life models from space science. A well-written paper on vehicle modeling and control for broad audience is by Ackermann [1]. In addition to [128] and [125], vehicle and tire dynamics modeling are investigated in detail by Nalecz [82], Nalecz and Bindemann [84, 83], and Smith and Starkey [107, 108].

Mayr writes on page 109 in [76], “It is still widely believed that the steam-engine governor is the oldest feedback device, and that James Watt had not only invented but also patented it. While both errors are easily refuted, we are still not able to reconstruct the history of this invention in all desired completeness”. Mayr gives examples of ancient feedback devices. Watt did not patent the governor. He did not invent it either. “But the application of the centrifugal pendulum in a system of speed regulation of steam engines was a new breakthrough for which the firm of Boulton & Watt, if not James Watt himself, clearly deserves the credit.” adds Mayr on page 112 of his book [76].

Sliding Modes

4.1 Simple Variable Structure Systems

The purpose of this section is to informally introduce the reader to variable structure sliding mode control systems. A formal analysis is found in the following sections.

A variable structure system is a dynamical system whose structure changes in accordance with the current value of its state. A variable structure system can be viewed as a system composed of independent structures together with a switching logic to switch between each of the structures. With appropriate switching logic a variable structure system can exploit the desirable properties of each of the structure the system is composed of. Even more, a variable structure system may have a property that is not a property of any of its structures. We illustrate the above ideas with two numerical examples. Example 4.1 Consider a dynamical system model

\{ \begin{matrix}ẋ_{1} & = & x_{2} \\ ẋ_{2} & = & −ax_{1}, \end{matrix}  (4.1)

having two structures corresponding to a = 1/4 and a = 4. The phase-plane portraits of the structures are shown in Figure 4.1 and Figure 4.2. The phase-plane portraits of the structures are families of ellipses. Neither of the structures is asymptotically stable—they are only stable. However, by choosing a suitable switching logic between the structures, we can make the resulting variable structure system asymptotically stable. Indeed, suppose the structure of the system is changed any time the system’s trajectory crosses either of the coordinate axes of the state plane, that is,

a = \{ \begin{matrix}1/4 & if & x_{1}x_{2} < 0 \\ 4 & if & x_{1}x_{2} > 0. \end{matrix}  (4.2)

106 CHAPTER 4. SLIDING MODES Figure 4.1: A phase-plane portrait of the system’s first structure corresponding to a=1/4. Figure 4.2: A phase-plane portrait of the system’s second structure corresponding to a=4.

Figure 4.3: A typical trajectory of the variable structure system.

A typical trajectory of the system (4.1) with the switching logic specified by (4.2) is shown in Figure 4.3.

Example 4.2 Consider the double integrator

\frac{d^{2}}{dt^{2}} x = u,

whose state-space model is

\{ \begin{matrix}ẋ_{1} & = & x_{2} \\ ẋ_{2} & = & u. \end{matrix}  (4.3)

Let u = ±2. Typical trajectories of the above system are shown in Figure 4.4. We thus have two independent structures, neither of them being stable. Let us now use the following switching logic between each of the structures:

u = −2sign (x_{1} + x_{2}) .  (4.4)

With the above switching logic the closed-loop system changes its structure anytime the state trajectory crosses the line

x_{1} + x_{2} = 0.

Figure 4.4: Typical trajectories of the double integrator system.

We introduce the notation,

σ(x) = σ(x_{1}, x_{2}) = x_{1} + x_{2}.

Then, we can describe the switching logic action as

u = \{ \begin{matrix}−2 & if & σ(x) > 0 \\ 2 & if & σ(x) < 0. \end{matrix}  (4.5)

Typical trajectories of the closed-loop system (4.3), (4.4) for the initial conditions located “close” to the origin of the state-plane are shown in Figure 4.5. Note that once the trajectories intercept the switching line they are confined to it thereafter. If we increase the integration step size, then we can observe oscillations, or chattering, of the trajectory about the switching surface as can be seen in Figure 4.6.

4.2 Sliding Mode

The motion of the system while confined to the switching line or a surface is referred to as sliding. A sliding mode will exist if in the vicinity of the switching surface the state velocity vectors are directed towards the surface. In other words, the switching surface attracts trajectories when they are in its vicinity. A surface σ(x) = 0 is attractive if

1. any trajectory starting on the surface remains there, and Figure 4.5: Typical trajectories of the closed-system (4.3), (4.4) for initial conditions “close” to the origin of the state-plane. The applied control law is bounded by 2.

Figure 4.6: An illustration of chattering of the double integrator variable structure sliding mode system for two different initial conditions. The applied control law is bounded by 1.

Figure 4.7: An illustration of the sufficient condition of the attractivity to the switching surface.

2. any trajectory starting outside the surface tends to it at least asymptotically

Thus, for a sliding motion to occur we need

lim_{σ→0^{+}} σ̇ < 0 and lim_{σ→0^{−}} σ̇ > 0.

The above conditions are illustrated in Figure 4.7. They ensure that the motion of the state trajectory x on either side of the switching surface σ(x) = 0 is towards the switching surface. The two conditions may be combined to give

σσ̇ < 0

in the neighborhood of the switching surface. Note, however, that the closed-loop system is now modeled by differential equations with discontinuous right-hand sides. The modeling differential equations are discontinuous on the switching surface σ(x) = 0. This is the reason we often refer to the switching surface as the discontinuity surface. A rigorous analysis of such systems cannot be carried out using methods of classical theory of differential equations. A solution to the differential equation ẋ = f(t, x) with a continuous right-hand side is a function x(t) that has a derivative and satisfies the differential equation everywhere on a given interval. We have to use more general methods that involve the theory of differential equations with discontinuous right-hand sides such as one developed by Filippov [30]. We will illustrate these points with a numerical example taken from Filippov [30, page 1].

Figure 4.8: Plots of x versus time for different initial conditions for the diferential equation (4.6).

Example 4.3 Consider the following differential equation with the discontinuous right-hand side,

ẋ = 1 − 2sign(x).  (4.6)

For x < 0, we have ẋ = 3, while for x > 0, ẋ = −1. In Figure 4.8, plots of x versus time for different initial conditions are given. As t increases, the solution x of (4.6)reaches the line x = 0 and stays there. The function x(t), however, does not satisfy (4.6) in the usual sense since if ẋ(t) = 0 and x(t) = 0, the right hand side of (4.6) becomes

1 − 2sign(0) = 1 ≠ 0,

where we defined sign(0) = 0.

We now briefly describe Filippov’s approach. Filippov [30, pages 2-3] recommends that the generalized solution to an ordinary differential equation with a discontinuous right-hand side should meet the following criteria:

(i) for a differential equation with a continuous right-hand side, the generalized solution should be equivalent with the usual one;

(ii) for the equation ẋ = f(t), the generalized solution should be of the form

x(t) = ∫ f(t)dt + constant;

(iii) for any initial condition x(t_{0}) = x_{0}, the solution must exist at least for t > t_{0} and should be continuable;

(iv) the generalized solution should be applicable to a broad class of equations that model dynamical systems;

(v) the limit of an uniformly convergent sequence of solutions should be a solution;

(vi) after applying to a solution a common change of variables, the result of the transformation should be a solution.

We next describe the proposed generalized solution in the sense of Filippov. We begin with an example. Suppose that we are given a single-input dynmical system model, ẋ = f(x, u) along with a switching surface σ(x) = 0. Let f^{−} = f(x, u^{−}) and f^{+} = f(x, u^{+}), where u = u^{−} if σ < 0 and u = u^{+} if σ > 0. We next join the velocity vectors f^{−} and f^{+}. The resulting velocity vector f^{0} is obtained by drawing the line tangent to the switching surface that intersects the line segment joining the velocity vectors f^{−} and f^{+}. Thus,

f^{0} = αf^{−} + (1 − α)f^{+},

where α ∈ [0, 1]. The above construction is illustrated in Figure 4.9. Observe that f^{0} belongs to the smallest convex set that contains f^{−} and f^{+}. Following Itkis [53] we now give a simple interpretation of the above construction. The number α may be interpreted as the time the system’s trajectory remains below the switching surface, while (1 − α) may be interpreted as the time the trajectory remains above the switching surface during the trajectory oscillation period about the switching surface. Specifically, if the representative point performs one oscillation about σ = 0 in a “small” time interval Δτ, then the representative point will traverse a distance ∥f^{−}∥αΔτ in time αΔτ with velocity f^{−}. During the remaining time of oscillation the representative point will traverse a distance ∥f^{+}∥(1−α)Δτ in time (1−α)Δτ with velocity f^{+}. The average displacement of the representative point during the time Δτ will then be ∥αf^{−} + (1 − α)f^{+}∥Δτ with the average velocity f^{0}.

Figure 4.9: An illustration of Filippov’s method in ℝ^{2}.

Suppose now that there are m discontinuity surfaces of the form

σ_{i} = {x : σ_{i}(x) = 0}, i = 1, 2, …, m.

Let

H = ⋃_{i=1}^{m} σ_{i}.

Then, we take a solution of the differential equation ẋ = f(x) with the discontinuous righthand side to be a solution of the differential inclusion

ẋ(t) ∈ G(x(t)),

where for each x, the set G(x) is the smallest closed convex set containing the cluster values of the function f(y) as y → x, y ∉ H, that is, a solution of the dynamical system ẋ = f(x) is an absolutely continuous function x(t), in the sense of Definition ??, defined on an interval, or segment, L for which ẋ(t) ∈ G(x(t)) almost everywhere on L (Filippov [30]). Observe that if the function f is continuous at a point x, that is, x ∉ H, then G(x) consists of a single point, namely f(x), and the solution satisfies ẋ = f(x) in the usual sense. However, if x ∈ H, then x lies on one or more of the surfaces σ_{1}, σ_{2}, …, σ_{m}. Suppose that x lies on the surfaces σ_{i_{1}}, σ_{i_{2}}, …, σ_{i_{k}}. It then follows that for a sufficiently small δ > 0 the surfaces σ_{i_{1}}, σ_{i_{2}}, …, σ_{i_{k}} partition the δ-neighborhood of x into ℓ ≤ 2^{k} regions R_{1}, R_{2}, …, R_{ℓ}, in each

Figure 4.10: An illustration of Filippov’s method in ℝ^{3}.

of which the function f is continuous. Let f_{j} denote the function f restricted to the region R_{j}. Then, G(x) is the smallest closed convex set containing the set of points

\{ lim_{{x^{}˜ →x \atop x^{}˜ ∈R_{j}}} f_{j}(x̃) : j = 1, 2, …, ℓ\} .  (4.7)

The velocity vector, ẋ = f^{0}, lies then on the intersection of the tangent plane to the surfaces of discontinuity and the convex set described by (4.7). This is illustrated in Figure 4.10.

There is a number of benefits when the system is in sliding. In particular,

• while in sliding the system is not affected by matched uncertainties, and

• while in sliding the system behavior is governed by a reduced set of differential equations. This is what we refer to as the order reduction. It is very useful in designing variable structure sliding mode controllers.

4.3 A Simple Sliding Mode Controller

In this section, we discuss the problem of designing sliding mode controllers for single-input plants. The design of a variable structure sliding mode controller consists of two phases.

They are:

• switching (sliding) surface design so as to achieve the desired system behavior, like the stability to the origin, when restricted to the surface, and

• selecting feedback gains of the controller so that the closed-system is stable to the sliding surface.

We now discuss the switching surface design phase for single-input plants. Without loss of generality, we can assume that the plant model is already in its controller companion form. Thus, the plant model we are given has the form

ẋ = [ \begin{matrix}0 & 1 & 0 & ⋯ & 0 \\ 0 & 0 & 1 & ⋯ & 0 \\ ⋮ & & & & ⋮ \\ 0 & 0 & 0 & ⋯ & 1 \\ −a_{0} & −a_{1} & −a_{2} & ⋯ & −a_{n−1} \end{matrix} ] x + [ 0 0 ⋮ 0 1 ] u.

Let the switching surface be

σ(x) = sx
= [ s_{1} s_{2} ⋯ s_{n−1} 1 ] x
= s_{1}x_{1} + s_{2}x_{2} + ⋯ + s_{n−1}x_{n−1} + x_{n}
= 0.

Note that, without loss of generality, we assumed s_{n} = 1. If this was not the case, we could divide both sides of sx = 0 by s_{n} ≠ 0 to ensure that s_{n} = 1. Using the equation of the switching surface, we can write

x_{n} = −s_{1}x_{1} − s_{2}x_{2} − ⋯ − s_{n−1}x_{n−1}.

We now combine the equations of the plant model and that of the switching surface. First note that

ẋ_{n−1} = x_{n}.

Hence, we can write

ẋ_{1} = x_{2}
ẋ_{2} = x_{3}
⋮
ẋ_{n−1} = −s_{1}x_{1} − s_{2}x_{2} − ⋯ − s_{n−1}x_{n−1},

or equivalently in matrix form,

ẋ = [ \begin{matrix}0 & 1 & 0 & ⋯ & 0 \\ 0 & 0 & 1 & ⋯ & 0 \\ ⋮ & & & & ⋮ \\ 0 & 0 & 0 & ⋯ & 1 \\ −s_{1} & −s_{2} & −s_{3} & ⋯ & −s_{n−1} \end{matrix} ] x.  (4.8)

The above equation describes the system dynamics in sliding. Observe the order reduction of the system dynamics in sliding. We now have n−m = n−1 first order differential equations. The poles of the reduced-order system are the roots of the equation

λ^{n−1} + s_{n−1}λ^{n−2} + ⋯ + s_{2}λ + s_{1} = 0.

The response of the system in sliding is completely specified by an appropriate choice of the components s_{1}, s_{2}, …, s_{n−1} of the switching surface.

Consider again Example 4.2. The switching surface for this example can be represented as

σ(x) = [ s_{1} 1 ] x = 0,

or equivalently

s_{1}x_{1} + x_{2} = 0.

The dynamics of the system in sliding are described by

ẋ_{1} = −s_{1}x_{1},

which is just a first order system. Note that to ensure the reduced-system stability we select s_{1} > 0.

After designing a switching surface, we construct a feedback controller. In this section, we discuss a controller design for single-input plants. The controller goal is to drive the plant state to the switching surface and maintain it on the surface for all subsequent time. We use a generalized Lyapunov approach in constructing the controller. Specifically, we use a generalized distance from the switching surface as a Lyapunov function candidate. Then, we select the controllers gains so that the time derivative of the chosen Lyapunov function candidate evaluated on the solution of the controlled system is negative definite with resepect to the switching surface, thus ensuring the motion of the state trajectory to the surface as it is illustrated in Figure 4.7. Our goal is to find u(x) so that σσ̇ < 0. The controller structure we consider has the form:

u = k_{1}^{±}x_{1} + k_{2}^{±}x_{2} + ⋯ + k_{n}^{±}x_{n},

where k_{i}^{±} are the feedback gains to be determined so that the condition σσ̇ < 0 is satisfied. To determine the feedback gains we substitute u into the expression σσ̇ to get

σσ̇ = σ [ s_{1} s_{2} ⋯ 1 ] ẋ
= σ (s_{1}ẋ_{1} + ⋯ + ẋ_{n−1} + ẋ_{n})
= σ (s_{1}x_{2} + ⋯ + s_{n−1}x_{n} + ẋ_{n})
= σ(s_{1}x_{2} + ⋯ + s_{n−1}x_{n} − a_{0}x_{1} − ⋯ − a_{n−1}x_{n} + u)
= σ(s_{1}x_{2} + ⋯ + s_{n−1}x_{n} − a_{0}x_{1} − ⋯ − a_{n−1}x_{n} + k_{1}^{±}x_{1} + ⋯ + k_{n}^{±}x_{n}).

Rearranging the terms yields

σσ̇ = σx_{1} (−a_{0} + k_{1}^{±}) + σx_{2} (−a_{1} + s_{1} + k_{2}^{±}) + ⋯ + σx_{n} (−a_{n−1} + s_{n−1} + k_{n}^{±}) .

A possible way to ensure that σσ̇ < 0 is to select the gains to force each term in σσ̇ to be negative, that is,

σx_{1} (−a_{0} + k_{1}^{±}) < 0,
σx_{2} (−a_{1} + s_{1} + k_{2}^{±}) < 0,
⋮
σx_{n} (−a_{n−1} + s_{n−1} + k_{n}^{±}) < 0.

Using the above, we can propose a possible set of gains that lead to satisfaction of the attractivity to the surface condition. A possible choice of gains is

k_{1}^{+} < −|a_{0}|, σx_{1} > 0
k_{1}^{−} > |a_{0}|, σx_{1} < 0
⋮
k_{n}^{+} < −(|a_{n−1}| + |s_{n−1}|), σx_{n} > 0
k_{n}^{−} > |a_{n−1}| + |s_{n−1}|, σx_{n} < 0.

We can also use the following controller gains:

k_{1}^{±} = −(|a_{0}| + ε)sign(x_{1}σ)
⋮
k_{n}^{±} = −(|a_{n−1}| + |s_{n−1}| + ε)sign(x_{n}σ),

where ε > 0 is a design parameter. The larger the value of ε the faster the trajectory converges to the switching surface.

4.4 Sliding in Multi-Input Systems

In this section, we describe a sliding mode in linear multi-input plants. When the number of the plant inputs is m, we can use switching surfaces of the form

{x : Sx = 0, S ∈ ℝ^{m×n}} .

We also use Sx = 0 to denote the above switching surface. In many applications nonlinear switching surfaces arise. We will use linear switching surfaces only, although one can use nonlinear switching surfaces as well. Let σ(x) = Sx. Then,

σ(x) = [ σ_{1}(x) ; σ_{2}(x) ; ⋮ ; σ_{m}(x) ] = [ s_{1}x ; s_{2}x ; ⋮ ; s_{m}x ] ,

where s_{i} ∈ ℝ^{1×n}. Recall, that the system is in sliding mode if σ(x(t)) = 0 for t ≥ t_{0}, where x is the state trajectory and t_{0} is a specified time. It follows that in a sliding mode the velocity vector ẋ(t) is tangent to the switching surface. Equivalently,

(∇σ_{i})^{T} ẋ(t) = s_{i}ẋ(t)
= 0, for i = 1, 2, …, m.

Hence

\frac{d}{dt}σ(x(t)) = Sẋ(t)
= 0.

We can thus characterize the system in sliding mode as

\begin{matrix}σ(x(t)) & = & 0 \\ σ̇(x(t)) & = & 0. \end{matrix} \}  (4.9)

Consider now the plant model ẋ(t) = Ax(t) + Bu(t). Combining Sẋ(t)) = 0 and the modeling equation of the plant we get

SAx + SBu = 0.

If the matrix SB is nonsingular, then we can solve the above algebraic equation for u to obtain

u = − (SB)^{−1} SAx.

Substituting the above into the plant model equation yields

ẋ = (I_{n} − B(SB)^{−1}S) Ax.  (4.10)

The above system is sometimes referred to as the equivalent system. Thus, we can describe the behavior of the system in sliding in an alternative way as

\begin{matrix}ẋ & = & (I_{n} − B(SB)^{−1}S) Ax \\ Sx & = & 0. \end{matrix} \}  (4.11)

Note that in sliding the system is governed by a reduced set of differential equations.

4.5 Sliding Mode and System Zeros

In this section, we show that the poles of the reduced-order system in sliding are the so called system zeros of the dynamical system described by

\{ \begin{matrix}ẋ & = & Ax + Bu \\ σ & = & Sx, \end{matrix}  (4.12)

where S ∈ ℝ^{m×n} and det SB ≠ 0. The system zeros are defined using the system matrix. The system matrix, P (s), corresponding to the plant (4.12) has the form

P (s) = [ \begin{matrix}sI_{n} − A & −B \\ S & O \end{matrix} ] .  (4.13)

The system matrix P (s) corresponding to the plant (4.12)is a square matrix of order n + m. The zeros of its determinant are the system zeros of (4.12). The system zeros are invariant under the following set of transformations:

1. nonsingular state variable transformation;

2. nonsingular transformation of the inputs;

3. nonsingular transformation of the outputs;

4. state-feedback to the inputs;

5. output-feedback to the rates of change of the states.

120 CHAPTER 4. SLIDING MODES We have

det P (s) = det ([ \begin{matrix}I_{n} & O \\ −S(sI_{n} − A)^{−1} & I_{m} \end{matrix} ] [ \begin{matrix}sI_{n} − A & −B \\ S & O \end{matrix} ])
= det [ \begin{matrix}sI_{n} − A & −B \\ O & S(sI_{n} − A)^{−1}B \end{matrix} ]
= det (sI_{n} − A) det (S(sI_{n} − A)^{−1}B) .  (4.14)

We now evaluate the characteristic polynomial of the equivalent system (4.10). While performing manipulations, we use the result of Exercise ?? to obtain

det (sI_{n} − A + B(SB)^{−1}SA)
= det(sI_{n} − A) det (I_{n} + (sI_{n} − A)^{−1}B(SB)^{−1}SA)
= det(sI_{n} − A) det (I_{m} + SA(sI_{n} − A)^{−1}B(SB)^{−1}) .  (4.15)

In our further manipulations, we use the identity

A (Adj(sI_{n} − A)) = s Adj(sI_{n} − A) − det(sI_{n} − A)I_{n}.  (4.16)

The above follows from the fact that

(sI_{n} − A)(sI_{n} − A)^{−1} = I_{n}.

which is equivalent to

(sI_{n} − A)Adj(sI_{n} − A) = det(sI_{n} − A)I_{n}.  (4.17)

We use (4.16) to get

det (I_{m} + SA(sI_{n} − A)^{−1}B(SB)^{−1})

= det (\frac{det(sI_{n} − A)I_{m} + S (s Adj(sI_{n} − A) − det(sI_{n} − A)I_{n}) B(SB)^{−1}}{det(sI_{n} − A)} )

= det (\frac{s (S Adj(sI_{n} − A)B) (SB)^{−1}}{det(sI_{n} − A)} )

= s^{m} det (S(sI_{n} − A)^{−1}B) det(SB)^{−1}. We now use (4.14) to represent (4.15) as

det (sI_{n} − A + B(SB)^{−1}SA)
= s^{m} det(SB)^{−1} det(sI_{n} − A) det (S(sI_{n} − A)^{−1}B) ,

and hence

det (sI_{n} − A + B(SB)^{−1}SA) = det(SB)^{−1}s^{m} det P (s)

(4.18)

We conclude that the dynamics of the system modeled by (4.12) in sliding along σ = Sx are determined by its system zeros. This fact leads to the observation that the switching surface design can be viewed as choosing an output matrix S so that the system (4.12) has a desired set of system zero locations, which in turn govern the dynamics of the systems in sliding along Sx = 0.

4.6 Properties of Equivalent Control

We consider a nonlinear dynamic system modeled by

ẋ = f(t, x) + B(t, x)u,  (4.19)

where x(t) ∈ ℝ^{n}, f(t, x(t)) ∈ ℝ^{n}, B(t, x(t)) ∈ ℝ^{n×m}, and u ∈ ℝ^{m}. The switching surface is {x : σ(x) = 0}, where σ(x(t)) ∈ ℝ^{m}. Let

S = \frac{∂σ}{∂x} ,

where we assume that SB is invertible. Once the system enters sliding mode, an appropriately constructed controller is switched at an infinite frequency to keep the system on the surface {σ = 0}. Thus, once the state trajectory hits the switching surface, say at t_{0}, then for t ≥ t_{0} we have σ = 0, and σ̇(x) = 0. Solving the algebraic equation

σ̇(x) = Sẋ
= S (f(t, x) + B(t, x)u)
= 0

for u yields

u = − (SB)^{−1} Sf.

The above expression for u is called the equivalent control and is denoted by u_{eq}. Substituting u_{eq} into to the modeling equation (4.19), we obtain

ẋ = f − B (SB)^{−1} Sf
= (I_{n} − B (SB)^{−1} S) f.

The above equation together with the condition σ = 0 describes the system in ideal sliding.

We now study a non-ideal sliding mode. In non-ideal sliding mode we no longer have σ̇(x) = 0. Hence, solving the equation

0 = σ̇(x)
= S (f(t, x) + B(t, x)u)

for u that we refer now to as ũ, yields

ũ = − (SB)^{−1} Sf + (SB)^{−1} σ̇.

Substituting the above into the plant dynamics gives

˙x̃ = f − B (SB)^{−1} Sf + B (SB)^{−1} σ̇.

The above describes the plant modeled by (4.19) in non-ideal sliding mode. The term B (SB)^{−1} σ̇ can be viewed as representing non-idealities. In real sliding the state trajectory x̃ will reside in some vicinity of the manifold {x : σ(x) = 0} so that

∥σ(x̃)∥ ≤ Δ,

where Δ > 0. We now show, following Utkin [116, pages 45–48], that if Δ → 0, then the real sliding mode behavior approaches that of the ideal sliding mode.

Theorem 4.1 Suppose that

1. a solution to ẋ = f(t, x) + B(t, x)ũ, on the interval [0, T], satisfies the condition

∥σ(x̃)∥ ≤ Δ;

2. there is a Lipschitz constant L for the right hand side of

ẋ = f − B (SB)^{−1} Sf + B (SB)^{−1} σ̇;

3. partial derivatives of the function

B (SB)^{−1}

exist and are bounded in any bounded domain of ℝ^{n};

4. for the function

f(t, x) + B(t, x)ũ

there exist nonnegative constants M and N such that

∥f(t, x) + B(t, x)ũ∥ ≤ M + N∥x∥.

Then, for any pair of state trajectories that are solutions to the ideal and non-ideal sliding models for which the initial conditions satisfy

∥x(0) − x̃(0)∥ ≤ P Δ, P ≥ 0,

there exists a nonnegative H such that

∥x(t) − x̃(t)∥ ≤ HΔ

for all t ∈ [0, T].

Proof The trajectory of the system in ideal sliding mode satisfies

x(t) = x(0) + ∫_{0}^{t} (f − B (SB)^{−1} Sf ) (x(s))ds,  (4.20)

while the trajectory of the system in non-ideal sliding satisfies

x̃(t) = x̃(0) + ∫_{0}^{t} (f − B (SB)^{−1} Sf ) (x̃(s))ds + ∫_{0}^{t} B (SB)^{−1} σ̇(x̃(s))ds

Integrating the last term of the above by parts gives

x̃(t) = x̃(0) + ∫_{0}^{t} (f − B (SB)^{−1} Sf ) (x̃(s))ds

+ B (SB)^{−1} σ(x̃(s))|_{0}^{t} − ∫_{0}^{t} (\frac{d}{ds}B (SB)^{−1}) σ(x̃(s))ds.

Subtracting the above from (4.20), taking the norms of both sides, and applying the triangle inequality yields

∥x − x̃∥ ≤ ∥x(0) − x̃(0)∥ + ‖B (SB)^{−1} σ(x̃(s))|_{0}^{t}‖

+ ∫_{0}^{t} ‖(\frac{d}{ds}B (SB)^{−1}) σ(x̃(s))‖ ds

+ ∫_{0}^{t} ‖(f − B (SB)^{−1} Sf ) (x(s)) − (f − B (SB)^{−1} Sf ) (x̃(s))‖ ds

Taking into account the hypothesis ∥x(0) − x̃(0)∥ ≤ P Δ and assumption 2, we obtain

∥x − x̃∥ ≤ P Δ + ‖B (SB)^{−1} σ(x̃(s))|_{0}^{t}‖

+ ∫_{0}^{t} ‖(\frac{d}{ds}B (SB)^{−1}) σ(x̃(s))‖ ds + ∫_{0}^{t} L∥x(s) − x̃(s)∥ds.

We next use assumption 1 to get

∥x − x̃∥ ≤ (P + ‖B (SB)^{−1} (x̃(t)) − B (SB)^{−1} (x̃(0))‖

+ ∫_{0}^{t} ‖(\frac{d}{ds}B (SB)^{−1}) (x̃(s))‖ ds) Δ + ∫_{0}^{t} L∥x(s) − x̃(s)∥ds.

The trajectory of the system in non-ideal sliding is bounded on the interval [0, T]. Indeed, using assumption 4 to the system ˙ x̃ = f(t, x̃) + B(t, x̃)ũ, we get

∥x̃(t)∥ ≤ ∥x̃(0)∥ + ∫_{0}^{t} ∥f(s, x̃) + B(s, x̃)ũ∥ ds
≤ ∥x̃(0)∥ + ∫_{0}^{t} (M + N \left∥{ \atop x}\right∥) ds
≤ ∥x̃(0)∥ + MT + ∫_{0}^{t} N \left∥{ \atop x}\right∥ ds.

Applying now the Bellman-Gronwall lemma, from page 25, to the above integral inequality, we obtain

∥x̃(t)∥ ≤ (∥x̃(0)∥ + MT) e^{Nt}
≤ (∥x̃(0)∥ + MT) e^{NT} , t ∈ [0, T].

The above means that x̃ is bounded on the interval [0, T]. By assumption 3, the function B (SB)^{−1} is bounded. Hence, for some S ≥ 0 that depends on the initial conditions, time T, and constant P, we have

∥x − x̃∥ ≤ SΔ + L ∫_{0}^{t} ∥x(s) − x̃(s)∥ ds.

Applying the Bellman-Gronwall lemma to the above inequality yields

∥x − x̃∥ ≤ SΔe^{LT} , t ∈ [0, T].

Let H = Se^{LT}. Then we can write

∥x − x̃∥ ≤ HΔ,

which completes the proof of the theorem.

□

4.7 Switching Surface Design

We now prove a sufficient condition for the existence of a switching surface σ(x) = Sx = 0 such that the plant model ẋ = Ax + Bu restricted to it has arbitrarily prescribed poles. A general method for designing a switching surface is contained implicitly in the proof of the following theorem that comes from Hui and Żak [51]. In our discussion, we use the following notation. For a full column rank matrix X, its left inverse will be denoted by X^{g}. Thus, X^{g}X = I, where I is the identity matrix whose size is equal to the number of columns of X. The imaginary part of a complex number z is denoted ℑ(z). The set of eigenvalues of a square matrix A, also called its spectrum, is denoted eig(A).

Theorem 4.2 Consider the plant model

ẋ = Ax + Bu,  (4.21)

where x(t) ∈ ℝ^{n}, u(t) ∈ ℝ^{m}, rank(B) = m, and the pair (A, B) is controllable. Let

{λ_{1}, λ_{2}, …, λ_{n−m}} ∈ ℂ,

where ℂ denotes the set of complex numbers, be an arbitrary set of n − m complex numbers such that any λ_{i} with ℑ(λ_{i}) ≠ 0 appears in this set in conjugate pair. Then, there exist full rank matrices W ∈ ℝ^{n×(n−m)} and W^{g} ∈ ℝ^{(n−m)×n} so that W^{g}W = I_{n−m}, W^{g}B = 0, and the spectrum of the matrix W^{g}AW is

eig(W^{g}AW) = {λ_{1}, λ_{2}, …, λ_{n−m}}.

Furthermore, there exists a matrix S ∈ ℝ^{m×n} such that SB = I_{m} and the system (4.21) restricted to the surface Sx = 0 has the poles λ_{1}, λ_{2}, …, λ_{n−m}.

Proof We first prove the existence of the desired matrices W and W^{g}. We can find T ∈ ℝ^{n×(n−m)} so that [T , B] is invertible because B has full rank. Let

[ T , B ]^{−1} = [ \begin{matrix}T & g \\ B & g \end{matrix} ] .

We now introduce a new coordinate z such that x = [T , B]z. Then, in the new coordinates the system ẋ = Ax + Bu has the form

ż = [ \begin{matrix}T^{g}AT & T^{g}AB \\ B^{g}AT & B^{g}AB \end{matrix} ] z + [ \begin{matrix}O \\ I & m \end{matrix} ] u.

Since the pair (A, B) is controllable, we can use the PBH rank test, given on page ??, to conclude that

rank [ λI_{n−m} − T^{g}AT , T^{g}AB ] = n − m for all λ ∈ ℂ,

which means that the pair (T^{g}AT , T^{g}AB) is controllable. The set {λ_{1}, λ_{2}, …, λ_{n−m}} is symmetric with respect to the real axis, therefore we can find F ∈ ℝ^{m×(n−m)} such that

eig (T^{g}AT − T^{g}ABF) = {λ_{1}, λ_{2}, …, λ_{n−m}}.

Let W^{g} = T^{g} and let W = T − BF. Then, since by construction T^{g}T = I_{n−m} and T^{g}B = 0, we have W^{g}W = I_{n−m} and W^{g}B = 0. Note that

W^{g}AW = T^{g}AT − T^{g}ABF .

Hence,

eig (W^{g}AW) = eig (T^{g}AT − T^{g}ABF) = {λ_{1}, λ_{2}, …, λ_{n−m}}.

We next prove the existence of the switching surface with the desired properties. We first note that the matrix [W , B] is invertible because W^{g}W = I_{n−m}, W^{g}B = O, and W, B are full rank. Hence, Range(W) ∩ Range(B) = {0} and we write

[ W , B ]^{−1} = [ \begin{matrix}W & g \\ S \end{matrix} ]

for some S ∈ ℝ^{m×n}. Note that SB = I_{m} and SW = 0. We now introduce new coordinates

z = [ \begin{matrix}z & 1 \\ z & 2 \end{matrix} ]
= [ \begin{matrix}W & g \\ S \end{matrix} ] x,

where z_{2} = Sx. In the new coordinates, the equivalent system ẋ = (I_{n} − B(SB)^{−1}S) Ax takes the form

ż = [ \begin{matrix}W & g \\ S \end{matrix} ] (I_{n} − B(SB)^{−1}S) A [ W , B ] z
= [ \begin{matrix}W^{g}AW & W^{g}AB \\ O & O \end{matrix} ] z.

The restriction to z_{2} = Sx = 0 is

ż_{1} = W^{g}AW z_{1},

and hence the system restricted to the switching surface Sx = 0 has the same eigenvalues as the matrix W^{g}AW. The proof of the theorem is now complete.

□

We use the results of the proof of the above theorem to propose the following switching surface design algorithm.

Algorithm 4.1 Given a controllable pair (A, B) with rank B = m, and the set of desired poles {λ_{1}, λ_{2}, …, λ_{m}} of the system restricted to a switching surface.

1. Find a matrix T ∈ ℝ^{(n−m)×n} such that [T , B] is invertible.

2. Compute

[ T , B ]^{−1} = [ \begin{matrix}T & g \\ B & g \end{matrix} ] .

3. Find a matrix F ∈ ℝ^{m×(n−m)} such that

eig (T^{g}AT − T^{g}ABF) = {λ_{1}, λ_{2}, …, λ_{n−m}}.

4. Take W = T − BF.

5. Compute

[ W , B ]^{−1} = [ \begin{matrix}W & g \\ S \end{matrix} ] .

Form state switching surface Sx = 0 using the last m rows of [ W , B ]^{−1}.

4.8 Bounded Control

In practice, a control input signal is obtained from a physical device, the actuator, which has a limited range, that is, the actuator is capable of providing only a limited amount of energy. We say that the actuator saturates or it has limited authority. A practical control system design must account for this fact. A controller that is designed to avoid reaching the saturation limits is usually considered as overdesigned, and hence inefficient, since the actuator has the capability that is not used. Friedland [33, page 96] compares such a design to a heavily overpowered automobile. In this section, we analyze sliding mode controllers for uncertain dynamical system with actuators of limited authority.

4.8.1 Closed-Loop System

We consider a class of uncertain, or nonlinear, dynamical systems modeled by

ẋ(t) = Ax(t) + Bu(t) + f(t, x(t)) + Bh(t, x(t), u(t)), x(t_{0}) = x_{0},  (4.22)

where x(t) ∈ ℝ^{n}, u(t) ∈ ℝ^{m}, the constant matrices A ∈ ℝ^{n×n} and B ∈ ℝ^{n×m} are known, and the function f and h model uncertainties, or nonlinearities, in the system. We refer to f as the unmatched uncertainty, while to h as the matched uncertainty. We assume that

1. ∥f(t, x)∥ ≤ α_{f}∥x∥ + β_{f},

2. ∥h(t, x, u)∥ ≤ γ_{h}∥u∥ + α_{h}∥x∥ + β_{h},

where α_{f}, β_{f}, γ_{h}, α_{h}, and β_{h} are known nonnegative constants, and ∥ · ∥ = ∥ · ∥_{2} denotes the Euclidean norm for vectors and the induced spectral norm for matrices. We also assume that

3. the pair (A, B) is controllable,

4. rankB = m.

We further assume

5. Existence of solutions. For each t_{0}, t_{1} ∈ ℝ, t_{0} < t_{1}, x_{0} ∈ ℝ^{n}, there exists a solution x(·) : [t_{0}, t_{1}) → ℝ^{n} for the above system, with x(t_{0}) = x_{0}.

6. Indefinite extension of solutions. All solutions can be continued over [t_{0}, ∞), that is, each solution x(·) : [t_{0}, t_{1}) → ℝ^{n}, with x(t_{0}) ∈ ℝ^{n}, has an extension x̃(·) : [t_{0}, ∞) → ℝ^{n}, where x̃(t) = x(t) for all t ∈ [t_{0}, t_{1}).

Our goal is to develop bounded state-feedback control strategies such that the zero state of the closed-loop system is uniformly asymptotically stable in the sense of Lyapunov or, the system is at least uniformly ultimately bounded with respect to an “acceptably” small neighborhood of the zero state in the sense of the following definition.

Definition 4.1 Let the set Γ ⊆ ℝ^{n} be a neighborhood of the origin. The system modeled by

ẋ(t) = F (t, x(t)),

where t ∈ ℝ, x(t) ∈ ℝ^{n} and F : ℝ × ℝ^{n} → ℝ^{n} is said to be uniformly ultimately bounded, or practically stable, with region of attraction Γ if there exists a compact set Σ ⊂ ℝ^{n}, Σ ⊂ Γ, containing the origin such that the following hold:

(i) Uniform boundedness of solutions. For each x_{0} ∈ ℝ^{n}, there exists a bound b(x_{0}) such that if x(·) : [t_{0}, ∞) → ℝ^{n} is any solution of the system ẋ(t) = F (t, x(t)) with x(t_{0}) = x_{0}, then ∥x(t)∥ ≤ b(x_{0}) for all t ≥ t_{0}.

(ii) Uniform stability to within Σ. For each neighborhood N of Σ, there exists a δ > 0 such that if x(·) : [t_{0}, ∞) → ℝ^{n} is any solution of the system ẋ(t) = F (t, x(t)) with ∥x(t_{0})∥ < δ, then x(t) ∈ N for all t ≥ t_{0}.

(iii) Uniform attractivity of Σ. For each neighborhood N of Σ and each x_{0} ∈ Γ, there exists T (N, x_{0}) such that if x(·) : [t_{0}, ∞) → ℝ^{n} is any solution of the system ẋ(t) = F (t, x(t)) with x(t_{0}) = x_{0}, then x(t) ∈ N for all t ≥ t_{0} + T (N, x_{0}).

Observe that if the system ẋ(t) = F (t, x(t)) satisfies the requirements of the above definition with Σ = {0}, then it is uniformly asymptotically stable with region of attraction Γ. Also note that if a given set Σ satisfies the requirements of the above definition, then we can say that this set is uniformly asymptotically stable.

Definition 4.2 The system ẋ(t) = F (t, x(t)) is globally practically stable if it is practically stable with ℝ^{n} as a region of attraction.

We now discuss the problem of designing controllers for the above class of uncertain system subject to hard bounds on control action, that is, saturating actuators. We propose to analyze three bounded control laws. They are:

u^{(1)}(x) = −η [ signσ_{1}(x) ; signσ_{2}(x) ; ⋮ ; signσ_{m}(x) ]

= −ηsignσ(x),

u^{(2)}(x) = \{ \begin{matrix}−η \frac{σ(x)}{∥σ(x)∥} & if & σ(x) ≠ 0 \\ 0 & if & σ(x) = 0, \end{matrix}

where η > 0 is a design parameter. The third control structure is labeled u^{∞}. To describe this third controller, we need the following notation. Let

θ = max_{i} |σ_{i}|,

Then, the i-th component of u^{∞} is

u_{i}^{(∞)}(x) = \{ \begin{matrix}−ηsignσ_{i}(x) & if & θ = |σ_{i}(x)| \\ 0 & if & θ > |σ_{i}(x)|. \end{matrix}

In other words,

u_{i}^{(∞)}(x) = \{ \begin{matrix}−ηsignσ_{i}(x) & if & |σ_{i}(x)| = ∥σ(x)∥_{∞} \\ 0 & otherwise, \end{matrix}

where ∥ · ∥_{∞} is the ∞ - norm of a vector. Let ∥ · ∥_{p} denotes the p-norm of a vector, where in our analysis p = 1, 2, ∞, and let ∇_{σ} denote the gradient operation with respect to σ. Then, in the regions where the gradients ∇_{σ}∥σ∥_{p}, p = 1, 2, ∞, are defined, the above controllers can be represented as

u^{(p)} = −η∇_{σ}∥σ∥_{p},

where

∇_{σ}∥σ∥_{p} = [ \frac{∂}{∂σ_{1}} ∥σ∥_{p} \frac{∂}{∂σ_{2}} ∥σ∥_{p} ⋯ \frac{∂}{∂σ_{n}} ∥σ∥_{p} ]^{T} .

Note that the above controllers are bounded. Indeed,

∥u^{(1)}∥_{2} ≤ η\sqrt{m}, ∥u^{(2)}∥_{2} ≤ η, and ∥u^{(∞)}∥_{2} ≤ η\sqrt{m}.

We can construct control laws for any p such that 1 ≤ p ≤ ∞ as follows

u^{(p)}(x) = −η∇_{σ}∥σ(x)∥_{p},

where

∥σ∥_{p} = (∑_{j=1}^{m} |σ_{j}|^{p})^{1/p} .

For such a generalization we refer the reader to Hui and Żak [51].

The closed-loop system that we are analyze has the form

ẋ = Ax + Bu + f + Bh
u^{(p)} = −η∇_{σ}∥σ(x)∥_{p}, p = 1, 2, ∞.

To facilitate the analysis of the above system, we use the following state variable transformation. Let

x̃ = Mx
= [ \begin{matrix}W & g \\ S \end{matrix} ] x,

where W^{g} and S are the matrices obtained from the algorithm in the previous section. Note that M is invertible with

M^{−1} = [ W B ] .

Let z = W^{g}x and σ = Sx. Then,

x̃ = [ \begin{matrix}z \\ σ \end{matrix} ] .

In the new coordinates the plant model (4.22) takes the form

˙x̃ = MAM^{−1}x̃ + MB(u + h) + f
= [ \begin{matrix}W & g \\ S \end{matrix} ] A [ W B ] x̃ + [ \begin{matrix}W & g \\ S \end{matrix} ] B(u + h) + f.

Let

MAM^{−1} = [ \begin{matrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix} ] ,

where A_{11} = W^{g}AW, A_{12} = W^{g}AB, A_{21} = SAW, and A_{22} = SAB. Noting that W^{g}B = O and SB = I_{m}, we obtain

x̃ = [ ż ; σ̇ ]
= [ \begin{matrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix} ] [ \begin{matrix}z \\ σ \end{matrix} ] + [ \begin{matrix}O \\ I & m \end{matrix} ] (u + h̃) + [ \begin{matrix}W & g & f̃ \\ S & f̃ \end{matrix} ] ,

where

f̃ = f(t, x)|_{x=M^{−1}x^{}}˜
= f̃(t, z, σ)
= f(t, W z + Bσ),

and

h̃ = h(t, x, u)|_{x=M^{−1}x^{}}˜
= h̃(t, z, σ, u)
= h(t, W z + Bσ, u),

because x = M^{−1}x̃ = W z + Bσ. From assumptions 1 and 2 we have

∥f̃∥ ≤ α_{f}∥W ∥∥z∥ + α_{f}∥B∥∥σ∥ + β_{f},

and

∥h̃∥ ≤ γ_{h}∥u∥ + α_{h}∥W ∥∥z∥ + α_{h}∥B∥∥σ∥ + β_{h}.

In the next section, using the above background results, we estimate stability regions of the closed-loop system.

4.8.2 Estimating Stability Regions

The presence of unmatched uncertainties has a deteriorating effect on the stability properties of a closed-loop system. The above combined with the imposed hard bounds on the control action may make it impossible to globally asymptotically stabilize the closed-loop system. The best it can be done in such a situation maybe uniform practical stability with some region of attraction Γ ⊂ ℝ^{n} and uniformly asymptotically stable set Σ containing the origin of ℝ^{n}. Our goal here is to estimate the “size” of the set Γ and that of Σ.

Stability of Motion on the Switching Surface

We first investigate the effect of the uncertainties on the stability of the system restricted to the switching surface. The system confined to the switching surface is described, in the new coordinates, by

ż = A_{11}z + W^{g}f̃(t, z, 0),

where ∥f̃(t, z, 0)∥ ≤ α_{f}∥W ∥∥z∥+β_{f}. We refer to the above system as the reduced-order system. The matrix A_{11} is asymptotically stable by construction. Therefore, for any symmetric and positive definite matrix Q ∈ ℝ^{(n−m)×(n−m)} the solution P to the Lyapunov equation

A_{11}^{T}P + P A_{11} = −2Q

is (symmetric) positive definite. Consider now the positive definite function:

V_{1} = (z^{T} P z)^{1/2} .

The matrix P is symmetric positive definite, hence it can be represented as

P = P^{\frac{1}{2}} P^{\frac{1}{2}} ,

where P^{\frac{1}{2}} ∈ ℝ^{(n−m)×(n−m)} is symmetric and nonsingular. We refer to P^{\frac{1}{2}} as a square root of P —see page 14 for a discussion of a matrix square root. Let

ζ = P^{\frac{1}{2}} z.

Then,

V_{1} = (z^{T} P z)^{1/2}
= (ζ^{T} ζ)^{1/2}
= ∥ζ∥.

We now evaluate the time derivative of V_{1} on the trajectories of the reduced-order system. Since P = P^{\frac{1}{2}} P^{\frac{1}{2}} and ζ = P^{\frac{1}{2}} z, then for z ≠ 0,

\frac{d}{dt}V_{1} = \frac{d}{dt} (z^{T} P z)^{1/2}
= \frac{d}{dt}∥ζ∥
= \frac{ζ^{T} ζ̇}{∥ζ∥}
= \frac{ζ^{T}}{∥ζ∥} (P^{\frac{1}{2}} A_{11}z + P^{\frac{1}{2}} W^{g}f̃(t, z, 0))
= \frac{z^{T}}{∥P^{\frac{1}{2}} z∥} (P A_{11}z + P W^{g}f̃(t, z, 0)) .

Note that

z^{T} P A_{11}z = \frac{1}{2}z^{T} (A_{11}P + P A_{11}) z
= −Q.

Hence,

\frac{d}{dt}∥ζ∥ = \frac{1}{∥P^{\frac{1}{2}} z∥} (−z^{T} Qz + z^{T} P W^{g}f̃(t, z, 0))
≤ \frac{−λ_{min}(Q)∥z∥^{2}}{∥P^{\frac{1}{2}} z∥} + \frac{λ_{max}(P)∥z∥}{∥P^{\frac{1}{2}} z∥}∥W^{g}∥∥f(t, z, 0)∥.

To proceed further, note that

λmin^{\frac{1}{2}}(P)∥z∥ ≤ ∥P^{\frac{1}{2}} z∥ ≤ λmax^{\frac{1}{2}}(P)∥z∥.

Taking the above into account, we get

\frac{d}{dt}∥ζ∥ ≤ −\frac{λ_{min}(Q)}{λmax^{\frac{1}{2}}(P)}∥z∥ + \frac{λ_{max}(P)}{λmin^{\frac{1}{2}}(P)}∥W^{g}∥∥f(t, z, 0)∥.

We now eliminate z from the right hand-side of the above inequality. For this observe that z = P^{− \frac{1}{2}} ζ. Hence,

\frac{1}{λmax^{\frac{1}{2}}(P)}∥ζ∥ ≤ ∥z∥ ≤ \frac{1}{λmin^{\frac{1}{2}}(P)}∥ζ∥.

Furthermore,

∥f̃(t, z, 0)∥ ≤ α_{f}∥W ∥∥z∥ + β_{f}
≤ \frac{α_{f}}{λmin^{\frac{1}{2}}(P)}∥W ∥∥ζ∥ + β_{f}.

134 CHAPTER 4. SLIDING MODES

Therefore,

\frac{d}{dt}∥ζ∥ ≤ −\frac{λ_{min}(Q)}{λ_{max}(P)}∥ζ∥ + α_{f} \frac{λ_{max}(P)}{λ_{min}(P)}∥W^{g}∥∥W ∥∥ζ∥

+β_{f} \frac{λ_{max}(P)}{λmin^{\frac{1}{2}}(P)}∥W^{g}∥.

Rearranging the above yields

\frac{d}{dt}∥ζ∥ ≤ − (\frac{λ_{min}(Q)}{λ_{max}(P)} − α_{f} \frac{λ_{max}(P)}{λ_{min}(P)}∥W^{g}∥∥W ∥) ∥ζ∥ + β_{f} \frac{λ_{max}(P)}{λmin^{\frac{1}{2}}(P)}∥W^{g}∥.

To simplify our presentation let

ρ = \frac{λ_{min}(Q)}{λ_{max}(P)} − α_{f} \frac{λ_{max}(P)}{λ_{min}(P)}∥W^{g}∥∥W ∥,

and

r = \frac{β_{f}}{ρ} \frac{λ_{max}(P)}{λmin^{\frac{1}{2}}(P)}∥W^{g}∥.

We assume that

7. ρ > 0. With the above notation, we can write

\frac{d}{dt}∥ζ∥ ≤ −ρ(∥ζ∥ − r).

In our stability analysis we use the following lemma. Lemma 4.1 Suppose ϕ(t), a, b ∈ ℝ, and a ≠ 0. If

ϕ̇(t) − aϕ(t) ≤ −b,

then, for t ≥ t_{0},

ϕ(t) ≤ \frac{b}{a} + (ϕ(t_{0}) − \frac{b}{a}) e^{a(t−t_{0})}.

Proof We multiply ϕ̇(t) − aϕ(t) ≤ −b by the integrating factor and represent the result in the form

\frac{d}{dt} (e^{−at}ϕ(t)) ≤ −be^{−at}.

Integrating the above from t_{0} to t yields the result.

□

Using the above results we can state and prove the following proposition concerning the stability of motion on the switching surface.

Proposition 4.1 The reduced-order system

ż = A_{11}z + W^{g}f̃(t, z, 0)

is globally practically stable, with the uniformly asymptotically stable set

Σ = \{z : ∥ζ∥ = ∥P^{\frac{1}{2}} z∥ ≤ r\} ⊂ ℝ^{n−m}.

Proof From the discussion before Lemma 4.1, it follows that any solution of the reducedorder system will asymptotically approach the set Σ because for z ≠ 0

\frac{d}{dt}|z^{T} P z|^{\frac{1}{2}} = \frac{d}{dt}∥P^{\frac{1}{2}} z∥
= \frac{d}{dt}∥ζ∥
≤ −ρ(∥ζ∥ − r)
= −ρ (∥P^{\frac{1}{2}} z∥ − r)
< 0

for all z outside the set Σ = \{z : ∥P^{\frac{1}{2}} z∥ ≤ r\}. Hence, all solutions of the reduced-order system are uniformly bounded, and the system is uniformly stable to within the set Σ. To prove the uniform attractivity of Σ, let N ⊂ ℝ^{n−m} be any neighborhood of the set Σ, and let R be a number such that R > r and \{z : ∥P^{\frac{1}{2}} z∥ ≤ R\} ⊂ N. By Lemma 4.1, any solution of the reduced-order system will enter the set \{z : ∥P^{\frac{1}{2}} z∥ ≤ R\}, and hence the set N, and will stay there thereafter, after time t_{0} + T_{r}(R, z_{0}), where

T_{r}(R, z_{0}) = \{ \begin{matrix}\frac{1}{ρ} ln (^{\frac{∥P^{\frac{1}{2}} z_{0}∥−r}{R−r}} ) & if & ∥P^{\frac{1}{2}} z_{0}∥ > R \\ 0 & if & ∥P^{\frac{1}{2}} z_{0}∥ ≤ R. \end{matrix}

Hence, the set Σ is uniformly attractive, and the proof is complete.

□

We note that the size of the uniformly asymptotically stable set Σ is determined solely by the unmatched uncertainty parameters α_{f} and β_{f}, and the eigenstructure of the nominal reduced-order system ż = A_{11}z reflected by the matrices W and W^{g}. The dynamics of the closed-loop system in a sliding mode are not affected by the matched uncertainty.

Closed-Loop System Stability Analysis

We now investigate the stability of the closed-loop system

ẋ = Ax + Bu + f + Bh
u^{(p)} = −η∇_{σ}∥σ(x)∥_{p}, for p = 1, 2, or ∞.

136 CHAPTER 4. SLIDING MODES To simplify the analysis we will be working in the following coordinate system

[ \begin{matrix}ζ \\ σ \end{matrix} ] = [ \begin{matrix}P & \frac{1}{2} & W & g \\ & S \end{matrix} ] x.

Note that

x = [ W P^{− \frac{1}{2}} B ] [ \begin{matrix}ζ \\ σ \end{matrix} ] .

In the new coordinates the system takes the form

[ ζ̇ ; σ̇ ] = [ \begin{matrix}P^{\frac{1}{2}} A_{11}P^{− \frac{1}{2}} & P^{\frac{1}{2}} A_{12} \\ A_{21}P^{− \frac{1}{2}} & A_{22} \end{matrix} ] [ \begin{matrix}ζ \\ σ \end{matrix} ] + [ \begin{matrix}O \\ I & m \end{matrix} ] (u + h̃) + [ \begin{matrix}P & \frac{1}{2} & W & g & f̃ \\ S & f̃ \end{matrix} ] .

We begin our analysis by obtaining estimates for \frac{d}{dt}∥σ∥ and \frac{d}{dt}∥ζ∥ evaluated on any solution of the closed-loop system. For σ ≠ 0, we have

\frac{d}{dt}∥σ∥ = \frac{σ^{T} σ̇}{∥σ∥}
= \frac{σ^{T}}{∥σ∥} (A_{21}P^{− \frac{1}{2}} ζ + A_{22}σ + u + h̃ + Sf̃) .

In further derivation, we use the fact that

sign(σ_{i}) = \frac{σ_{i}}{|σ_{i}|}
= \frac{|σ_{i}|}{σ_{i}},

and hence

σ^{T} sign(σ) = ∑_{i=1}^{m} |σ_{i}|.

For u = u^{(1)}, we have

\frac{d}{dt}∥σ∥ ≤ ∥A_{21}P^{− \frac{1}{2}} ∥∥ζ∥ + ∥A_{22}∥∥σ∥ − η \frac{∑_{i=1}^{m} |σ_{i}|}{∥σ∥} + ∥h̃∥ + ∥Sf̃∥.

Then, since ∑_{i=1}^{m} |σ_{i}| ≥ ∥σ∥, and taking into account norm bounds on h̃ and f̃, we get

\frac{d}{dt}∥σ∥ ≤ ∥A_{21}P^{− \frac{1}{2}} ∥∥ζ∥ + ∥A_{22}∥∥σ∥ − η \frac{∥σ∥}{∥σ∥}
+γ_{h}∥u^{(1)}∥ + α_{h}∥W P^{− \frac{1}{2}} ∥∥ζ∥ + α_{h}∥B∥∥σ∥ + β_{h}
+α_{f}∥S∥∥W P^{− \frac{1}{2}} ∥∥ζ∥ + α_{f}∥S∥∥B∥∥σ∥ + β_{f}∥S∥
= (∥A_{21}P^{− \frac{1}{2}} ∥ + α_{h}∥W P^{− \frac{1}{2}} ∥ + α_{f}∥S∥∥W P^{− \frac{1}{2}} ∥) ∥ζ∥
+ (∥A_{22}∥ + α_{h}∥B∥ + α_{f}∥S∥∥B∥) ∥σ∥
− (η(1 − γ_{h}\sqrt{m}) − β_{h} − β_{f}∥S∥) .

4.8. BOUNDED CONTROL 137 Let

a_{21} = ∥A_{21}P^{− \frac{1}{2}} ∥ + α_{h}∥W P^{− \frac{1}{2}} ∥ + α_{f}∥S∥∥W P^{− \frac{1}{2}} ∥,
a_{22} = ∥A_{22}∥ + α_{h}∥B∥ + α_{f}∥S∥∥B∥,
μ^{(1)} = η(1 − γ_{h}\sqrt{m}) − β_{h} − β_{f}∥S∥,

then we can write

\frac{d}{dt}∥σ∥ ≤ a_{21}∥ζ∥ + a_{22}∥σ∥ − μ^{(1)}.

When applying u = u^{(1)}, we assume that γ_{h} < 1/\sqrt{m}, and

η > \frac{β_{f}λ_{max}(P)∥W^{g}∥a_{21}}{λmin^{\frac{1}{2}}(P)ρ(1 − γ_{h}\sqrt{m})} + \frac{β_{h} + β_{f}∥S∥}{1 − γ_{h}\sqrt{m}}.

We now evaluate \frac{d}{dt}∥ζ∥ on any trajectory of the closed-loop system. For ζ ≠ 0, we have

\frac{d}{dt}∥ζ∥ = \frac{ζ^{T} ζ̇}{∥ζ∥}
= \frac{ζ^{T}}{∥ζ∥} (P^{\frac{1}{2}} A_{11}P^{− \frac{1}{2}} ζ + P^{\frac{1}{2}} A_{12}σ + P^{\frac{1}{2}} W^{g}f̃) .

Recall that

∥f̃∥ ≤ α_{f}∥W ∥∥z∥ + α_{f}∥B∥∥σ∥ + β_{f}.

Combining the above with the derivations from the previous subsection yields

\frac{d}{dt}∥ζ∥ ≤ −ρ∥ζ∥ + (∥P^{\frac{1}{2}} A_{12}∥ + α_{f}∥P^{\frac{1}{2}} W^{g}∥∥B∥) ∥σ∥ + β_{f}∥P^{\frac{1}{2}} W^{g}∥.

Let

a_{12} = ∥P^{\frac{1}{2}} A_{12}∥ + α_{f}∥P^{\frac{1}{2}} W^{g}∥∥B∥,
β̃_{f} = β_{f}∥P^{\frac{1}{2}} W^{g}∥,

then, we can write

\frac{d}{dt}∥ζ∥ ≤ −ρ∥ζ∥ + a_{12}∥σ∥ + β̃_{f}.

Note that since the pair (A, B) is controllable, we have

∥P^{\frac{1}{2}} A_{12}∥ = ∥P^{\frac{1}{2}} W^{g}AB∥
≠ 0,

and therefore a_{12} ≠ 0. This claim can easily be verified using the PBH test.

When u = u^{(2)}, we obtain the same estimates except now

μ^{(2)} = η(1 − γ_{h}) − β_{h} − β_{f}∥S∥

because

∥u^{(2)}∥ = ∥η \frac{σ}{∥σ∥}∥
= η.

When applying u = u^{(2)}, we assume that γ_{h} < 1, and

η > \frac{β_{f}λ_{max}(P)∥W^{g}∥a_{21}}{λmin^{\frac{1}{2}}(P)ρ(1 − γ_{h})} + \frac{β_{h} + β_{f}∥S∥}{1 − γ_{h}}.

While calculating estimates for u = u^{(∞)}, we take into account the fact that

\frac{σ^{T}}{∥σ∥}∇_{σ}∥σ∥_{∞} ≥ \frac{∥σ∥_{∞}}{∥σ∥}
≥ \frac{1}{\sqrt{m}}

because

∥σ∥ = ∥σ∥_{2}
≤ \sqrt{m}∥σ∥_{∞}.

The above implies that

μ^{(∞)} = \frac{η}{\sqrt{m}}(1 − γ_{h}\sqrt{m}) − β_{h} − β_{f}∥S∥.

When applying u = u^{(∞)}, we assume that γ_{h} < 1/\sqrt{m}, and

η > \sqrt{m}\frac{β_{f}λ_{max}(P)∥W^{g}∥a_{21}}{λmin^{\frac{1}{2}}(P)ρ(1 − γ_{h}\sqrt{m})} + \frac{β_{h} + β_{f}∥S∥}{1 − γ_{h}\sqrt{m}}.

The other parameters are the same as in the previous cases.

Note that the assumptions on the controllers’ gains imply that

μ^{(p)} > \frac{β_{f}λ_{max}(P)∥W^{g}∥a_{21}}{λmin^{\frac{1}{2}}(P)ρ} , p = 1, 2, ∞.

To proceed further, we need the following lemma.

4.8. BOUNDED CONTROL 139 Lemma 4.2 If a_{21} ≠ 0, then the region

Δ = \{(ζ, σ) : ∥ζ∥ < \frac{μ^{(p)}}{a_{21}} , σ = 0\}

is a sliding domain for the closed-loop system. Proof For σ ≠ 0 we have

σ^{T} σ̇ ≤ (a_{21}∥ζ∥ + a_{22}∥σ∥ − μ^{(p)}) ∥σ∥

because

\frac{d}{dt}∥σ∥ = \frac{σ^{T} σ̇}{∥σ∥}
≤ a_{21}∥ζ∥ + a_{22}∥σ∥ − μ^{(p)}.

Hence, for σ ≠ 0 we have σ^{T} σ̇ ≤ 0 in a neighborhood of Δ, and therefore Δ is a sliding domain for the closed-loop system.

□

We now prove a result concerning uniform practical stability of the closed-loop system. Theorem 4.3 Let a_{21} ≠ 0 and

Γ = {(ζ, σ) : α∥ζ∥ + β∥σ∥ < γ} ,

where

α = \frac{βa_{21}}{β + ρ}
β = \frac{1}{2} (a_{22} − ρ + \sqrt{(a_{22} + ρ)^{2} + 4a_{12}a_{21}})
γ = μ^{(p)} − \frac{αβ̃_{f}}{β} .

Then, the closed-loop system is practically stable with respect to the uncertainty region

Σ × {0} = {(ζ, 0) : ∥ζ∥ ≤ r}

with region of attraction Γ. Proof Since a_{21} > 0, the parameters α and β are also positive, and hence the region Γ is a bounded neighborhood of the origin of ℝ^{n}. We show that γ > 0 later in the proof. For an

Figure 4.11: Illustration of the region of attraction and the uncertainty region.

illustration of the above mentioned region see Figure 4.11. We first show that all solutions of the closed-loop system starting in Γ stay in Γ, that is, the solutions are uniformly bounded. We consider the Lyapunov function candidate V (ζ, σ) = α∥ζ∥ + β∥σ∥. Note that V is a positive definite function. The time derivative of V evaluated on a solution of the closed-loop system, for ζ ≠ 0, σ ≠ 0, is

\frac{d}{dt}V (ζ, σ) = α\frac{d∥ζ∥}{dt} + β \frac{d∥σ∥}{dt}
≤ α (−ρ∥ζ∥ + a_{12}∥σ∥ + β̃_{f} ) + β (a_{21}∥ζ∥ + a_{22}∥σ∥ − μ^{(p)})
= (−αρ + βa_{21}) ∥ζ∥ + (αa_{12} + βa_{22}) ∥σ∥ + αβ̃_{f} − βμ^{(p)}.

By assumption

γ = μ^{(p)} − \frac{αβ̃_{f}}{β} ,

and hence,

βγ = βμ^{(p)} − αβ̃_{f}.

Taking the above into account, we write

\frac{d}{dt}V (ζ, σ) ≤ (−αρ + βa_{21}) ∥ζ∥ + (αa_{12} + βa_{22}) ∥σ∥ − βγ.

Substituting the expressions for α and β into the above inequality yields

\frac{d}{dt}V (ζ, σ) ≤ β (α∥ζ∥ + β∥σ∥ − γ)

= β (V (ζ, σ) − γ) .

It follows from the above that any solution of the closed-loop system with initial condition in Γ cannot escape Γ through the region

{(ζ, σ) : V (ζ, σ) = γ, ζ ≠ 0, σ ≠ 0} .

The only possibility of such a solution escaping Γ is while moving along either of the manifolds {(ζ, σ) : ζ = 0} or {(ζ, σ) : σ = 0}. We now analyze the above two special cases one by one.

We first show that it is not possible for a solution of the closed-loop system to escape Γ while moving along the manifold {(ζ, σ) : ζ = 0}. For the closed-loop system confined to {(ζ, σ) : ζ = 0} the inequality

\frac{d}{dt}∥σ∥ ≤ a_{22}∥σ∥ − μ^{(p)}

holds when evaluated on any solution of the system. We note that β > a_{22} and γ ≤ μ^{(p)}. Hence,

\frac{d}{dt}∥σ∥ ≤ β∥σ∥ − γ,

which means that a solution of the closed-loop system with initial condition in Γ cannot escape Γ while moving along the manifold {(ζ, σ) : ζ = 0}.

We now show that a solution of the closed-loop system with initial condition in Γ cannot escape Γ while moving along the manifold {(ζ, σ) : σ = 0}. It follows from Proposition 4.1 that a solution of the closed-loop system confined to the manifold {(ζ, σ) : σ = 0} asymptotically approaches the uncertainty region Σ×{0} = {(ζ, σ) : ∥ζ∥ ≤ r, σ = 0}. Thus, a solution of the closed-loop system cannot escape Γ while moving along the manifold {(ζ, σ) : σ = 0} if

\frac{γ}{α} ≥ r.

Recall that μ^{(p)} > β_{f}λ_{max}(P)∥W^{g}∥a_{21}/ (λmin^{\frac{1}{2}}(P)ρ). Since α = βa_{21}/(β + ρ), we have

μ^{(p)} > \frac{β_{f}λ_{max}(P)∥W^{g}∥a_{21}}{λmin^{\frac{1}{2}}(P)ρ}
= \frac{β_{f}λ_{max}(P)∥W^{g}∥α(β + ρ)}{λmin^{\frac{1}{2}}(P)ρβ}
= \frac{β_{f}λ_{max}(P)∥W^{g}∥α}{λmin^{\frac{1}{2}}(P)ρ} + \frac{β_{f}λ_{max}(P)∥W^{g}∥α}{λmin^{\frac{1}{2}}(P)β}.

Observe that

β̃_{f} = β_{f}∥P^{\frac{1}{2}} W^{g}∥
≤ β_{f}λmax^{\frac{1}{2}}(P)∥W^{g}∥
≤ β_{f} \frac{λ_{max}(P)}{λmin^{\frac{1}{2}}(P)}∥W^{g}∥.

Then, using the definition of γ, we get

γ = μ^{(p)} − \frac{αβ̃_{f}}{β}
> \frac{β_{f}λ_{max}(P)∥W^{g}∥α}{λmin^{\frac{1}{2}}(P)ρ}.

Hence,

\frac{γ}{α} > \frac{β_{f}λ_{max}(P)∥W^{g}∥}{λmin^{\frac{1}{2}}(P)ρ}
= r,

which means that any solution of the closed-loop system with initial condition in Γ cannot leave Γ while moving along the manifold {(ζ, σ) : σ = 0}. Thus, all solutions of the closedloop system are uniformly bounded.

We now show that the closed-loop system is uniformly stable to within the uncertainty set Σ × {0} = {(ζ, 0) : ∥ζ∥ ≤ r}. Clearly Σ × {0} is compact and contains the origin of ℝ^{n}. Let N be any neighborhood of Σ × {0}, and let ν be a number such that 0 < ν ≤ γ, and Γ(ν) ⊂ N, where Γ(ν) = {(ζ, σ) : V (ζ, σ) < ν}. We consider now a solution of the closedloop system with initial condition in Γ(ν). If ν/α ≥ r, then it follows from the first part of this proof that such a solution cannot escape Γ(ν), and thus cannot escape N. If ν/α < r, then again it follows from the first part of this proof that such a solution may escape Γ(ν) only through the region Σ × {0}. However, such a solution cannot escape Σ × {0}, and thus cannot escape N. This is because μ^{(p)} > β_{f}λ_{max}(P)∥W^{g}∥a_{21}/ (λmin^{\frac{1}{2}}(P)ρ) implies

\frac{μ^{(p)}}{a_{21}} > \frac{β_{f}λ_{max}(P)∥W^{g}∥}{λmin^{\frac{1}{2}}(P)ρ}
= r,

and hence Σ × {0} ⊂ Δ. Thus, by Proposition 4.1 and Lemma 4.2 any solution of the closed-loop system that enters the region Σ × {0} is thereafter confined to it. We conclude that a solution of the closed-loop system with initial condition in Γ(ν) cannot escape N, which means that the closed-loop system is uniformly stable to within Σ × {0}.

We now show that the region Σ×{0}is uniformly attractive in the sense of Definition 4.2. Let R be a number such that r < R < μ^{(p)}/a_{21} and Σ(R) × {0} ⊂ N, where Σ(R) = {ζ : ∥ζ∥ ≤ R}. We will now show that a solution of the closed-loop system with initial condition in Γ is confined to Σ(R) × {0} after a finite time. Since R < μ^{(p)}/a_{21}, we have Σ(R) × {0} ⊂ Δ. By Proposition 4.1 and Lemma 4.2 any solution of the closed-loop system that enters the region Σ(R) × {0} is thereafter confined to it. Thus, it remains to show that such a solution, with initial condition in Γ, reaches this region in finite time. From the first part of the proof, we have

\frac{d}{dt}∥σ∥ ≤ β∥σ∥ − γ

on any solution that is confined to the region {(ζ, σ) : ζ = 0, σ ≠ 0}. Hence, for such a solution

\frac{d}{dt}V (ζ, σ) ≤ β (V (ζ, σ) − γ) .

Thus, by Lemma 4.1, for a solution that does not reside on the switching surface

V (ζ(t), σ(t)) ≤ γ + (V (ζ(t_{0}), σ(t_{0})) − γ) e^{β(t−t_{0})}.

The above implies that a solution with initial condition in Γ that never moves along the switching surface is guaranteed to reach Σ(R) × {0} by time t_{0} + T (R, ζ(t_{0}), σ(t_{0})), where

T (R, ζ(t_{0}), σ(t_{0})) = \{ \begin{matrix}\frac{1}{β} ln (\frac{γ}{γ−α∥ζ(t_{0})∥−β∥σ(t_{0})∥}) & if & (ζ(t_{0}), σ(t_{0})) ∉ Σ(R) × \{0\} \\ 0 & if & (ζ(t_{0}), σ(t_{0})) ∈ Σ(R) × \{0\}. \end{matrix}

By Proposition 4.1 a closed-loop system solution confined to the switching surface reaches Σ(R) × {0} by time t_{0} + T_{r}(R, ζ(t_{0})). Combining the above, we conclude that a solution with initial condition in Γ is guaranteed to reach the region Σ(R) × {0}, and thereafter is confined to it, by time t_{0} + T (R, ζ(t_{0}), σ(t_{0})) + T_{r}(R, ζ^{∗}), where ∥ζ^{∗}∥ = ∥ζ(t_{0})∥ + \frac{β}{α}∥σ(t_{0})∥. Hence, Σ × {0} is uniformly attractive, and the proof is complete.

□

The following corollaries are concerned with two special cases.

Corollary 4.1 If a_{21} = 0 and a_{22} ≠ 0, then the region

{(ζ, σ) : σ = 0}

is a sliding domain of the closed-loop system, and the system is practically stable with respect to the region Σ × {0}, and with region of attraction

Γ = \{(ζ, σ) : ∥σ∥ ≤ \frac{μ^{(p)}}{a_{22}} \} .

Corollary 4.2 If a_{21} = 0 and a_{22} = 0, then the region

{(ζ, σ) : σ = 0}

is a sliding domain of the closed-loop system, and the system is globally practically stable.

If β_{f} = 0, then we can replace the word “practically” with the word “asymptotically” in Theorem 4.3, Corollary 4.1, and Corollary 4.2 because β_{f} = 0 implies that solutions of the reduced-order system satisfy

\frac{d}{dt}∥ζ∥ ≤ −ρ∥ζ∥,

which in turn reduces the uncertainty set Σ × {0} to the origin of ℝ^{n}.

4.9 State Estimation of Uncertain Systems

4.9.1 Discontinuous Estimators

So far in our discussion of variable structure sliding mode control system design we assumed that the entire state vector of the plant to be controlled is available to us through measurement. If the state vector cannot be measured, then we can try to determine its suitable estimate that can be used by the controller instead of the true state vector. If we use a state vector estimate for the inaccessible state, then we split a control design problem into three phases. The first phase is the design of a controller assuming the availability of the entire state vector of the plant. In the second phase we construct a state estimator based on the plant input and output. In the final, third, phase we combine the controller and the estimator into the combined controller-estimator compensator.

We now discuss the problem of constructing a state estimator for a class of dynamical systems with matched uncertainties modeled by

ẋ = Ax + B (u + ξ)
y = Cx,

where C ∈ ℝ^{p×n}, and p ≥ m, which means that the number of input channels does not exceed the number of output channels. The vector function ξ models the lumped uncertainties or nonlinearities in the plant. We assume that there is a nonnegative real-valued function ρ = ρ (y) such that

∥ξ∥ ≤ ρ.

We assume that the pair (A, C) is observable. Our goal is to construct the plant state estimator. Let x̂ be the estimate of x. We denote the estimation error by e, that is,

e(t) = x̂(t) − x(t).

The observability of (A, C) implies the existence of a matrix L ∈ ℝ^{n×p} such that A − LC has prescribed (symmetric with respect to the real axis) eigenvalues in the open left half plane. Since A − LC is asymptotically stable, by the Lyapunov theorem for any Q = Q^{T} > 0, there is a unique P = P^{T} > 0 such that

(A − LC)^{T} P + P (A − LC) = −Q.

We choose Q, if possible, so that for some F ∈ ℝ^{m×p} we obtain

F C = B^{T} P .

We need this technical condition for the construction of the state estimator. This condition has a nice system theoretic interpretation. Its satisfaction means that the transfer function matrix

G(s) = F C (sI − A + LC)^{−1} B

is positive real, that is, the elements of G(s) are analytic in the open right half plane and G(s) + G^{∗}(s) ≥ 0 in the open right half plane. Here the superscript asterisk denotes the conjugate transpose. To proceed, define the vector function

E (e, η) = \{ \begin{matrix}\frac{F Ce}{∥F Ce∥}η & for & F Ce ≠ 0 \\ \{v : v ∈ ℝ^{m}, ∥v∥ ≤ η\} & for & F Ce = 0, \end{matrix}

where η ≥ 0 is a design parameter. Note that if we have a single-input single-output plant, then we can write

E(e, η) = η sign (F Ce) .

We now introduce the state estimator proposed by Walcott and Żak [119]. It has the form

˙x̂ = (A − LC) x̂ − BE(e, η) + Ly + Bu

If η ≥ ρ, the function x̂ is an asymptotic estimate of the state x, that is,

lim_{t→∞} e(t) = lim_{t→∞} (x̂(t) − x(t)) = 0.

We can prove the above statement using Lyapunov’s type of arguments. We first construct the differential equation satisfied by the error

ė = ˙ x̂ − ẋ
= (A − LC) e − BE(e, η) − Bξ.

We then show that

\frac{d}{dt} (e^{T} P e) = −e^{T} Qe < 0,

which implies

lim_{t→∞} e(t) = 0.

We conclude that the estimation error is insensitive to matched uncertainty.

4.9.2 Boundary Layer Estimators

We can view the error differential equation ė = (A − LC) e − BE(e, η) − Bξ as a model of a closed-loop system, with matched uncertainties, controlled by −E(e, η). The controller is discontinuous and varies its structure with respect to the switching surface {e : F Ce = 0}. In our analysis, we assumed that E(e, η) can switch with infinitely high frequency. In practice, very high frequencies may excite unmodeled high frequency modes and degrade the system performance. The alleviate this problem, we smooth out the discontinuous part of the estimator by introducing a boundary layer around the surface {e : F Ce = 0}. This results in replacing E(e, η) by its continuous approximation Ẽ(e, η). An example of such a continuous approximation is

Ẽ (e, η) = \{ \begin{matrix}\frac{F Ce}{∥F Ce∥}η & for & ∥F Ce∥ ≥ ϵ \\ \frac{F Ce}{ϵ}η & for & ∥F Ce∥ < ϵ, \end{matrix}

where ϵ is the width of the boundary layer. However, the error equation, or more precisely e = 0, is not anymore asymptotically stable when a boundary layer estimator is used. To describe the behavior of the error differential equation resulting from the application of a boundary layer estimator, we recall the following definitions.

Definition 4.3 The ball B_{δ}(0) = {e : ∥e∥ < δ} is uniformly stable if for all θ > δ, there is r(θ) > 0 so that if ∥e(t_{0})∥ ≤ r(θ), then ∥e(t)∥ ≤ θ for t ≥ t_{0}.

Definition 4.4 The equilibrium e = 0 is uniformly bounded if for any given r > 0, there exists d(r) < ∞ so that for each solution e(t) = e(t; t_{0}, e_{0}) with ∥e_{0}∥ ≤ r, we have ∥e(t)∥ ≤ d(r) for all t ≥ t_{0}.

We next define the uniform ultimate boundedness, or for short, practical stability.

Definition 4.5 The error is uniformly ultimately bounded, or practically stable, with respect to a closed ball B̄ if for every r > 0, there is T (r) ≥ t_{0} such that for each solution e(t) with ∥e(t_{0})∥ ≤ r, we have e(t) ∈ B for t ≥ T (r).

We can show that the error of the boundary layer estimator is uniformly ultimately bounded. Specifically, the estimation error of the boundary layer estimator ends in a ball whose radius depends on ϵ. The ball radius tends to zero with ϵ tending to zero. To see this consider, as before, the Lyapunov function candidate V (e) = e^{T} P e. Performing simple manipulations yields

\frac{d}{dt}V (e) = −e^{T} Qe − 2e^{T} P BẼ(e, η) − 2e^{T} P Bξ
≤ \{ \begin{matrix}−e^{T} Qe & for & ∥F Ce∥ ≥ ϵ \\ −e^{T} Qe + 2ϵρ & for & ∥F Ce∥ < ϵ. \end{matrix}

To obtain the last inequality, we used the definition of Ẽ and the fact that

e^{T} P BF Ce = ∥F Ce∥^{2} ≥ 0.

Since

−e^{T} Qe + 2ϵρ ≤ −λmin(Q)∥e∥^{2} + 2ϵρ,

we conclude that \frac{d}{dt}V (e) < 0 for

∥e∥ > \sqrt{\frac{2ϵρ}{λmin(Q)}} =^{def} R.

We illustrate the regions used in the discussion of the boundary layer state estimator in Figure 4.12. Referring to Figure 4.12 and Definition 4.4 we have

d(r) = \sqrt{\frac{λmax(P)}{λmin(P)}} R.

The estimation error is uniformly ultimately bounded, that is, practically stable, with respect to any ball with a radius greater than

\sqrt{\frac{λmax(P)}{λmin(P)}} R.

148 CHAPTER 4. SLIDING MODES

Figure 4.12: A two-dimensional illustration of an uniformly ultimately bounded error.

4.10 Notes

Sliding Modes in Solving Optimization Problems

In this chapter, we use sliding modes to analyze dynamics of an analog circuit that can solve convex optimization problems with linear constraints. The network is modeled by a set of nonlinear ordinary differential equations with discontinuous right-hand sides. In carrying out the analysis, concepts from the theory of differential equations with discontinuous right-hand sides and Lyapunov stability theory are used. We show that the equilibrium points of the differential equations modeling the network coincide with the optimizers of the underlying optimization problem. We then prove that irrespective of the initial state of the network, its state converges to the solution set of the optimization problem.

The idea of using analog circuits to solve constrained optimization problems first appeared in the literature in 1956 when Pyne [92] proposed a method for solving linear programming problems using an electronic analog computer. Soon after, other methods were proposed for solving various mathematical programming problems on an analog computer—see, for example, Rybashov [100, 101, 102, 103], and Karpinskaya [57], among others. More recently, many new analog architectures have been proposed for solving constrained optimization problems. In particular, Chua and Lin [18] developed a canonical nonlinear programming circuit for solving general nonlinear programming problems. Tank and Hopfield [114] later proposed an analog circuit for solving linear programming problems. The results of Tank and Hopfield were then extended to nonlinear programming problems by Kennedy and Chua [58]. Rodríguez-Vázquez et al. [95], and Cichocki and Unbehauen [19] proposed analog networks that use switched-capacitor integrated circuit technology for solving certain constrained optimization problems. The design of dynamical systems for solving optimization problems proposed by Rodríguez-Vázquez et al. [95] is based on concepts from the penalty function method and results in analog dynamical systems modeled by differential equations with discontinuous right-hand sides. This class of optimizers is particularly attractive for two reasons. First, their design does not require the calculation of penalty parameters. Second, these networks can be realized using switched-capacitor technology, and thus are suitable for monolithic implementation.

5.1 Problem Statement

We consider the constrained optimization problem

minimize f(x)
subject to Ax ≤ b,

(5.1)

where x ∈ ℝ^{n}, f : ℝ^{n} → ℝ, A ∈ ℝ^{m×n}, and b ∈ ℝ^{m}. Let

Ω = {x : Ax ≤ b}.  (5.2)

denote the feasible region for the above optimization problem. The feasible set Ω is the intersection of m closed half-spaces and is therefore a convex polytope. Let

h_{i} = {x : a_{i}x = b_{i}} , i = 1, 2, …, m,

denote the hyperplanes associated with the above m closed half-spaces, where a_{i} is the i-th row of A and b_{i} is the i-th component of the vector b. Also, we define

H = ⋃_{i=1}^{m} h_{i}.

We let Γ denote the collection of all relative minimizers of the optimization problem (5.1). Lastly, we introduce the index sets

I(x) = {i : a_{i}x = b_{i}}

and

J(x) = {i : a_{i}x > b_{i}} .

We make the following assumption: A1. The objective function f is convex over ℝ^{n} and has continuous first partial derivatives on ℝ^{n}, that is, f ∈ C^{1}; A2. The feasible set Ω is a nonempty, bounded, and closed polyhedron, which means that Ω is a compact set; A3. Each point in Ω is a regular point of the constraints, that is, for any x ∈ Ω the vectors a_{i}, i ∈ I(x), are linearly independent. Remark 1 It follows from Assumption A1 that the optimization problem (5.1) is a convex programming problem. Indeed, we are asked to minimize a convex objective function f over a polyhedron Ω. Remark 2 Problem (5.1) is well posed in the sense that the feasible set Ω is nonempty and the optimal value of f is not −∞ because by assumptions A1 and A2 the set f(Ω) is compact.

We now present two well known results related to problem (5.1). We use these results in our analysis.

Lemma 5.1 Any relative minimizer of the optimization problem (5.1) is a global minimizer.

Furthermore, the Karush-Kuhn-Tucker condition for optimality is both necessary and sufficient for problem (5.1). Specifically, we have the following lemma:

Lemma 5.2 Suppose that x^{∗} ∈ Ω. Then, x^{∗} ∈ Γ if and only if there exist non-negative constants μ_{1}, μ_{2}, …, μ_{m} such that

∇f(x^{∗}) + ∑_{i∈I(x^{∗})} μ_{i}a_{i}^{T} = 0.

The network applied to the optimization problem (5.1) has the form

τẋ(t) = g(x(t))
= \{ \begin{matrix}−μ ∑_{i∈J(x(t))} a_{i}^{T} & if & x(t) ∉ Ω \\ −∇f(x(t)) & if & x(t) ∈ Ω, \end{matrix}  (5.3)
x(0) = x_{0},

where τ and μ are positive design constants.

5.2 Background Results

An approach commonly used in developing analog optimizers is to first convert the constrained optimization problem into an associated unconstrained optimization problem, and then design an analog network that solves the unconstrained problem. Such a network is typically an implementation of the dynamical gradient system for minimizing the objective function of the associated unconstrained problem. One method of converting a constrained optimization problem into an associated unconstrained optimization problem is the penalty function method. The idea behind the penalty function method is to replace the constrained optimization problem

minimize f(x)
subject to x ∈ Ω,

where Ω ⊂ ℝ^{n} is the feasible set, with an associated unconstrained problem of the form

minimize f(x) + ζp(x),

where the objective function f : ℝ^{n} → ℝ is continuous on ℝ^{n} and ζ is a positive constant. The function p : ℝ^{n} → ℝ is continuous on ℝ^{n} and satisfies

(i) p(x) ≥ 0 for all x ∈ ℝ^{n},

and

(ii) p(x) = 0 if and only if x ∈ Ω.

For more details on the subject of the penalty method see, for example, Luenberger [70]. If for a finite value of the penalty parameter ζ the solution of the unconstrained problem coincides with the solution of the constrained problem, then the penalty function p is said to be exact. Bertsekas [11] has shown that except for trivial cases, an exact penalty function cannot be everywhere differentiable. Quite often we can find an exact penalty function that results in the function f(x) + ζp(x) being piecewise smooth. However, the dynamical gradient system for such a function will have a discontinuous right-hand side. As such, a rigorous analysis of its behavior cannot be carried out using only methods derived from the classical theory of differential equations. Rather, the analysis of the system must be carried out using other methods, for example those reported by Filippov [30] and Aubin and Cellina [3]. Such an approach was undertaken, already in 1967 by Karpinskaya [57], and later by Korovin and Utkin [62], Utkin [117], Flåm [32], Flåm and Zowe [31], and Żak et al. [131], among others. A differential geometric approach to the analysis of dynamical systems that solve optimization problems is reported by Faybusovich [28] and Helmke and Moore [46].

System (5.3) can be viewed as a dynamical gradient system for the unconstrained problem

minimize κ_{0}f(x) + μ ∑_{i=1}^{m} κ_{i} (a_{i}x − b_{i}) ,

where

κ_{i} = \{ \begin{matrix}0 & if & a_{i}x ≤ b_{i} \\ 1 & if & a_{i}x > b_{i}, \end{matrix}

and

κ_{0} = \{ \begin{matrix}1 & if κ_{i} = 1 for all i = 1, 2, …, m; \\ 0 & otherwise, \end{matrix}

that is, κ_{0} = ∏_{i=1}^{m} (1 − κ_{i}). Using the above notation, we can represent system (5.3) as

τẋ(t) = g(x(t)) = −κ_{0}∇f(x(t)) − μ ∑_{i=1}^{m} κ_{i}a_{i}^{T} , x(0) = x_{0}.  (5.4)

The function g(x) is discontinuous on the surfaces h_{1}, h_{2}, … h_{m} and is continuous elsewhere. A phenomenon commonly occurring in systems such as (5.3) is the so-called sliding mode that we discussed in Section 4.2.

A detailed discussion of using τẋ(t) ∈ G(x(t)) to analyze the behavior of the dynamical systems modeled by differential equations with discontinuous right-hand sides, as well as theorems guaranteeing the existence of solutions, is provided in Filippov [30, Chapter 2], and in Aubin and Cellina [3]. Applications of differential inclusions to the analysis of ordinary differential equations for solving convex programs are discussed in Flåm [32], and Flåm and Zowe [31].

5.3 Neural Network Analysis

We will now show that every trajectory of system (5.3), irrespective of the initial state converges to the solution set of the optimization problem (5.1). We do this by showing that every trajectory of system (5.3) has the following properties:

1. Irrespective of the initial state x_{0}, the resulting trajectory reaches the feasible set Ω in finite time and is thereafter confined to Ω. This portion of the trajectory is hereafter referred to as the reaching phase.

2. While confined to the feasible set Ω, the trajectory converges asymptotically to the solution set Γ. This portion of the trajectory is hereafter referred to as the convergence phase.

5.3.1 Reaching Phase

We begin by introducing the function V : ℝ^{n} → ℝ of the form

V (x) = ∑_{i=1}^{m} max {a_{i}x − b_{i}, 0} ,

or equivalently

V (x) = \{ \begin{matrix}∑_{i∈J(x)} (a_{i}x − b_{i}) & if & x ∉ Ω \\ 0 & if & x ∈ Ω. \end{matrix}  (5.5)

Observe that V (x) > 0 for all x ∉ Ω, and V (x) = 0 for all x ∈ Ω. We can view V (x) as a distance measure from the point x to the feasible set Ω. We show that V is strictly decreasing when evaluated along any trajectory of system (5.3) outside Ω. Specifically, we prove the existence of a positive constant, η, with the property that

\frac{d}{dt}V (x(t)) ≤ −η

when evaluated along any trajectory of system (5.3) outside Ω. In doing so we use the following result:

Lemma 5.3 Suppose x̂ ∉ Ω and β_{1}, β_{2}, …, β_{m} are non-negative constants. If β_{j} > 0 and

j ∈ J(x̂), then ∑_{I(x^{}ˆ )∪J(x^{}ˆ )} β_{i}a_{i} ≠ 0.

Proof Let the function q : ℝ^{n} → ℝ be defined as

q(x) = ∑_{I(x^{}ˆ )∪J(x^{}ˆ )} β_{i} (a_{i}x − b_{i}) .

Clearly q ∈ C^{1} and

∇^{T} q(x) = ∑_{I(x^{}ˆ )∪J(x^{}ˆ )} β_{i}a_{i}.

Since β_{1}, β_{2}, …, β_{m} are all non-negative, it follows that

1. the function q is convex over ℝ^{n}, and

2. q(x) ≤ 0 for all x ∈ Ω.

Furthermore, since β_{j} > 0 for j ∈ J(x̂), we also have q(x̂) > 0. Hence, we must have

∇^{T} q(x̂) = ∑_{I(x^{}ˆ )∪J(x^{}ˆ )} β_{i}a_{i} ≠ 0.

□

We must consider two cases when analyzing the behavior of system (5.3). The first case being when the motion of the system is not confined to any of the surfaces h_{1}, h_{2}, …, h_{m}. The second case being when the system is in sliding mode on one or more of the surfaces h_{1}, h_{2}, …, h_{m}. We only need to consider these two cases, since every trajectory of system (5.3) is composed of these two types of motion. That is, every trajectory x(t) of system (5.3) can be broken into a countable number of intervals, (t_{0}, t_{1}), …, (t_{l−1}, t_{l}), …, on each of which both index sets I(x(t)) and J(x(t)) are constant. If I(x(t)) is not empty, then the system is in a sliding mode on that interval, and if I(x(t)) is empty, then the system is not in a sliding mode. Case 1 Let x(t) denote any particular trajectory of system (5.3), and let t′ ∈ (t_{l−1}, t_{l}). Suppose that on the interval (t_{l−1}, t_{l}) the trajectory x(t) does not intersect Ω or any of the surfaces h_{1}, h_{2}, …, h_{m}, that is,

x(t) ∉ Ω ∪ H for all t ∈ (t_{l−1}, t_{l}).  (5.6)

Denoting the index set J(x(t′)) by J′, it follows from (5.6) and the fact that x(t) is absolutely continuous that I(x(t)) = ∅ and J′ ≠ ∅ for all t ∈ (t_{l−1}, t_{l}). This in turn implies that on the interval (t_{l−1}, t_{l}) the trajectory x(t) must satisfy (5.3) in the usual sense. Hence, we have

τẋ(t) = −μ ∑_{i∈J′} a_{i}^{T}

almost everywhere on the interval (t_{l−1}, t_{l}). Also, it follows from (5.5) that

V (x(t)) = ∑_{i∈J′} (a_{i}x(t) − b_{i})  (5.7)

for all t ∈ (t_{l−1}, t_{l}). Applying the chain rule to (5.7), we obtain

\frac{d}{dt}V (x(t)) = ∇^{T} V (x(t))ẋ(t)
= −\frac{μ}{τ} ‖∑_{i∈J′} a_{i}‖_{2}^{2}

almost everywhere on the interval (t_{l−1}, t_{l}). However, Lemma 5.3 along with the fact that there is a finite number of constraints, that is, m is finite, implies the existence of a positive constant, η_{1}, with the property that

η_{1} ≤ ‖ ∑_{i∈J(x)} a_{i}‖_{2}^{2}

for all x ∉ Ω. Hence,

\frac{d}{dt}V (x(t)) ≤ −\frac{μη_{1}}{τ} < 0  (5.8)

almost everywhere on the interval (t_{l−1}, t_{l}), which concludes our analysis of the first case.

Case 2 Suppose now that on an interval (t_{l−1}, t_{l}) a particular trajectory x(t) is confined to one or more of the surfaces h_{1}, h_{2}, …, h_{m} and does not intersect Ω. Specifically, suppose that there exist nonempty index sets I′ and J′ with the property that

I(x(t)) = I′ for all t ∈ (t_{l−1}, t_{l}),  (5.9)

and

J(x(t)) = J′ for all t ∈ (t_{l−1}, t_{l}).  (5.10)

Let i_{1}, i_{2}, …, i_{k} denote the elements of I′, and let S = ⋂_{i∈I′} h_{i} denote the surface to which the trajectory x(t) is confined on the interval (t_{l−1}, t_{l}), and let

ã = ∑_{i∈J′} a_{i}^{T} .

It follows from (5.9) and (5.10) that for a sufficiently small δ > 0 the surfaces h_{i_{1}}, …, h_{i_{k}} partition the δ-neighborhood of x(t′) into ℓ ≤ 2^{k} regions, R_{1}, R_{2}, …, R_{ℓ}, in each of which the function g is continuous. Then, G(x(t′)) is the set of all vectors v of the form

v = −μ ∑_{j=1}^{ℓ} α_{j} (ã + ∑_{i∈I′} U(i, j)a_{i}^{T} ) ,  (5.11)

where

α_{j} ≥ 0, j = 1, 2, …, ℓ,  (5.12)
∑_{j=1}^{ℓ} α_{j} = 1,  (5.13)

and

U(i, j) = \{ \begin{matrix}1 & if a_{i}x > b_{i} & for all & x ∈ R_{j} \\ 0 & if a_{i}x ≤ b_{i} & for all & x ∈ R_{j}. \end{matrix}

Using (5.13), we rewrite (5.11) as

v = −μã − μ ∑_{j=1}^{ℓ} α_{j} (∑_{i∈I′} U(i, j)a_{i}^{T} ) .  (5.14)

For notational simplicity, we let

r = ∑_{j=1}^{ℓ} α_{j} (∑_{i∈I′} U(i, j)a_{i}^{T} ) .

Then, (5.14) becomes

v = −μã − μr.  (5.15)

Let T (x) denote the tangent plane to the surface S at the point x. Observe that since the trajectory x(t) is confined to the surface S on the interval (t_{l−1}, t_{l}), then τẋ(t) ∈ T (x(t)) almost everywhere on the interval (t_{l−1}, t_{l}). Thus, x(t) is an absolutely continuous function that satisfies

τẋ(t) ∈ G(x(t)) ∩ T (x(t))

almost everywhere on the interval (t_{l−1}, t_{l}). We note that since the trajectory x(t) is confined to the surface S on the interval (t_{l−1}, t_{l}), the set G(x(t))∩T (x(t)) is nonempty for all (t_{l−1}, t_{l}). In particular, G(x(t′)) ∩ T (x(t′)) is the set of all v of the form (5.15) that also lie on the tangent plane T (x(t′)). However,

r ∈ Span {a_{i_{1}}^{T}, a_{i_{2}}^{T}, …, a_{i_{k}}^{T}} ,

and thus r is orthogonal to T (x(t′)). Therefore, the set G(x(t′)) ∩ T (x(t′)) consists of exactly one element, the orthogonal projection of the vector −μã onto the tangent plane T (x(t′)). Hence, for (t_{l−1}, t_{l}) the set G(x(t)) ∩ T (x(t)) consists of exactly one element, and thus ẋ(t) = −(μ/τ)P ã is uniquely determined almost everywhere on the interval (t_{l−1}, t_{l}), where P is the orthogonal projector onto the tangent plane T (x(t)). Note that

P = I_{n} − B (B^{T} B)^{−1} B^{T} ,

where I_{n} denotes the n × n identity matrix and

B = [a_{i_{1}}^{T}, a_{i_{2}}^{T}, …, a_{i_{k}}^{T}] .

By (5.5) and (5.10),

V (x(t)) = ∑_{i∈J′} (a_{i}x(t) − b_{i})  (5.16)

for all t ∈ (t_{l−1}, t_{l}). Applying the chain rule to (5.16) yields

\frac{d}{dt}V (x(t)) = ∇^{T} V (x(t))ẋ(t)
= −\frac{μ}{τ}ã^{T} P ã  (5.17)

almost everywhere on the interval (t_{l−1}, t_{l}). Observing that P = P^{T} = P^{2}, we rewrite (5.17) as

\frac{d}{dt}V (x(t)) = −\frac{μ}{τ} ∥P ã∥_{2}^{2} .  (5.18)

We now show that P ã ≠ 0. Suppose, on the contrary, that P ã = 0, which is true if and only ã = Bλ for some λ ∈ ℝ^{k}. Note that ã = ∇V (x). This implies that the convex function V is minimized subject to the active constraints outside of Ω, which is not true. Hence, P ã ≠ 0. Combining this result with the fact that there are finitely many constraints, we conclude that there exists a positive constant, η_{2}, such that

η_{2} ≤ ∥P ã∥_{2}^{2} for all x ∈ H \ Ω.

Hence,

\frac{d}{dt}V (x(t)) ≤ −\frac{μη_{2}}{τ} < 0  (5.19)

almost everywhere on the interval (t_{l−1}, t_{l}), which concludes our analysis of the second case. Using the above results, we can now state and prove the following theorem.

Theorem 5.1 Every trajectory of system (5.3) reaches Ω in finite time, and is thereafter confined to Ω.

Proof Let x(t) be any particular trajectory of system (5.3) and let

η = min \{\frac{μη_{1}}{τ} , \frac{μη_{2}}{τ} \} .

It follows directly from (5.8) and (5.19) that

\frac{d}{dt}V (x(t)) ≤ −η

for almost all t when x(t) ∉ Ω. Hence, x(t) ∈ Ω for all t ≥ V (x(0))/η.

□

5.3.2 Convergence Phase

We now analyze dynamical behavior of system (5.3) when its motion is confined to the feasible set Ω. We begin by introducing the function F : ℝ^{n} → ℝ such that

F (x) = f(x) − f^{∗},

where f^{∗} is the optimal value of f for the optimization problem (5.1). Clearly F ∈ C^{1}. Furthermore, we see from Lemma 5.1 that F (x) = 0 for all x ∈ Γ and F (x) > 0 for all x ∈ Ω \ Γ.

Similar to the previous subsection, we consider two cases when analyzing the behavior of system (5.3) in the region Ω. We first consider the case when the motion of system (5.3) is confined to the interior of Ω. Then, we consider the case when the motion of system (5.3) is confined to the boundary of Ω, that is, when system (5.3) is in a sliding mode on one or more of the surfaces h_{1}, …, h_{m}. Case 1 Let x(t) be any particular trajectory of system (5.3). Suppose that on an interval (t_{l−1}, t_{l}) the trajectory x(t) is confined to the interior of Ω. Then, I(x(t)) = ∅ and J(x(t)) = ∅ for all t ∈ (t_{l−1}, t_{l}). This implies that on the interval (t_{l−1}, t_{l}), the trajectory x(t) satisfies τẋ(t) = −∇f(x(t)) almost everywhere on the interval (t_{l−1}, t_{l}) in the usual sense. Applying the chain rule to F (x(t)), we obtain

\frac{d}{dt}F (x(t)) = ∇^{T} F (x(t))ẋ(t)
= −\frac{1}{τ} ∥∇f(x(t))∥_{2}^{2}  (5.20)

almost everywhere on the interval (t_{l−1}, t_{l}). Using the fact that I(x(t)) = ∅ for all t ∈ (t_{l−1}, t_{l}), we see from Lemma 5.2 that on the interval (t_{l−1}, t_{l}), we have ∇f(x(t)) = 0 if and only if x(t) ∈ Γ. Hence, we have

\frac{d}{dt}F (x(t)) = 0 if and only if x(t) ∈ Γ,  (5.21)

and

\frac{d}{dt}F (x(t)) < 0 if and only if x(t) ∉ Γ,  (5.22)

almost everywhere on the interval (t_{l−1}, t_{l}). This concludes our analysis of the first case. Case 2 Let x(t) denote any particular trajectory of system (5.3) and let t ∈ (t_{l−1}, t_{l}). Suppose that on the interval (t_{l−1}, t_{l}) the trajectory x(t) is confined to the boundary of Ω. Thus,

1. there exists a nonempty index set I′ with the property that I(x(t)) = I′ for all t ∈ (t_{l−1}, t_{l}), and

2. J(x(t)) = ∅ for all t ∈ (t_{l−1}, t_{l}).

Let i_{1}, …, i_{k} denote the elements of I′, and let S = ⋂_{i∈I′} h_{i} denote the surface to which the trajectory x(t) is confined on the interval (t_{l−1}, t_{l}). Let B ∈ ℝ^{n×k} be the matrix with columns a_{i_{1}}^{T}, …, a_{i_{k}}^{T}. By assumption A3 the vectors a_{i_{1}}^{T}, …, a_{i_{k}}^{T} are linearly independent, and hence the matrix B has full rank.

It follows from the above that for a sufficiently small δ > 0 the surfaces h_{i_{1}}, …, h_{i_{k}} partition the δ-neighborhood of x(t′) into 2^{k} regions, in each of which the function g is continuous. Using similar argument to the one in the previous subsection, we can show that the set G(x(t′)) is the set of all v of the form

v = −α_{0}∇f(x(t′)) − μ ∑_{j=1}^{2^{k}−1} α_{j}Bu_{j},  (5.23)

where

α_{j} ≥ 0, j = 0, 1, …, 2^{k} − 1,  (5.24)
∑_{j=0}^{2^{k}−1} α_{j} = 1,  (5.25)

and u_{j} ∈ ℝ^{k}, j = 1, …, 2^{k} − 1 are defined as

u_{1} = [ 1 0 0 ⋯ 0 0 0 ]^{T} ,
u_{2} = [ 0 1 0 ⋯ 0 0 0 ]^{T} ,
u_{3} = [ 1 1 0 ⋯ 0 0 0 ]^{T} ,
⋮
u_{2^{k}−3} = [ 1 1 1 ⋯ 1 0 1 ]^{T} ,
u_{2^{k}−2} = [ 1 1 1 ⋯ 1 1 0 ]^{T} ,
u_{2^{k}−1} = [ 1 1 1 ⋯ 1 1 1 ]^{T} .

Let T (x) denote the tangent plane to the surface S at the point x. Observe that ẋ(t) ∈ T (x(t)) almost everywhere on the interval (t_{l−1}, t_{l}) because the trajectory x(t) is confined to the surface S on the interval (t_{l−1}, t_{l}). Thus, x(t) is an absolutely continuous function that satisfies

τẋ(t) ∈ G(x(t)) ∩ T (x(t))

almost everywhere on the interval (t_{l−1}, t_{l}). We note that since the trajectory x(t) is confined to the surface S on the interval (t_{l−1}, t_{l}), the set G(x(t))∩T (x(t)) is by assumption nonempty for all t ∈ (t_{l−1}, t_{l}). In particular, we see that G(x(t′)) ∩ T (x(t′)) is the set of all v that have the form (5.23) and also lie on the tangent plane T (x(t′)). Let

r = ∑_{j=1}^{2^{k}−1} α_{j}Bu_{j}.

Then, we represent (5.23) as

v = −α_{0}∇f(x(t′)) − μr.  (5.26)

Obviously, r ∈ Span {a_{i_{1}}^{T}, …, a_{i_{k}}^{T}}, and thus r is orthogonal to T (x(t′)). However, v ∈ T (x(t′)) by assumption. Therefore, v is the orthogonal projection of the vector −α_{0}∇f(x(t′)) onto the tangent plane T (x(t′)), that is,

v = −α_{0}P ∇f(x(t′)),  (5.27)

where P = I_{n} − B (B^{T} B)^{−1} B^{T}.

Note that by virtue of assumptions A1 and A2, the set ∇f(Ω) is compact. This implies the existence of a non-negative constant, M, with the property that

M ≥ ∥∇f(x)∥_{2} for all x ∈ Ω.

Let

γ = \frac{μ}{μ + M\sqrt{k} ∥(B^{T} B)^{−1} B^{T} ∥_{2}} .

Proposition 5.1

γ ≤ α_{0} ≤ 1.

Proof It follows from (5.24) and (5.25) that α_{0} ≤ 1. Thus, it remains to show that α_{0} ≥ γ. It follows from (5.26) and (5.27) that μr is the orthogonal projection of the vector −α_{0}∇f(x(t′)) onto the subspace T (x(t′))^{⊥}, where T (x(t′))^{⊥} is the orthogonal complement of T (x(t′)) in ℝ^{n}. Therefore,

−α_{0} (I_{n} − P) ∇f(x(t′)) = μ ∑_{j=1}^{2^{k}−1} α_{j}Bu_{j}.  (5.28)

Let U ∈ ℝ^{k×(2^{k}−1)} be the matrix with columns u_{1}, …, u_{2^{k}−1}, and α ∈ ℝ^{2^{k}−1} the column vector with the components α_{1}, …, α_{2^{k}−1}. Then, we can represent (5.28) as

−\frac{α_{0}}{μ} (I_{n} − P) ∇f(x(t′)) = BUα.  (5.29)

Premultiplying both side of (5.29) by (B^{T} B)^{−1} B^{T}, we obtain

−\frac{α_{0}}{μ} (B^{T} B)^{−1} B^{T} (I_{n} − P) ∇f(x(t′)) = Uα,

and hence

\frac{α_{0}}{μ} ∥(B^{T} B)^{−1} B^{T} (I_{n} − P) ∇f(x(t′))∥_{2} = ∥Uα∥_{2} .  (5.30)

Applying the Cauchy-Schwarz inequality to (5.30), and using the fact that ∥I_{n} − P ∥ = 1, we have

\frac{Mα_{0}}{μ} ∥(B^{T} B)^{−1} B^{T} ∥_{2} ≥ ∥Uα∥_{2} ≥ \frac{1}{\sqrt{k}} ∥Uα∥_{1}

since for z ∈ ℝ^{k}, we have ∥z∥_{2} ≥ \frac{1}{\sqrt{k}}∥z∥_{1}. However, using (5.24) and (5.25), we see that

∥Uα∥_{1} ≥ ∑_{j=1}^{2^{k}−1} α_{j} = 1 − α_{0}.

Thus,

\frac{Mα_{0}\sqrt{k}}{μ} ∥(B^{T} B)^{−1} B^{T} ∥_{2} ≥ 1 − α_{0}.  (5.31)

It follows from (5.31) that α_{0} > 0. Dividing both sides of (5.31) by α_{0} yields

\frac{M\sqrt{k}}{μ} ∥(B^{T} B)^{−1} B^{T} ∥_{2} ≥ \frac{1 − α_{0}}{α_{0}} = \frac{1}{α_{0}} − 1.

Hence,

α_{0} ≥ \frac{1}{1 + \frac{M\sqrt{k}}{μ} ∥(B^{T} B)^{−1} B^{T} ∥_{2}} = γ.

The proof of the proposition is complete.

□

We have the following two corollaries. Corollary 5.1

G(x(t′)) ∩ T (x(t′)) ⊂ {−αP ∇f(x(t′)) : γ ≤ α ≤ 1} .

Proof This follows directly from (5.27) and Proposition 5.1.

□

Corollary 5.2

P ∇f(x(t′)) = 0 if and only if 0 ∈ G(x(t′)) ∩ T (x(t′)) .

Proof If P ∇f(x(t′)) = 0, then 0 ∈ G(x(t′)) ∩ T (x(t′)) since, by Corollary 5.1,

P ∇f(x(t′)) ∈ G(x(t′)) ∩ T (x(t′)) .

Conversely, suppose that 0 ∈ G(x(t′)) ∩ T (x(t′)). By Corollary 5.1, −αP ∇f(x(t′)) = 0 for some α ∈ [γ, 1]. Since γ > 0 for all μ > 0, −αP ∇f(x(t′)) = 0 is possible only if P ∇f(x(t′)) = 0, which completes the proof.

□

Proposition 5.2

P ∇f(x(t′)) = 0 if and only if x(t′) ∈ Γ.

Proof We first prove sufficiency part of the proposition. Suppose P ∇f(x(t′)) = 0. It follows from (5.27) and (5.23) that

α_{0}∇f(x(t′)) + μ ∑_{j=1}^{2^{k}−1} α_{j}Bu_{j} = 0.  (5.32)

Let

d = \frac{μ}{α_{0}} ∑_{j=1}^{2^{k}−1} α_{j}u_{j}.

Note that since α_{0} ≠ 0, d is well defined. It then follows from (5.32) that

∇f(x(t′)) + Bd = 0.

We see from (5.24) and the definition of u_{1}, …, u_{2^{k}−1} that each element of d is non-negative. By Lemma 5.2, the point x(t′) satisfies the Karush-Kuhn-Tucker conditions for problem (5.1) and hence x(t′) ∈ Γ. The sufficiency condition is thus proven.

We now prove the necessity condition. Suppose that x(t′) ∈ Γ. By Lemma 5.2, the point x(t′) satisfies the Karush-Kuhn-Tucker optimality conditions for problem (5.1). This implies the existence of non-negative constants μ_{1}, …, μ_{k} with the property that

∇f(x(t′)) + Bμ = 0,  (5.33)

where μ = [μ_{1}, …, μ_{k}]^{T}. Let

ν = 1 + \frac{1}{μ} ∑_{i=1}^{k} μ_{i},  (5.34)
α̃_{0} = \frac{1}{ν},  (5.35)
α̃_{2^{i−1}} = \frac{μ_{i}}{μν} , i = 1, …, k,  (5.36)
α̃_{j} = 0, j = 1, …, 2^{k} − 1, j ≠ 2^{i−1} for i = 1, …, k,  (5.37)
ṽ = −α̃_{0}∇f(x(t′)) − μ ∑_{j=1}^{2^{k}−1} α̃_{j}Bu_{j}.  (5.38)

It follows from (5.34)–(5.37) that

α̃_{j} ≥ 0, j = 0, 1, …, 2^{k} − 1, and ∑_{j=0}^{2^{k}−1} α̃_{j} = 1.

Therefore, ṽ ∈ G(x(t′)). Note that

ṽ = −\frac{μ}{μ + ∑_{i=1}^{k} μ_{i}} (∇f(x(t′)) + Bμ) .

Using (5.33), we conclude that ṽ = 0, and thus ṽ ∈ T (x(t′)). Hence,

0 ∈ G(x(t′)) ∩ T (x(t′)).

Using Corollary 5.2, we get P ∇f(x(t′)) = 0, which completes the proof of the necessary condition and the proof of the proposition.

□

Applying now the chain rule to F, taking into account (5.27), Proposition 5.1, and the fact that P = P^{T} = P^{2}, we obtain

\frac{d}{dt}F (x(t)) ≤ −\frac{γ}{τ} ∥P ∇f(x(t))∥_{2}^{2}  (5.39)

almost everywhere on the interval (t_{l−1}, t_{l}). By Proposition 5.2,

\frac{d}{dt}F (x(t)) = 0 if and only if x(t) ∈ Γ,  (5.40)

and

\frac{d}{dt}F (x(t)) < 0 if and only if x(t) ∉ Γ,  (5.41)

almost everywhere on the interval (t_{l−1}, t_{l}), which concludes our analysis for the second case.

We now show that any trajectory of system (5.3) that is confined to Ω converges asymptotically to the region Γ.

Theorem 5.2 Let x(t) be any trajectory of system (5.3). If x(t) ∈ Ω for all t, then the trajectory x(t) converges asymptotically to Γ.

Proof Recall that F (x) = 0 if x ∈ Γ and F (x) > 0 if x ∈ Ω \ Γ. Therefore, to prove the theorem it is enough to show that

lim_{t→∞} F (x(t)) = 0,

or equivalently, that for any ε > 0 there exists a finite time, T (x(0), ε), such that F (x(t)) < ε for all t ≥ T (x(0), ε). We see from (5.22) and (5.41) that

\frac{d}{dt}F (x(t)) < 0

for almost all t when x(t) ∉ Γ. Consequently F (x(t)) ≤ F (x(0)) for all t ≥ 0. Therefore, we only need to consider the case when F (x(0)) ≥ ε. Let

θ(ε) = \{x : x ∈ Ω, F (x) ≥ \frac{ε}{2}\} .

Since F (x(0)) ≥ ε, we have x(0) ∈ θ(ε) and θ(ε) ⊂ Ω \ Γ. Furthermore, since the function F is continuous, the set θ(ε) is compact. We will now show that there exists a positive constant, ρ(ε), such that

\frac{d}{dt}F (x(t)) ≤ −ρ(ε)

for almost all t when x(t) ∈ θ(ε), thereby guaranteeing that F (x(t)) ≤ ε/2 for all

t ≥ \frac{F (x(0)) − ε/2}{ρ(ε)}.

We analyze two cases. The first case is when the trajectory x(t) is in θ(ε) and is not in a sliding mode on any of the surfaces h_{i}. Then, we analyze the case when the trajectory is in θ(ε) in a sliding mode on a surface that is an intersection of some of the hyperplanes h_{i}. Case 1 Since the function f is convex and f ∈ C^{1}, we have ∇f(x) ≠ 0 for all x ∈ Ω \ Γ. Hence,

−\frac{1}{τ} ∥∇f(x)∥_{2}^{2} < 0 for all x ∈ θ(ε)  (5.42)

because θ ⊂ Ω \ Γ and ∇f(x) ≠ 0 in θ(ε). The fact that the function −1/τ ∥∇f(x)∥_{2}^{2} is continuous on the compact set θ(ε) implies the existence of a positive constant, ρ_{1}(ε), such that

−\frac{1}{τ} ∥∇f(x)∥_{2}^{2} ≤ −ρ_{1}(ε) for all x ∈ θ(ε).

Combining the above with (5.20) yields

\frac{d}{dt}F (x(t)) ≤ −ρ_{1}(ε)

for almost all t when x(t) ∈ θ(ε), which concludes our analysis of the first case. Case 2 Let i_{1}, …, i_{k} denote the elements of I′, and let S = ⋂_{i∈I′} h_{i} denote the surface to which the trajectory x(t) is confined on the interval (t_{l−1}, t_{l}). Suppose that S ∩ θ(ε) ≠ ∅. Clearly,

S ∩ θ(ε) ⊂ Ω \ Γ.

By Proposition 5.2,

−\frac{1}{τ} ∥P ∇f(x)∥_{2}^{2} < 0 for all x ∈ S ∩ θ(ε),  (5.43)

where P is the orthogonal projector onto the tangent plane to the surface S. By the above and the fact that the function −\frac{1}{τ} ∥P ∇f(x)∥_{2}^{2} is continuous on the compact set S ∩ θ(ε), there exists a positive constant, ρ̂(ε), such that

−\frac{1}{τ} ∥P ∇f(x)∥_{2}^{2} < −ρ̂(ε) for all x ∈ S ∩ θ(ε).

It then follows directly from (5.39) that there exists a positive constant γ̂ such that

\frac{d}{dt}F (x(t)) ≤ −γ̂ρ̂(ε)

for almost all t when x(t) ∈ S ∩ θ(ε). The above along with the fact that there are finitely many constraints implies the existence of a positive constant, ρ_{2}(ε), such that

\frac{d}{dt}F (x(t)) ≤ −ρ_{2}(ε)  (5.44)

for almost all t when x(t) ∈ S ∩ θ(ε). This concludes our analysis for the second case.

Let

ρ(ε) = min {ρ_{1}(ε), ρ_{2}(ε)} .

Then, using (5.43) and (5.44) yields

\frac{d}{dt}F (x(t)) ≤ −ρ(ε)

for almost all t when x(t) ∈ θ(ε). Hence, F (x(t)) ≤ ε/2 < ε for all

t ≥ \frac{F (x(0)) − ε/2}{ρ(ε)}.

Therefore,

T (x(0), ε) = \{ \begin{matrix}0 & if & F (x(0)) < ε \\ \frac{F (x(0))−ε/2}{ρ(ε)} & if & F (x(0)) ≥ ε. \end{matrix}

The proof of the theorem is complete.

□

Theorem 5.3 Every trajectory of system (5.3), irrespective of the initial condition, x(0), converges to a solution of problem (5.1).

Proof This statement follows directly from Theorems 5.1 and 5.2.

□

Figure 5.1: Phase-plane portrait for the system in the Example.

Example 5.1 We consider the quadratic programming problem

minimize \frac{1}{2}x^{T} Qx + c^{T} x, x ∈ ℝ^{2}
subject to Ax ≤ b,

where

Q = [ \begin{matrix}4 & 1 \\ 1 & 2 \end{matrix} ] , c = [ \begin{matrix}−12 \\ −10 \end{matrix} ] , A = [ \begin{matrix}5 & 1 \\ −5 & 1 \\ −1 & −2 \end{matrix} ] , and b = [ \begin{matrix}0 \\ 10 \\ 10 \end{matrix} ] .

The above optimization problem clearly satisfies A1-A4, and one can easily verify that the point x^{∗} = [ −19/22 95/22 ]^{T} is a unique minimizer for the problem, that is, Γ = {x^{∗}}. Note that x^{∗} ∈ h_{1}. A phase-plane portrait for system (5.3) that solves the above optimization problem is shown in Figure 5.1. Observe that each of the trajectories in the phase-plane portrait converges to the point x^{∗} while sliding along the surface h_{1}.

5.4 Notes

168 CHAPTER 5. SLIDING MODES IN SOLVING OPTIMIZATION PROBLEMS

Neural Networks

This chapter is concerned with applying the Lyapunov’s method to the stability analysis of a class of dynamical systems called artificial neural networks. The motivation to study artificial neural networks, commonly referred to as neural networks, or just neural nets, is in attempt to build computers whose construction would be inspired by the brain. The brain is a highly complex information processing system. It is capable of organizing its building blocks, called neurons, to perform complex computations in a very efficient way. A drawing of a more or less typical biological neuron is given in Figure 6.1. The neuron collects signals from other neurons through its dendrites. The signals have the form of electrical pulses. These pulses travel to the neuron’s cell body where they are being processed. We note that the surface of the cell body of the neuron is dotted with numerous receptor sites that receive messages form other neurons in a similar fashion as the dendrites do. The neuron may produce its own pulses in response to excitation from other neurons. The neuron’s own pulses travel through a long, thin strand called an axon, which splits into many end branches. The neuron’s message to the other neurons is delivered via the end branches. The key to the transmission of the neuron’s message is the synapse, whose drawing is shown in Figure 6.2. The end branches of the neuron’s axon terminate in synaptic knobs. The synaptic knob of the first neuron is separated from the receptor site of another neuron by a microscopic distance. The cell body of the first neuron produces chemical substances called neurotransmitters, which are delivered down to the synaptic vesicles. The neurotransmitters are stored in the synaptic vesicles until the neuron fires in which case a burst of neurotransmitters is released by the vesicles. The neurotransmitters then flow across the synaptic cleft and act on the second neuron. The neurotransmitters of the first neuron may stimulate or inhibit the second neuron activity. The first neuron has many other synaptic knobs elsewhere, while the second neuron has many other receptor sites that receive messages from many other neurons. In 170 CHAPTER 6. NEURAL NETWORKS

Figure 6.1: A typical biological neuron. Figure 6.2: The synapse is a junction point where neuron meets neuron. fact each neuron is receiving messages from hundreds or thousands of other neurons. The neuron sets off the nervous pulse, that is, it fires off its own nervous pulse only when it is properly stimulated. It was observed that in general the neuron will not fire in response to a single message. To fire, the neuron needs many messages arriving simultaneously or in quick succession from other neurons. In addition, the stimulating messages to fire must outweigh the messages that inhibit the neuron from firing.

In the following sections, we analayze two different classes of neural networks. We begin with the so called radial-basis function (RBF) neural networks and their applications to identification of dynamical systems.

6.1 Radial-Basis Function Networks

In this section, we present radial-basis function (RBF) neural networks and their applications to interpolation and then to identification of unknown nonlinear dynamical systems using only the systems’ inputs and outputs data. We first show how an RBF network can be used for interpolation purposes. Then, we will concern ourselves with the problem of identification of dynamical systems. Our exposition of using RBF neural networks to identify dynamical systems is based on the paper by Liu, Kadirkamanathan, and Billings [69]. The RBF neural identifier is an adaptive system for which the learning algorithm is developed using Lyapunov techniques. The algorithm allocates resources, by adding new nodes, to obtain the desired identification accuracy.

6.1.1 Interpolation Using RBF Networks

A diagram of a radial-basis function (RBF) network is shown in Figure 6.3. The functions ϕ (∥x − x^{(i)}∥), i = 1, 2, …, N, are called the radial-basis functions, where ∥ · ∥ denotes a p-norm that is usually the Euclidean 2-norm. The real parameters w_{i}, i = 1, 2, …, N are the weights. A common choice for ϕ is

ϕ (∥x − x^{(i)}∥) = exp (−\frac{∥x − x^{(i)}∥^{2}}{2σ_{i}^{2}} ) ,  (6.1)

where x^{(i)} is the center of the basis function and σ_{i} is sometimes called the radius of the radial basis function. The function given by (6.1) is called the Gaussian radial basis function.

The RBF networks are used to perform interpolation. The interpolation problem can be formulated as:

172 CHAPTER 6. NEURAL NETWORKS

Figure 6.3: A two-input radial-basis function (RBF) network.

For a given set of N different points,

{x^{(i)} : x^{(i)} ∈ ℝ^{n}, i = 1, 2, …, N}

and a corresponding set of N real numbers

{d_{i} : d_{i} ∈ ℝ, i = 1, 2, …, N}

find a function

F : ℝ^{n} → ℝ

such that

F (x^{(i)}) = d_{i}, i = 1, 2, …, N.  (6.2)

The function that the RBF network implements is

F (x) = ∑_{j=1}^{N} w_{j}ϕ (∥x − x^{(j)}∥) .  (6.3)

Substituting the interpolation conditions given by (6.2) into (6.3) gives

[ \begin{matrix}ϕ_{11} & ϕ_{12} & ⋯ & ϕ_{1N} \\ ϕ_{21} & ϕ_{22} & ⋯ & ϕ_{2N} \\ ⋮ & ⋮ & ... & ⋮ \\ ϕ_{N1} & ϕ_{N2} & ⋯ & ϕ_{NN} \end{matrix} ] [ w_{1} ; w_{2} ; ⋮ ; w_{N} ] = [ d_{1} ; d_{2} ; ⋮ ; d_{N} ] ,  (6.4)

where ϕ_{ij} = ϕ (∥x^{(i)} − x^{(j)}∥). Let w = [ w_{1} w_{2} ⋯ w_{N} ], d = [ d_{1} d_{2} ⋯ d_{N} ], and

Φ = [ \begin{matrix}ϕ_{11} & ϕ_{12} & ⋯ & ϕ_{1N} \\ ϕ_{21} & ϕ_{22} & ⋯ & ϕ_{2N} \\ ⋮ & ⋮ & ... & ⋮ \\ ϕ_{N1} & ϕ_{N2} & ⋯ & ϕ_{NN} \end{matrix} ] .

Then, using the above notation, we represent (6.4) as

Φw = d.  (6.5)

The matrix Φ is called the interpolation matrix. If the interpolation matrix is nonsingular, then we can solve (6.5) for w to obtain

w = Φ^{−1}d.  (6.6)

Example 6.1 We construct an RBF network for the interpolation data given in Table 6.1. In this example, ϕ_{ij} = exp (−\frac{(x^{(i)}−x^{(j)})^{2}}{2σ_{i}^{2}} ). We set σ_{i} = 1, i = 1 …, 5, and evaluate (6.6) to

Table 6.1: Interpolation data for Example 6.1.

\begin{matrix}x^{(i)} & d_{i} \\ 0 & 0.8 \\ 6 & 0.2 \\ 7 & 0.4 \\ 8 & 0.6 \\ 10 & 0.3 \end{matrix}

obtain

w = [ 0.8000 0.1459 −0.0390 0.5737 0.2227 ]^{T} .

The interpolation result is shown in Figure 6.4. In Figure (6.4)(a), we used solid line to show the result of interpolating using the Gaussian radial-radial basis functions, while broken line indicates the result of the straight line interpolation. The plots of the basis functions that contribute to the interpolating curve are shown in Figure (6.4)(b).

Figure 6.4: Results of interpolating using radial-basis functions.

6.1.2 Identification of a Single-Input Single-State System

We first discuss the identification of the single-input single-state dynamical system using neural network. Let the dynamical system be modeled by

ẋ(t) = f(x(t), u(t)), x(0) = x_{0},  (6.7)

where f : ℝ × ℝ → ℝ is an unknown, possibly nonlinear, function of x and u to be identified by a neural network. Adding and subtracting ax(t), where a > 0 is a design parameter, to the right-hand side of (6.7) gives

ẋ(t) = −ax(t) + ax(t) + f(x(t), u(t))
= −ax(t) + g(x(t), u(t)),  (6.8)

where g(x(t), u(t)) = ax(t) + f(x(t), u(t)). The reason for the above manipulations will become apparent when we discuss the learning rules. We will now proceed with the construction of a neural identifier of the dynamical system (6.7). The identifier will be described by

˙x̂(t) = −ax̂(t) + ĝ(x(t), u(t), m_{k}, σ_{k}), x̂(0) = x̂_{0},  (6.9)

where x̂ is the state of the identification network, m_{k} and σ_{k} are adjustable parameters of the identifier. The function ĝ has the form,

ĝ(x(t), u(t), m_{k}, σ_{k}) = ∑_{k=1}^{K} w_{k}ϕ(x, u, m_{k}, σ_{k}),  (6.10)

where for k = 1, 2, …, K, we have m_{k} = [ m_{xk} m_{uk} ], and

ϕ(x, u, m_{k}, σ_{k}) = exp (−\frac{1}{2σ_{k}^{2}} ((x(t) − m_{xk})^{2} + (u(t) − m_{uk})^{2})) .

Thus, we postulate that the function ĝ be generated using an RBF network. Suppose now that the basis functions are given to us. The problem of choosing the basis functions will be discussed later. Suppose also that we know the “optimal” weights w_{k}^{∗}, k = 1, 2, …, K, that result in the least identification error between g in (6.8) and ĝ in (6.9), where the identification error is defined as

e(t) = g(x(t), u(t)) − ∑_{k=1}^{K} w_{k}^{∗}ϕ(x(t), u(t), m_{k}, σ_{k})  (6.11)

Using the above notation, we represent the dynamical system model (6.8) as

ẋ(t) = −ax(t) + ∑_{k=1}^{K} w_{k}^{∗}ϕ(x(t), u(t), m_{k}, σ_{k}) + e(t).  (6.12)

The neural identifier, on the other hand can be represented as

˙x̂(t) = −ax̂(t) + ∑_{k=1}^{K} w_{k}ϕ(x(t), u(t), m_{k}, σ_{k})  (6.13)

A schematic of the above identification architecture is depicted in Figure 6.5.

6.1.3 Learning Algorithm for the RBF Identifier

We assume that both the state x and the input u of the dynamical system model (6.7), and hence that of (6.8), are bounded. We next define the state estimation error to be

e_{x} = x − x̂,

while the weights’ estimation errors,

ξ_{k} = w_{k}^{∗} − w_{k}, k = 1, …, K.

Figure 6.5: A configuration for the radial-basis function neural identifier.

We assume that the state estimation error, e_{x}, is bounded by some finite constant e_{K}, that is, |e_{x}| ≤ e_{K}. However, e_{K} is unknown to us. Next, we use (6.12) and (6.13) to form the differential equation describing the state error dynamics. For simplicity of notation, we omit the argument t of the time-varying functions. We have

ė_{x} = ẋ − ˙ x̂
= −ax + ∑_{k=1}^{K} w_{k}^{∗}ϕ(x, u, m_{k}, σ_{k}) + e + ax̂ − ∑_{k=1}^{K} w_{k}ϕ(x, u, m_{k}, σ_{k})
= −ae_{x} + ∑_{k=1}^{K} ξ_{k}ϕ(x, u, m_{k}, σ_{k}) + e.  (6.14)

Our goal now is to use the second method of Lyapunov to devise a learning law for the RBF identifier that would yield stable identification scheme in the sense that the state error e_{x} decreases as the time increases and the weights do not diverge. For this purpose, we will use the following Lyapunov function candidate,

V (e_{x}, ξ) = \frac{1}{2} (e_{x}^{2} + \frac{1}{α} ∑_{k=1}^{K} ξ_{k}^{2}) ,  (6.15)

where α > 0 is a design parameter and ξ = [ ξ_{1} ξ_{2} ⋯ ξ_{K} ]^{T}. We now evaluate the time derivative of V on the trajectories of (6.14),

V̇ = e_{x}ė_{x} + \frac{1}{α} ∑_{k=1}^{K} ξ_{k}ξ̇_{k}

= e_{x} (−ae_{x} + ∑_{k=1}^{K} ξ_{k}ϕ(x, u, m_{k}, σ_{k}) + e) + \frac{1}{α} ∑_{k=1}^{K} ξ_{k}ξ̇_{k}
= −ae_{x}^{2} + \frac{1}{α} ∑_{k=1}^{K} ξ_{k} (αe_{x}ϕ(x, u, m_{k}, σ_{k}) + ξ̇_{k}) + e_{x}e.  (6.16)

Let

ξ̇_{k} = −αe_{x}ϕ(x, u, m_{k}, σ_{k}).  (6.17)

Then (6.16) simplifies to

V̇ = −ae_{x}^{2} + e_{x}e.  (6.18)

Since ẇ_{k} = −ξ̇_{k}, we can represent (6.17) as

ẇ_{k} = αe_{x}ϕ(x, u, m_{k}, σ_{k})  (6.19)

We further evaluate (6.18) to obtain

V̇ ≤ −ae_{x}^{2} + |e_{x}| e_{K}
= −a |e_{x}| (|e_{x}| − \frac{e_{K}}{a} )  (6.20)

It follows from the above that if there is no state estimation error, that is, e_{K} = 0, then V̇ is negative semi-definite in the [ e_{x} ξ^{T} ]^{T} -space. In such a case the origin of the [ e_{x} ξ^{T} ]^{T} - space is stable, which guarantees boundedness of the weights w_{k}s. In the presence of the state estimation error, if |e_{x}| < e_{K}/a, then it is possible that V̇ > 0, which implies that the weights w_{k}s may drift to infinity with time. To avoid this problem, Liu, Kadirkamanathan, and Billings [69] propose to set ẇ_{k} = 0 if |e_{x}| < e_{K}/a. Also since e_{K} is unknown, a bound is imposed on the weights as well as on the error. The learning rule becomes

ẇ_{k} = \{ \begin{matrix}αe_{x}ϕ(x, u, m_{k}, σ_{k}) & if & |e_{x}| ≥ e_{0} and ∥w∥ ≤ M \\ 0 & & otherwise \end{matrix} (6.21)

where the bounds e_{0} and M are design parameters, and w = [ w_{1} ⋯ w_{K} ]^{T}. With the above modified learning law, whenever |e_{x}| ≥ e_{0} ≥ e_{K}/a, the Lyapunov derivative V̇ is always semi-definite. However, when e_{0} ≤ |e_{x}| ≤ e_{K}/a, the weights may increase with time because it is possible that V̇ > 0. But the weights, as well as the estimation error, e_{x}, are upper-bounded. When any of the two bounds is reached, more basis functions maybe needed to improve the approximation of g. New basis functions are added to reduce the error until V̇ < 0. We now describe an algorithm for adding radial-basis functions to the identification network.

6.1.4 Growing RBF Network

As we mentioned above, when V̇ > 0, the identification scheme is no longer stable, but the learning rule (6.21) guarantees that the adjustable parameters are bounded. Once the parameter adjustments can no longer reduce the error and the bounds are reached, additional basis functions in the RBF network should be added. This continues until the approximation leads to the error reduction so that V̇ < 0.

In the proposed identification scheme, we use the Gaussian radial-basis functions, whose areas of “influence” are determined by the functions’ centers and widths. As the function to be approximated moves further away from a particular basis function, the contribution of that basis function decreases. This is due to the exponential form of the Gaussian basis functions. As z increases exp(−z) decreases. The maximum contribution of a particular basis function to approximating a given function occurs when the center of the basis function and the approximated function are at the same point, since in this case we have exp(0) = 1. This heuristic argument leads to the following rule for determining the centers of new basis functions to be added at a time instance t,

m_{K+1} = [ round (x(t)/δ_{x}) δ_{x} round (u(t)/δ_{u}) δ_{u} ]^{T} ,  (6.22)

where the round operator rounds the elements on which it operates to the nearest integer. For example, let x = [ −1.8 −0.3 3.7 ]. Then, round(x) = [ −2 0 4 ]. The parameters, δ_{x} and δ_{u} represent the spacing of the centers of the basis functions. We can think as having a grid over the input-state space where the spacing between the vertices of the grid are determined by the parameters δ_{x} and δ_{u}. Each vertex of the grid represents the potential center of a basis function. The radius of the new basis function and the adjusted radii are determined from the relation,

σ_{k} = κ ∥m_{k} − m_{k}^{∗}∥ ,  (6.23)

where κ > 0 is a design parameter and

m_{k}^{∗} = arg min_{{i=1,...,K \atop m_{i}≠m_{k}}} \{∥m_{k} − m_{k}^{∗}∥\} .  (6.24)

Thus, m_{k}^{∗} is the nearest center of all basis functions to the center of the k-th basis function. With the above background, we can now present the rules for adding a new basis function to the RBF network. Initially, the network contains no basis functions. Then, once the following two conditions are satisfied, a new basis function is added. The first condition is:

min_{k=1,...,K} |x(t) − m_{xk}| > δ_{x}/2 or min_{k=1,...,K} |x(t) − m_{uk}| > δ_{u}/2.  (6.25)

The above will be satisfied if the state or input, at time t, is closer to an available grid vertex than to any of the existing centers of the basis functions. The basis function at the new vertex would be the closest to the existing state or input and would have the greatest influence in the approximation of g. The second condition is

|x(t) − x̂(t)| > e_{max},  (6.26)

where e_{max} is the maximum allowable state estimation error. If condition (6.26) is satisfied, a new basis function is added to reduce the state estimation error.

Example 6.2 The above technique for identification of dynamical systems using growing radial-basis function network is illustrated with the following system,

ẋ = 1.1 (1 − x + 2xu − x^{2}) exp (−0.5x^{2} − u^{2}) , x(0) = 0,  (6.27)

where u = cos(t/100). In our simulations, we discretized the above differential equation using the Euler method with the step size dt = 0.01 sec. The initial weights and the initial state of the identifier were set to zero. We use the following parameter values: e_{0} = 0.001, e_{max} = 0.005, α = 5, κ = 3. In Figure 6.6, plots of the true state and its estimate versus time are shown. A plot of the error, e_{x}, between the true state and its estimate is shown in Figure 6.7. Initially, the error exceeds the bound e_{max}. Adding additional basis functions drives the error back down. The plots shown in Figure 6.8 depict the true state and its estimate versus the input u. Finally, in Figure 6.9, plots of the derivative of the true state versus the true state and the derivative of the estimated state versus the state estimate are shown.

6.1.5 Identification of Multivariable Systems

We now extend the above concepts to multi-input multi-state dynamical systems. In our derivation of the update law for the weights of the RBF network, we use a result from matrix algebra that we present next. Recall the definition of the trace of a square matrix as given by (1.32). If Γ ∈ ℝ^{m×n}, then by (1.67) its Frobenius norm satisfies

∥Γ∥_{F}^{2} = trace (Γ^{T} Γ) = ∑_{i=1}^{m} ∑_{j=1}^{n} γ_{ij}^{2}.

If the elements, γ_{ij}, of Γ are functions of time, then

\frac{d}{dt} (trace (Γ^{T} Γ)) = 2 ∑_{i=1}^{m} ∑_{j=1}^{n} γ_{ij}γ̇_{ij} = 2 trace (Γ^{T} Γ̇) ,  (6.28)

180 CHAPTER 6. NEURAL NETWORKS Figure 6.6: Plots of the true state and its estimate versus time. The estimated state “hugs” closely the actual state.

Figure 6.7: A plot of the error, e_{x}, between the true state and its estimate.

6.1. RADIAL-BASIS FUNCTION NETWORKS 181

Figure 6.8: A plot of the true and its estimate versus the input u.

Figure 6.9: Plots of the derivative of the true state versus the true state and the derivative of the estimated state versus the state estimate. where Γ̇ = \frac{dΓ}{dt}. With the above background, we can proceed with the analysis of the RBF neural identifier for the multivariable systems.

The multivariable dynamical systems that we consider are modeled by

ẋ(t) = f(x(t), u(t)), x(0) = x_{0},  (6.29)

where u(t) ∈ ℝ^{m}, x(t) ∈ ℝ^{n}, and f : ℝ^{n} × ℝ^{m} → ℝ^{n} is the nonlinear function to be identified. Let A ∈ ℝ^{n×n} be a Hurwitz matrix, that is, the matrix whose eigenvalues are all in the right-half plane. Since A, then by the theorem of Lyapunov, which is Theorem 2.1, for any given real symmetric positive definite (r.s.p.d.) matrix Q the solution P to the continuous Lyapunov matrix equation,

A^{T} P + P A = −Q  (6.30)

is also (real symmetric) positive definite. We use this fact later when we deal with a learning law for the RBF neural identifier.

We represent the model (6.29) in an equivalent form,

ẋ(t) = Ax(t) + g(x(t), u(t)), x(0) = x_{0},  (6.31)

where

g(x(t), u(t)) = −Ax(t) + f(x(t), u(t)).

Let the k-th basis function have the form

ϕ_{k}(x, u) = exp (−\frac{1}{2σ_{k}^{2}} ‖[ \begin{matrix}x \\ u \end{matrix} ] − m_{k}‖^{2}) ,

and let

Φ_{K} = [ ϕ_{1}(x, u) ϕ_{2}(x, u) ⋯ ϕ_{K}(x, u) ]^{T}

and

ĝ(x, u, m_{k}, σ_{k}) = W_{K}Φ_{K},

where W_{K} is the n × K weight matrix and m_{k} = [ m_{xk}^{T} m_{uk}^{T} ]^{T} is the center of the k-th radial basis function in the [ x^{T} u^{T} ]^{T} -space. The RBF neural identifier has the form,

˙x̂(t) = Ax̂(t) + ĝ(x, u, m_{k}, σ_{k}), x̂(0) = x_{0}.  (6.32)

Let W _{K}^{∗} be the optimal weight matrix that minimizes the modeling error,

e(t) = g(x(t), u(t)) − W _{K}^{∗}Φ_{K}(x(t), u(t)).

Using the above definition, we represent the system model (6.31) as

ẋ(t) = Ax(t) + W _{K}^{∗}Φ_{K}(x(t), u(t)) + e(t).  (6.33)

We define the state estimation error and the weight estimation error as

ex = x − x̂ and Γ_{K} = W _{K}^{∗} − W_{K}.

Next, we form the differential equation describing the state estimation error dynamics. We have

ėx = ẋ − ˙ x̂
= Ax + W _{K}^{∗}Φ_{K}(x, u) + e − Ax̂ − ĝ(x, u, m_{k}, σ_{k})
= A (x − x̂) + W _{K}^{∗}Φ_{K}(x, u) − W_{K}Φ_{K}(x, u) + e
= Aex + Γ_{K}Φ_{K}(x, u) + e.  (6.34)

We use the following Lyapunov function candidate,

V = V (ex, Γ_{K}) = \frac{1}{2} (ex^{T}P ex + \frac{1}{α}trace (Γ_{K}Γ_{K}^{T})) .  (6.35)

Using (6.28), we evaluate the time derivative of V on the solutions of the state estimation error equation to get

V̇ = ex^{T}P ėx + \frac{1}{α}trace (Γ_{K} Γ̇_{K}^{T})
= ex^{T} (P Aex + P Γ_{K}Φ_{K}(x, u) + P e) + \frac{1}{α}trace (Γ_{K} Γ̇_{K}^{T})
= \frac{1}{2}ex^{T} (A^{T} P + P A) ex + ex^{T}P Γ_{K}Φ_{K}(x, u) + \frac{1}{α}trace (Γ_{K} Γ̇_{K}^{T}) + ex^{T}P e.(6.36)

Note that by (1.33) for non-square matrices, we have

ex^{T}P Γ_{K}Φ_{K}(x, u) = trace (ex^{T}P Γ_{K}Φ_{K}(x, u))
= trace (Γ_{K}Φ_{K}(x, u)ex^{T}P ) .  (6.37)

Observe that

Ẇ_{K} = −Γ̇_{K}

Using the above and (6.37) along with the Lyapunov equation (6.30), we write

V̇ = −ex^{T}Qex + \frac{1}{α} (trace (αΓ_{K}Φ_{K}(x, u)ex^{T}P ) − trace (Γ_{K} Ẇ _{K}^{T})) + ex^{T}P e. (6.38)

We propose to choose the following learning rule,

Ẇ_{K} = αP exΦ_{K}^{T}(x, u)  (6.39)

With the above weight matrix update law, the Lyapunov derivative (6.38) evaluates to

V̇ = −ex^{T}Qex + ex^{T}P e
≤ −λ_{min}(Q)∥ex∥^{2} + λ_{max}(P)∥e∥∥ex∥
= −λ_{min}(Q) (∥ex∥ − \frac{λ_{max}(P)}{λ_{min}(Q)}∥e∥) ∥ex∥.  (6.40)

It follows from the above that if the magnitude of the state estimation error is such that ∥ex∥ < \frac{λ_{max}(P)}{λ_{min}(Q)}∥e∥, then it may happen that V̇ > 0. This in turn may lead to the weights’ drift to infinity over time. To prevent this, we modify (6.39) following the analysis of the single-input single-state case. The modified weight update law is

Ẇ_{K} = \{ \begin{matrix}αP exΦ_{K}^{T}(x, u) & if & ∥ex∥ ≥ e_{0} and ∥W_{K}∥ ≤ M \\ O & & otherwise \end{matrix} (6.41)

6.2 Hopfield Neural Network Model

The analog Hopfield neural network is a dynamical system whose schematic is shown in Figure 6.10. It is a nonlinear interconnected system proposed by Hopfield [48] as a model of biological neural networks. Each node, or neuron, in Figure 6.10 is represented by a circle. The structure of a neuron is depicted in Figure 6.11. The nonlinear amplifier in Figure 6.11 has an input-output characteristic described by a function g : ℝ → [0, 1]. We assume that g is continuously differentiable and strictly monotonically increasing, that is, g(u) > g(u′) if u > u′. The function g is sometimes called, the activation function. A typical choice of g is the sigmoid function given by

x = g(u) = \frac{1}{1 + e^{−u/θ}} ,  (6.42)

where the positive parameter θ controls the slope of g. A circuit implementation of the Hopfield network is shown in Figure 6.12. To derive equations that model the Hopfield neural network, we use an equivalent representation of the single node of the Hopfield network. This equivalent representation is shown in Figure 6.13. Applying Kirchhoff’s current law at the input of each amplifier of the equivalent node representation and taking into account that the current into the amplifier is negligibly small due to the amplifier’s high input resistance, 6.2. HOPFIELD NEURAL NETWORK MODEL 185

Figure 6.10: Hopfield neural network model.

Figure 6.11: Model of a neuron in the Hopfield neural network.

we get

C_{i} \frac{du_{i}}{dt} + \frac{u_{i}}{R_{i}} = \frac{x_{1} − u_{i}}{R_{i1}} + \frac{x_{2} − u_{i}}{R_{i2}} + ⋯ + \frac{x_{1} − u_{n}}{R_{in}} + I_{i}, i = 1, 2, …, n,  (6.43)

where x_{i} = g(u_{i}). Let

T_{ij} = \frac{1}{R_{ij}} , \frac{1}{r_{i}} = ∑_{j=1}^{n} T_{ij} + \frac{1}{R_{i}} .  (6.44)

Substituting (6.44) into (6.43) yields

C_{i} \frac{du_{i}}{dt} + \frac{u_{i}}{r_{i}} = ∑_{j=1}^{n} T_{ij}x_{j} + I_{i}, i = 1, 2, …, n  (6.45)

The system n first-order nonlinear differential equations given by (6.45) constitutes a model of the Hopfield neural network. The Hopfield neural network can be viewed as an associative memory. This is because it can have asymptotically stable states that attract the neighboring states to develop in time. This behavior can be interpreted as an evolution of an imperfect, corrupted pattern towards the correct, stored pattern—hence the term associative memory.

In further analysis, we will use an equivalent representation of (6.45). To obtain this equivalent representation of the Hopfield model, we first note that the activation function 6.2. HOPFIELD NEURAL NETWORK MODEL 187

Figure 6.12: A circuit realization of the Hopfield neural network.

Figure 6.13: Equivalent representation of a single node in the circuit realization of the Hopfield neural network.

g is, by assumption, strictly increasing and differentiable. Therefore g^{−1} exists and is also differentiable—see the Inverse Function Theorem on page ??. In Figure 6.14, we show plots of x = g(u) and u = g^{−1}(x) for θ = 0.2. We re-write equations (6.45) as

C_{i} \frac{dg^{−1}(x_{i})}{dx_{i}} \frac{dx_{i}}{dt} + \frac{g^{−1}(x_{i})}{r_{i}} = ∑_{j=1}^{n} T_{ij}x_{j} + I_{i}, i = 1, 2, …, n  (6.46)

Let T ∈ ℝ^{n×n} be the matrix whose elements are T_{ij} and v = [ I_{1} I_{2} ⋯ I_{n} ]^{T} ∈ ℝ^{n}. Using this notation, we write (6.45) in matrix form,

[ \begin{matrix}C_{1} \frac{dg^{−1}(x_{1})}{dx_{1}} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & C_{n} \frac{dg^{−1}(x_{n})}{dx_{n}} \end{matrix} ] ẋ(t) = T x + v − [ ⋮_{\frac{g^{−1}(x_{n})}{r_{n}}}^{\frac{g^{−1}(x_{1})}{r_{1}}} ] .  (6.47)

Associated with the Hopfield neural network is its computational energy function, E : ℝ^{n} → ℝ, given by

E(x) = −\frac{1}{2} ∑_{i=1}^{n} ∑_{j=1}^{n} T_{ij}x_{i}x_{j} − ∑_{i=1}^{n} I_{i}x_{i} + ∑_{i=1}^{n} ∫_{0}^{x_{i}} \frac{g^{−1}(s)}{r_{i}}ds
= −\frac{1}{2}x^{T} T x − v^{T} x + ∑_{i=1}^{n} ∫_{0}^{x_{i}} \frac{g^{−1}(s)}{r_{i}}ds.  (6.48)

Figure 6.14: Plots of x = g(u) and u = g^{−1}(x).

Suppose now that T = T^{T}. Then, the gradient of E is

∇E(x) = −T x − v + [ \frac{g^{−1}(x_{1})}{r_{1}} ⋯ \frac{g^{−1}(x_{n})}{r_{n}} ]^{T} .  (6.49)

Let

S(x) = [ \begin{matrix}C_{1} \frac{dg^{−1}(x_{1})}{dx_{1}} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & C_{n} \frac{dg^{−1}(x_{n})}{dx_{n}} \end{matrix} ] .  (6.50)

We now combine (6.49) and (6.50) to obtain an equivalent form of the Hopfield network modeling equations (6.47). We have

S(x)ẋ(t) = −∇E(x(t))  (6.51)

We will analyze the dynamics of the Hopfield network modeled by (6.51) using the computational energy function. We will first show that E is bounded. Taking norms of (6.48) yields

|E(x)| ≤ \frac{1}{2} ∥T ∥ ∥x∥^{2} + ∥v∥∥x∥ + ∑_{i=1}^{n} \frac{1}{r_{i}} |∫_{0}^{x_{i}} g^{−1}(s)ds| .  (6.52)

The components of the vector x are the outputs of the nonlinear amplifiers and they take values from the interval [0, 1]. Hence, ∥x∥ = ∥x∥_{2} ≤ \sqrt{n}. Thus, to show that E(x) is 190 CHAPTER 6. NEURAL NETWORKS bounded it is enough to show that the integral ∫_{0}^{x_{i}} g^{−1}(s)ds is bounded. We first find the expression for g^{−1}. Recall that g(u) is given by (6.42). Therefore,

\frac{1}{x} = 1 + e^{−u/θ},

or

\frac{1}{x} − 1 = e^{−u/θ},  (6.53)

Taking logarithms of both sides of the above yields

ln (\frac{1 − x}{x}) = −\frac{u}{θ}.

Hence,

u = g^{−1}(x) = θ (ln(x) − ln(1 − x)) .  (6.54)

Using the symmetry property of g^{−1}, we obtain

|∫_{0}^{x_{i}} g^{−1}(s)ds| ≤ ∫_{0}^{x_{i}} |g^{−1}(s)| ds
≤ −2 ∫_{0}^{1/2} g^{−1}(s)ds.  (6.55)

Recall that

∫ ln(s)ds = s ln(s) − s.

Hence,

∫ ln(1 − s)ds = −(1 − s) ln(1 − s) − s.

Using the above, we obtain

∫_{0}^{1/2} g^{−1}(s)ds = ∫_{0}^{1/2} θ (ln(s) − ln(1 − s)) ds
= θ (s ln(s) − s + (1 − s) ln(1 − s) + s)|_{0}^{1/2}
= θ (s ln(s) + (1 − s) ln(1 − s))|_{0}^{1/2} .  (6.56)

Note that

s ln(s) = \frac{ln(s)}{1/s}.

Applying L’Hospital’s rule to the above yields

lim_{s→0} \frac{ln(s)}{1/s} = lim_{s→0} \frac{1/s}{−1/s^{2}} = 0.

Applying the above to (6.56), we get

∫_{0}^{1/2} g^{−1}(s)ds = θ ln(1/2).  (6.57)

Combining (6.55) and (6.57) yields

|∫_{0}^{x_{i}} g^{−1}(s)ds| ≤ −2θ ln(1/2),  (6.58)

which completes the proof of the statement that E is bounded.

Let x(t) denote the trajectory of the Hopfield network model (6.51). We will show that if T = T^{T}, then

\frac{d}{dt}E(x(t)) ≤ 0

on the Hopfield network trajectories. Using the chain rule gives

\frac{d}{dt}E(x(t)) = (∇E(x(t)))^{T} ẋ(t).  (6.59)

It follows from (6.51) that

∇E(x(t)) = −S(x(t))ẋ(t).  (6.60)

Substituting (6.60) into (6.59), we obtain

\frac{d}{dt}E(x(t)) = −ẋ^{T} (t)S(x(t))x(t)
= − ∑_{i=1}^{n} C_{i} \frac{dg^{−1}(x_{i}(t))}{dx_{i}}ẋ_{i}^{2},  (6.61)

where C_{i} > 0 and

\frac{dg^{−1}(x_{i})}{dx_{i}} = θ (1/x_{i} + 1/(1 − x_{i})) > 0

because x_{i} ∈ (0, 1). Hence,

\frac{d}{dt}E(x(t)) ≤ 0

on the trajectories of the Hopfield model (6.51).

6.3 Analog-to-Digital Converter

In this section, we show how the Hopfield neural network can be used to solve some engineering problems. Specifically, we discuss the construction of an analog-to-digital (A/D) converter using the Hopfield neural network. The idea was put forward by Tank and Hopfield [114]. In A/D conversion, an analog value is transformed into its corresponding binary representation. The conversion, we discuss, consists of two steps. The first step is to round the given analog value to the nearest integer. The second step is to convert the integer into its binary equivalent. For example, 3.4 is first rounded to 3, then 3 is converted into its binary equivalent “11” if we have a 2-bit converter, “011” if we have a 3-bit converter, or “0011” if we have a 4-bit converter.

The first step of rounding a number to its nearest integer by a neural network can be organized as solving an optimization problem by the neural network. The second step of converting the integer representation from integer to binary can be accomplished using a neural network with a suitable nonlinear activation function mapping real numbers into the interval [0, 1]. Therefore, an N-bit A/D converter can be viewed as a device that assigns to a given analog input a corresponding vertex of the hypercube [0, 1]^{N}.

Since A/D conversion is viewed as an optimization problem, we need to formulate a suitable cost, or penalty, function that we will minimize. In fact finding a suitable cost function will be the first step in constructing a neural network based A/D converter.

To better understand the idea behind solving A/D conversion problems using neural networks, we first consider constructing a 2-bit A/D converter. Let V_{1} be the most significant bit (MSB), V_{0} be the least significant bit (LSB) and x an analog value to be converted. In the actual neural network, V_{1} and V_{0} are the outputs of the two neurons in the neural network based A/D converter. We first consider a very simple cost function,

E = \frac{1}{2} (V_{1}2^{1} + V_{0}2^{0} − x)^{2} .  (6.62)

A plot of the above function, for an analog input of x = 1.5, is shown in Figure 6.15(a). Note that E is defined over the two-dimensional hypercube [0, 1]^{2}. Observe that minimizing (6.62) corresponds to choosing V_{1} and V_{0} in a least-squares sense. However, constructing a circuit minimizing the cost function (6.62) would not yield good results since V_{0} and V_{1} would take value from the interval [0, 1] rather than the set {0, 1}. To overcome the problem, we add to (6.62) two terms whose purpose is to penalize for non-integer values of V_{0} and V_{1}. The modified cost function is

E = \frac{1}{2} (V_{1}2^{1} + V_{0}2^{0} − x)^{2} + aV_{1}(V_{1} − 1) + bV_{0}(V_{0} − 1),  (6.63)

where a and b are weighting coefficients. Tank and Hopfield [114] suggest selecting a and b so that the quadratic terms V_{0}^{2} and V_{1}^{2} are cancelled out. In particular, if we set a = −2 and b = −0.5, then (6.63) becomes

E = \frac{1}{2}x^{2} + 2V_{0}V_{1} − 2V_{1}x − V_{0}x + 2V_{1} + \frac{1}{2}V_{0}.  (6.64)

6.3. ANALOG-TO-DIGITAL CONVERTER 193 Figure 6.15: Constructing the cost function for a 2-bit A/D converter; plots of the cost function and its components.

The term used to cancel the quadratic terms is plotted in Figure 6.15(b). The final cost function given by (6.64) is plotted in Figure 6.15(c). When an analog input x is converted to its binary equivalent, the cost function (6.64) is minimized. One can interpret this process as follows. A given analog input x defines a corresponding cost function E. We can view E as a landscape. Now a ball placed somewhere on this landscape will move towards the lowest point of this landscape, which should correspond to one of the vertices of the hypercube [0, 1]^{2}.

Our next task is to construct differential equations that model a neural network implementing an A/D converter. The parameters of the state equations of the network will be chosen so that the time derivative of E evaluated on the solutions of the equations describing the network is negative. We consider a general case of an N-bit A/D converter. After this, we will derive state equations for a 2-bit A/D converter. Note that

\frac{dE}{dt} = \frac{∂E}{∂V_{0}} V̇_{0} + \frac{∂E}{∂V_{1}} V̇_{1} + ⋯ + \frac{∂E}{∂V_{N−1}} V̇_{N−1},

where V_{i} = g_{i}(u_{i}). Hence, similarly as in (6.61), it is enough to set

\frac{∂E}{∂V_{i}} = −C_{i}u̇_{i} = −C_{i} \frac{dg_{i}^{−1}(V_{i})}{dV_{i}} V̇_{i}, i = 0, 1, …, N − 1  (6.65)

to obtain

\frac{dE}{dt} = − ∑_{i=0}^{N−1} C_{i} \frac{dg_{i}^{−1}(V_{i})}{dV_{i}} V̇_{i}^{2} ≤ 0.  (6.66)

Thus, if the state equations of an N-bit A/D converter satisfy (6.65), then dE/dt ≤ 0 on the trajectories of the converter. Using (6.65), we can write the state equations that model the 2-bit A/D converter consisting of two neurons. Let V_{0} = g_{0}(u_{0}) and V_{1} = g_{1}(u_{1}). Then, we have

\begin{matrix}C_{0}u̇_{0} & = & −\frac{∂E}{∂V_{0}} = − (2V_{1} − x + 0.5) = −(2g_{1}(u_{1}) − x + 0.5) \\ C_{1}u̇_{1} & = & −\frac{∂E}{∂V_{1}} = −2 (V_{0} − x + 1) = −2(g_{0}(u_{0}) − x + 1) \end{matrix} \}  (6.67)

For simplicity, we assume that C_{0} = C_{1} = 1 and that g_{0} = g_{1} = g. Note that since C is present in both state equations, assigning C_{0} = C_{1} = 1 does not distort the dynamic behavior of the network; it only changes the time scale of both equations by the same factor. Observing that for i = 1, 2,

u̇_{i} = \frac{dg_{i}^{−1}(V_{i})}{dV_{i}} V̇_{i},

we represent (6.67) as

\begin{matrix}V̇_{0} & = & − (2V_{1} − x + 0.5) (\frac{dg^{−1}(V_{0})}{dV_{0}})^{−1} \\ V̇_{1} & = & −2 (V_{0} − x + 1) (\frac{dg^{−1}(V_{1})}{dV_{1}})^{−1} \end{matrix} \}  (6.68)

Figure 6.16: Plot of the cost function and a phase portrait of the 2-bit A/D converter for the analog value x = 1.5.

We display a plot of the cost function E for the analog value x = 1.5 in Figure (6.16)(a) and a phase portrait of the 2-bit A/D converter in Figure (6.16)(b). We used the activation function

g(V_{i}) = \frac{1}{1 + e^{−4V_{i}}} .

Thus,

\frac{dg^{−1}(V_{i})}{dV_{i}} = \frac{1}{4V_{i}(1 − V_{i})}.

Note that for x = 1.5, the cost function has two minimizers in the hypercube [0, 1]^{2} that are separated by a “ridge”. Initial conditions on each side of this ridge yield a different final value of the state vector [V_{0}, V_{1}]. Depending on the initial conditions of the state vector of the A/D converter, the analog value x = 1.5 will either be converted to the binary “01” or “10”, which are both valid solutions for x = 1.5. Thus, initial conditions influence the dynamic behavior of the A/D converter. Also, the activation nonlinearity influences the dynamical behavior of the A/D converter via the multiplicative term (\frac{dg^{−1}(V_{i})}{dV_{i}})^{−1} present in the state equations (6.68). One can see that the multiplicative factor influences the direction in which the converter state trajectory evolves.

Using the above technique, we can easily construct higher order A/D converters. For example, to construct a 3-bit A/D converter, we begin with the cost function

E = \frac{1}{2} (V_{2}2^{2} + V_{1}2^{1} + V_{0}2^{0} − x)^{2} + aV_{2}(V_{2} − 1) + bV_{1}(V_{1} − 1) + cV_{0}(V_{0} − 1).

To cancel the quadratic terms, select a = −8, b = −2, c = −1/2. The state equations of the converter are obtained from the equations

V̇_{i} = −\frac{∂E}{∂V_{i}} (\frac{dg_{i}^{−1}(V_{i})}{dV_{i}})^{−1} .

In our analysis of A/D neural converters, we observed that while the 2-bit converter yielded correct results, the 3-bit and 4-bit converters showed some deficiency in their performance. The errors observed for these two converters were quite significant and increased with the number of bits and consequently with network size.

6.4 Hopfield Network Stability Analysis

We apply Lyapunov’s second method to analyze stability in the sense of Lyapunov of the Hopfield neural network. The neural network is viewed as a large interconnected dynamical system, that is, as an interconnection of many neurons. We first analyze stability properties of a single neuron, and then proceed with the stability analysis of the whole network in terms of the properties of the single neurons. Interconnection information is used in conjunction with properties of the individual neurons to obtain sufficient conditions for asymptotic stability of the network.

6.4.1 Hopfield Network Model Analysis

Recall the equations describing the Hopfield neural net. They are given by (6.45), that is,

C_{i} \frac{du_{i}}{dt} + \frac{u_{i}}{r_{i}} = ∑_{j=1}^{n} T_{ij}x_{j} + I_{i}, i = 1, 2, …, n,

where x_{i} = g_{i}(u_{i}). In our further analysis, we assume that the activation functions g_{i} : ℝ → (−1, 1), i = 1, 2, …, n, are continuously differentiable and strictly monotonically increasing. We also assume that

u_{i}g_{i}(u_{i}) > 0 for u_{i} ≠ 0,

and

g_{i}(0) = 0.

To proceed further, we divide both sides of the equations describing the Hopfield network by C_{i} > 0 to get

u̇_{i} = −b_{i}u_{i} + ∑_{j=1}^{n} A_{ij}g_{j}(u_{j}) + U_{i}, i = 1, 2, …, n,  (6.69)

where b_{i} = \frac{1}{r_{i}C_{i}}, A_{ij} = \frac{T_{ij}}{C_{i}}, and U_{i} = \frac{I_{i}}{C_{i}}. We are interested in the behavior of solutions of the above model near its equilibrium points. If U_{i} = 0, the equilibrium points are obtained by solving the set of algebraic equations

−b_{i}u_{i} + ∑_{j=1}^{n} A_{ij}g_{j}(u_{j}) = 0, i = 1, 2, …, n,

for u = [u_{1}, u_{2}, …, u_{n}]^{T} ∈ ℝ^{n}.

In some applications the external inputs U_{i} are held constant, that is, U_{i} = c_{i} = constant. In such a case the equilibrium points are obtained by solving for u_{i} the set of the following algebraic equations

−b_{i}u_{i} + ∑_{j=1}^{n} A_{ij}g_{j}(u_{j}) + c_{i} = 0, i = 1, 2, …, n.

However, we can always transform the algebraic equations with non-zero constant external inputs into the algebraic equations where the external inputs are set to zero. Indeed, let for j = 1, 2, …, n,

u_{j} = y_{j} + \frac{c_{j}}{b_{j}} ,

and

G_{j}(y_{j}) = g_{j} (y_{j} + \frac{c_{j}}{b_{j}} ) .

Substituting the above into (6.69) and assuming that U_{i} = c_{i} yields

ẏ_{i} = −b_{i}y_{i} + ∑_{j=1}^{n} A_{ij}G_{j}(y_{j}), i = 1, 2, …, n.  (6.70)

The equilibrium points of the above system of differential equations are obtained by solving the following algebraic equations,

−b_{i}y_{i} + ∑_{j=1}^{n} A_{ij}G_{j}(y_{j}) = −b_{i}y_{i} + ∑_{j=1}^{n} A_{ij}g_{j}(y_{j} + \frac{c_{j}}{b_{j}})
= 0, i = 1, 2, …, n.

Thus, after the transformation, the algebraic equations with external constant inputs used for computing the equilibrium points assume the same form as the algebraic equations corresponding to zero external inputs. Hence, without loss of generality we can assume that the external inputs are zero and that the network is modeled by the equations,

u̇_{i} = −b_{i}u_{i} + ∑_{j=1}^{n} A_{ij}g_{j}(u_{j}), i = 1, 2, …, n,  (6.71)

Throughout, we assume that an isolated equilibrium point of interest is located at the origin of the state-space ℝ^{n}. If this is not the case, then by an appropriate change of variables we can arrange for the equilibrium point of interest to be translated to the origin of ℝ^{n}. Indeed, suppose u^{∗} ≠ 0 is an isolated equilibrium point of interest of (6.69). Let for i = 1, 2, …, n

x_{i} = u_{i} − u_{i}^{∗},
h_{i}(u_{i}) = h_{i}(x_{i} + u_{i}^{∗}) = g_{i}(u_{i}) − g_{i}(u_{i}^{∗}).

Substituting the above into (6.71) yields

ẋ_{i} = −b_{i}x_{i} − b_{i}u_{i}^{∗} + ∑_{j=1}^{n} A_{ij} (h_{j}(x_{j} + u_{j}^{∗}) + g_{j}(u_{j}^{∗}))
= −b_{i}x_{i} + ∑_{j=1}^{n} A_{ij}h_{j}(x_{j} + u_{j}^{∗}) + {−b_{i}u_{i}^{∗} + ∑_{j=1}^{n} A_{ij}g_{j}(u_{j}^{∗})} .

Note that, by the definition of the equilibrium point, the expression in the curly brackets vanishes. Thus we have

ẋ_{i} = −b_{i}x_{i} + ∑_{j=1}^{n} A_{ij}h_{j}(x_{j} + u_{j}^{∗}).

Let

G_{j}(x_{j}) = h_{j}(x_{j} + u_{j}^{∗}) = g_{j}(x_{j} + u_{j}^{∗}) − g_{j}(u_{j}^{∗}).

Then, the above differential equations can be written as

ẋ_{i} = −b_{i}x_{i} + ∑_{j=1}^{n} A_{ij}G_{j}(x_{j}) + U_{i}, i = 1, 2, …, n,  (6.72)

where G_{j} : ℝ → (c_{1}, c_{2}) ⊆ (−2, 2), c_{1} < 0 < c_{2}, G_{j}(0) = 0, G_{j} is continuously differentiable and strictly monotonically increasing in x_{j}. Furthermore x_{j}G_{j}(x_{j}) > 0 for all x_{j} ≠ 0. Observe that the above system of differential equations has an equilibrium point at the origin of x^{∗} = 0, U^{∗} = 0. We assume that there is no other equilibrium point in a neighborhood of the origin, that is, x^{∗} = 0 is an isolated equilibrium state of the above system corresponding to U = 0. We now represent the above model of the Hopfield neural network in the form

ẋ_{i} = −b_{i}x_{i} + A_{ii}G_{i}(x_{i}) + ∑_{j≠i^{j=1}}^{n} A_{ij}G_{j}(x_{j}), i = 1, 2, …, n,

We shall view the terms

∑_{j≠i^{j=1}}^{n} A_{ij}G_{j}(x_{j})

as the interconnections between the i-th neuron and the remaining neurons in the network. We shall refer to the equations

ẋ_{i} = −b_{i}x_{i} + A_{ii}G_{i}(x_{i}), i = 1, 2, …, n  (6.73)

as models of free isolated neurons. In what follows we analyze the stability of an isolated single neuron.

6.4.2 Single Neuron Stability Analysis

A single neuron in the Hopfield network, when the external input is set to zero, is modeled by a first order differential equation

ẋ_{i} = −b_{i}x_{i} + A_{ii}G_{i}(x_{i})  (6.74)

Consider now, as in Michel et al. [78], the following positive definite Lyapunov function candidate

V_{i}(x_{i}) = \frac{1}{2}x_{i}^{2}.

Its time derivative evaluated along the trajectories of the single neuron model is

V̇_{i}(x_{i}(t)) = x_{i}ẋ_{i}
= x_{i} (−b_{i}x_{i} + A_{ii}G_{i}(x_{i})) .

If there exists an r_{i} > 0 such that for all x_{i} ∈ (−r_{i}, r_{i})

\{ \begin{matrix}−b_{i}x_{i} + A_{ii}G_{i}(x_{i}) > 0 & if & x_{i} < 0 \\ −b_{i}x_{i} + A_{ii}G_{i}(x_{i}) = 0 & if & x_{i} = 0 \\ −b_{i}x_{i} + A_{ii}G_{i}(x_{i}) < 0 & if & x_{i} > 0, \end{matrix}

Figure 6.17: Characteristics of a single neuron model whose null solution is uniformly asymptotically stable.

then for x_{i} ≠ 0 and x_{i} ∈ (−r_{i}, r_{i}),

V̇_{i}(x_{i}(t)) < 0

and V̇_{i}(0) = 0. Hence the null solution x_{i} = 0 is uniformly asymptotically stable. Figure 6.17 illustrates this case. Now, if there exists an r_{i} > 0 such that for x_{i} ∈ (−r_{i}, r_{i}) we have

\{ \begin{matrix}−b_{i}x_{i} + A_{ii}G_{i}(x_{i}) < 0 & if & x_{i} < 0 \\ −b_{i}x_{i} + A_{ii}G_{i}(x_{i}) = 0 & if & x_{i} = 0 \\ −b_{i}x_{i} + A_{ii}G_{i}(x_{i}) > 0 & if & x_{i} > 0, \end{matrix}

then in addition to V_{i} being positive definite,

V̇_{i}(x_{i}(t)) < 0

for x_{i} ≠ 0 and x_{i} ∈ (−r_{i}, r_{i}), and V̇_{i}(0) = 0. Thus the null solution x_{i} = 0 is unstable. Figure 6.18 illustrates the case when the null solution of a single neuron is unstable. Assume now that for some r_{i} > 0 there exist constants σ_{i1} > 0 and σ_{i2} > 0 such that for all x_{i} ∈ (−r_{i}, r_{i}), x_{i} ≠ 0, we have

σ_{i1}x_{i}^{2} < x_{i}G_{i}(x_{i}) < σ_{i2}x_{i2}^{2},

or equivalently

σ_{i1} < \frac{G_{i}(x_{i})}{x_{i}} < σ_{i2}.

6.4. HOPFIELD NETWORK STABILITY ANALYSIS 201

Figure 6.18: Characteristics of single neuron model whose null solution is unstable. If in addition to the sufficient conditions for asymptotic stability, shown above, we also have

−b_{i} + A_{ii}δ_{i} < 0,

where

δ_{i} = \{ \begin{matrix}σ_{i1} & if & A_{ii} < 0 \\ σ_{i2} & if & A_{ii} > 0, \end{matrix}

then the null solution of the single neuron model is exponentially stable. Indeed, for x_{i} ≠ 0 we have

V̇_{i}(x_{i}(t)) = x_{i}ẋ_{i}
= x_{i} (−b_{i}x_{i} + A_{ii}G_{i}(x_{i}))
= (−b_{i} + A_{ii} \frac{G_{i}(x_{i})}{x_{i}}) x_{i}^{2}
≤ (−b_{i} + A_{ii}δ_{i}) x_{i}^{2}
= (\frac{−b_{i} + A_{ii}δ_{i}}{2} ) V_{i}(x_{i}(t)).

Applying Theorem ?? yields

V_{i}(x_{i}(t)) = \frac{x_{i}^{2}(t)}{2}
≤ exp \{\frac{−b_{i} + A_{ii}δ_{i}}{2} (t − t_{0})\} V_{i}(x_{i}(t_{0}))

= exp \{\frac{−b_{i} + A_{ii}δ_{i}}{2} (t − t_{0})\} \frac{x_{i}(t_{0})}{2}.

Hence

|x_{i}(t)| ≤ exp \{\frac{−b_{i} + A_{ii}δ_{i}}{4} (t − t_{0})\} |x_{i}(t_{0})|.

From the above we conclude that the null solution is not only asymptotically stable but even more, it is exponentially stable.

6.4.3 Stability Analysis of the Network

Lyapunov’s second method in conjunction with interconnection information will now be used in the stability analysis of the Hopfield neural model viewed as an interconnected dynamical system. Applications of Lyapunov’s second method to a general class of interconnected systems were investigated by Bailey [4]. Here, we use Lyapunov’s second method to investigate the stability of the null solution of the neural network described by

ẋ_{i} = −b_{i}x_{i} + ∑_{j=1}^{n} A_{ij}G_{j}(x_{j}), i = 1, 2, …, n  (6.75)

We impose the following assumptions on the interconnections: A1: There exist positive constants r_{i}, r_{j}, and a_{ij} such that for |x_{i}| < r_{i} and |x_{j}| < r_{j}, i, j = 1, 2, …, n, we have

x_{i}A_{ij}G_{j}(x_{j}) ≤ |x_{i}|a_{ij}|x_{j}|.

A2: Let S be an n × n real matrix whose elements are

s_{ij} = \{ \begin{matrix}α_{i} (−b_{i} + a_{ii}) & if & i = j \\ \frac{1}{2} (α_{i}a_{ij} + α_{j}a_{ji}) & if & i ≠ j. \end{matrix}

We assume that there exists a vector α = [α_{1}, α_{2}, …, α_{n}]^{T}, where α_{i} > 0 for i = 1, 2, …, n such that the matrix S is negative definite.

We now state and prove a theorem concerned with the exponential stability of the neural network model. This theorem can be found in Michel et al. [78].

Theorem 6.1 The null solution of the neural network model (6.75) is exponentially stable if the assumptions A1 and A2 are satisfied.

Proof Consider the following Lyapunov function candidate

V (x) = \frac{1}{2} ∑_{i=1}^{n} α_{i}x_{i}^{2}

= \frac{1}{2}x^{T} [ \begin{matrix}α_{1} & & … & 0 \\ & α_{2} \\ ⋮ & & ... & ⋮ \\ 0 & & … & α_{n} \end{matrix} ] x,

where α_{i} > 0, i = 1, 2, …, n are defined in assumption A2. Note that V is real symmetric positive definite. The time derivative of V evaluated along the trajectories of the neural model is

V̇ (x(t)) = ∑_{i=1}^{n} α_{i}x_{i}ẋ_{i}
= ∑_{i=1}^{n} α_{i}x_{i} (−b_{i}x_{i} + ∑_{j=1}^{n} A_{ij}G_{j}(x_{j}))
≤ ∑_{i=1}^{n} α_{i} (−b_{i}x_{i}^{2} + |x_{i}| ∑_{j=1}^{n} a_{ij}|x_{j}|)
= |x|^{T} R|x|,

where |x| = [|x_{1}|, |x_{2}|, …, |x_{n}|]^{T}, and the matrix R ∈ ℝ^{n×n} has its elements defined as

r_{ij} = \{ \begin{matrix}α_{i} (−b_{i} + a_{ii}) & if & i = j \\ α_{i}a_{ij} & if & i ≠ j. \end{matrix}

Note that

|x|^{T} R|x| = |x|^{T} (\frac{R + R^{T}}{2} ) |x|
= |x|^{T} S|x|
≤ λ_{max}(S)∥x∥^{2},

where the matrix S is defined in A2. By assumption A2 the matrix S is real symmetric negative definite. Hence λ_{max}(S) is real and negative. Thus

V̇ (x(t)) ≤ λ_{max}(S)∥x(t)∥^{2} < 0

in a neighborhood of x = 0 of radius

r = min_{1≤i≤n} r_{i}.

Since V is a quadratic Lyapunov function, the null solution is exponentially stable.

□

Note that in our stability analysis of the Hopfield neural network we do not require that the parameters A_{ij}, and hence the interconnections T_{ij}, be symmetric. The symmetry requirement on the interconnections was usually imposed in the early stability analyses of this network.

Note also that a necessary condition for the matrix S to be negative definite is that its diagonal elements be negative, that is,

−b_{i} + a_{ii} < 0, i = 1, 2, …, n.

However this condition is not sufficient. The above necessary condition, though, can shed some light on the influence of the stability properties of the individual neurons on the stability of the whole network. To show how the stability of the individual neurons may affect the stability of the whole network, we set i = j in A1 to obtain

x_{i}A_{ii}G_{i}(x_{i}) ≤ a_{ii}|x_{i}|^{2}.

Applying the sector condition σ_{i1}x_{i}^{2} < x_{i}G_{i}(x_{i}) < σ_{i2}x_{i2}^{2} to the above inequalities yields

a_{ii} = A_{ii}δ_{i},

where

δ_{i} = \{ \begin{matrix}σ_{i2} & if & A_{ii} > 0 \\ σ_{i1} & if & A_{ii} < 0 \end{matrix}

Therefore for i = 1, 2, …, n, we have

−b_{i} + A_{ii}δ_{i} < 0

since −b_{i} + a_{ii} < 0 and a_{ii} = A_{ii}δ_{i}. Thus A2 can only be satisfied if the null solution of each individual neuron is exponentially stable.

We now present another sufficiency condition for the exponential stability of the Hopfield neural network model. Before we state the condition we need to perform some algebraic manipulations. We first combine A2 with the sector condition σ_{i1}x_{i}^{2} < x_{i}G_{i}(x_{i}) < σ_{i2}x_{i2}^{2}. It follows from the sector condition that for x_{j} ≠ 0

\frac{G_{j}(x_{j})}{x_{j}} < σ_{j2}.

Applying the above when i ≠ j yields

|x_{i}A_{ij}G_{j}(x_{j})| ≤ |A_{ij}||x_{i}G_{j}(x_{j})|
= |A_{ij}||x_{i} \frac{G_{j}(x_{j})}{x_{j}}x_{j}|
≤ |A_{ij}||x_{i}||x_{j}| σ_{j2}.

When i = j we obtain

x_{i}A_{ii}G_{i}(x_{i}) ≤ δ_{i}A_{ii}x_{i}^{2},

where δ_{i} is defined as above. We use the results of the above manipulations in the following assumption: A3: There exist positive constants r_{i} and r_{j} such that for |x_{i}| < r_{i} and |x_{j}| < r_{j} the interconnections of the neural network model ẋ_{i} = −b_{i}x_{i} + ∑_{j=1}^{n} A_{ij}G_{j}(x_{j}), i = 1, 2, …, n satisfy the conditions

x_{i}A_{ii}G_{i}(x_{i}) ≤ δ_{i}A_{ii}x_{i}^{2} for i = j
|x_{i}A_{ij}G_{j}(x_{j})| ≤ |x_{i}||A_{ij}||x_{j}|σ_{j2} for i ≠ j.

We shall use another assumption. A4: There exists a vector α = [α_{1}, α_{2}, …, α_{n}]^{T} ∈ ℝ^{n}, α_{i} > 0, such that the matrix S ∈ ℝ^{n×n} is negative definite, where the elements of S are defined as

s_{ij} = \{ \begin{matrix}α_{i}(−b_{i} + δ_{i}A_{ii}) & if & i = j \\ \frac{1}{2} (α_{i}|A_{ij}|σ_{j2} + α_{j}|A_{ji}|σ_{j2}) & if & i ≠ j. \end{matrix}

We are now ready to state and prove an alternative sufficiency condition for the Hopfield neural network to be exponentially stable. This theorem can be found in Michel et al. [78].

Theorem 6.2 The null solution of the neural network model (6.75) is exponentially stable if the assumptions A3 and A4 are satisfied.

Proof Consider the following quadratic positive definite Lyapunov function candidate

V (x) = \frac{1}{2} ∑_{i=1}^{n} α_{i}x_{i}^{2},

where α_{i} > 0, i = 1, 2, …, n, are defined in A4. Taking into account the assumptions A1 and A2 we evaluate the time derivative of V along the trajectories of the neural network to get

V̇ (x(t)) = ∑_{i=1}^{n} α_{i}x_{i} (−b_{i}x_{i} + ∑_{j=1}^{n} A_{ij}G_{j}(x_{j}))

= ∑_{i=1}^{n} α_{i}x_{i} (−b_{i}x_{i} + A_{ii}G_{i}(x_{i}) + ∑_{j≠i^{j=1}}^{n} A_{ij}G_{j}(x_{j}))

≤ ∑_{i=1}^{n} α_{i} ((−b_{i} + δ_{i}A_{ii}) x_{i}^{2} + ∑_{j≠i^{j=1}}^{n} |A_{ij}||x_{i}||x_{j}| σ_{j2})
= \frac{1}{2}|x|^{T} S|x|
< 0 for x ≠ 0.

Thus the null solution of the neural network model is exponentially stable since V is a quadratic Lyapunov function.

□

6.5 Notes

This chapter offers a glimpse at artificial neural networks from a dynamical system perspective. The disclaimer that we found in the book by Hertz, Krogh, and Palmer [47, page xix] also applies to this chapter as well as to the whole field of artificial neural networks, “The adjective “neural” is used because much of the inspiration for such networks comes from neuroscience, not because we are concerned with networks of real neurons. Brain modelling is a different field and, though we sometimes describe biological analogies, our prime concern is with what artificial networks can do, and why.” The readers interested in principles underlying the organization of biological neurons into circuits and networks within different regions of the brain may consult the book edited by Shepherd [105]. For textbooks with an engineering approach to neural networks, we recommend Hassoun [44], Hagan, Demuth, and Beale [39], Zurada [133], Haykin [45], Fausett [27], Khanna [60], or Harvey [43]. Applications of artificial neural networks to signal processing and solving optimization problems are presented in the book by Cichocki and Unbehauen [19]. A psychological slant to the subject is presented in the book by Rumelhart, McClelland, and the PDP Reserach Group [99], and Anderson [2]. Impatient readers wishing to get a feel of artificial neural networks may consult tutorials by Lippmann [68], and by Hush and Horne [52].

According to Haykin [45, page 38], “The modern area of neural networks began with the pioneering work of McCulloch and Pitts (1943). McCulloch was a psychiatrist and neuroanatomist by training; he spent some 20 years thinking about the representation of an event in the nervous system. Pitts was a mathematical prodigy, who joined McCulloch in 1942.” In their paper [77], McCulloch and Pitts, assuming that the activity of the neuron is an “all-or-none” process, developed a logical calculus of neural events. This paper influenced John von Neumann who used idealized switch-delay elements derived from the McCulloch-Pitts neuron model in the construction of the EDVAC (Electronic Discrete Variable Automatic Computer) that was developed out of the ENIAC (Electronic Numerical Integrator and Computer). The ENIAC was the first general purpose computer and it was invented by J. Presper Eckert and John Mauchly.

208 CHAPTER 6. NEURAL NETWORKS

Fuzzy Systems

7.1 Motivation and Basic Definitions

Conventional control algorithms require a mathematical model of the system to be controlled. The mathematical model is used to synthesize a controller. In many practical situations, however, it is not always feasible to obtain an accurate mathematical model of the system to be controlled. Fuzzy logic is used to implement linguistic, non-formally expressed control laws derived from expert knowledge. This approach is especially successful in the case of processes such as washing machines, camera focusing, or drying processes that are being controlled well by human operators. The control of such processes can be captured in the form of rules

IF (process state) THEN (control action).

Thus, by a rule we will understand an IF/THEN statement that defines the set of facts that must be true (the IF part) before a set of actions (the THEN part) can be executed. The basic notion on which fuzzy logic is based is fuzzy set. An ordinary set, or just a set, is a collection of objects. Ordinary sets have two membership states: inclusion or exclusion. An object is either in the set or it is not. Thus the degree to which an object is a member of a set is either 0 or 1. For example, a car is fast if it is driven faster than 65 mph, and slow if it is driven slower than 65 mph—see Figure 7.1. This is the reason we sometimes refer to ordinary sets as crisp sets. Fuzzy sets, on the other hand, allow for partial membership in a set. The degree to which an object is a member of a fuzzy set can vary between 0 and 1. With a fuzzy set, we have a gradual transition from membership to nonmembership. For example, a car moving with velocity of 65 mph is a member of the fuzzy set FAST to degree 0.5. A car being driven 57.5 mph is fast to degree 0.25—see Figure 7.2. We next illustrate 210 CHAPTER 7. FUZZY SYSTEMS

Figure 7.1: An example of a crisp set—fast cars. Figure 7.2: A fuzzy set FAST

Figure 7.3: Crisp sets for SLOW and FAST.

the crisp and fuzzy versions of the sets SLOW and FAST. Observe the sudden change of membership in the crisp SLOW and FAST sets at a speed of 65 mph in Figure 7.3. The fuzzy SLOW and FAST sets allow for a gradual change from SLOW to FAST for a car moving between 50 and 80 mph as it is illustrated in Figure 7.4. Notice that a car moving at a speed between 50 and 80 mph has partial membership in SLOW and partial membership in FAST at the same time.

With the above examples in mind, we now define a fuzzy set. For this, we need to introduce some notation. Let U be a set. We use u ∈ U to denote that u is an element of U. We will call U the universe of discourse, or just the universe. We first recall the definition of a function. A function A from a set X to a set Y is a set of ordered pairs in X × Y with the property that if (x, y) and (x, z) are elements of A, then y = z. Thus a function is a set of ordered pairs.

Definition 7.1 (Fuzzy set) A fuzzy set A on U is a function that maps U into the interval [0, 1]. Equivalently,

A = {(u, μ_{A}(u)) : u ∈ U},

where μ_{A} is commonly called the membership function of the fuzzy set A.

Note that the concept of a fuzzy set generalizes that of a set in the following way. Let V ⊂ U. Define μ_{V} : U → [0, 1] by μ_{V} (u) = 1 if u ∈ V and μ_{V} (u) = 0 otherwise. Then V

Figure 7.4: Fuzzy sets for SLOW and FAST.

can be identified with the fuzzy set associated with μ_{V}. In other words, the subsets of U are identified with the functions from U to the two point set {0, 1}.

Example 7.1 Consider the following universe U = {4, 5, 6, 7, 8}. Let

A = {(4, 0.0), (5, 0.5), (6, 1.0), (7, 0.5), (8, 0.0)}.

Then A is a fuzzy set. One interpretation of A is closeness to 6.

Let A and B in U be fuzzy sets on U. We define the union, intersection, and complement using the following memebership functions:

(union): μ_{A∪B}(u) = max{μ_{A}(u) , μ_{B}(u)} for all u ∈ U;

(intersection) μ_{A∩B}(u) = min{μ_{A}(u) , μ_{B}(u)} for all u ∈ U;
(complement) μĀ(u) = 1 − μ_{A}(u) for all u ∈ U.

Example 7.2 Consider another fuzzy set B, “number close to 4”, given by

B = {(4, 1.0), (5, 0.8), (6, 0.5), (7, 0.2), (8, 0.0)}.

Then,

A ∪ B = {(4, 1.0), (5, 0.8), (6, 1.0), (7, 0.5), (8, 0.0)}.

The intersection of the two fuzzy sets A and B is

A ∩ B = {(4, 0.0), (5, 0.5), (6, 0.5), (7, 0.2), (8, 0.0)}.

The complement of the fuzzy set A is

Ā = {(4, 1.0), (5, 0.5), (6, 0.0), (7, 0.5), (8, 1.0)}.

Definition 7.2 (Fuzzy number) A fuzzy number is a fuzzy set F on a universe U ⊂ ℝ such that

max_{u∈U} μ_{F} (u) = 1,

and

μ_{F} (λu_{1} + (1 − λ)u_{2}) ≥ min(μ_{F} (u_{1}), μ_{F} (u_{2})),

for all u_{1}, u_{2} ∈ U and all λ ∈ [0, 1].

A fuzzy set that satisfies the conditions max_{u∈U} μ_{F} (u) = 1 and μ_{F} (λu_{1} + (1 − λ)u_{2}) ≥ min(μ_{F} (u_{1}), μ_{F} (u_{2})) is called normal and convex, respectively. Thus, a fuzzy number is a fuzzy set that is both normal and convex. Examples of convex and non-convex fuzzy sets are given in Figure 7.5.

7.2 Standard Additive Model

Let X and Y be nonempty sets and let I and J be nonempty index sets. Suppose we are given collections of fuzzy sets {A_{α} : α ∈ I} and {B_{β} : β ∈ J} on X and Y, respectively, and rules of the form

IF A_{α} THEN B_{β}.

The IF-THEN rules along with the collections of fuzzy sets form a function R from {A_{α} : α ∈ I} to {B_{β} : β ∈ J}, which we call the fuzzy system associated with the rules.

The support of a function into ℝ is the subset of the domain where the function is nonzero. The support of the fuzzy sets A_{α} and B_{β} in the IF-THEN rules describe subsets, or patches, in the Cartesian product space X × Y, as it is illustrated in Figure 7.6. Suppose the support 214 CHAPTER 7. FUZZY SYSTEMS

Figure 7.5: The first fuzzy set is a fuzzy number, while the second one is not. Figure 7.6: Function approximation using IF-THEN rules.

Figure 7.7: Generating output fuzzy sets from THEN-part sets of IF-THEN rules.

of each A_{α} and each B_{b}eta is a one point set, say x_{α} ∈ X and y_{β} ∈ Y, respectively. If X = ⋃{x_{α} : α ∈ I}, then R can be identified with a function R̃ from X to Y by

R̃(x_{α}) = y_{α}

if

R(A_{α}) = B_{α}.

One can think of this case as having complete knowledge and the patches are just points.

A fuzzy system can be thought of as a “fuzzified” function from X to Y with fuzzy sets replacing points in the domain. The size of a patch is usually inversely proportional to the knowledge or confidence of the situation. A point x ∈ X is usually in the support of more than one A_{α} with confidence represented by the values of the membership functions μ_{α}. The fuzzy set R(A_{α}) represents the desired action corresponding to A_{α}. This is illustrated in Figure 7.7. Suppose we are given the following two rules:

IF A_{3} THEN B_{3},
IF A_{4} THEN B_{2}.

The point x_{0} is contained in the support of A_{3} and A_{4}. Thus, according to the given rules, both B_{3} and B_{2} need to be taken into account. The process by which we combine these actions is called defuzzification. The process of defuzzification is also known as fuzzy Figure 7.8: Single-input single-output standard additive model schematic representation.

reasoning in the literature. The outcome of defuzzification is a numerical value y ∈ Y. An example of a defuzzification algorithm is a centroidal defuzzifier of the form

y = \frac{∑_{α} μ_{A_{α}}(x)ξ(x, A_{α}, R(A_{α}))}{∑_{α} μ_{A_{α}}(x)},  (7.1)

where ξ(x, A_{α}, R(A_{α})) is a function into the real numbers. We now give an example of ξ.

Let ξ(x, A_{α}, R(A_{α})) be the centroid of the support of R(A_{α}). For example, the centroids of the support of R(A_{3}) = B_{3} is c_{3} and that of R(A_{4}) = B_{2} is c_{2}. A fuzzy system with the centroidal defuzzifier is called the standard additive model (SAM). The term standard additive model was coined by Kosko [63]. A schematic representation of a SAM is shown in Figure 7.8. In the above description of the SAM, we were concerned with a single-input single-output case. The above construction can be extended to a many-input case. Suppose we are given IF-THEN rules of the form

IF A_{i} AND B_{i} THEN C_{i},

The SAM corresponding to this case is illustrated in Figure 7.9. We could use a defuzzification algorithm of the form

z = \frac{∑_{l=1}^{M} μ_{A_{l}}(x)μ_{B_{l}}(y)c_{l}}{∑_{l=1}^{M} μ_{A_{l}}(x)μ_{B_{l}}(y)} ,  (7.2)

where M is the number of IF-THEN rules, and c_{l} is the centroid of the fuzzy set C_{l}, l = 1, 2, …, M. This type of defuzzifier is called the center average defuzzifier. Another type of

Figure 7.9: Two-input single-output standard additive model representation.

defuzzifier that we could use is

z = \frac{∑_{l=1}^{M} min \{μ_{A_{l}}(x), μ_{B_{l}}(y)\} c_{l}}{∑_{l=1}^{M} min \{μ_{A_{l}}(x)μ_{B_{l}}(y)\}} .  (7.3)

An illustration of fuzzy reasoning using defuzzification scheme (7.3) is given in Figure 7.10. In general, for q inputs, the center average defuzzifier takes the form

z = \frac{∑_{l=1}^{M} ∏_{j=1}^{q} μ_{A_{j}^{l}}(x_{j})c_{l}}{∑_{l=1}^{M} ∏_{j=1}^{q} μ_{A_{j}^{l}}(x_{j})} .  (7.4)

7.3 Fuzzy Logic Control

A closed-loop system with a fuzzy logic controller (FLC) is shown in Figure 7.11. A typical fuzzy logic control architecture is depicted in Figure 7.12. The controller uses the control error e and the change in the control error ė(t) as its inputs. A typical fuzzy logic control architecture has a structure of the SAM and hence it is composed of two basic components. They are: a decision making logic module also referred to as an inference engine, and a defuzzifier. The inference engine processes the inputs using a knowledge base. The outputs of the inference engine are converted into crisp values, defuzzified, by a defuzzifier. Designing a fuzzy logic controller can be summarized in the following six-step algorithm:

218 CHAPTER 7. FUZZY SYSTEMS Figure 7.10: An illustration of fuzzy reasoning in a two-input fuzzy system. Figure 7.11: A diagram of a closed-loop system with a fuzzy logic controller.

Figure 7.12: Standard additive model as a fuzzy logic controller.

• Identify ranges of the controller inputs;

• Identify ranges of the controller outputs;

• Create fuzzy sets for each input and output;

• Translate the interaction of the inputs and outputs into IF-THEN rules, and then form the rules matrix;

• Decide on the fuzzifier, the inference engine, and the defuzzifier, and then apply them to crisp inputs to produce crisp outputs;

• Implement the controller, test it, and modify it if necessary.

As an example, suppose we use five membership functions for the inputs and the output. We then generate the rules. A possible set of rules is shown in Table 7.1, where LN denotes large negative, N means negative, ZE—zero, P—positive, and LP refers to large positive. The next step involves choosing an inference engine and a defuzzifier. We can use the product inference rule and the modified center average defuzzifier to get

u(ė, e) = \frac{∑_{k=1}^{25} p_{k}m_{k}^{o}/s_{k}^{o}}{∑_{k=1}^{25} p_{k}/s_{k}^{o}} ,

Table 7.1: The rules matrix.

CHANGE-IN-ERROR LN N ZE P LP

ERROR

LN  LN LN LN N ZE
N  LN LN N ZE P
ZE  N N ZE P P
P  N ZE P LP LP
LP  ZE P LP LP LP

where for k = 1, 2, …, 25, following the rules shown in Table 7.1, we compute

p_{k} = μ_{i}^{e}μ_{j}^{ce}, i, j = 1, 2, …, 5.

The functions μ_{i}^{e}, μ_{j}^{ec} are the membership functions corresponding to the linguistic variables ERROR and CHANGE-IN-ERROR, respectively. The parameters m_{k}^{o} and s_{k}^{o} are the m and s parameters of the controller output membership functions. The modified center average defuzzifier takes into account the shape of the membership functions, and in particular their “spread”. As it is pointed out by Wang [121], the sharper the shape of μ, the stronger is our belief that the value should be closer to its m. This observation is reflected in the formula for the defuzzifier we use, where we divide the weights p_{k} by s_{k}^{o}—the spread of the corresponding controller output membership function. We can use triangular membership functions whose equations are given next. The expression for the function shown in Figure 7.13 is

μ(x, m, s) = max (0, 1 − \frac{|x − m|}{s}) .

The expression for the function shown in Figure 7.14 is

μ(x, m, s) = min (1, max (0, 1 + \frac{x − m}{s})) .

The expression for the function shown in Figure 7.15 is

μ(x, m, s) = min (1, max (0, 1 − \frac{x − m}{s})) .

Example 7.3 Consider the vehicle model given in Subsection 2.4.2. Let the input signal be u = δ_{f}—the front wheel steer angle, and δ_{r} = 0. We will design a simple fuzzy logic 7.3. FUZZY LOGIC CONTROL 221

Figure 7.13: A triangular shaped fuzzy set. Figure 7.14: A fuzzy set at the right end of a universe of discourse.

Figure 7.15: A fuzzy set at the left end of a universe of discourse.

controller for the above vehicle model. The range for the error is [−0.5, 0.5]. The range for the change-in-error is [−0.06, 0.06]. The range for the controller output is [−3 deg, 3 deg]. The parameters of the membership functions are given in Table 7.2.

Plots of the error, the change-in-error, and controller’s output membership functions are given in Figures 7.16, 7.17, and 7.18, respectively. We created a nonlinear mapping from the controller’s input space into its output space and formed a look-up table that implements the obtained nonlinear mapping representing fuzzy logic controller. We used Simulink’s the 2-D Look-Up Table block to implement the controller. The look-up table size was 22 × 22. The output of the look-up table was scaled by the factor −2π/180 before using it as the

Table 7.2: The m and s-parameters of the membership functions.

error  change-in-error  output
m  σ  m  σ  m  σ
m_{1} = −0.4 s_{1} = 0.1 m_{1} = −0.05 s_{1} = 0.025 m_{1} = −2.5 s_{1} = 1
m_{2} = −0.2 s_{2} = 0.2 m_{2} = −0.025 s_{2} = 0.025 m_{2} = −1 s_{2} = 1
m_{3} = 0 s_{3} = 0.2 m_{3} = 0 s_{3} = 0.025 m_{3} = 0 s_{3} = 1
m_{4} = 0.2 s_{4} = 0.2 m_{4} = 0.025 s_{4} = 0.025 m_{4} = 1 s_{4} = 1
m_{5} = 0.4 s_{5} = 0.1 m_{5} = 0.05 s_{5} = 0.025 m_{5} = 2.5 s_{5} = 1

7.3. FUZZY LOGIC CONTROL 223

Figure 7.16: Fuzzy numbers of the linguistic variable ERROR. Figure 7.17: Fuzzy numbers of the linguistic variable CHANGE-IN-ERROR.

Figure 7.18: Fuzzy numbers of the linguistic variable CONTROL.

vehicle’s steering input. The negative sign is to take into account the opposite directions of the lateral axes of the vehicle coordinate system and the fixed system. The factor π/180 converts degrees into radians. We multiplied the output of the look-up table by 2 to increase the control authority. This allows to use the same controller to track step inputs with large amplitudes. A plot of the control surface is given in Figure 7.19. Next, a reference signal generator was constructed. The transfer function of the signal generator is

\frac{R(s)}{H(s)} = \frac{1}{s^{3} + 1.75s^{2} + 2.15s + 1},

where R(s) = L(r(t)), H(s) = L(6h(t)), and h(t) is the unit step function. The closed-loop system was simulated using Simulink. Plots Y and r versus Xare shown in Figure 7.20.

Example 7.4 This example is based on Example 6.1, pages 266–272, found in Driankov, Hellendoorn, and Reinfrank [24]. Consider a nonlinear dynamical system modeled by the differential equation,

ẍ + a_{1}(1 − x^{2})ẋ + a_{2}x = bu,  (7.5)

7.3. FUZZY LOGIC CONTROL 225

Figure 7.19: A plot of the control surface u = f(ė, e). Figure 7.20: Plots of Y and r versus X.

Figure 7.21: A phase-plane portrait of the uncontrolled system in Example 7.4.

where |u| ≤ 15, a_{1} = 2.2165, a_{2} = b = 12.7388. Let x_{1} = x and x_{2} = ẋ_{1}. The system model in state space format has the form

[ ẋ_{1} ; ẋ_{2} ] = [ \begin{matrix}x_{1} \\ −a_{2}x_{1} − a_{1} (1 − x_{1}^{2}) x_{2} \end{matrix} ] + [ \begin{matrix}0 \\ b \end{matrix} ] u.

A phase-plane portrait of the uncontrolled system is shown in Figure 7.21. Plots of x_{1} and x_{2} versus time for the initial conditions x_{1}(0) = −1 and x_{2}(0) = 0 are shown in Figure 7.22. The relevant membership functions shown in Figure 7.23, 7.24, 7.25, and 7.26. The following MATLAB function can be used used to generate all the relevant membership functions.

function y = trap_a(x,a1,a2,c,s)
%TRAP_A is an asymmetric trapezoid type membership function,
% also called a ``pi'' function,
% where x is the function domain, a1 is the length of the
% left trapezoid's side
% at the base, a2 is the length of the right side at the base,
% c is the trapezoid's center, and s is a half of the
% trapezoid upper side. Thus the length of the trapezoid upper
% side is 2*s. The length of the trapezoid base is then 2*s+a1+a2.

7.3. FUZZY LOGIC CONTROL 227

Figure 7.22: Plots of x_{1} and x_{2} versus time of the uncontrolled system. Figure 7.23: Membership functions of x_{1}.

228 CHAPTER 7. FUZZY SYSTEMS Figure 7.24: Membership functions of x_{2}. Figure 7.25: Membership functions of u.

Figure 7.26: Magnification of the membership functions of u on the interval [−1.5, 1.5].

% Set s=0 to get a triangular membership function.
y1=min(0.5,(max((-0.5),(0.5 +((x-c+s)./a1)))));
y2=min(0.5,(max((-0.5),(0.5 -((x-c-s)./a2)))));
y=y1+y2;

We use the rules matrix given in Table 7.3 to construct a nonlinear mapping u = Φ(x_{1}, x_{2}) and form a look-up table that implements the obtained mapping representing a fuzzy logic controller. The 2-D Look-Up Table block to implement the controller is then used. The look-up table size is 22×22. A plot of u = Φ(x_{1}, x_{2}) is shown in Figure 7.27. A SIMULINK diagram of the closed-loop system is given in Figure 7.28. A phase-portrait of the closed-loop system is depicted in Figure 7.29. Plots x_{1} and x_{2} versus time for x_{1}(0) = −1 and x_{2}(0) = 0 for the closed-loop system are shown in Figure 7.30.

Figure 7.27: A plot of u = Φ(x_{1}, x_{2}).

Table 7.3: The rules matrix.

x_{1} LN  N ZE P  LP

x_{2}

LN PVB PM PM PM PVB
N PB  P PS PS PM
ZE P PVS ZE NVS N
P NM NS NS N  NB
LP NVB NM NM NM NVB

7.3. FUZZY LOGIC CONTROL 231 Figure 7.28: A SIMULINK diagram of the closed-loop system. Figure 7.29: A phase-plane portrait of the closed-loop system.

Figure 7.30: Plots x_{1} and x_{2} versus time for x_{1}(0) = −1 and x_{2}(0) = 0 for the closed-loop system in Example 7.4.

7.4 Stabilization Using Fuzzy Models

In this section, a Lyapunov based stabilizing control design method for uncertain nonlinear dynamic systems using fuzzy models is presented. The controller is constructed using a design model of the dynamic process to be controlled. The design model is obtained from the truth model using a fuzzy modeling approach. The truth model represents a detailed description of the process dynamics. The truth model is used in a simulation experiment to evaluate performance of the controller design. A method for generating local models that constitute the design models is proposed. Sufficient conditions for stability and stabilizability of fuzzy models using fuzzy state-feedback controllers are given. The results obtained are illustrated with a numerical example involving a four dimensional nonlinear model of a stick balancer.

7.4.1 Fuzzy Modeling

In many situations there may be human experts that can provide a linguistic description, in terms of IF-THEN rules, of the process to be controlled. Combining the available mathematical description of the process with its linguistic description results in a fuzzy system model. Such an approach to modeling was proposed by Takagi and Sugeno [111] and further developed by Sugeno and Kang [110]. This type of model is called the Takagi-Sugeno (T-S) or Takagi-Sugeno-Kang (TSK) fuzzy model. The discrete-time version of this model was analyzed by Tanaka and Sugeno [113], and Tanaka [112]. Wang, Tanaka, and Griffin [120], Cao, Rees, and Feng [16], and Feng et al. [29] used TSK fuzzy models to design stabilizing controllers for nonlinear systems. One can construct a TSK model if local description of the dynamic plant to be controlled is available in terms of local linear models

ẋ(t) = A_{i}x(t) + B_{i}u(t), i = 1, 2, …, r,

where the state vector x(t) ∈ ℝ^{n}, the control input u(t) ∈ ℝ^{m}, and the matrices A_{i} and B_{i} are of appropriate dimensions. The above information is then fused with the available IF-THEN rules, where the ith rule can have the form:

Rule i : IF x_{1}(t) is F_{1}^{i} AND ⋯ AND x_{n}(t) is F_{n}^{i},
THEN ẋ(t) = A_{i}x(t) + B_{i}u(t),

where F_{j}^{i}, j = 1, 2, …, n, is the jth fuzzy set of the ith rule. Let μ_{j}^{i}(x_{j}) be the membership function of the fuzzy set F_{j}^{i}, and let

w^{i} = w^{i}(x) = ∏_{j=1}^{n} μ_{j}^{i}(x_{j}).

Then, given a pair (x(t), u(t)), the resulting fuzzy system model is inferred as the weighted average of the local models, and has the form

ẋ = \frac{∑_{i=1}^{r} w^{i} (A_{i}x + B_{i}u)}{∑_{i=1}^{r} w^{i}}
= ∑_{i=1}^{r} α_{i} (A_{i}x + B_{i}u)
= (∑_{i=1}^{r} α_{i}A_{i}) x + (∑_{i=1}^{r} α_{i}B_{i}) u
= A(α)x + B(α)u,  (7.6)

where for i = 1, 2, …, r,

α_{i} = \frac{w^{i}}{∑_{i=1}^{r} w^{i}} .

Note that, for i = 1, 2, …, r,

α_{i} ≥ 0, ∑_{i=1}^{r} α_{i} = 1 and α = [ α_{1} α_{2} ⋯ α_{r} ]^{T} ∈ [0, 1]^{r}.  (7.7)

We will combine the truth model of the process with its linguistic description to form a fuzzy design model. The class of the truth models that we consider is given by

ẋ = f(x) + G(x)(u + h(t, x)),  (7.8)

where f : ℝ^{n} → ℝ^{n}, G : ℝ^{n} → ℝ^{n×m}, and the vector-valued function h(t, x) represents modeling uncertainties. The only information about uncertainties available to the designer are their bounds. Specifically, we assume that the uncertainty modeling function h is bounded by a non-negative function η = η(t, x), that is,

∥h(t, x)∥ ≤ η(t, x),  (7.9)

where ∥ · ∥ denotes the vector p-norm. A detailed discussion of condition (7.9) is included in Section 7.4.4. Observe that h affects the system dynamics via the input matrix B(α) in a similar fashion as the control input does. For this reason, we say that the uncertainty h satisfies the matching condition.

The resulting fuzzy design model has the form

ẋ = A(α)x + B(α) (u + h(t, x)) .  (7.10)

Our goal, in this section, is to give methods for constructing stabilizing state-feedback controllers for nonlinear systems modeled by( 7.8). We construct stabilizing controller using fuzzy design model of the form (7.10) and a Lyapunov based approach. The fuzzy design model is obtained using the truth model and available linguistic description of the process. The controller performance is tested on the simulation model.

7.4.2 Constructing the Design Model

In the stabilization problem of a nonlinear plant, we are concerned with constructing a controller so that starting from an arbitrary point, in some neighborhood of the operating point, the controller forces the closed-loop system trajectory to converge to the operating point. On the other hand, if the starting point coincides with the operating point, the closed-loop system trajectory is expected to stay at this point for all subsequent time. It is clear from the above description that an operating point satisfying the above requirements must be an asymptotically stable equilibrium state of the closed-loop system. When designing a controller, we use a design model constructed from the truth model. We combine the mathematical description of the process in terms of the truth model and a linguistic description of the process to obtain a fuzzy design model. The fuzzy design model is of the form of the TSK fuzzy model. The essential components of the TSK model are linear local models. These linear local models describe the plant dynamical behavior at its different operating points. We now show that using common linearization approach to constructing local models may result in a fuzzy design model different from what the designer had in his or her mind. For example, suppose that the truth model of a plant has the form

ẋ = f(x) + G(x)u.  (7.11)

For simplicity of notation, let

F (x, u) = f(x) + G(x)u.

Then, we can represent model (7.11) as

ẋ = F (x, u).  (7.12)

Expanding F by means of a Taylor series around (x_{0}, u_{0}) yields

ẋ = F (x_{0}, u_{0}) + \frac{∂F}{∂x} |_{{x=x_{0} \atop u=u_{0}}} (x − x_{0}) + \frac{∂F}{∂u} |_{{x=x_{0} \atop u=u_{0}}} (u − u_{0}) + higher order terms, (7.13)

where

F (x_{0}, u_{0}) = f(x_{0}) + G(x_{0})u_{0}.  (7.14)

To write expressions for the second and the third terms of (7.13) as functions of f and G, let g_{ij} be the (i, j)-th element of the matrix G. Then,

\frac{∂F}{∂x} |_{{x=x_{0} \atop u=u_{0}}} = \frac{∂f}{∂x}|_{x=x_{0}} + H(x_{0}, u_{0}),  (7.15)

where the (i, j)-th element of the n × n matrix H has the form

∑_{k=1}^{m} u_{k} \frac{∂g_{ik}(x)}{∂x_{j}} |_{{x=x_{0} \atop u=u_{0}}} .

Finally,

\frac{∂F}{∂u} |_{{x=x_{0} \atop u=u_{0}}} = G(x_{0}).  (7.16)

A point (x_{0}^{T} , u_{0}^{T})^{T} ∈ ℝ^{n+m} is an equilibrium point of (7.12) if F (x_{0}, u_{0}) = 0, that is, if at (x_{0}, u_{0}) we have ẋ = 0. Let δx = x − x_{0} and δu = u − u_{0}. Note that

\frac{dx_{0}}{dt} = 0.

Then, the linearized model about the equilibrium (x_{0}, u_{0}) is obtained by neglecting higher order terms and observing that for the equilibrium point F (x_{0}, u_{0}) = 0. The linearized model has the form

\frac{d}{dt}δx = Aδx + Bδu,  (7.17)

where

A = \frac{∂F}{∂x} |_{{x=x_{0} \atop u=u_{0}}} and B = \frac{∂F}{∂u} |_{{x=x_{0} \atop u=u_{0}}} .

Suppose now that we wish to construct a stabilizing controller for the truth model ẋ = f(x) + G(x)u, where f(0) = 0, so that (x = 0, u = 0) is an asymptotically stable equilibrium point of the closed-loop system. Our first step is to construct a fuzzy design model. We start with generating linear local models describing plant’s behavior at selected operating points. It is natural to first construct a local linear model that describes plant’s behavior about the equilibrium state x = 0. We can obtain this first local linear model by using the above described linearization technique. The resulting model is given by

ẋ = A_{1}x + B_{1}u.

Next, we construct local linear models describing plant’s behavior at the remaining operating points. Suppose that x = x_{j} is the next operating state of interest. The result of Taylor’s linearization of a nonlinear model about an operating non-equilibrium point is an affine, rather than a linear model. Even if the operating point is an equilibrium point, Taylor series linearization, in general, will not yield a local model that is linear. Indeed, suppose that the operating point (x_{j}, u_{j}) is an equilibrium point, that is,

f(x_{j}) + G(x_{j})u_{j} = 0.  (7.18)

By (7.13), the resulting linearized model is

\frac{d}{dt}(x − x_{j}) = f(x_{j}) + G(x_{j})u_{j} + A_{j}(x − x_{j}) + B_{j}(u − u_{j})  (7.19)
= A_{j}(x − x_{j}) + B_{j}(u − u_{j}).  (7.20)

We can represent model (7.20) in the form

ẋ = A_{j}x + B_{j}u − (A_{j}x_{j} + B_{j}u_{j}) .  (7.21)

The term (A_{j}x_{j} + B_{j}u_{j}) does not have to be equal to zero, and hence model (7.21) is not a linear model, but rather an affine model. We illustrate on a numerical example that the term (A_{j}x_{j} + B_{j}u_{j}) is indeed in general not equal to zero.

Example 7.5 We consider a subsystem of model (A.33) of the form

[ ẋ_{1} ; ẋ_{2} ] = [ x_{2}_{\frac{g sin(x_{1})−mlax_{2}^{2} sin(2x_{1})/2}{4l/3−mla cos^{2}(x_{1})}} ] + [ \begin{matrix}0 \\ −\frac{a cos(x_{1})}{4l/3−mla cos^{2}(x_{1})} \end{matrix} ] u,  (7.22)

where g = 9.8 m/sec^{2}, m = 2 kg, M = 8 kg, a = 1/(m + M), l = 0.5 m. We linearize model (7.22) about x_{1} = π/4. For x_{1} = π/4 to be a component of an equilibrium state, we have to have x_{2} = 0. Hence, the equilibrium state is

x_{e} = [ π/4 0 ]^{T} .

We next compute u = u_{e} so that equation (7.18) holds. Performing simple calculations, we get u_{e} = 98. Then, using (7.15) we obtain

A = [ \begin{matrix}0 & 1 \\ 22.4745 & 0 \end{matrix} ] .  (7.23)

Next, using (7.16) gives

B = [ \begin{matrix}0 \\ −0.1147 \end{matrix} ] .  (7.24)

Note that

Ax_{e} + Bu_{e} = [ \begin{matrix}0 \\ 6.4108 \end{matrix} ] ≠ 0.

The above example illustrates the fact that the system being linear in δx = x − x_{0} and δu = u − u_{0} is in general only affine, and not linear in x and u because of the presence of the offset term − (Ax_{0} + Bu_{0}). The designer must than be careful when constructing local models using Taylor series linearization, because this approach in general will not yield a local model that is linear in x and u. We note that Taylor’s linearization will yield a linear system in x and u if the equilibrium point (x_{0}^{T} , u_{0}^{T}) = (0^{T} , 0^{T}).

We now describe our method for constructing local models using the truth model of the system to be controlled. We assume that the truth model has the form given by (7.11). Suppose that we are given an operating state x_{0} that does not have to be an equilibrium state of (7.11). Our goal is to construct a linear model in x and u that approximates the behavior of (7.11) in the vicinity of the operating state x_{0}, that is, we wish to find constant matrices A and B such that in a neighborhood of x_{0},

f(x) + G(x)u ≈ Ax + Bu for any u,  (7.25)

and

f(x_{0}) + G(x_{0})u = Ax_{0} + Bu for any u.  (7.26)

Since u is arbitrary, we have to have

G(x_{0}) = B.  (7.27)

Thus, we are left with finding a constant matrix A such that in a neighborhood of x_{0},

f(x) ≈ Ax,  (7.28)

and

f(x_{0}) = Ax_{0}.  (7.29)

Let a_{i}^{T} denote the i-th row of the matrix A. Then, for the purpose of further analysis, we represent condition (7.28) as

f_{i}(x) ≈ a_{i}^{T} x, i = 1, 2, …, n,  (7.30)

and we represent (7.29) as

f_{i}(x_{0}) = a_{i}^{T} x_{0}, i = 1, 2, …, n,  (7.31)

where the i-th component of f is f_{i} : ℝ^{n} → ℝ. Expanding the left hand side of (7.30) about x_{0} and neglecting second and higher order terms yields

f_{i}(x_{0}) + ∇^{T} f_{i}(x_{0})(x − x_{0}) ≈ a_{i}^{T} x,  (7.32)

where ∇f_{i}(x) : ℝ^{n} → ℝ^{n} is the gradient (a column vector) of f_{i} at x. Now, using (7.31), we represent (7.32) as

∇^{T} f_{i}(x_{0})(x − x_{0}) ≈ a_{i}^{T} (x − x_{0}) ,  (7.33)

where x is arbitrary but “close” to x_{0}. Our task now is to determine a constant vector a_{i} that is as close as possible to ∇f_{i}(x_{0}) and satisfies the constraint a_{i}^{T} x_{0} = f_{i}(x_{0}). Let

E = \frac{1}{2} ∥∇f_{i}(x_{0}) − a_{i}∥_{2}^{2} .

Then, we can formulate our task as a constrained optimization problem of the form

minimize E ; a_{i} subject to a_{i}^{T} x_{0} = f_{i}(x_{0}). }  (7.34)

We note that (7.34) is a convex constrained optimization problem. This means that the first-order necessary condition for a minimum of E is also sufficient, see, for example, Luenberger [70, Section 6.5] or Chong and Żak [17, Chapter 21]. The first-order conditions for the optimization problem given by (7.34) are

∇_{a_{i}}E + λ∇_{a_{i}} (a_{i}^{T} x_{0} − f_{i}(x_{0})) = 0,  (7.35)
a_{i}^{T} x_{0} = f_{i}(x_{0}),  (7.36)

where λ is the Lagrange multiplier and the subscript a_{i} in ∇_{a_{i}} indicates that the gradient, ∇ is computed with respect to a_{i}. Performing the required differentiation in (7.35) yields

a_{i} − ∇f_{i}(x_{0}) + λx_{0} = 0,  (7.37)
a_{i}^{T} x_{0} = f_{i}(x_{0}).  (7.38)

Recall, that we consider now the case when x_{0} ≠ 0. Premultiplying (7.37) by x_{0}^{T} and substituting (7.38) into the resulting equation yields

λ = \frac{x_{0}^{T} ∇f_{i}(x_{0}) − f_{i}(x_{0})}{∥x_{0}∥^{2}}.  (7.39)

Substituting λ given by (7.39) into (7.37), we obtain

a_{i} = ∇f_{i}(x_{0}) + \frac{f_{i}(x_{0}) − x_{0}^{T} ∇f_{i}(x_{0})}{∥x_{0}∥^{2}}x_{0}, x_{0} ≠ 0  (7.40)

Example 7.6 We illustrate the above procedure for constructing linear local models on the nonlinear model (7.22) that we analyzed in Example 7.5. This model was also considered by Wang et al. [120, page 20]. Suppose that, as in Wang et al. [120], we wish to construct two local models; one describing the plant operation about x_{1} = 0 and the other model when x_{1} is about ±π/2. In both models, we assume that x_{2} = 0 and u is arbitrary. Note that the nonlinear model (7.22) is in the format given by (7.11). We first construct a local model about the operating state x_{1} = x_{2} = 0. The local model for this case can be obtained using Taylor’s linearization about the equilibrium point (x_{0}, u_{0}) = (0, 0).

We next construct the second local model that corresponds to x_{1} = 88^{∘}π/180^{∘} and x_{2} = 0. Let β = cos(88^{∘}). Then, the matrix B can be obtained from (7.22) using (7.27). We have

B = [ \begin{matrix}0 \\ −\frac{aβ}{4l/3−mlaβ^{2}} \end{matrix} ] .

We next compute the matrix A of the second local model using (7.40). We first compute the first row A. We note that f_{1}(x) = x_{2}, and hence

∇f_{1} = [ 0 1 ]^{T} .

The operating state is x_{0} = [ 88^{∘}π/180^{∘} 0 ]^{T}. We compute a_{1}^{T}, the first row of A, using (7.40) to get

a_{1}^{T} = ∇^{T} f_{1}(x_{0}) + [ 0 0 ]
= [ 0 1 ] .

We again use (7.40) to compute a_{2}^{T}, the second row of the matrix A, of the second local model. We get

a_{2}^{T} = [ f_{2}(x_{0})/(88^{∘}π/180^{∘}) 0 ]
≈ [ \frac{g}{4l/3−mlaβ^{2}} (\frac{2}{π}) 0 ] .

The above obtained local models are the same as the ones that were proposed by Wang et al. [120] using a heuristic approach.

7.4.3 Stabilizability of Fuzzy Models

In this subsection, we first present sufficient conditions for asymptotic stability in the large of unforced continuous-time fuzzy model

ẋ = ∑_{i=1}^{r} α_{i}A_{i}x,  (7.41)

where α_{i} = α_{i}(x(t)) ≥ 0 for i = 1, 2, …, r, and ∑_{i=1}^{r} α_{i} = 1. We next discuss the stabilizability problem of forced fuzzy models using fuzzy state-feedback control strategies.

It is straightforward to show that a sufficient condition for asymptotic stability in the large of the equilibrium state x = 0 of fuzzy system (7.41) is that there exists a symmetric positive definite matrix P such that for i = 1, 2, …, r,

A_{i}^{T} P + P A_{i} < 0.  (7.42)

It is obvious that a necessary condition for the existence of a common symmetric positive definite P satisfying (7.42) is that each A_{i} be asymptotically stable, that is, the eigenvalues of each A_{i} be in the open left-hand complex plane. We now give another necessary condition for the existence of a common P that satisfies (7.42).

Theorem 7.1 Suppose that each A_{i}, i = 1, 2, …, r in fuzzy model (7.41) is asymptotically stable and there exists a symmetric positive definite P such that conditions (7.42) hold. Then, the matrices

∑_{k=1}^{s} A_{i_{k}},

where i_{k} ∈ {1, 2, …, r} and s = 2, 3, …, r are asymptotically stable.

Proof Suppose conditions (7.42) hold. Let

A_{i_{k}}^{T}P + P A_{i_{k}} = −Q_{i_{k}},  (7.43)

where Q_{i_{k}} = Q_{i_{k}}^{T} > 0 for i_{k} ∈ {1, 2, …, r}. Adding both sides of (7.43) yields

(∑_{k=1}^{s} A_{i_{k}}^{T}) P + P (∑_{k=1}^{s} A_{i_{k}}) = − ∑_{k=1}^{s} Q_{i_{k}}.  (7.44)

By assumption P = P^{T} > 0, and since Q_{i_{k}} = Q_{i_{k}}^{T} > 0 for i_{k} = {1, 2, …, r}, the symmetric matrices

∑_{k=1}^{s} Q_{i_{k}}, s = 2, 3, …, r,

are also positive definite. Hence, by the Lyapunov Theorem the matrices ∑_{k=1}^{s} A_{i_{k}}, s = 2, 3, …, r are asymptotically stable.

□

The following corollary can be viewed as a continuous-time counterpart of Theorem 4.3 of Tanaka and Sugeno [113, pages 146–147].

Corollary 7.1 Suppose that each A_{i}, i = 1, 2, …, r in fuzzy model (7.41) is asymptotically stable and there exists a symmetric positive definite P such that conditions (7.42) hold. Then, the matrices

A_{i} + A_{j}, i, j = 1, 2, …, r,

are asymptotically stable.

We mention that Tanaka and Sugeno’s Theorem 4.3 can be generalized to correspond to Theorem 7.1 rather than to the above corollary.

The above corollary can be restated in an equivalent way as follows. If there exist i and j such that a matrix A_{i} + A_{j} is not asymptotically stable, then there is no positive definite matrix P such that A_{i}^{T} P + P A_{i} < 0 for i = 1, 2, …, r. The following example illustrates this fact.

Example 7.7 Given the unforced fuzzy system model described by (7.41), where r = 2 and

A_{1} = [ \begin{matrix}−1 & 4 \\ 0 & −2 \end{matrix} ] , A_{2} = [ \begin{matrix}−1 & 0 \\ 4 & −2 \end{matrix} ] .

Obviously each A_{i}, i = 1, 2, is asymptotically stable. However, there is no common positive definite P such that A_{i}^{T} P + P A_{i} < 0 for i = 1, 2 because the matrix

A_{1} + A_{2} = [ \begin{matrix}−1 & 4 \\ 0 & −2 \end{matrix} ] + [ \begin{matrix}−1 & 0 \\ 4 & −2 \end{matrix} ] = [ \begin{matrix}−2 & 4 \\ 4 & −4 \end{matrix} ]

is unstable since its eigenvalues are

{1.1231, −7.1231}.

Theorem 7.1 can be used to check if there does not exist a positive definite matrix P satisfying conditions (7.42) because this theorem gives us a sufficient condition for non-existence of a common P that satisfies (7.42).

We now present a sufficiency condition for asymptotic stabilizability in the large of fuzzy model given by (7.6) and (7.7) using fuzzy state feedback control law

u = − ∑_{j=1}^{r} α_{j}F_{j}x.  (7.45)

The closed-loop system has the form

ẋ = ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j} (A_{i} − B_{i}F_{j}) x  (7.46)

Suppose that the gain matrices, F_{i}, where chosen so that the matrices

A_{i} − B_{i}F_{i}, i = 1, 2, …, r,

have their eigenvalues in the open left-hand complex plane, that is, they are asymptotically stable. Let

G_{ij} = (A_{i} − B_{i}F_{j}) + (A_{j} − B_{j}F_{i}) for i < j ≤ r.

We now present sufficient conditions of Wang et al. [120] for the closed-loop system modeled by (7.46) to be asymptotically stable in the large.

Theorem 7.2 If for some symmetric positive definite P the following conditions are satisfied:

(A_{i} − B_{i}F_{i})^{T} P + P (A_{i} − B_{i}F_{i}) < 0, i = 1, 2, …, r  (7.47)

and

G_{ij}^{T}P + P G_{ij} < 0, i < j ≤ r,  (7.48)

then the closed-loop system modeled by (7.46) to be asymptotically stable in the large.

Proof Let P be such that (7.47) and (7.48) are satisfied. We evaluate the time derivative of x^{T} (t)P x(t) on the trajectories of (7.46) to get

\frac{d}{dt} (x^{T} (t)P x(t)) = 2x^{T} P ẋ
= 2x^{T} P (∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j} (A_{i} − B_{i}F_{j}) x)
= ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j}x^{T} ((A_{i} − B_{i}F_{j})^{T} P + P (A_{i} − B_{i}F_{j})) x
= ∑_{r=1}^{r} α_{i}^{2}x^{T} ((A_{i} − B_{i}F_{i})^{T} P + P (A_{i} − B_{i}F_{i})) x
+ ∑_{i=1}^{r} ∑_{j>i}^{r} α_{i}α_{j}x^{T} (G_{ij}^{T}P + P G_{ij}) x.  (7.49)

Hence, when (7.47) and (7.48) are satisfied, the positive definite quadratic function x^{T} P x becomes a Lyapunov function for the system (7.46), which implies asymptotic stability in the large of (7.46). The proof of the theorem is complete.

□

We now give a sufficient condition for the stability of system (7.46) for the case when conditions (7.48) are not satisfied. To proceed, we need the following notation. Suppose that there is a positive definite P such that conditions (7.47) are satisfied. Represent conditions (7.47) as

(A_{i} − B_{i}F_{i})^{T} P + P (A_{i} − B_{i}F_{i}) = −Q_{i}, i = 1, 2, …, r,  (7.50)

where each Q_{i} is symmetric positive definite. Let λ_{i} denote the minimum eigenvalue of Q_{i}. Since Q_{i} = Q_{i}^{T} > 0, we have

λ_{i} > 0, i = 1, 2, …, r.  (7.51)

Now, let

G_{ij}^{T}P + P G_{ij} = −Q_{ij}, i < j ≤ r,  (7.52)

where now we do not require Q_{ij} to be positive definite. Note that Q_{ij} = Q_{ij}^{T}, and hence the eigenvalues of Q_{ij} are real. Let λ_{ij} denote the minimum eigenvalue of Q_{ij}. We are now ready to state and prove the following theorem.

Theorem 7.3 Suppose that each (A_{i} − B_{i}F_{i}), i = 1, 2, …, r, in (7.46) is asymptotically stable and there exists a symmetric positive definite P such that conditions (7.47) hold. Then, the closed-loop fuzzy model (7.46) is asymptotically stable in the large if the matrix

Λ = [ \begin{matrix}λ_{1} & λ_{12}/2 & ⋯ & λ_{1r}/2 \\ λ_{12}/2 & λ_{2} & & λ_{2r}/2 \\ ⋮ & & ... & ⋮ \\ λ_{1r}/2 & λ_{2r}/2 & ⋯ & λ_{r} \end{matrix} ]  (7.53)

is positive definite.

Proof Let

W = W (x(t)) = x^{T} (t)P x(t).

be the Lyapunov function candidate for the closed-loop system modeled by (7.46). We then evaluate the time derivative of W on the trajectories of (7.46) to get

Ẇ = 2x^{T} P ẋ
= 2x^{T} P (∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j} (A_{i} − B_{i}F_{j}) x)
= ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j}x^{T} ((A_{i} − B_{i}F_{j})^{T} P + P (A_{i} − B_{i}F_{j})) x
= − ∑_{i=1}^{r} α_{i}^{2}x^{T} Q_{i}x − ∑_{i=1}^{r} ∑_{j>i}^{r} α_{i}α_{j}x^{T} Q_{ij}x.  (7.54)

We now use that fact that for any symmetric matrix M = M^{T}, we have

λ_{min}(M)∥x∥^{2} ≤ x^{T} Mx,

where λ_{min}(M) is the smallest eigenvalue of M. Using this fact in (7.54) yields

Ẇ ≤ − (∑_{i=1}^{r} α_{i}^{2}λ_{i} + ∑_{i=1}^{r} ∑_{j>i}^{r} α_{i}α_{j}λ_{ij}) ∥x∥^{2}

= − ([ α_{1} α_{2} ⋯ α_{r} ] [ \begin{matrix}λ_{1} & λ_{12}/2 & ⋯ & λ_{1r}/2 \\ λ_{12}/2 & λ_{2} & & λ_{2r}/2 \\ ⋮ & & ... & ⋮ \\ λ_{1r}/2 & λ_{2r}/2 & ⋯ & λ_{r} \end{matrix} ] [ α_{1} ; α_{2} ; ⋮ ; α_{r} ]) ∥x∥^{2}

= − (α^{T} Λα) ∥x∥^{2}.  (7.55)

The Lyapunov derivative is negative definite if the matrix Λ is positive definite, which completes the proof.

□

7.4.4 A Lyapunov Based Stabilizer

In this subsection, we apply a Lyapunov control approach to the fuzzy control design for uncertain dynamic systems. Some of the Lyapunov based control strategies, using nonfuzzy controllers, for uncertain systems are nicely documented in Zinober [132]. The method of constructing stabilizing controllers, presented in this section, requires that each of the matrices A_{i}, i = 1, 2, …, r is already asymptotically stable, and that there exists a common symmetric positive definite P = P^{T} > 0 such that, for i = 1, 2, …, r,

A_{i}^{T} P + P A_{i} < 0.  (7.56)

If this is not the case, we break the design of stabilizing control law, u, into two parts u = u_{1} + u_{2}. We first design u_{1} of the form

u_{1} = − ∑_{j=1}^{r} α_{j}F_{j}x  (7.57)

so that the system

ẋ = ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j} (A_{i} − B_{i}F_{j}) x

is asymptotically stable in the large and there exists a common P such that conditions (7.47) and (7.48) or conditions of Theorem 7.3 are satisfied. Note that the system modeled by (7.10) and driven by (7.57) has the form

ẋ = ∑_{i=1}^{r} α_{i}A_{i}x − ∑_{i=1}^{r} α_{i}B_{i} (∑_{j=1}^{r} α_{j}F_{j}x) + ∑_{i=1}^{r} α_{i}B_{i} (u_{2} + h)
= ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j} (A_{i} − B_{i}F_{j}) x + ∑_{i=1}^{r} α_{i}B_{i} (u_{2} + h) .  (7.58)

We next proceed with the design of u_{2}. The role of u_{2} is to reject the uncertainty h. Then, u = u_{1} +u_{2} will asymptotically stabilize in the large the uncertain system (7.10). We propose a family of fuzzy controllers u_{2} that reject the uncertainty h. These controllers are obtained by fuzzifying the controllers proposed by Hui and Żak [51]. To describe this family we need the following notation. Let

μ = [ μ_{1} μ_{2} … μ_{m} ]^{T} = ∑_{i=1}^{r} α_{i}B_{i}^{T} P x.  (7.59)

Let ∥·∥_{p} denote the p-norm of a vector, that is, if a vector v ∈ ℝ^{m}, then its p-norm is defined as

∥v∥_{p} = (|v_{1}|^{p} + |v_{2}|^{p} + ⋯ + |v_{m}|^{p})^{1/p} , p ≥ 1.

A well-known result concerning p-norms is the Hölder inequality

|v^{T} w| ≤ ∥v∥_{p}∥w∥_{q}, \frac{1}{p} + \frac{1}{q} = 1.

All norms on ℝ^{m} are equivalent in the sense that if ∥ · ∥_{α} and ∥ · ∥_{β} are norms on ℝ^{m}, then there exist positive constants C_{1} and C_{2} such that

C_{1}∥v∥_{α} ≤ ∥v∥_{β} ≤ C_{2}∥v∥_{α} for all v ∈ ℝ^{m}.  (7.60)

Finally, the gradient of ∥μ∥_{p}, where it is defined, has the form

∇∥μ∥_{p} = [ \frac{∂∥μ∥_{p}}{∂μ_{1}}  \frac{∂∥μ∥_{p}}{∂μ_{2}} … \frac{∂∥μ∥_{p}}{∂μ_{m}} ]^{T} , p ≥ 1.  (7.61)

Applying the chain rule, we compute

∇∥μ∥_{p} = \frac{1}{∥μ∥_{p}^{p−1}} [ |μ_{1}|^{p−1} sign(μ_{1}) |μ_{2}|^{p−1} sign(μ_{2}) ⋮ |μ_{m}|^{p−1} sign(μ_{m}) ] .  (7.62)

Let ∥h(t, x)∥_{q} ≤ η_{q}. We then show that when

u_{2} = −η_{q}∇∥μ∥_{p}, p ≥ 1, and \frac{1}{p} + \frac{1}{q} = 1,  (7.63)

then the closed-loop system driven by u = u_{1} + u_{2} is asymptotically stable in the large for any uncertainty h such that ∥h∥_{q} ≤ η_{q}.

Theorem 7.4 Given fuzzy system model (7.10). Suppose that u_{1}, given by (7.57), is such that there exists a common P satisfying conditions (7.47) and (7.48) or conditions of Theorem 7.3. Then, the closed-loop system driven by u = u_{1} + u_{2}, where u_{2} is given by (7.63), is asymptotically stable in the large for any uncertainty h such that ∥h∥_{q} ≤ η_{q}.

Proof Suppose that u_{1}, given by (7.57), was constructed such that (7.47) and (7.48), or conditions of Theorem 7.3 hold. System (7.10) driven by u_{1} takes the form (7.58), that is,

ẋ = ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j} (A_{i} − B_{i}F_{j}) x + ∑_{i=1}^{r} α_{i}B_{i} (u_{2} + h) .

We now show that

W = x^{T} P x,  (7.64)

is the Lyapunov function for the closed-loop system, where P = P^{T} > 0 satisfies (7.47) and (7.48) or conditions of Theorem 7.3. For this, we evaluate the Lyapunov derivative of W, that is, we evaluate \frac{d}{dt}W on the trajectories of the closed-loop system, and show that the Lyapunov derivative of W is negative definite. We have

\frac{d}{dt}W = 2x^{T} P (∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j} (A_{i} − B_{i}F_{j}) x + ∑_{i=1}^{r} α_{i}B_{i} (u_{2} + h))
< −2η_{q}μ^{T} ∇∥μ∥_{p} + 2μ^{T} h
≤ −2η_{q}∥μ∥_{p} + 2 |μ^{T} h| .  (7.65)

Applying Hölder’s inequality to the above yields

\frac{d}{dt}W < −2η_{q}∥μ∥_{p} + 2∥μ∥_{p}∥h∥_{q}
< −2η_{q}∥μ∥_{p} + 2∥μ∥_{p} η_{q}
< 0.  (7.66)

By Lyapunov’s theorem, the closed loop-system is asymptotically stable in the large. □ Example 7.8 Using system model (A.34), we propose the following two rules describing the plant dynamics:

Rule 1 : IF x_{1}(t) is about 0

THEN ẋ(t) = A_{1}x(t) + B_{1}(u(t) + h),

and

Rule 2 : IF x_{1}(t) is about ± π/4
THEN ẋ(t) = A_{2}x(t) + B_{2}(u(t) + h),

where h = −f_{c} + mlx_{2}^{2} sin(x_{1}). Linear local models are constructed using the procedure described in the previous section, and in particular relations (7.27) and (7.40). We have

A_{1} = [ \begin{matrix}0 & 1 & 0 & 0 \\ \frac{g}{4l/3−aml} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ −\frac{amg}{4/3−am} & 0 & 0 & 0 \end{matrix} ] , B_{1} = [ 0 ; −\frac{a}{4l/3−aml} ; 0 ; \frac{4a/3}{4/3−ma} ] .

The matrices A_{2} and B_{2} of the local model corresponding to the second rule are:

A_{2} = [ \begin{matrix}0 & 1 & 0 & 0 \\ \frac{g sin(x_{1})}{4l/3−mla cos^{2}(x_{1})} (\frac{1}{x_{1}} ) & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \frac{−mag sin(2x_{1})/2}{4/3−ma cos^{2}(x_{1})} (\frac{1}{x_{1}} ) & 0 & 0 & 0 \end{matrix} ]_{|x_{1}=±π/4}

and

B_{2} = [ 0 \frac{−a cos(x_{1})}{4l/3−mla cos^{2}(x_{1})} 0 \frac{4a/3}{4/3−ma cos^{2}(x_{1})} ]_{|x_{1}=±π/4} .

The parameter values are the same as in Example 7.5. We use membership functions of the form:

w_{1}(x_{1}) = \frac{1 − 1/ (1 + exp(−14(x_{1} − π/8)))}{1 + exp(−14(x_{1} + π/8))}
w_{2}(x_{1}) = 1 − w_{1}(x_{1}).

The corresponding fuzzy system model is

ẋ = (α_{1}A_{1} + α_{2}A_{2}) x + (α_{1}b_{1} + α_{2}b_{2}) (u − μ_{c}sign(x_{4}) + mlx_{2}^{2} sin(x_{1})),  (7.67)

where α_{i} = w_{i}, i = 1, 2, because w_{1} + w_{2} = 1 for all t, and the friction coefficient μ_{c} = 0.005. We then proceed to construct a stabilizing controller. We first construct

u_{1} = − (α_{1}F_{1} + α_{2}F_{2}) x

such that the system

ẋ = ∑_{i=1}^{2} ∑_{j=1}^{2} α_{i}α_{j} (A_{i} − B_{i}F_{j}) x  (7.68)

is asymptotically stable in the large. To better test our stabilization algorithm, we randomly chose the set of desired poles for (A_{1} − B_{1}F_{1}) to be in the interval [−4, −1], and we got

{−1.0970, −2.1263, −2.5553, −3.9090}.

The desired poles for (A_{2} − B_{2}F_{2}) were also chosen in a random fashion to be in the interval [−4, −1]. In this case we got the set

{−1.5794, −1.6857, −2.7908, −3.6895}.

For this choice of the closed-loop poles of the local models, we obtain the following feedback gain matrices:

F_{1} = [ −294.8766 −73.1208 −13.4726 −27.3362 ] ,

and

F_{2} = [ −440.3915 −118.5144 −19.1611 −35.5575 ] .

Let

G_{12} = A_{1} − B_{1}F_{2} + A_{2} − B_{2}F_{1},

and

P = [ \begin{matrix}54.9580 & 15.6219 & 6.9389 & 12.1165 \\ 15.6219 & 4.5429 & 2.1011 & 3.5488 \\ 6.9389 & 2.1011 & 1.3972 & 1.7978 \\ 12.1165 & 3.5488 & 1.7978 & 2.9375 \end{matrix} ] .

Then, it is easy to check that sufficient conditions, given by (7.47) and (7.48), for asymptotic stability in the large of system (7.68) are satisfied. This completes the design of u_{1}.

We now construct u_{2}. We use (7.59) to construct

μ = (α_{1}B_{1}^{T} P + α_{2}B_{2}^{T} P ) x.

Figure 7.31: Plots of x_{1} versus time for two different initial values of x_{1}: x_{1}(0) = 1.2 and x_{1}(0) = 0.6. In both cases x_{2}(0) = x_{3}(0) = x_{4}(0) = 0.

In this example, the number of inputs is m = 1, and hence (7.63) reduces to

u_{2} = −η_{q}sign(μ),

where we can take η_{q} = |h|. We combine u_{1} and u_{2} to obtain the final form of stabilizing controller. This controller has the form

u = u_{1} + u_{2}
= − (α_{1}F_{1} + α_{2}F_{2}) x − η_{q}sign(μ).  (7.69)

The above controller globally stabilizes the design model. In our simulations we connected the controller to the truth model. Typical responses are shown in Figure 7.31 and Figure 7.32. The controller locally stabilizes the truth model. We observed that when |x_{1}(0)| > 1.3 rads, and x_{2}(0) = x_{3}(0) = x_{4}(0) = 0, then the equilibrium state x = 0 of the closed-loop system consisting of the truth model driven by (7.69) becomes unstable.

Figure 7.32: Plots of x_{3} versus time for two different initial values of x_{1}: x_{1}(0) = 1.2 and x_{1}(0) = 0.6. In both cases x_{2}(0) = x_{3}(0) = x_{4}(0) = 0.

A common approach to the control system design is to use a design model to construct a controller. The design model is a simplified version of the truth model and usually neglects high frequency dynamics of the plant. Thus, the designer has to cope with an incomplete mathematical model of the process to be controlled. To account for modeling uncertainties, we propose to use a two-part controller. The first part stabilizes the nominal plant, that is, the plant model that does not include modeling uncertainties. The role of the second part of the controller is to reject modeling uncertainties. In both cases fuzzy state-feedback control strategies were used. In order to be able to apply the proposed controllers in real-life systems, further research is needed to account for limited authority of actuators, unmatched uncertainties, as well as inaccessibility of state variables.

7.5 Stability of Discrete Fuzzy Models

We analyze the model of the form

x(k + 1) = ∑_{i=1}^{r} w_{i}(x(k))A_{i}x(k),  (7.70)

where

w_{i}(x(k)) ≥ 0 and ∑_{i=1}^{r} w_{i}(x(k)) = 1.

To proceed, we need the following notation. Given two n × n matrices A and B. Then, A < B means that

x^{T} Ax < x^{T} Bx for all x ∈ ℝ^{n}, x ≠ 0,

that is, the quadratic form x^{T} (A − B) x is negative definite.

We now prove the following matrix inequality lemma that is useful in the stability analysis of discrete fuzzy models.

Lemma 7.1 Let A, B, P ∈ ℝ^{n×n}. If P = P^{T} > 0 such that

A^{T} P A − P < 0 and B^{T} P B − P < 0,

then

A^{T} P B + B^{T} P A − 2P < 0.

Proof We first represent A^{T} P B + B^{T} P A − 2P as

A^{T} P B + B^{T} P A − 2P = −(A − B)^{T} P (A − B) + A^{T} P A + B^{T} P B − 2P
= −(A − B)^{T} P (A − B) + (A^{T} P A − P)
+(B^{T} P B − P).

The first term, −(A − B)^{T} P (A − B), is non-positive definite because by assumption P = P^{T} > 0, while the second and the third terms are negative definite by assumption. Hence,

A^{T} P B + B^{T} P A − 2P < 0,

and the lemma is proven.

□

Using Lemma 7.1, we now prove a sufficient condition for global asymptotic stability of fuzzy model (7.70).

Theorem 7.5 The equilibrium state, x_{e} = 0, of fuzzy model (7.70) is globally asymptotically stable if there exists a positive definite matrix P = P^{T} > 0 such that for i = 1, 2, …, r,

A_{i}^{T} P A_{i} − P < 0.  (7.71)

Proof Suppose that there exists a symmetric positive definite matrix P = P^{T} > 0 such that for i = 1, 2, …, r inequalities given by (7.71) are satisfied. Consider then the Lyapunov function candidate

V (x(k)) = x^{T} (k)P x(k).  (7.72)

To prove the theorem, it is enough to show that

ΔW (x(k)) < 0.

We have

ΔV (x(k)) = V (x(k + 1)) − V (x(k))
= x^{T} (k + 1)P x(k + 1) − x^{T} (k)P x(k)
= (∑_{i=1}^{r} w_{i}x^{T} A_{i}^{T} ) P (∑_{i=1}^{r} w_{i}A_{i}x) − x^{T} P x
= x^{T} ((∑_{i=1}^{r} w_{i}A_{i}^{T} ) P (∑_{i=1}^{r} w_{i}A_{i}) − P ) x.  (7.73)

To proceed, we represent (7.73) in an equivalent form as

ΔV (x) = x^{T} ((∑_{i=1}^{r} w_{i}A_{i}^{T} ) P (∑_{j=1}^{r} w_{j}A_{j}) − P ) x.  (7.74)

By assumption

∑_{i=1}^{r} w_{i} = 1,

and hence

∑_{i=1}^{r} w_{i} ∑_{j=1}^{r} w_{j} = ∑_{i=1}^{r} ∑_{j=1}^{r} w_{i}w_{j} = 1.  (7.75)

Substituting (7.75) into (7.74) yields

ΔV (x) = x^{T} ((∑_{i=1}^{r} w_{i}A_{i}^{T} ) P (∑_{j=1}^{r} w_{j}A_{j}) − ∑_{i=1}^{r} ∑_{j=1}^{r} w_{i}w_{j}P ) x
= x^{T} (∑_{i=1}^{r} ∑_{j=1}^{r} w_{i}w_{j} (A_{i}^{T} P A_{j} − P )) x.  (7.76)

We rearrange (7.76) to get

ΔV (x(k)) = ∑_{i=1}^{r} w_{i}^{2}x^{T} (A_{i}^{T} P A_{i} − P ) x + ∑_{i=1}^{r} ∑_{{j=1 \atop j≠i}}^{r} w_{i}w_{j}x^{T} (A_{i}^{T} P A_{j} − P ) x

= ∑_{i=1}^{r} w_{i}^{2}x^{T} (A_{i}^{T} P A_{i} − P ) x
+ ∑_{i=1}^{r} ∑_{j>i}^{r} w_{i}w_{j}x^{T} (A_{i}^{T} P A_{j} + A_{j}^{T} P A_{i} − 2P ) x.  (7.77)

The first term in (7.77) is negative definite because of the assumption and the fact that at least one of the weights, w_{i}, is positive since w_{i} ≥ 0 and ∑_{i=1}^{r} w_{i} = 1. The second term in (7.77) is nonpositive definite by virtue of Lemma 7.1. Hence, V = x^{T} P x is a Lyapunov function for the fuzzy model (7.70), and by the Lyapunov theorem the equilibrium state x_{e} = 0 of the model is globally asymptotically stable.

□

The above theorem gives no method for finding a common positive definite matrix P that satisfies r inequalities (7.71). All of the matrices A_{i}, i = 1, 2, …, r, are asymptotically stable if there exists a common positive definite P that satisfies (7.71). However, the asymptotic stability of all the matrices A_{i} is only necessary but not sufficient for the existence of a common symmetric positive definite P that satisfies (7.71). The following theorem provides a necessary, but not sufficient, condition for the existence of positive definite P satisfying (7.71).

Theorem 7.6 Given r square n × n matrices A_{i} that are asymptotically stable and nonsingular. If there exists a symmetric positive definite matrix P = P^{T} > 0 such that for i = 1, 2, …, r,

A_{i}^{T} P A_{i} − P < 0,  (7.78)

then for i, j = 1, 2, …, r, the matrices A_{i}Aj are asymptotically stable.

Proof By assumption, there exists a symmetric positive definite matrix P such that for j = 1, 2, …, r,

A_{j}^{T} P A_{j} − P < 0.  (7.79)

Premultiplying the above by (A_{j}^{T} )^{−1} and postmultiplying by A_{j}^{−1} yields

P − (A_{j}^{T} )^{−1} P A_{j}^{−1} < 0.  (7.80)

Hence,

P < (A_{j}^{T} )^{−1} P A_{j}^{−1}.  (7.81)

On the other hand, using (7.78) we get

A_{i}^{T} P A_{i} < P .  (7.82)

Combining (7.81) and (7.82), we obtain

A_{i}^{T} P A_{i} < (A_{j}^{T} )^{−1} P A_{j}^{−1}.  (7.83)

Premultiplying both sides of (7.83) by A_{j}^{T} and postmultiplying by A_{j}, we get, for i, j = 1, 2, …, r,

(A_{i}A_{j})^{T} P (A_{i}A_{j}) − P < 0.  (7.84)

Therefore, for i, j = 1, 2, …, r, the matrices A_{i}A_{j} are asymptotically stable, which completes the proof.

□

The above theorem can be stated in an equivalent way as follows. Assume that for i = 1, 2, …, r, the matrices A_{i} are asymptotically stable. If for some i, j = 1, 2, …, r, the matrix A_{i}A_{j} is unstable, then there does not exist a common positive definite P such that

A_{i}^{T} P A_{i} − P < 0 for i = 1, 2, …, r.

Example 7.9 Let

A_{1} = [ \begin{matrix}1 & −0.5 \\ 1 & 0 \end{matrix} ] and A_{2} = [ \begin{matrix}−1 & −0.5 \\ 1 & 0 \end{matrix} ] .

Both matrices, A_{1} and A_{2}, are asymptotically stable. Indeed, the eigenvalues of A_{1} are located at {0.5+j0.5, 0.5−j0.5}, while the eigenvalues of A_{2} are located at {−0.5+j0.5, −0.5− j0.5}. However, the matrix A_{1}A_{2} has its eigenvalues located at {−1.866, −0.134}, and hence it is unstable. Thus, it follows from Theorem 7.6, that there is no positive definite P such that

A_{i}^{T} P A_{i} − P < 0 for i = 1, 2.  (7.85)

7.6 Fuzzy Estimator

As we mentioned in Subsection 7.4.1, one of the essential elements of the control problem is a model of a dynamical system to be controlled. In many situations a mathematical model of the system is not available or is incomplete. However, there may be experts who can provide a linguistic description, in terms of IF-THEN rules, of the system. Combining the available mathematical description of the system with its linguistic description results in a fuzzy system model. An example of a fuzzy system model is the so called Takagi-Sugeno (T-S), also known as the Takagi-Sugeno-Kang (TSK), fuzzy model. One can construct a TSK model if local description of the dynamical system to be controlled is available in terms of linear models

ẋ(t) = A_{i}x(t) + B_{i}u(t), i = 1, 2, …, r,

where the state vector x(t) ∈ ℝ^{n}, the control input u(t) ∈ ℝ^{m}, and the matrices A_{i} and B_{i} are of appropriate dimensions. The above information is then fused with the available IF-THEN rules, where the i-th rule can have the form:

Rule i : IF z_{1}(t) is F_{1}^{i} AND ⋯ AND z_{q}(t) is F_{q}^{i},
THEN \{ \begin{matrix}ẋ(t) & = & A_{i}x(t) + B_{i}u(t) \\ y & = & C_{i}x(t), \end{matrix}

where F_{j}^{i}, j = 1, 2, …, q, is the jth fuzzy set of the ith rule, and z_{1}(t), …, z_{q}(t) are measurable system variables. Let μ_{j}^{i}(x_{j}) be the membership function of the fuzzy set A_{j}^{i}, and let

w^{i} = w^{i}(z) = ∏_{j=1}^{n} μ_{j}^{i}(z_{j}),

where r is the number of fuzzy rules. Then, given a pair (x(t), u(t)), the resulting fuzzy system model is inferred as the weighted average of the local models, and has the form

ẋ = \frac{∑_{i=1}^{r} w^{i} (A_{i}x + B_{i}u)}{∑_{i=1}^{r} w^{i}}
= ∑_{i=1}^{r} α_{i} (A_{i}x + B_{i}u)
= (∑_{i=1}^{r} α_{i}A_{i}) x + (∑_{i=1}^{r} α_{i}B_{i}) u
= A(α)x + B(α)u,  (7.86)

where for i = 1, 2, …, r,

α_{i} = \frac{w^{i}}{∑_{i=1}^{r} w^{i}} .

Similarly,

y = \frac{∑_{i=1}^{r} w^{i}C_{i}x}{∑_{i=1}^{r} w^{i}}
= (∑_{i=1}^{r} α_{i}C_{i}) x
= C(α)x.  (7.87)

7.6. FUZZY ESTIMATOR 257 Note that

α_{i} ≥ 0, ∑_{i=1}^{r} α_{i} = 1 and α = [ α_{1} α_{2} ⋯ α_{r} ]^{T} ∈ [0, 1]^{r}.  (7.88)

A fuzzy state estimator for the system modeled by (7.86) and (7.87) can be represented as

˙x̃ = A(α)x̃ + B(α)u + L(α) (y − ỹ) ,  (7.89)

where

L(α) = ∑_{i=1}^{r} α_{i}L_{i}.

The state estimator (7.89) can be represented equivalently as

˙x̃ = ∑_{i=1}^{r} α_{i}A_{i}x̃ + ∑_{i=1}^{r} α_{i}B_{i}u + ∑_{i=1}^{r} α_{i}L_{i} (y − ỹ)  (7.90)

We note that the state estimator (7.90) can be viewed as a result of implementing the following r fuzzy rules:

Rule i : IF z_{1}(t) is F_{1}^{i} AND ⋯ AND z_{q}(t) is F_{q}^{i},

THEN \{ \begin{matrix}˙x̃ & = & A_{i}x̃ + B_{i}u + L_{i} (y − ỹ) \\ ỹ & = & C_{i}x̃. \end{matrix}

We use the state estimate in the controller implementation to get

u = − ∑_{j=1}^{r} K_{j}x̃ + v.  (7.91)

Substituting the above in to the plant model yields

ẋ = ∑_{i=1}^{r} α_{i}A_{i}x − ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j}B_{i}K_{j}x̃ + ∑_{i=1}^{r} α_{i}B_{i}v.  (7.92)

Substituting (7.87) and (7.91) into the estimator equation (7.90), we obtain

˙x̃ = ∑_{i=1}^{r} α_{i}A_{i}x̃ − ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j}B_{i}K_{j}x̃ + ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j}L_{i}C_{j} (x − x̃) + ∑_{i=1}^{r} α_{i}B_{i}v
= ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j} (A_{i} − B_{i}K_{j}) x̃ + ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j}L_{i}C_{j} (x − x̃) + ∑_{i=1}^{r} α_{i}B_{i}v.  (7.93)

Equations (7.92) and (7.93) constitute the closed-loop system driven by the combined fuzzy controller-estimator compensator. We represent the above two equations modeling the closed-loop system in matrix form as

[ ẋ ; ˙x̃ ] = [ \begin{matrix}∑_{i=1}^{r} α_{i}A_{i} & − ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j}B_{i}K_{j} \\ ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j}L_{i}C_{j} & ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j} (A_{i} − B_{i}K_{j} − L_{i}C_{j}) \end{matrix} ] [ x ; x̃ ]
+ [ \begin{matrix}∑_{i=1}^{r} α_{i}B_{i} \\ ∑_{i=1}^{r} α_{i}B_{i} \end{matrix} ] v  (7.94)
y = [ ∑_{i=1}^{r} α_{i}C_{i} O ] [ x ; x̃ ] .  (7.95)

7.6.1 The Comparison Method for Linear Systems

We will now present the Comparison Method for linear systems. This method will then be used in investigating the stability of the closed-loop system consisting of a fuzzy model of a plant and a fuzzy controller-estimator compensator. The closed-loop system is modeled by (7.94) and (7.95). Let A be a constant n × n matrix and e^{At} denote the matrix exponential. We will now give a necessary and sufficient condition for the matrix exponential to have all its elements non-negative for t ≥ 0. This result appears in [9, page 171].

Lemma 7.2 A necessary and sufficient condition that all elements of e^{At} be non-negative for t ≥ 0 is that

a_{ij} ≥ 0 i ≠ j.  (7.96)

Proof We follow the arguments of Bellman [9, page 172], see also [8]. We first prove the necessary condition. Since

e^{At} = I_{n} + At + \frac{1}{2}A^{2}t^{2} + ⋯

it is clear that (7.96) is necessary for the result to hold for small t, whch establishes the necessity condition

To prove the sufficiency, let c_{1} be a scalar so that all the elements of A + c_{1}I_{n} are nonnegative. Then, obviously, all the elements of e^{A+c_{1}I_{n}} are non-negative. Also, the elements of e^{−c_{1}I_{n}t} are non-negative since exponentials are always non-negative. The matrices A+c_{1}I_{n} and −c_{1}I_{n} commute. Hence, we can write

e^{At} = e^{(A+c_{1}I_{n})t−c_{1}I_{n}t}
= e^{(A+c_{1}I_{n})t}e^{−c_{1}I_{n}t},

which establishes the desired non-negativity condition and the sufficiency condition.

□

Using the above lemma, we can prove the following lemma due to Bellman [10]. We use the following notation: x ≤ y implies component-by-component majorization, that is, x_{i} ≤ y_{i} for i = 1, 2, …, n, where x_{i} and y_{i} are the i-th components of x and y, respectively.

Lemma 7.3 If a_{ij} ≥ 0, i ≠ j, then

\frac{dx}{dt} ≤ Ax, x(0) = c,  (7.97)

implies x ≤ y, where

\frac{dy}{dt} = Ay, y(0) = c.  (7.98)

Proof We can represent (7.97) as

\frac{dx}{dt} = Ax − f(t), x(0) = c,  (7.99)

where f(t) is a vector-valued function whose components are non-negative for all t ≥ 0. The solution of (7.99) can be represented as

x = e^{At}c − ∫_{0}^{t} e^{A(t−s)}f(s)ds
= y − ∫_{0}^{t} e^{A(t−s)}f(s)ds.  (7.100)

Using Lemma 7.2, we conclude that the components of the integrand of the second term in (7.100) are all non-negative for t ≥ s. Hence,

∫_{0}^{t} e^{A(t−s)}f(s)ds ≥ 0, for t ≥ 0,

which implies

x ≤ y.

The proof is complete.

□

Equation (7.98) is referred as the equation of comparison for (7.97).

7.6.2 Stability Analysis of the Closed-Loop System

We now perform a stability analysis of the closed-loop system given by (7.94) and (7.95). We first apply a state variable transformation to bring the system into a form convenient for further analysis. The transformation that we will use has the form,

[ \begin{matrix}I_{n} & O \\ −I_{n} & I_{n} \end{matrix} ] [ x ; x̃ ] .  (7.101)

The closed-loop system model in the new coordinates is

[ ẋ ; ˙x̃ − x ] = [ \begin{matrix}∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j} (A_{i} − B_{i}K_{j}) & − ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j}B_{i}K_{j} \\ O & ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j} (A_{i} − L_{i}C_{j}) \end{matrix} ] [ \begin{matrix}x \\ x̃ − x \end{matrix} ]
+ [ \begin{matrix}∑_{i=1}^{r} α_{i}B_{i} \\ O \end{matrix} ] v  (7.102)
y = [ ∑_{i=1}^{r} α_{i}C_{i} O ] [ x ; x̃ ] .  (7.103)

Let e = x̃−x. We will now present a sufficient condition for the unforced closed-loop system to be globally uniformly asymptotically stable, that is, we will give a condition for uniform asymptotic stability in the large of the system

[ ẋ ; ė ] = [ \begin{matrix}∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j} (A_{i} − B_{i}K_{j}) & − ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j}B_{i}K_{j} \\ O & ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j} (A_{i} − L_{i}C_{j}) \end{matrix} ] [ \begin{matrix}x \\ e \end{matrix} ] (7.104)

This condition was proposed by Ma, Sun and He [71].

Theorem 7.7 If there exist two scalar functions V (x) : ℝ^{n} → ℝ and Ṽ (e) : ℝ^{n} → ℝ and positive real numbers γ_{i}, γ̃_{i}, i = 1, …, 4 such that

γ_{1}∥x∥^{2} ≤ V (x) ≤ γ_{2}∥x∥^{2}, γ̃_{1}∥e∥^{2} ≤ Ṽ (e) ≤ γ̃_{2}∥e∥^{2},  (7.105)
\begin{matrix}\frac{∂V (x)}{∂x} ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j} (A_{i} − B_{i}K_{j}) x ≤ −γ_{3}∥x∥^{2} \\ \frac{∂Ṽ (e)}{∂e} ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j} (A_{i} − L_{i}C_{j}) e ≤ −γ̃_{3}∥e∥^{2}, \end{matrix} \}  (7.106)

and

‖\frac{∂V (x)}{∂x}‖ ≤ γ_{4}∥x∥, ‖\frac{∂Ṽ (e)}{∂e}‖ ≤ γ̃_{4}∥e∥,  (7.107)

then the system (7.104) is globally uniformly asymptotically stable.

Proof We first evaluate the time derivative of V on the trajectories of the system (7.104). We obtain

V̇ (x) = \frac{∂V (x)}{∂x}ẋ

= \frac{∂V (x)}{∂x} ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j} (A_{i} − B_{i}K_{j}) x − \frac{∂V (x)}{∂x} ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j}B_{i}K_{j}e
≤ −γ_{3}∥x∥^{2} + ‖\frac{∂V (x)}{∂x}‖ ∑_{i=1}^{r} ∑_{j=1}^{r} ∥B_{i}K_{j}∥ ∥e∥
≤ −\frac{γ_{3}}{2} ∥x∥^{2} − \frac{γ_{3}}{2} ∥x∥^{2} + γ_{4} ∑_{i=1}^{r} ∑_{j=1}^{r} ∥B_{i}K_{j}∥ ∥e∥∥x∥.  (7.108)

To proceed, we observe that

\frac{γ_{3}}{2} ∥x∥^{2} − γ_{4} ∑_{i=1}^{r} ∑_{j=1}^{r} ∥B_{i}K_{j}∥ ∥e∥∥x∥ + \frac{γ_{4}^{2}}{2γ_{3}} (∑_{i=1}^{r} ∑_{j=1}^{r} ∥B_{i}K_{j}∥)^{2} ∥e∥^{2} ≥ 0.  (7.109)

Indeed, we can represent the left-hand side of the above as

[ ∥x∥ ∥e∥ ] [ \begin{matrix}\frac{γ_{3}}{2} & −\frac{γ_{4}}{2} ∑_{i=1}^{r} ∑_{j=1}^{r} ∥B_{i}K_{j}∥ \\ −\frac{γ_{4}}{2} ∑_{i=1}^{r} ∑_{j=1}^{r} ∥B_{i}K_{j}∥ & \frac{γ_{4}^{2}}{2γ_{3}} (∑_{i=1}^{r} ∑_{j=1}^{r} ∥B_{i}K_{j}∥)^{2} \end{matrix} ] [ \begin{matrix}∥x∥ \\ ∥e∥ \end{matrix} ] (7.110)

It is easy to check that the above quadratic form is positive semi-definite. We use this fact and (7.105) to represent (7.109) as

V̇ (x) ≤ −\frac{γ_{3}}{2} ∥x∥^{2} + \frac{γ_{4}^{2}}{2γ_{3}} (∑_{i=1}^{r} ∑_{j=1}^{r} ∥B_{i}K_{j}∥)^{2} ∥e∥^{2}
≤ −\frac{γ_{3}}{2γ_{2}} V (x) + \frac{γ_{4}^{2}}{2γ_{3}γ̃_{1}} (∑_{i=1}^{r} ∑_{j=1}^{r} ∥B_{i}K_{j}∥)^{2} Ṽ (e).  (7.111)

Let a = (∑_{i=1}^{r} ∑_{j=1}^{r} ∥B_{i}K_{j}∥)^{2}. Then, we represent (7.111) as

V̇ (x) ≤ −\frac{γ_{3}}{2γ_{2}} V (x) + \frac{aγ_{4}^{2}}{2γ_{3}γ̃_{1}} Ṽ (e).  (7.112)

We now evaluate the time derivative of Ṽ (e) on the trajectories of (7.104) to obtain

˙Ṽ (e) = \frac{∂Ṽ (e)}{∂e}ė
≤ \frac{∂Ṽ (e)}{∂e} ∑_{i=1}^{r} ∑_{j=1}^{r} α_{i}α_{j} (A_{i} − L_{i}C_{j}) e
≤ −γ̃_{3}∥e∥^{2}
≤ −\frac{γ̃_{3}}{γ̃_{2}} Ṽ (e).  (7.113)

We combine (7.112) and (7.113) into a single matrix differential inequality of the form,

[ \begin{matrix}V̇ (x) \\ ˙Ṽ (e) \end{matrix} ] ≤ [ \begin{matrix}−\frac{γ_{3}}{2γ_{2}} & \frac{aγ_{4}^{2}}{2γ_{3}γ̃_{1}} \\ 0 & −\frac{γ̃_{3}}{γ̃_{2}} \end{matrix} ] [ \begin{matrix}V (x) \\ Ṽ (e) \end{matrix} ]
= A [ \begin{matrix}V (x) \\ Ṽ (e) \end{matrix} ] .  (7.114)

We then refer to Lemma 7.3 to conclude that the system (7.104) is globally uniformly asymptotically stable. Indeed, since the equation of comparison for (7.114) is globally uniformly asymptotically stable, we have to have

V (x) → 0 and Ṽ (e) → 0 for t → 0.

Since V and Ṽ are both positve definite decrescent and radially ubounded functions, the above can hold provided x → 0 and e → 0 uniformly with respect to t and irrespective of the initial conditions x(0) and e(0). Therefore, the closed-loop system is globally uniformly asymptotically stable.

□

7.7 Notes

The reader is encouraged to peruse a text on classical logic to become familiar with basic concepts of logic before undertaking his or her further study of fuzzy logic. An excellent introduction to logic is Copi [20]. For an introduction to fuzzy set theory the reader is referred to Klir, St.Clair and Yuan [61], while for an introduction to fuzzy control, see Driankov, Hellendoorn, and Reinfrank [24] and Palm, Driankov and Hellendoorn [88]. Kosko [63] is the first book on fuzzy engineering. Graduate texts on fuzzy systems and control are Pedrycz [89] and Wang [122]. Paradigms and practice of fuzzy modeling are the topics of the edited book by Pedrycz [90]. Lee and Lin [67], and Jang, Sun and Mizutani [54] cover applications of fuzzy systems as well as issues of fusing fuzzy logic and neural networks.

The first chapter of Kosko’s book [63] contains a thrilling account of the beginnings of fuzzy logic and fuzzy engineering. According to Kosko [63, page 5], “Fuzziness began as vagueness in the late nineteenth century.” A concept is vague in case it has blurred boundaries. For example, the concept mountain is vague because we do not know where a mountain ends and a hill begins. It was Bertrand Russell who in 1923 published a paper in which he described the concept of vagueness in terms of symbolic logic. In the 1920s, Polish mathematician Jan Łukasiewicz invented the multivalued logic. Max Black applied multivalued logic to sets of objects in the 1930s . In the 1950s, a number of logicians worked out a vague set algebra. In 1965 Zadeh [130] used the term fuzzy to mean vague. “Zadeh’s 1965 paper applied Lukasiewicz’s logic to each object in a set to work out a complete fuzzy set algebra and to extend the convex separation theorem to pattern recognition.” [63, page 8] The beginning of fuzzy engineering, on the other hand, can be marked by the work of Mamdani and Assilian [74] who first used fuzzy calculus to construct a fuzzy logic controller.

264 CHAPTER 7. FUZZY SYSTEMS

Appendix A

Models

A.1 Ground Vehicle Model

In this appendix, we present a model of ground vehicle in a turning maneuver, for example, a step lane change. During such a maneuver, the vehicle undergoes translational as well as rotational motion. When modeling a moving vehicle it is convenient to use reference coordinate frame attached to and moving with the vehicle. This is because with respect to the vehicle coordinate frame moments of inertias of the vehicle are constant. We use the Society of Automotive Engineers (SAE) standard coordinate system as shown in Figure A.1. Our modeling follows that of Wong [128] and Will and Żak [125]. We use lower case letters to label axes of the vehicle coordinate system. Following the SAE convention, we define the coordinates as follows:

• the x-coordinate is directed forward and on the longitudinal plane of symmetry of the vehicle,

• the y-coordinate has a lateral direction out the right side of the vehicle,

• the z-coordinate is directed downward with respect to the vehicle,

• the yaw velocity about the z-axis is denoted θ̇.

In developing our simple model, we neglect the effects of roll and pitch. The origin of the vehicle coordinate system is located at the vehicle center of gravity (CG). We also use a global, or fixed coordinate system. Its axes are labeled using upper case letters as shown in Figure A.2. To form the equations of motion of the vehicle, we need an expression for the Figure A.1: The Society of Automotive Engineers (SAE) coordinate system attached to the vehicle.

Figure A.2: Analyzing vehicle motion using coordinate frame attached to the vehicle; snapshots of the vehicle velocity at times t and t + Δt.

Figure A.3: Computing the change in the longitudinal and lateral components of the vehicle velocity.

vehicle CG acceleration. We first compute the change in the longitudinal component of the velocity. Using Figure A.2 and Figure A.3, and assuming small angles, we obtain

(v_{x} + Δv_{x}) cos Δθ − (v_{y} + Δv_{y}) sin Δθ − v_{x} ≈ Δv_{x} − v_{y}Δθ
= Δẋ − ẏΔθ.

We form the Newton quotient using the above. Then, taking the limit, we obtain the total longitudinal acceleration component a_{x},

a_{x} = lim_{Δt→0} \frac{Δv_{x} − v_{y}Δθ}{Δt}
= v̇_{x} − v_{y}θ̇
= ẍ − ẏω_{z},  (A.1)

where ω_{z} = θ̇. Using similar argument to the one above, we obtain the lateral acceleration component a_{y} to be

a_{y} = v̇_{y} + v_{x}θ̇
= ÿ + ẋω_{z}.  (A.2)

We assume that the vehicle is symmetric with respect to the x-z plane. We use the so called bicycle model that is shown in Figure A.4, where δ_{f} denotes the front wheels’ steering angle

Figure A.4: A simplified, bicycle model of a vehicle.

and δ_{r} is the rear wheels’ steering angle. We denote by 2F_{xf} lumped longitudinal force at the front axle and by 2F_{xr} lumped longitudinal force at the rear axle. The lumped lateral force at the front axle is denoted 2F_{yf}, while lumped lateral force at the rear axle is denoted 2F_{yr}. Summing the forces along the x axis and assuming small angles, we obtain

m (ẍ − ẏω_{z}) = 2F_{xf} cos δ_{f} − 2F_{yf} sin δ_{f} + 2F_{xr} cos δ_{r} + 2F_{yr} sin δ_{r}
≈ 2F_{xf} + 2F_{xr}.  (A.3)

Next, summing the forces along the y axis and assuming small angles yields

m (ÿ + ẋω_{z}) = 2F_{xf} sin δ_{f} + 2F_{yf} cos δ_{f} + 2F_{yr} cos δ_{r} − 2F_{xr} sin δ_{r}
≈ 2F_{yf} + 2F_{yr}.  (A.4)

The third equation of motion is obtained by summing the torques with respect to the z-axis. Let I_{z} denote the moment of inertia of the vehicle about the z axis. Then, referring again to Figure A.4 and assuming small angles, we obtain:

I_{z}ω̇_{z} = 2l_{1}F_{yf} cos δ_{f} + 2l_{1}F_{xf} sin δ_{f} − 2l_{2}F_{yr} cos δ_{r} + 2l_{2}F_{xr} sin δ_{r}
≈ 2l_{1}F_{yf} − 2l_{2}F_{yr}.  (A.5)

In our further analysis, we assume v_{x} = ẋ = constant, that is, the driver neither is accelerating nor braking the vehicle. This means that we need to use only the last two equations of

Figure A.5: A typical cornering characteristic.

motion in our model. To proceed, we need to define tire slip angles and cornering stiffness coefficients. The tire slip angle is the difference between the tire’s actual travel direction and the tire’s heading direction, which is the steer angle. In other words, the the tire slip angle is the angle between the desired direction of motion of the tire specified by the steer angle and the actual direction of travel of the center of the tire contact. We dnote by C_{αf} the cornering stiffness coefficient of the front tires and by C_{αr} the cornering stiffness coefficient of the rear tires. The coefficients C_{αf} and C_{αr} are measured experimentally. A typical relation between a tire slip angle and a cornering force is shown in Figure A.5. It is a nonlinear relation, where the cornering coefficient is

C_{αw} = \frac{dF_{yw}}{dα_{w}} ,

where w = f or w = r. The lateral forces F_{yf} and F_{yr} that appear in (A.3)–(A.5) are functions of the corresponding tire slip angles α_{f} and α_{r} and cornering stiffness coefficients C_{αf} and C_{αr},In our simple linearized model, we assume that C_{αf} = C_{αr} = constant, and model dependencies between the lateral fopces and tire slip angles as

F_{yf} = C_{αf}α_{f},  (A.6)

and

F_{yr} = C_{αr}α_{r}.  (A.7)

270 APPENDIX A. MODELS

Figure A.6: A diagram of a simplified vehicle model for computing tire slip angles. We formulate expressions for the tire slip angles using Figure A.6. Assuming small angles, we form an expression for the front tire slip angle

α_{f} = δ_{f} − tan^{−1} (\frac{l_{1}ω_{z} + v_{y}}{v_{x}} )
≈ δ_{f} − \frac{l_{1}ω_{z} + v_{y}}{v_{x}}
= δ_{f} − \frac{l_{1}ω_{z} + ẏ}{ẋ} .  (A.8)

The rear tire slip angle is

α_{r} = δ_{r} − tan^{−1} (−\frac{l_{2}ω_{z} − v_{y}}{v_{x}} )
= δ_{r} + tan^{−1} (\frac{l_{2}ω_{z} − v_{y}}{v_{x}} )
≈ δ_{r} + \frac{l_{2}ω_{z} − v_{y}}{v_{x}}
= δ_{r} + \frac{l_{2}ω_{z} − ẏ}{ẋ} .  (A.9)

Hence

F_{yf} = C_{αf} (δ_{f} − \frac{l_{1}ω_{z} + ẏ}{ẋ} ) ,  (A.10)

and

F_{yr} = C_{αr} (δ_{r} + \frac{l_{2}ω_{z} − ẏ}{ẋ} ) .  (A.11)

Substituting the above two equations into (A.4), and assuming small angles yields

mÿ + mẋω_{z} = 2C_{αf}δ_{f} − 2C_{αf} \frac{l_{1}ω_{z} + ẏ}{v_{x}} + 2C_{αr}δ_{r} + 2C_{αr} \frac{l_{2}ω_{z} − ẏ}{v_{x}} .

Rearranging terms and performing some manipulations, we get

ÿ = − (\frac{2C_{αf} + 2C_{αr}}{mv_{x}} ) ẏ + (−v_{x} − \frac{2C_{αf}l_{1} − 2C_{αr}l_{2}}{mv_{x}} ) ω_{z} + \frac{2C_{αf}}{m} δ_{f} + \frac{2C_{αr}}{m} δ_{r}. (A.12)

Substituting the expressions for F_{yf} and F_{yr} into (A.5) and assuming small angles, we obtain

I_{z}ω̇_{z} = 2l_{1}C_{αf} (δ_{f} − \frac{l_{1}ω_{z} + ẏ}{v_{x}} ) δ_{f} − 2l_{2}C_{αr} (δ_{r} + \frac{l_{2}ω_{z} − ẏ}{v_{x}} ) .

Rearranging and performing manipulations yields

ω̇_{z} = − (\frac{2l_{1}C_{αf} − 2l_{2}C_{αr}}{I_{z}v_{x}} ) ẏ − (\frac{2l_{1}^{2}C_{αf} + 2l_{2}^{2}C_{αr}}{I_{z}v_{x}} ) ω_{z} + \frac{2l_{1}C_{αf}}{I_{z}} δ_{f} − \frac{2l_{2}C_{αr}}{I_{z}} δ_{r}. (A.13)

Later, we will analyze a vehicle performance in the fixed coordinates. In particular, we will be interested in an emergency lane change maneuver. This requires that we transform the vehicle lateral variable into the fixed coordinates. Assuming small angles and referring to Figure A.7, we obtain

Ẏ = −ẋ sin ψ − ẏ cos ψ
≈ −v_{x}ψ − ẏ.  (A.14)

Note that

Ẋ = ẋ cos ψ − ẏ sin ψ.

For the step lane change, the yaw rate of the vehicle is equal to the yaw rate error, that is,

ω_{z} = ψ̇.

We further assume that

θ = ψ.  (A.15)

We can now represent the vehicle modeling equations in a state space format. Let

x_{1} = ẏ, x_{2} = θ, x_{3} = ω_{z}, x_{4} = Y.

272 APPENDIX A. MODELS

Figure A.7: Computing the velocity of a vehicle with respect to the fixed coordinates. Then, using (A.12), (A.13), (A.15) and (A.14), we obtain

[ ẋ_{1} ; ẋ_{2} ; ẋ_{3} ; ẋ_{4} ] = [ \begin{matrix}−\frac{2C_{αf} +2C_{αr}}{mv_{x}} & 0 & −v_{x} − \frac{2C_{αf} l_{1}−2C_{αr}l_{2}}{mv_{x}} & 0 \\ 0 & 0 & 1 & 0 \\ −\frac{2l_{1}C_{αf} −2l_{2}C_{αr}}{I_{z}v_{x}} & 0 & −\frac{2l_{1}^{2}C_{αf} +2l_{2}^{2}C_{αr}}{I_{z}v_{x}} & 0 \\ −1 & −v_{x} & 0 & 0 \end{matrix} ] [ \begin{matrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix} ]

+ [ \begin{matrix}\frac{2C_{αf}}{m} & \frac{2C_{αr}}{m} \\ 0 & 0 \\ \frac{2l_{1}C_{αf}}{I_{z}} & −\frac{2l_{2}C_{αr}}{I_{z}} \\ 0 & 0 \end{matrix} ] [ \begin{matrix}δ_{f} \\ δ_{r} \end{matrix} ]

= Ax + Bu.  (A.16)

A possible vehicle parameters are given in Table A.1.

A.2 Stick Balancer Model

In this appendix, we present a model of the stick balancer, also known as the cart with an inverted pendulum. In our derivation of the model, we use Newton’s laws, in a similar Table A.1: Typical vehicle data.

C_{αf} = C_{αr} 30, 000.00 N/rad
l_{1}  1.20 m
l_{2}  1.22 m
m  1280 kg
I_{z}  2500 kg·m^{2}
v_{x}  18.3 m/sec

fashion as in Kwakernaak and Sivan [64], and Ogata [86], where linearized models of the stick balancer were developed. An alternative derivation, using the method of D’Alembert, can be found in Cannon [93, Section 22.4].

A free body diagram of a cart on which a stick, an inverted pendulum, is mounted is shown in Figure A.8. We denote by H = H(t) and V = V (t) horizontal and vertical reaction forces, respectively. The x and y are the coordinates of the fixed, nonrotating coordinate frame xy. The angular displacement of the stick from the vertical position is denoted by θ = θ(t). The mass of the cart is denoted by M, while the mass of the stick by m. The length of the stick is 2l, and its center of gravity is at its geometric center. The control force applied to the cart is labeled by u. We assume that the wheels of the cart do not slip. We model the frictional force of the cart wheels on the track by

f_{c} = μ_{c}sign(ẋ),  (A.17)

where μ_{c} is the cart friction coefficient. Friedland [33, page 201] refers to this model of friction as the classical Coulomb friction model. Let (x_{G}, y_{G}) be the coordinates of the center of gravity of the stick. Then,

\{ \begin{matrix}x_{G} & = & x + l sin(θ) \\ y_{G} & = & l cos(θ). \end{matrix}  (A.18)

We are now ready to write down equations modeling the system. The equation that describes the rotational motion of the stick about its center of gravity is obtained by applying the rotational version of Newton’s second law. Summing the moments about the center of gravity of the stick, we obtain

I \frac{d^{2}θ}{dt^{2}} = V l sin(θ) − Hl cos(θ),  (A.19)

274 APPENDIX A. MODELS Figure A.8: Free body diagram of a cart with an inverted pendulum. where

I = ∫_{−l}^{l} r^{2}dm = ml^{2}/3,  (A.20)

is the moment of inertia of the stick with respect to its center of gravity.

We next write the equation that describes the horizontal motion of the center of gravity of the stick. Applying Newton’s second along the x axis yields

m\frac{d^{2}}{dt^{2}} (x + l sin(θ)) = H.  (A.21)

Performing the required differentiation, we get

m (ẍ + l (−θ̇^{2} sin(θ) + θ̈cos(θ))) = H.  (A.22)

The equation that describes the vertical motion of the center of gravity of the stick is obtained by applying Newton’s second law along the y axis. We have

m\frac{d^{2}}{dt^{2}} (l cos(θ)) = V − mg.  (A.23)

Performing the needed differentiation yields

ml (−θ̇^{2} cos(θ) − θ̈sin(θ)) = V − mg.  (A.24)

Finally, we apply Newton’s second law to the cart to get

M \frac{d^{2}x}{dt^{2}} = u − H − f_{c}.  (A.25)

Substituting (A.25) into (A.22) gives

mẍ + mlθ̈cos(θ) − mlθ̇^{2} sin(θ) + f_{c} = u − Mẍ.  (A.26)

Substituting now equations (A.24) and (A.25) into (A.19) yields

Iθ̈ = (mg − mlθ̇^{2} cos(θ) − mlθ̈sin(θ)) l sin(θ) + f_{c}l cos(θ) − (u − Mẍ) l cos(θ). (A.27)

We next substitute u − Mẍ from (A.26) into (A.27) and perform manipulations to get

Iθ̈ = mgl sin(θ) − ml^{2}θ̈ − mẍl cos(θ).  (A.28)

Let

a = \frac{1}{m + M} .

Then, we can represent (A.26) as

ẍ = −malθ̈cos(θ) + malθ̇^{2} sin(θ) − af_{c} + au.  (A.29)

We substitute (A.29) into (A.28) to obtain

θ̈ = \frac{mgl sin(θ) − m^{2}l^{2}aθ̇^{2} sin(2θ)/2 + mal cos(θ)f_{c} − mal cos(θ)u}{I − m^{2}l^{2}a cos^{2}(θ) + ml^{2}}.  (A.30)

Let x_{1} = θ and x_{2} = θ̇. Using the expression for I, given by (A.20), we represent (A.30) in state space format as follows

\begin{matrix}ẋ_{1} & = & x_{2} \\ ẋ_{2} & = & \frac{g sin(x_{1})−mlax_{2}^{2} sin(2x_{1})/2+a cos(x_{1})f_{c}}{4l/3−mla cos^{2}(x_{1})} − \frac{mal cos(x_{1})u}{4ml^{2}/3−m^{2}l^{2}a cos^{2}(x_{1})}. \end{matrix} \}  (A.31)

We next substitute θ̈, obtained from (A.28), into (A.29) to get

ẍ = \frac{−mag sin(2x_{1})/2 + ax_{2}^{2} sin(x_{1})4ml/3 + (u − f_{c})4a/3}{4/3 − ma cos^{2}(x_{1})}.  (A.32)

Let x_{3} = x and x_{4} = ẋ. Then, combining (A.31) and (A.32) we obtain a state space model of the inverted pendulum on a cart of the form,

[ ẋ_{1} ; ẋ_{2} ; ẋ_{3} ; ẋ_{4} ] = [ x_{2} \frac{g sin(x_{1})−mlax_{2}^{2} sin(2x_{1})/2}{4l/3−mla cos^{2}(x_{1})} x_{4} \frac{−mag sin(2x_{1})/2+alx_{2}^{2} sin(x_{1})4m/3}{4/3−ma cos^{2}(x_{1})} ] + [ 0 \frac{−a cos(x_{1})}{4l/3−mla cos^{2}(x_{1})} 0 \frac{4a/3}{4/3−ma cos^{2}(x_{1})} ] (u − f_{c}).  (A.33)

While designing a controller for the stick balancer system, we will use an equivalent representation of (A.33),

[ ẋ_{1} ; ẋ_{2} ; ẋ_{3} ; ẋ_{4} ] = [ x_{2} \frac{g sin(x_{1})}{4l/3−mla cos^{2}(x_{1})} x_{4} \frac{−mag sin(2x_{1})/2}{4/3−ma cos^{2}(x_{1})} ] + [ 0 \frac{−a cos(x_{1})}{4l/3−mla cos^{2}(x_{1})} 0 \frac{4a/3}{4/3−ma cos^{2}(x_{1})} ] (u − f_{c} + mlx_{2}^{2} sin(x_{1})).  (A.34)

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