ECE 602 Fall 2010

Constructing Robust Controllers Using Lyapunov’s

Theorem

by

Stanislaw H. Żak

October 20, 2010

One of the applications of the Lyapunov theorem is to construct robust state-feedback controllers. A basic issue in the control of dynamic systems is the effect of uncertainties, or neglected nonlinearities, on the stability of the closed-loop system.

A controller is viewed robust if it maintains stability for all uncertainties in an expected range. We consider a class of dynamic systems modeled by

ẋ(t) = Ax(t) + Bu(t) + f(t, x(t)) + Bh(t, x(t), u(t)),  (1)

where x(t) ∈ ℝ^{n}, u(t) ∈ ℝ^{m}, and the functions h and f model uncertainties, or nonlinearities, in the system. We refer to h as the matched uncertainty because it affects the system dynamics via the input matrix B in the same fashion as the input u does. In other words, the uncertainty h matches the system input u. The vector f models the unmatched uncertainty. We assume that the uncertain elements h and f satisfy the following norm bounds:

1. ∥h(t, x, u)∥ ≤ γ_{h}∥u∥ + α_{h}∥x∥,

2. ∥f(t, x)∥ ≤ α_{f}∥x∥,

where γ_{h}, α_{h}, and α_{f} are known nonnegative constants. We further assume that the matrix A is asymptotically stable. If this is not the case, we apply a preliminary state-feedback controller u = −Kx such that A − BK is asymptotically stable. Such a feedback exists provided that the pair (A, B) is stabilizable. Our goal is to construct a linear state-feedback controller that would make the closed-loop system asymptotically stable for arbitrary f and h that satisfy the above norm bounds.

Theorem 1 [1, p. 168] Suppose that A is asymptotically stable, and P = P^{⊤} > 0 is the solution to the Lyapunov matrix equation A^{⊤}P + P A = −2Q for some Q = Q^{⊤} > 0. Suppose also that

α_{f} < \frac{λ_{min}(Q)}{λ_{max}(P)} and γ_{h} < 1.

Then, the state-feedback controller u = −γB^{⊤}P x, where

γ > \frac{α_{h}^{2}}{4 (λ_{min}(Q) − α_{f}λ_{max}(P)) (1 − γ_{h})}

stabilizes the uncertain system model (1) for arbitrary f and h that satisfy the norm bounds.

Proof The time derivative of the positive definite function V = x^{⊤}P x evaluated on any trajectory of the closed-loop system is

V̇ = 2x^{⊤}P ẋ
= −2x^{⊤}Qx − 2γx^{⊤}P BB^{⊤}P x + 2x^{⊤}P f + 2x^{⊤}P Bh.

Our goal is to determine γ̃ > 0 such that if γ > γ̃, then V̇ < 0, which in turn implies the asymptotic stability of the closed-loop system. To proceed further, recall that if Q is a symmetric matrix, then

λ_{min}(Q)∥x∥^{2} ≤ x^{⊤}Qx ≤ λ_{max}(Q)∥x∥^{2},

and therefore

−x^{⊤}Qx ≤ −λ_{min}(Q)∥x∥^{2}.

Furthermore,

x^{⊤}P BB^{⊤}P x = x^{⊤}P B (x^{⊤}P B)^{⊤} = ∥x^{⊤}P B∥^{2}.

For a symmetric positive definite matrix P, its induced 2-norm is

∥P ∥ = λ_{max}(P).

Taking the above relations into account, using the norm bounds on the uncertain elements, and performing some manipulations yields

V̇ ≤ −2λ_{min}(Q)∥x∥^{2} − 2γ∥x^{⊤}P B∥^{2} + 2α_{f}λ_{max}(P)∥x∥^{2}
+2∥x^{⊤}P B∥ (α_{h}∥x∥ + γ_{h}γ∥x^{⊤}P B∥) .  (2)

Let

Q̃ = [ \begin{matrix}λ_{min}(Q) − α_{f}λ_{max}(P) & −\frac{α_{h}}{2} \\ −\frac{α_{h}}{2} & γ(1 − γ_{h}) \end{matrix} ] ∈ ℝ^{2×2}.  (3)

Then, we can represent (2) as

V̇ ≤ −2 [ ∥x∥ ∥x^{⊤}P B∥ ] Q̃ [ \begin{matrix}∥x∥ \\ ∥x^{⊤}P B∥ \end{matrix} ] .  (4)

For V̇ to be negative it is enough that the leading principal minors of the matrix Q̃ given by (3) be positive. The first-order leading principal minor is positive by assumption. For the second-order leading principal minor to be positive it is sufficient that γ_{h} < 1 and

γ > \frac{α_{h}^{2}}{4 (λ_{min}(Q) − α_{f}λ_{max}(P)) (1 − γ_{h})}.

The proof is now complete.

□

Note that if γ_{h} = 0, then the closed-loop system is asymptotically stable if

α_{f} < \frac{λ_{min}(Q)}{λ_{max}(P)} and γ > \frac{α_{h}^{2}}{4 (λ_{min}(Q) − α_{f}λ_{max}(P))}.

Example 1 Consider a dynamic system model

ẋ = Ax + b(u + h(x, u))
= [ \begin{matrix}−2 & 0 \\ 0 & −1 \end{matrix} ] x + [ \begin{matrix}1 \\ 1 \end{matrix} ] (u + h(x, u)).

Let P be the solution of the matrix Lyapunov equation A^{⊤}P +P A = −2I_{2}. We will design a linear state-feedback controller u = −γb^{⊤}P x so that the origin of the closed-loop system is uniformly asymptotically stable for any h(x, u) such that

|h(x, u)| ≤ 4∥x∥ + \frac{1}{2}|u|.  (5)

In particular, we will find a bound γ̃ > 0 so that for any γ > γ̃ the closed-loop system is uniformly asymptotically stable in the face of uncertainties satisfying the norm bound (5).

Solving the Lyapunov equation A^{⊤}P + P A = −2I_{2} yields

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

Hence,

u = −γb^{⊤}P x = −γ (\frac{1}{2}x_{1} + x_{2}) .

We obtain a bound on γ by evaluating \frac{d}{dt} (x^{⊤}P x) on the trajectories of the closed-loop system. First, we evaluate

\frac{d}{dt} (x^{⊤}P x) = 2x^{⊤}P ẋ
= 2x^{⊤}P (Ax + b(u + h(x, u)))
= −2x^{⊤}x − 2γx^{⊤}P bb^{⊤}P x + 2x^{⊤}P bh.

Taking the norms and performing manipulations yields

\frac{d}{dt} (x^{⊤}P x) ≤ −2x^{⊤}x − 2γ |x^{⊤}P b|^{2} + 2 |x^{⊤}P b| (4∥x∥ + \frac{1}{2}γ |x^{⊤}P b|)
= −2∥x∥^{2} − γ |x^{⊤}P b|^{2} + 8 |x^{⊤}P b| ∥x∥
= −2 [ ∥x∥ |x^{⊤}P b| ] [ \begin{matrix}1 & −2 \\ −2 & γ/2 \end{matrix} ] [ \begin{matrix}∥x∥ \\ | & x & ⊤ & P & b & | \end{matrix} ] .

For \frac{d}{dt} (x^{⊤}P x) to be negative definite it is enough that the matrix

[ \begin{matrix}1 & −2 \\ −2 & γ/2 \end{matrix} ]

be positive definite. This is the case whenever

γ > γ̃ = 8.

References

[1] S. H. Żak. Systems and Control. Oxford University Press, New York, 2003.