\chapter{Dynamical System Control}
\section{What Is a System?}
\label{sec:whatisasys}
A {\em 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:
\begin{enumerate}
\item interrelations between the components that are contained within the system, and
\item the system boundaries that separate the components within the system from the components outside.
\end{enumerate}
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 {\em 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 {\em inputs} to the system. The condition or the state of the system are described by the {\em 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 {\em 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 measured. The system quantities whose behavior can be measured or observed are referred to as the system {\em outputs}. In Figure~\ref{fig:system}, we schematically represent system's boundaries using a box. The input signal is processed by the system to provide an output signal variable.
\begin{figure}
\setlength{\unitlength}{.1in}
\begin{center}
\begin{picture}(60,20)
\thicklines
\put(14,9){\vector(1,0){8}}
\put(16,10){Input}
\put(22,5){\framebox(10,8){System}}
\put(32,9){\vector(1,0){9}}
\put(33,10){Output}
\end{picture}
\end{center}
\caption{Schematic representation of a system.}
\label{fig:system}
\end{figure} 
  
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 {\em controlling} the system states or outputs. This is accomplished by means of a {\em controller\/} whose task is to produce the required system's inputs. An interconnection of the system and a controller is called a {\em control system}. Constructing a controller is a part of the {\em control problem}. The essential elements of the control problem, as described, for example, by Owens~\cite[page~181]{owens81}, are:
\begin{enumerate}
\item a specified objective for the system,
\item a model of a dynamical system to be controlled,
\item a set of admissible controllers, and
\item a means of measuring the performance of any given control strategy to evaluate its effectiveness.
\end{enumerate}
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 {\em truth model\/} of the dynamics of the process to be controlled. The truth model is a {\em 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~\cite{friedland96} as the {\em 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:
\begin{eqnarray*}
\dot{\mb{\bm$x$}}(t) &=& \mb{\bm$f$}(t,\mb{\bm$x$}(t),\mb{\bm$u$}(t)),\quad \mb{\bm$x$}(t_0)=\mb{\bm$x$}_0\\
\mb{\bm$y$}(t) &=& \mb{\bm$h$}(t,\mb{\bm$x$}(t),\mb{\bm$u$}(t)),
\end{eqnarray*}
where the state $\mb{\bm$x$}\in \mb{\Bb R}^n$, the input $\mb{\bm$u$}\in \mb{\Bb R}^m$, the output $\mb{\bm$y$}\in \mb{\Bb R}^p$, $\mb{\bm$f$}$ 
and $\mb{\bm$h$}$ are vector valued functions with $\mb{\bm$f$}:\mb{\Bb R} \times \mb{\Bb R}^n \times 
\mb{\Bb R}^m \rightarrow \mb{\Bb R}^n$ and $\mb{\bm$h$}:\mb{\Bb R} \times \mb{\Bb R}^n \times \mb{\Bb R}^m \rightarrow \mb{\Bb R}^p$. Another common model of a dynamical system is the finite set of difference equations,
\begin{eqnarray*}
\mb{\bm$x$}(k+1) &=& \b f(k,\b x(k),\b u(k)), \quad \mb{\bm$x$}(k_0)=\mb{\bm$x$}_0\\
{\b y}(k) &=& {\b h}(k,{\b x}(k),{\b u}(k)),
\end{eqnarray*}
where $\mb{\bm$x$}(k)=\mb{\bm$x$}(kh)$, $\mb{\bm$u$}(k)=\mb{\bm$u$}(kh)$, $h$ is the sampling interval, and $k\ge 0$ 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 $(\mb{\bm$f$}(t,\mb{\bm$x$},\mb{\bm$u$}), \mb{\bm$h$}(t,\mb{\bm$x$},\mb{\bm$u$}))$ has little engineering significance until constraints on the state $\mb{\bm$x$}$, the control $\mb{\bm$u$}$, and the output $\mb{\bm$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.

\section{Open-Loop Versus Closed-Loop}
We distinguish between two types of control systems. They are:
\begin{itemize}
\item open-loop control systems, and
\item closed-loop control systems.
\end{itemize}
An open-loop control system usually contains:
\begin{enumerate}
\item A process to be controlled, labeled {\em plant},
\item The controlling variable of the plant, called the {\em plant input}, or just {\em input} for short,
\item The controlled variable of the plant, called the {\em plant output}, or just {\em output} for short,
\item A {\em reference input}, which dictates the desired value of the output,
\item A {\em 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.
\end{enumerate}
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~\ref{op-loop}.
\begin{figure}
\setlength{\unitlength}{.08in}
\begin{center}
\begin{picture}(60,20)
\thicklines
\put(15,5){\framebox(10,8){Controller}}
\put(5,9){\vector(1,0){10}}
\put(5,10){\shortstack{Reference\\input}}
\put(6,5){\shortstack{Desired\\output}}
\put(25,9){\vector(1,0){12}}
\put(26,10){\shortstack{Controller\\output}}
\put(28,5){\shortstack{Plant\\input}}
\put(37,5){\framebox(10,8){Plant}}
\put(47,9){\vector(1,0){5}}
\put(48,10){\shortstack{Actual\\output}}
\end{picture}
\end{center}
\caption{Schematic representation of an open-loop control system.}
\label{op-loop}
\end{figure}
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, 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 open-loop 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:
\begin{enumerate}
\setcounter{enumi}{5}
\item The {\em feedback loop} where the output signal is measured with a sensor and then the measured signal is fed back to the {\em summing junction},
\item The summing junction, where the measured output signal is subtracted from the reference, command, input signal in order to generate an {\em error signal}, also labeled as an {\em actuating signal}.
\end{enumerate}
\begin{figure}
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\begin{center}
\begin{picture}(70,25)
\thicklines
\put(0,18){\vector(1,0){11}}
\put(0,19){\shortstack{Desired\\output}}
\put(0,14){\shortstack{Reference\\input}}
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\put(14,18){\line(1,1){2.12}}
\put(14,18){\line(-1,-1){2.12}}
\put(14,18){\line(-1,1){2.12}}
\put(14,18){\line(1,-1){2.12}}
\put(11,17){$+$}
\put(13,16){$-$}
\put(17,18){\vector(1,0){11}}
\put(17,19){\shortstack{Actuating\\signal}}
\put(28,14){\framebox(10,8){Controller}}
\put(38,18){\vector(1,0){12}}
\put(39,19){\shortstack{Controller\\output}}
\put(41,14){\shortstack{Plant\\input}}
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\put(60,18){\vector(1,0){10}}
\put(61,19){\shortstack{Actual\\output}}
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\put(24,7){\shortstack{Measured\\output}}
\put(14,6){\vector(0,1){9}}
\end{picture}
\end{center}
\caption{Schematic representation of a closed-loop control system.}
\label{cl_loop}
\end{figure}
A schematic of a closed-loop control system is shown in Figure~\ref{cl_loop}. 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 {\em compensator}, can be placed either in the {\em forward loop}, or in the {\em feedback loop\/}.

If the goal of the controller is to maintain the output at a constant value, then we have a {\em 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~\ref{fig:watts}.
\begin{figure}
\vspace{4.5in}
\special{bmp:c:/mybook/watts.bmp x=6in y=4.5in}
\caption{A simplified schematic of the flyball governor.}
\label{fig:watts}
\end{figure}
The task of centrifugal speed regulator 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~\ref{cl_loop}. 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 \cite{macfarlane79}. 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~\cite{macfarlane79}. 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~\ref{sec:watt_stability} 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 {\em servomechanism\/} or {\em 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 {\em 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~\ref{subsec:fuzzymodeling}.

\section{Mathematical Modeling}
The goal of mathematical modeling is to provide a mathematical description of the inter-relations 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 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~\ref{fig:mat_mod}.
\begin{figure}
\vspace{3.5in}
\hskip1.5in\special{bmp:c:/mybook/mat_mod.bmp x=4.6in y=3.5in}
\caption{Schematic diagram of mathematical modeling process.}
\label{fig:mat_mod}
\end{figure}
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~\cite[pages 213--220]{pontryagin62}. For an alternative approach to modeling the governor see Maxwell~\cite{maxwell68} and Fuller~\cite{fuller96}. Some, like for example MacFarlane~\cite[page~250]{macfarlane79}, claim 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~\cite{fuller96} does a wonderful job elucidating, correcting, and expanding Maxwell's paper. Mayr~\cite[Chapter~X]{mayr70} 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~\ref{fig:watts}, while forces acting on the governor are shown in Figure~\ref{fig_watt_s}.
\begin{figure}
\vspace{4.5in}
\special{bmp:c:/mybook/watt_s.bmp x=6in y=4.5in}
\caption{Forces acting on the weights in the governor.}
\label{fig_watt_s}
\end{figure}
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 $\omega_{gov}$, then the arms form an angle $\varphi$ 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, $v_{weight}=\omega_{gov}r$. Note that $r=\sin \varphi$. The resulting centrifugal force acting on each weight is
\begin{eqnarray*}
m\frac{v_{weight}^2}{r} &=& m\omega_{gov}^2r\\
&=& m\omega_{gov}^2\sin \varphi.
\end{eqnarray*}
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 $\dot{\varphi}$ with the friction coefficient $b$. Summing forces along the axis perpendicular to an arm yields an equation of motion of the weight
\begin{eqnarray}
m\ddot{\varphi}L &=& m\omega_{gov}^2\sin\varphi\cos\varphi - mg\sin\varphi - b\dot{\varphi}\nonumber\\
&=& \frac{1}{2}m\omega_{gov}^2\sin 2\varphi - mg\sin\varphi - b\dot{\varphi}.
\label{eq:eformoofwe}
\end{eqnarray}
Note that $m\ddot{\varphi}L=m\ddot{\varphi}$ since by assumption $L=1$. The governor is connected to the engine via a gear with the gear ratio $N$ such that
\begin{equation}
\omega_{gov}=N\omega,
\label{eq:gearratengo}
\end{equation}
where $\omega$ is the angular velocity of the flywheel. Substituting~(\ref{eq:gearratengo}) into~(\ref{eq:eformoofwe}) gives
\[
m\ddot{\varphi}=\frac{1}{2}mN^2\omega^2\sin 2\varphi - mg\sin\varphi - b\dot{\varphi}.
\]
Let $I$ be the moment of inertia of the flywheel, $\tau_1$ the torque delivered by the engine, and $\tau_l$ the load torque. Then, the equation modeling the engine is
\begin{equation}
I\dot{\omega}=\tau_1-\tau_l.
\label{eq:emotheeng}
\end{equation}
The collar of the governor is connected with the valve so that at the nominal value of $\varphi=\varphi_0$ the engine delivers the nominal torque $\tau_d$ that results in the desired angular velocity $\omega=\omega_d$ of the flywheel. We model the interaction of the collar and the valve using the equation 
\begin{equation}
\tau_1=\tau_d+\kappa(\cos\varphi-\cos\varphi_0),
\label{eq:collavav}
\end{equation}
where $\kappa>0$ is a proportionality constant that depends on the governor. Substituting~(\ref{eq:collavav}) into~(\ref{eq:emotheeng}) yields
\begin{eqnarray*}
I\dot{\omega} &=& \tau_1-\tau_l\\
&=&\tau_d+\kappa(\cos\varphi-\cos\varphi_0)-\tau_l\\
&=& \kappa\cos\varphi-\tau,
\end{eqnarray*}
where $\tau=\tau_l-\tau_d+\kappa \cos\varphi_0$. We thus have two equations that model the governor. They are:
\begin{eqnarray*}
m\ddot{\varphi} &=& \frac{1}{2}mN^2\omega^2\sin 2\varphi - mg\sin\varphi - b\dot{\varphi},\\
I\dot{\omega} &=& \kappa\cos\varphi-\tau.
\end{eqnarray*}
We use the state variables
\[
x_1=\varphi,\quad x_2=\dot{\varphi},\quad\mb{and}\quad x_3=\omega
\]
to represent the above model in state space format,
\begin{equation}
\left.\begin{array}{ccl}
\dot{x}_1 &=& x_2\\
\dot{x}_2 &=& \frac{1}{2}N^2x_3^2\sin 2x_1-g\sin x_1 - \frac{b}{m}x_2\\
\dot{x}_3 &=& \frac{\kappa}{I}\cos x_1-\frac{\tau}{I}.
\end{array}\right\}
\label{eq:watt_model}
\end{equation}




\section{Centrifugal Governor Stability Analysis}

We first linearize the model of Watt's governor that we derived in Appendix~\ref{apppend:watt}. Recall that the equations modeling the governor have the form:
\begin{eqnarray*}
\dot{x}_1 &=& x_2\\
\dot{x}_2 &=& \frac{1}{2}N^2x_3^2\sin 2x_1-g\sin x_1 - \frac{b}{m}x_2\\
\dot{x}_3 &=& \frac{\kappa}{I}\cos x_1-\frac{\tau}{I}.
\end{eqnarray*}
We assume that the equilibrium state has the form
\[
\b x_e=\left[\begin{array}{ccc}
x_{1e}&0&x_{3e}
\end{array}\right]^T.
\]
Equating the right-hand sides of the engine-governor equations to zero yields
\begin{eqnarray*}
x_{2e} &=& 0\\
N^2\,x_{3e}^2 &=& \frac{g}{\cos x_{1e}}\\
\cos x_{1e} &=& \frac{\tau}{\kappa}.
\end{eqnarray*}
The Jacobian matrix of the engine-governor model evaluated at the equilibrium state is
\[
\left.\frac{\partial \b f}{\partial \b x}\right\arrowvert_{\b x=\b x_e}=\left[\begin{array}{ccc}
0 & 1 & 0\\
N^2x_{3e}^2\cos 2x_{1e}-g\cos x_{1e} & -\frac{b}{m} & N^2x_{3e}\sin 2x_{1e}\\
-\frac{\kappa}{I}\sin x_{1e} & 0 & 0
\end{array}\right].
\]
We use the trigonometric identities
\[
\cos 2\alpha=\cos ^2\alpha - \sin^2 \alpha,\quad \sin 2\alpha=2\sin \alpha \,\cos \alpha,
\]
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
\[
\left.\frac{\partial \b f}{\partial \b x}\right\arrowvert_{\b x=\b x_e}=\left[\begin{array}{ccc}
0 & 1 & 0\\
-\frac{g \sin^2 x_{1e}}{\cos x_{1e}} & -\frac{b}{m} & \frac{2g\sin\,x_{1e}}{x_{3e}}\\
-\frac{\kappa}{I}\sin x_{1e} & 0 & 0
\end{array}\right].
\]
Hence, the linearized model of the engine-governor system can be represented as
\begin{equation}
\left[\begin{array}{c}
\Delta \dot{x}_1\\
\Delta \dot{x}_2\\
\Delta \dot{x}_3
\end{array}\right]=\left[\begin{array}{ccc}
0 & 1 & 0\\
-\frac{g \sin^2 x_{1e}}{\cos x_{1e}} & -\frac{b}{m} & \frac{2g\sin\,x_{1e}}{x_{3e}}\\
-\frac{\kappa}{I}\sin x_{1e} & 0 & 0
\end{array}\right]\left[\begin{array}{c}
\Delta {x_1}\\
\Delta {x_2}\\
\Delta {x_3}
\end{array}\right].
\label{eq:linmoofwgo}
\end{equation}

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~(\ref{eq:linmoofwgo}). The model's characteristic equation is
\begin{eqnarray}
\det\left[\begin{array}{ccc}
s & -1 & 0\\
\frac{g \sin^2 x_{1e}}{\cos x_{1e}} & s+\frac{b}{m} & -\frac{2g\sin\,x_{1e}}{x_{3e}}\\
\frac{\kappa}{I}\sin x_{1e} & 0 & s
\end{array}\right] &=& s^3+\frac{b}{m}s^2+\frac{g\sin^2 x_{1e}}{\cos x_{1e}}s+\frac{2g\kappa \sin^2 x_{1e}}{I x_{3e}}\nonumber\\
&=&s^3+a_2s^2 + a_1 s+a_0.
\label{eq:chareqofwg}
\end{eqnarray}
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~(\ref{eq:chareqofwg}) to get,
\begin{equation}
\begin{array}{c|cc}
s^3& 1& a_1\\
s^2 &a_2 & a_0\\
s^1&\frac{1}{a_2}\left( a_2 a_1-a_0\right)\\
s^0& a_0
\end{array}
\label{eq:rharfwg}
\end{equation}
Since the characteristic polynomial coefficients are all positive, the linearized model is asymptotically if and only if
\begin{equation}
a_2 a_1-a_0=\frac{b}{m}\frac{g\sin^2 x_{1e}}{\cos x_{1e}}-\frac{2g\kappa \sin^2 x_{1e}}{I x_{3e}}>0.
\label{eq:stacofowg}
\end{equation}
Let
\[
\nu=\frac{x_{3e}}{2\kappa \cos x_{1e}}.
\]
Using the above notation, we express~(\ref{eq:stacofowg}) as
\begin{equation}
\frac{b I}{m}\nu>1.
\label{eq:stcoforwnf}
\end{equation}
It follows from condition~(\ref{eq:stcoforwnf}) that
\begin{enumerate}
\item an increase of the mass $m$ of the weights of the governor has a deteriorating effect on the governor's stability;
\item a decrease of the friction coefficient $b$ has also a harmful effect on stability;
\item a decrease of the moment of inertia $I$ of the flywheel harms stability;
\item a decrease of the coefficient $\nu$ harms stability. 
\end{enumerate}
The above deductions are attributed by Pontryagin~\cite[page~219]{pontryagin62} to the Russian engineer Vyshnegradskiy. Vyshnegradskiy used these observations to propose, in~1876, design rules for governors to regulate engines without engine speeds fluctuating.


\section{Robust Controller Design}
\label{sec:discconuncs}
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
\begin{equation}
\dot{\mb{\bm$x$}}(t)=\mb{\bm$f$}(t,\mb{\bm$x$}(t))+\mb{\bm$G$}(t,\mb{\bm$x$}(t))\left(\mb{\bm$u$}(t)+\mb{\bm$\xi$}(t,\mb{\bm$x$}(t))\right),\qquad\mb{\bm$x$}(t_0)=\mb{\bm$x$}_0,
\label{eq:uncertmode}
\end{equation}
where $\mb{\bm$x$}(t)\in \mb{\Bb R}^n$, $\mb{\bm$u$}(t)\in \mb{\Bb R}^m$, and $\mb{\bm$\xi$}(t,\mb{\bm$x$}(t))\in \mb{\Bb R}^m$ represents the lumped uncertainties. We assume that the uncertain element $\mb{\bm$\xi$}$ is bounded by a known continuous non-negative function $\rho$, that is,
\[
\|\mb{\bm$\xi$}(t,\mb{\bm$x$}(t))\| \le \rho(t,\mb{\bm$x$}(t)).
\]
We also assume that the equilibrium state $\mb{\bm$x$}=\mb{\bm$0$}$ of the uncontrolled nominal system 
\begin{equation}
\dot{\mb{\bm$x$}}(t)=\mb{\bm$f$}(t,\mb{\bm$x$}(t))
\label{eq:nonunconguas}
\end{equation}
is globally uniformly asymptotically stable, and that we were able to construct a Lyapunov function $W:\mb{\Bb R} \times \mb{\Bb R}^n \to \mb{\Bb R}$ corresponding to this equilibrium state. The Lyapunov function $W$ is assumed to satisfy the following conditions: there exist three continuous and strictly increasing functions $\psi_i$, $i=1,2,3$, such that
\begin{equation}
\left.\begin{array}{rcl}
\psi_i(0) &=& 0,\quad i=1,2,3,\\
\lim_{r \to \infty} \psi_1(r)&=& \infty,\\
\psi_1(\|\mb{\bm$x$}\|) &\le& W(t,\mb{\bm$x$}) \le \psi_2(\|\mb{\bm$x$}\|),\\
\psi_3\left(\| \mb{\bm$x$}\| \right) &\le& - \dot{W}(t,\mb{\bm$x$})
\end{array}\right\}
\label{eq:thconspsis}
\end{equation}
for all $t$ and all $\mb{\bm$x$}$. We will show that if we apply the control strategy
\begin{equation}
\mb{\bm$u$}(t,\mb{\bm$x$})=\left\{\begin{array}{ccc}
-\frac{\mb{\bm$\mu$}(t,\mb{\bm$x$})}{\|\mb{\bm$\mu$}(t,\mb{\bm$x$})\|} \rho(t,\mb{\bm$x$})&\mb{if}& (t,\mb{\bm$x$}) \notin \Omega\\
\in \left\{\mb{\bm$p$}\in \mb{\Bb R}^m : \|\mb{\bm$p$}\| \le \rho(t,\mb{\bm$x$})\right\} &\mb{if}& (t,\mb{\bm$x$})\in \Omega,
\end{array} \right.
\label{eq:minmaxcs}
\end{equation}
where $\mb{\bm$\mu$}(t,\mb{\bm$x$})=\mb{\bm$G$}^T(t,\mb{\bm$x$}) \nabla_{\mb{\bm$x$}}W(t,\mb{\bm$x$})$, and $\Omega=\left\{(t,\mb{\bm$x$}):\mb{\bm$\mu$}=\mb{\bm$0$}\right\}$, then the equilibrium state $\mb{\bm$x$}=\mb{\bm$0$}$ 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
\begin{eqnarray}
\dot{W}(t,\mb{\bm$x$}) &=& \frac{\partial W}{\partial t}+\nabla_{\mb{\bm$x$}}^T W \left( \mb{\bm$f$}(t,\mb{\bm$x$})+\mb{\bm$G$}(t,\mb{\bm$x$})\left(\mb{\bm$u$}+\mb{\bm$\xi$}(t,\mb{\bm$x$})\right)\right)\nonumber\\
&\le& -\psi_3(\| \mb{\bm$x$} \|)+ \nabla_{\mb{\bm$x$}}^T W \mb{\bm$G$}(t,\mb{\bm$x$}) \mb{\bm$u$} + \nabla_{\mb{\bm$x$}}^T W \mb{\bm$G$}(t,\mb{\bm$x$})\mb{\bm$\xi$}(t,\mb{\bm$x$})\nonumber\\
&\le& -\psi_3(\| \mb{\bm$x$} \|)+ \b \mu^T(t,\mb{\bm$x$}) \mb{\bm$u$} + \| \mb{\bm$\mu$}(t,\mb{\bm$x$})\|\rho(t,\mb{\bm$x$}).
\label{eq:lyapderfmimx}
\end{eqnarray}
For $(t,\b x)\notin \Omega$,
\begin{eqnarray}
\dot{W}(t,\mb{\bm$x$}) &\le& -\psi_3(\| \mb{\bm$x$} \|)- \b \mu^T(t,\b x) \frac{\mb{\bm$\mu$}(t,\mb{\bm$x$})}{\|\mb{\bm$\mu$}(t,\mb{\bm$x$})\|} \rho(t,\mb{\bm$x$}) + \| \mb{\bm$\mu$}(t,\mb{\bm$x$})\|\rho(t,\mb{\bm$x$})\nonumber\\
&=& -\psi_3(\| \mb{\bm$x$} \|)-\frac{\|\mb{\bm$\mu$}(t,\mb{\bm$x$})\|^2}{\|\mb{\bm$\mu$}(t,\mb{\bm$x$})\|} \rho(t,\mb{\bm$x$}) + \| \mb{\bm$\mu$}(t,\mb{\bm$x$})\|\rho(t,\mb{\bm$x$})\nonumber\\
&=& -\psi_3(\| \mb{\bm$x$} \|)- \| \mb{\bm$\mu$}(t,\mb{\bm$x$})\|\rho(t,\mb{\bm$x$})+\| \mb{\bm$\mu$}(t,\mb{\bm$x$})\|\rho(t,\mb{\bm$x$})\nonumber\\
&=& -\psi_3(\| \mb{\bm$x$} \|)\nonumber\\
&<&0.
\label{eq:noinommimx}
\end{eqnarray}
For $(t,\b x)\in \Omega$, we have $\b \mu(t,\b x)=\b 0$, and using~(\ref{eq:lyapderfmimx}), we obtain
\begin{eqnarray}
\dot{W}(t,\mb{\bm$x$}) &\le& -\psi_3(\| \mb{\bm$x$} \|)\nonumber\\
&<& 0.
\label{eq:inmumimxc}
\end{eqnarray}
Thus, $W$ is also a Lyapunov function for the closed-loop system, which means that the uncertain system~(\ref{eq:uncertmode}) driven by the control law~(\ref{eq:minmaxcs}) is indeed globally uniformly asymptotically stable.
\begin{example}\rm
For the uncertain system model
\[
\dot{\b x}=\b A\b x+\b \xi_1(\b x)+\b b\left(u+\xi_2(\b x)\right),
\]
where
\[
\b A=\left[\begin{array}{cc}
-2 & 1\\
0 & -3
\end{array}\right],\quad \b b=\left[\begin{array}{c}
0\\
1
\end{array}\right],\quad \left\Arrowvert \b \xi_1(\b x) \right\Arrowvert \le 1.75 \|\b x\|,\quad\mb{and}\quad |\xi_2|\le 10\|\b x\|+0.5|u|,
\]
design stabilizing control strategy of the form~(\ref{eq:minmaxcs}).

The uncontrolled nominal system $\dot{\b x}=\b A\b x$ is asymptotically stable. Let
\[
\b Q=\left[\begin{array}{cc}
4 & -1\\
-1 & 6
\end{array}\right].
\]
The eigenvalues of $\b Q$ are $\lambda_{\min}(\b Q)=3.5858$ and $\lambda_{\max}(\b Q)=6.4142$. Solving the Lyapunov matrix equation $\b A^T\b P+\b P\b A=-\b Q$ yields
\[
\b P=\b I_2.
\]
The proposed controller has the form
\begin{equation}
u=-\rho(\b x)\frac{\mu(\b x)}{|\mu(\b x)|}=-\rho(\b x)\frac{\b b^T \b P\b x}{|\b b^T\b P\b x|}=-\rho(\b x)\frac{\b b^T \b x}{| \b b^T \b x|}=-\rho(\b x)\mb{sign}\left(\b b^T \b x\right).
\label{eq:conmimxex}
\end{equation}
Our goal is to determine a bound $\rho^*(\b x)>0$ such that for any $\rho(\b x)\ge\rho^*(\b x)$ the uncertain closed-loop system is globally uniformly asymptotically stable. To begin, consider
\[
V(\b x)=\b x^T\b P\b x=\b x^T\b x
\]
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
\begin{eqnarray}
\frac{d}{dt}V(\b x) &=& 2\b x^T\dot{\b x}\nonumber\\
&=&2\b x^T\left( \b A\b x+\b \xi_1(\b x)+\b b\left(u+\xi_2(\b x)\right)\right)\nonumber\\
&=&-\b x^T\b Q\b x+2\b x^T\b \xi_1(\b x)+2\b x^T\b b u(\b x)+2\b x^T\b b\xi_2(\b x).
\label{eq:lyapderforsime}
\end{eqnarray}
Substituting into the above the expression for the controller given by~(\ref{eq:conmimxex}), taking the norms, and using bounds on the uncertain elements, we obtain
\begin{eqnarray} 
\frac{d}{dt}V(\b x)&\le&-\lambda_{\min}(\b Q)\|\b x\|^2+2\|\b x\|\|\b \xi_1(\b x)\|-2\rho(\b x)\frac{\left\arrowvert\b x^T\b b\right\arrowvert^2}{\left\arrowvert\b x^T\b b\right\arrowvert} +2\left\arrowvert\b x^T\b b\right\arrowvert \left\arrowvert\xi_2(\b x)\right\arrowvert\nonumber\\
&\le&(-3.5858+3.5)\|\b x\|^2-2\rho(\b x)\left\arrowvert\b x^T\b b\right\arrowvert+2\left( 10\|\b x\|+0.5\rho(\b x)\right)\left\arrowvert\b x^T\b b\right\arrowvert .
\label{eq:afsubuantr}
\end{eqnarray}
In the above, we used the fact that
\[
|u|=\rho(\b x).
\]
Collecting the terms in~(\ref{eq:afsubuantr}) yields
\begin{equation}
\frac{d}{dt}V(\b x)\le(-3.5858+3.5)\|\b x\|^2+\left( 20\|\b x\|-\rho(\b x) \right)\left\arrowvert\b x^T\b b\right\arrowvert
\label{eq:aftcolltr}.
\end{equation}
Therefore, $\frac{d V}{dt}<0$ for any
\begin{equation}
\rho(\b x)\ge\rho^*(\b x)=20\|\b x\|.
\label{eq:rhgtrs}
\end{equation}
In summary, the uncertain closed-loop system will be globally uniformly asymptotically stable if the controller gain $\rho(\b x)$ is chosen to satisfy~(\ref{eq:rhgtrs}).
\end{example}\markex

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~\ref{fig:refm}.
\begin{figure}
\vspace{4in}
\hskip.5in\special{bmp:c:/mybook/refm.bmp x=5.33in y=4in}
\caption{Model-reference control system block diagram with input scaling.}
\label{fig:refm}
\end{figure}
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
\begin{equation}
\dot{\b x}=\b A\b x+\b B\b u,
\label{eq:liplamotc}
\end{equation}
where $\b A\in \mb{\Bb R}^{n\times n}$ and $\b B\in \mb{\Bb R}^{n\times m}$. The model-reference signal generator is described by
\begin{equation}
\dot{\b x}_d = \b A_d \b x_d + \b B_d \b r
\end{equation}
where $\b A_d \in \mb{\Bb R}^{n\times n}$, $\b B_d \in\mb{\Bb R}^{n\times m}$, and $\b r\in \mb{\Bb R}^m$ is a command signal that is chosen so that $\b x_d$ represents a desired state trajectory of the plant. The matrix $\b A_d$ of the reference system is chosen from the set
\begin{equation}
\left\{\b A_d:\b A_d=\b A-\b B\b K_d, \b K_d\in \mb{\Bb R}^{m\times n}\right\},
\end{equation}
and $\b B_d=\b B$. Thus, the reference system is state-feedback equivalent with the plant model~(\ref{eq:liplamotc}). We assume that the pair $(\b A, \b B)$ is controllable. Therefore, we can select $\b A_d$ to have its eigenvalues in prescribed locations in the open-left hand plane. To proceed further, we define the tracking error $\b e$ as 
\begin{equation}
\b e(t) = \b x(t) - \b x_d(t).
\end{equation}
Oup goal is to construct a state-feedback control law so that
\begin{equation}
\lim_{t \to \infty}\|\b e(t)\|=0.
\end{equation}
We now form the differential equation that describes the error dynamics. For this we evaluate the time derivative of the error vector to get
\begin{eqnarray}
\dot{\b e}(t) &=& \dot{\b x}(t) - \dot{\b x}_d(t)\nonumber\\
&=& \b A \b x(t) + \b B \b u - \b A_d \b x_d(t) - \b B \b r(t).
\label{eq:ermorefve}
\end{eqnarray}
Recall that $\b A_d = \b A - \b B \b K_d$, hence $\b A = \b A_d + \b B\b K_d$. Substituting this expression for $\b A$ into the error equation~(\ref{eq:ermorefve}), we obtain
\begin{eqnarray}
\dot{\b e}(t) &=& \b A_d(\b x(t) - \b x_d(t)) + \b B(\b u +\b K_d \b x(t) - \b r(t))\nonumber\\
&=& \b A_d \b e(t) + \b B(\b u+\b K_d\b x(t) - \b r(t)).
\label{eq:erreqftrc}
\end{eqnarray}
We will use a Lyapunov function approach in our further analysis that will result in a control law forcing the error $\b e(t)$ to go asymptotically to zero. As a Lyapunov function candidate for~(\ref{eq:erreqftrc}), we take
\begin{equation}
V(\b e)=\b e^T\b P\b e,
\end{equation}
where $\b P$ solves the Lyapunov equation
\begin{equation}
\b A_d^T\b P+\b P\b A_d=-\b Q
\label{eq:sotlaetco}
\end{equation}
for some $\b Q=\b Q^T>0$. Since by construction $\b A_d$ is asymptotically stable, the solution to the Lyapunov equation~(\ref{eq:sotlaetco}) is positive definite, that is, $\b P=\b P^T>0$. We next evaluate the time derivative of $V(\b e(t))$ on the trajectories of~(\ref{eq:erreqftrc}). We obtain
\begin{eqnarray}
\dot{V}(\b e) &=& 2 \b e^T \b P \dot{\b e}\nonumber\\
&=& 2 \b e^T\b P (\b A_d \b e + \b B(\b u + \b K_d \b x - \b r))\nonumber\\
&=& \b e^T(\b A_d^T\b P + \b P\b A_d) \b e +  2 \b e^T \b P \b B (\b u + \b K_d \b x - \b r).
\label{eq:evldtcomi}
\end{eqnarray} 
Using~(\ref{eq:sotlaetco}) and taking norms of the above yields
\begin{equation}
\dot{V}(\b e) \leq -\b e^T\b Q\b e + 2\b e^T \b P \b B\b u + 2\left\Arrowvert\b e^T \b P \b B\right\Arrowvert \left\Arrowvert\b K_d\b x\right\Arrowvert + 2\left\Arrowvert\b e^T\b P \b B\right\Arrowvert \|\b r\|.
\label{eq:doftrcmimx}
\end{equation}
If $\b e$ is such that $\b B^T \b P\b e=\b 0$, then
\begin{eqnarray}
\dot{V}(\b e) &\leq& -\b e^T\b Q\b e\nonumber\\
&<&0.
\label{eq:fctcmimx}
\end{eqnarray}
We now analyze the case when $\b B^T \b P\b e\ne\b 0$. Let the proposed control law be
\begin{equation}
\b u=-\rho(\b x,\b r)\frac{\b B^T\b P\b e}{\left\Arrowvert\b B^T\b P\b e\right\Arrowvert},
\label{eq:trcomimx}
\end{equation}
where $\rho(\b x,\b r)=\|\b K_d\b x\|+\|\b r\|$. Substituting~(\ref{eq:trcomimx}) into~(\ref{eq:doftrcmimx}), we obtain
\begin{eqnarray*}
\dot{V}(\b e) &\le& -\b e^T\b Q\b e -2\left\Arrowvert\b e^T\b P\b B\right\Arrowvert\left(\rho(\b x,r)-\left\Arrowvert\b K_d\b x\right\Arrowvert-\|\b r\|\right)\\
&=& -\b e^T\b Q\b e\\
&<& 0.
\end{eqnarray*}
Since for all $\b e \in \mb{\Bb R}^n$, we have $V(\b e)>0$ and $\dot{V}(\b e)<0$, the equilibrium point $\b e=\b 0$ of the error equation~(\ref{eq:erreqftrc}) is globally uniformly asymptotically stable, which means that $\b x(t)\to \b x_d(t)$.
\begin{example}\rm
\label{ex:ummxveh}
We now apply the above tracking controller to the vehicle model derived in the Subsection~\ref{subs:vehmod}, where $\delta_r=0$. 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:
\begin{equation} 
\{-4-2j,-4+2j,-4.5,-5.0\}.
\end{equation}
The pair $(\b A,\b b)$ is controllable, hence we can find $\b k_d$ such that the eigenvalues of the matrix $\b A_d = \b A - \b b \b k_d$ are at the above selected locations. The feedback gain is
\begin{equation} 
\b k_d =
\left[\begin{array}{cccc}
0.0329 &2.2854& 0.2429& -0.1653
\end{array} \right],
\end{equation}
and the resulting model reference plant matrix is
\begin{equation}
\b A_d =
\left[\begin{array}{cccc}
-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{array}
\right].
\end{equation}
The next step is to find $\b P$ that solves the Lyapunov equation, $\b A_d^T\b P + \b P\b A_d = -\b Q$. We take $\b Q=\b I_4$. Solving the Lyapunov equation, we get 
\begin{equation}
\b P = 
\left[\begin{array}{cccc}
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{array}\right].
\end{equation}
Now that we have $\b P$, we are ready to construct the controller. We have
\begin{eqnarray}
u &=& -\rho(\b x,r)\frac{\b b^T\b P\b e}{\left\arrowvert\b b^T\b P\b e\right\arrowvert}\nonumber\\
&=&-(|\b k_d\b x|+|r|)\mb{sign}(\b b^T\b P(\b x-\b x_d))\nonumber\\
&=&-\left(\left|\left[\begin{array}{c}
0.0329\\
2.2854\\
0.2429\\
-0.1653 
\end{array}\right]^T\b x \right| + |r| \right)\mbox{sign}\left(\left[\begin{array}{c}
0.1571\\
26.9931\\
6.3324\\
-3.0250\end{array}\right]^T(\b x-\b x_d)\right).
\label{eq:ummxvehtr}
\end{eqnarray}
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
\begin{equation}
\eta=\left( \lim_{s\to 0}\b c(s\b I_4-\b A_d)^{-1}\b b\right)^{-1},
\end{equation}
where $\b c=\left[\begin{array}{cccc}
0 & 0 & 0 & 1
\end{array}\right]$. Note that $\eta$ is well defined because the matrix $\b A_d$ is asymptotically stable. In our example, $\eta=-0.1653$. Thus, we have $r=\eta u_r$. The reference input is $u_r=4h(t-1\,\mb{sec})$, where $h(t)$ is the unit step function.

The performance of the controller, for zero initial conditions, is shown in Figures~\ref{fig:ummxlad} and~\ref{fig:ummxveh}. In Figure~\ref{fig:ummxlad}, 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~\ref{fig:ummxveh}. 
\begin{figure}
\vspace{3.5in}
\hskip1in\special{bmp:c:/mybook/ummxlad.bmp x=4.67in y=3.5in}
\caption{Plots of the actual output of the closed-loop system and the desired output.}
\label{fig:ummxlad}
\end{figure}
\begin{figure}
\vspace{3.5in}
\hskip1.1in\special{bmp:c:/mybook/ummxveh.bmp x=4.67in y=3.5in}
\caption{Plot of the controller action versus time.}
\label{fig:ummxveh}
\end{figure}
\end{example}\markex
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.

\section{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 {\em 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 closed-loop system driven by a boundary layer controller may only be {\em 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 {\em practical stability}. The notions we are concerned with in this section are spelled out in the following three definitions.
\begin{defn}
The closed ball $\bar{B}_q(\b 0)=\left\{ \mb{\bm$x$}\,:\, \|\mb{\bm$x$}\| \le q \right\}$ is uniformly stable if for any $\varepsilon > q$, there is a $\delta=\delta(\varepsilon) > 0$ so that if $\|\mb{\bm$x$}(t_0) \| \le \delta$, then $\|\mb{\bm$x$}(t)\| \le \varepsilon$ for $t \ge t_0$.
\end{defn}
For the benefit of the reader we now rephrase the definition of uniform boundedness from Section~\ref{ch:stability}.
\begin{defn}
The solutions of $\dot{\b x}=\b f(t, \b x)$ are uniformly bounded if for any given $d>0$, there exists $b=b(d)< \infty$ so that for each solution $\mb{\bm$x$}(t) = \mb{\bm$x$}(t;t_0,\mb{\bm$x$}_0)$ with $\|\mb{\bm$x$}_0\| \le d$, we have $\|\mb{\bm$x$}(t)\| \le b$ for all $t\ge t_0$.
\end{defn}
We next define the uniform ultimate boundedness of a system, or for short, practical stability.
\begin{defn}
The solutions of $\dot{\b x}=\b f(t, \b x)$ are uniformly ultimately bounded, or practically stable, with respect to the ball $\bar{B}_q(\b 0)$ if for every $d>0$, there is $T(q,d)$ such that for each solution $\mb{\bm$x$}(t)=\b x(t;t_0,\b x_0)$ with $\|\mb{\bm$x$}_0\| \le d$, we have $\mb{\bm$x$}(t) \in \bar{B}_q(\b 0)$ for $t \ge t_0+T(q,d)$.
\end{defn}

We now consider a dynamical system model~(\ref{eq:uncertmode}),
\[
\dot{\mb{\bm$x$}}(t)=\mb{\bm$f$}(t,\mb{\bm$x$}(t))+\mb{\bm$G$}(t,\mb{\bm$x$}(t))\left(\mb{\bm$u$}(t)+\mb{\bm$\xi$}(t,\mb{\bm$x$}(t))\right),\qquad\mb{\bm$x$}(t_0)=\mb{\bm$x$}_0,
\]
where $\mb{\bm$x$}(t)\in \mb{\Bb R}^n$, $\mb{\bm$u$}(t)\in \mb{\Bb R}^m$, and $\mb{\bm$\xi$}(t,\mb{\bm$x$}(t))\in \mb{\Bb R}^m$ represents the lumped uncertainties. We assume that the uncertain element $\mb{\bm$\xi$}$ is bounded by a known continuous non-negative function $\rho$, that is, $\|\mb{\bm$\xi$}(t,\mb{\bm$x$}(t))\| \le \rho(t,\mb{\bm$x$}(t))$. As in Section~\ref{sec:discconuncs}, we assume that the equilibrium state $\mb{\bm$x$}=\mb{\bm$0$}$ of the uncontrolled nominal system $\dot{\mb{\bm$x$}}(t)=\mb{\bm$f$}(t,\mb{\bm$x$}(t))$ is globally uniformly asymptotically stable, and that we were able to construct a Lyapunov function $W:\mb{\Bb R} \times \mb{\Bb R}^n \to \mb{\Bb R}$ satisfying the conditions~(\ref{eq:thconspsis}) for all $t$ and all $\mb{\bm$x$}$.

We showed in the previous section, that if we apply the discontinuous control strategy
\[
\mb{\bm$u$}(t,\mb{\bm$x$})=\left\{\begin{array}{ccc}
-\frac{\mb{\bm$\mu$}(t,\mb{\bm$x$})}{\|\mb{\bm$\mu$}(t,\mb{\bm$x$})\|} \rho(t,\mb{\bm$x$})&\mb{if}& (t,\mb{\bm$x$}) \notin \Omega\\
\in \left\{\mb{\bm$p$}\in \mb{\Bb R}^m : \|\mb{\bm$p$}\| \le \rho(t,\mb{\bm$x$})\right\} &\mb{if}& (t,\mb{\bm$x$})\in \Omega,
\end{array} \right.
\]
where $\mb{\bm$\mu$}(t,\mb{\bm$x$})=\mb{\bm$G$}^T(t,\mb{\bm$x$}) \nabla_{\mb{\bm$x$}}W(t,\mb{\bm$x$})$, and $\Omega=\left\{(t,\mb{\bm$x$}):\mb{\bm$\mu$}\rho(t,\mb{\bm$x$})=\mb{\bm$0$}\right\}$, then the equilibrium state $\mb{\bm$x$}=\mb{\bm$0$}$ is a globally uniformly asymptotically stable solution of the closed-loop system. However, the above controller is discontinuous in $\mb{\bm$x$}$. Following Corless and Leitmann~\cite{corless81}, we modify it to make it continuous by introducing a boundary layer. The proposed boundary layer controller is
\begin{equation}
\mb{\bm$u$}(t,\mb{\bm$x$})=\left\{\begin{array}{ccc}
-\frac{\mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})}{\| \mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})\|} \rho(t,\mb{\bm$x$}) &\mb{if}& \| \mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$}) \|> \nu\\
\|\mb{\bm$p$}(t,\mb{\bm$x$})\| \le \rho(t,\mb{\bm$x$}) &\mb{if}& \|\mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})\| \le \nu,
\end{array}\right.
\label{eq:boundlayerge}
\end{equation}
where $\nu$ is the thickness of the boundary layer about the naturally defined switching surface $\left\{(t,\mb{\bm$x$}): \| \mb{\bm$\mu$}(t,\mb{\bm$x$}) \rho(t,\mb{\bm$x$}) \|=0 \right\}$, and $\b p$ is a continuous function  such that
\[
\b p=-\frac{\mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})}{\| \mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})\|} \rho(t,\mb{\bm$x$}),
\]
when $\|\mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})\|=\nu$. A particular example of the above type of a controller is
\begin{equation}
\mb{\bm$u$}(t,\mb{\bm$x$})=\left\{\begin{array}{ccc}
-\frac{\mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})}{\| \mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})\|} \rho(t,\mb{\bm$x$}) &\mb{if}& \| \mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$}) \|> \nu\\
-\frac{\mb{\bm$\mu$}(t,\mb{\bm$x$})}{\nu}\rho^2(t,\mb{\bm$x$} ) &\mb{if}& \| \mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$}) \| \le \nu.
\end{array}\right.
\label{eq:bounlaspcs}
\end{equation}
We shall now show that the closed-loop system driven by the boundary layer controller~(\ref{eq:boundlayerge}) is practically stable with respect to a neighborhood of the zero state. This neighborhood can be made arbitrarily small by letting $\nu \to 0$. 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
\begin{eqnarray*}
\dot{W}(t,\mb{\bm$x$}) &=& \frac{\partial W}{\partial t}+\nabla_{\mb{\bm$x$}}^T W \left( \mb{\bm$f$}(t,\mb{\bm$x$})+\mb{\bm$G$}(t,\mb{\bm$x$})\left(\mb{\bm$u$}+\mb{\bm$\xi$}(t,\mb{\bm$x$})\right)\right)\\
&\le& -\psi_3(\| \mb{\bm$x$} \|)+ \nabla_{\mb{\bm$x$}}^T W \mb{\bm$G$}(t,\mb{\bm$x$}) \mb{\bm$u$} + \| \mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})\|.
\end{eqnarray*}
We now consider two cases. The first case is when $\|\mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})\|>\nu$. Taking into account the controller structure for this case yields
\begin{eqnarray*}
\dot{W}(t,\mb{\bm$x$}) &\le& -\psi_3(\|\mb{\bm$x$}\|) - \nabla_{\mb{\bm$x$}}^T W \mb{\bm$G$}(t,\mb{\bm$x$})\frac{\mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})}{\| \mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})\|} \rho(t,\mb{\bm$x$}) + \| \mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})\|\\
&=& -\psi_3(\|\mb{\bm$x$}\|) - \frac{\|\mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})\|^2}{\|\mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})\|} + \|\mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})\|\\
&=& -\psi_3(\|\mb{\bm$x$}\|).
\end{eqnarray*}
If, on the other hand, $\|\mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})\| \le \nu$, then using~(\ref{eq:boundlayerge}), we obtain
\begin{eqnarray*}
\dot{W}(t,\mb{\bm$x$}) &\le& -\psi_3(\|\mb{\bm$x$}\|) + \nabla_{\mb{\bm$x$}}^T W \mb{\bm$G$}(t,\mb{\bm$x$})\mb{\bm$u$} + \|\mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})\|\\
&\le& -\psi_3(\|\mb{\bm$x$}\|) + \|\nabla_{\mb{\bm$x$}}^T W \mb{\bm$G$}(t,\mb{\bm$x$})\|\,\|\mb{\bm$u$}\| + \|\mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})\|\\
&\le& -\psi_3(\|\mb{\bm$x$}\|) + 2\|\mb{\bm$\mu$}(t,\mb{\bm$x$})\rho(t,\mb{\bm$x$})\|\\
&\le& -\psi_3(\|\mb{\bm$x$}\|) + 2\nu.
\end{eqnarray*}
We conclude that for all $t$ and all $\mb{\bm$x$}$, we have
\[
\dot{W}(t,\mb{\bm$x$}) \le -\psi_3(\|\mb{\bm$x$}\|)+ 2\nu.
\]
To proceed further we need an assumption on the function $\psi_3$.\\
If $\psi_3$ is bounded and
\begin{equation}
\lim_{r \to \infty}\psi_3(r)=l<\infty
\label{eq:psithrllti}
\end{equation}
then we assume
\begin{equation}
2\nu < l.
\label{eq:twonulel}
\end{equation}
\begin{proposition}
\label{prop:prac_ubd}
The closed-loop system driven by the boundary layer controller is uniformly bounded.
\end{proposition}
{\bf Proof}\quad Let $R$ be such that
\[
\psi_3(R)=2\nu.
\]
Note that $R$ is well defined by virtue of~(\ref{eq:twonulel}) and the fact that $\psi_3$ is strictly increasing. Since $\psi_3$ is strictly increasing, we can write
\[
R=\psi^{-1}_3(2\nu).
\]
Figure~\ref{fig:psi_the} illustrates the above construction.
\begin{figure}
\vspace{3in}
\hskip1.5in\special{bmp:c:/mybook/psi_the.bmp x=4in y=3in}
\caption{Finding $R$ for a given $\nu$.}
\label{fig:psi_the}
\end{figure}

Given $d>0$, where $\| \mb{\bm$x$}_0\| \le d$. Let
\[
\tilde{d}=\max \{d,R\}.
\]
We claim that
\[
b=b(d)=\psi^{-1}_1 (\psi_2(\tilde{d})),
\]
that is, the system, or equivalently its null solution, is uniformly bounded. In other words, if $\| \mb{\bm$x$}_0 \| \le d$ then $\|\mb{\bm$x$}(t;t_0,\mb{\bm$x$}_0)\|\le b$ for $t \ge t_0$. First note that $\|\mb{\bm$x$}_0 \| \le \tilde{d}$ and that $\psi_1(b)=\psi_2(\tilde{d})$. Hence $\tilde{d} \le b$---see Figure~\ref{fig:uni_bdd} for an illustration of this construction.
\begin{figure}
\vspace{3in}
\hskip1.5in\special{bmp:c:/mybook/uni_bdd.bmp x=4in y=3in}
\caption{Finding $b$ for a given $d$.}
\label{fig:uni_bdd}
\end{figure}
We have
\begin{eqnarray}
\dot{W}(\mb{\bm$x$}(t)) &\le& -\psi_3 ( \| \mb{\bm$x$}(t) \| )+2\nu\nonumber\\
&=& -\psi_3 ( \| \mb{\bm$x$}(t) \| )+\psi_3(R).
\label{eq:wdolemptptr}
\end{eqnarray}
For $\| \mb{\bm$x$}_0 \| > R$ and $\|\b x(t)\|>R$, $\dot{W}<0$ and therefore
\begin{eqnarray*}
\psi_1(b) &=& \psi_2(\tilde{d})\\
&\ge& W(t_0;t_0,\mb{\bm$x$}_0)\\
&\ge& W(t;t_0,\mb{\bm$x$}_0)\\
&\ge& \psi_1(\| \mb{\bm$x$}(t)\|),
\end{eqnarray*}
which implies that $\| \mb{\bm$x$}(t;t_0,\mb{\bm$x$}_0)\| \le b$ for $t \ge t_0$.

If, on the other hand, $\| \mb{\bm$x$}_0 \| \le R$, then since $\dot{W}(\mb{\bm$x$}(t)) \le -\psi_3 ( \| \mb{\bm$x$}(t) \| )+2\nu$, the solution may reach the sphere $\| \mb{\bm$x$} \| = R$ at some time $t_2 \ge t_0$. However, by~(\ref{eq:wdolemptptr}), $\dot{W} < 0$ outside $\| \mb{\bm$x$} \| = R$. Therefore, the solution cannot leave the ball $\| \mb{\bm$x$} \| \le b$. Thus the null solution is uniformly bounded.\\
\hspace*{5.5in} $\Box$\\
We illustrate Proposition~\ref{prop:prac_ubd} with Figure~\ref{fig:unib_2d}.
\begin{figure}
\vspace{3in}
\hskip1.5in\special{bmp:c:/mybook/unib_2d.bmp x=4in y=3in}
\caption{A two-dimensional illustration of an uniformly ultimately bounded solution.}
\label{fig:unib_2d}
\end{figure}
The initial condition in Figure~\ref{fig:unib_2d} satisfies $\| \mb{\bm$x$}_0 \|< d$, and $d<R$. The trajectory is confined to stay within the ball of radius $b=\psi_1^{-1}(\psi_2(R))$.

We now show, using arguments similar to those of Corless and Leitmann~\cite{corless81}, that the closed-loop system is uniformly ultimately bounded with respect to the ball of radius $\tilde{b}>\psi^{-1}_1(\psi_2(R))$.
\begin{thm}
If $\mb{\bm$x$}(t)=\mb{\bm$x$}(t;t_0,\mb{\bm$x$}_0)$ is a solution for $[t_0,\infty)$ with $\| \mb{\bm$x$}_0 \| \le d$, then for a given
\begin{equation}
\tilde{b} > \psi_1^{-1}(\psi_2(R)),
\label{eq:btilpsopst}
\end{equation}
$\|\mb{\bm$x$}(t;t_0,\mb{\bm$x$}_0)\| \le \tilde{b}$ for all $t \ge t_0 + T(\tilde{b},d)$, where
\[
T(\tilde{b},d)=\left\{\begin{array}{ccc}
0 & \mb{if} & d \le \tilde{R}\\
\frac{\psi_2(d)-\psi_1(\tilde{R})}{\psi_3(\tilde{R})-2\nu}& \mb{if}& d > \tilde{R},
\end{array}\right.
\]
and
\begin{equation}
\psi_2\left(\tilde{R}\right) = \psi_1(\tilde{b}).
\label{eq:pstrtipso}
\end{equation}
\end{thm}
{\bf Proof}\quad From assumption~(\ref{eq:btilpsopst}),
\[
\psi_1(\tilde{b}) > \psi_2(R),
\]
and by~(\ref{eq:pstrtipso}),
\[
\psi_1(\tilde{b}) = \psi_2(\tilde{R}).
\]
Hence,
\begin{equation}
\tilde{R} > R,
\label{eq:rtistrbir}
\end{equation}
by~(\ref{eq:thconspsis}), where we assumed that $\psi_2$ is strictly increasing. We consider two cases. The first case is when $d \le \tilde{R}$. Then by uniform boundedness, for all $t \ge t_0$,
\[
\| \mb{\bm$x$}(t) \| < b,
\]
for any $b$ such that $\psi_1(b) > \psi_2(\tilde{R})$. We let $\tilde{b}=b(d)$. Hence in this case $T(\tilde{b},d)=0$.

We now consider the case when $d>\tilde{R}$. We show, by contradiction, that for some $t_1 \in [t_0,t_0+T(\tilde{b},d)]$
\[
\| \mb{\bm$x$}(t_1) \| \le \tilde{R}.
\]
From this we shall conclude the result by referring to the previous case. So suppose that
\[
\| \mb{\bm$x$}(t) \| > \tilde{R}
\]
for all $t \in \left[t_0, t_0 + T(\tilde{b},d) \right]$. Then,
\begin{eqnarray*}
\psi_1\left( \| \mb{\bm$x$}\left(t_0+T(\tilde{b},d);t_0,\mb{\bm$x$}_0\right)\| \right) &\le& W \left( t_0+T(\tilde{b},d),\mb{\bm$x$}\left(t_0+T(\tilde{b},d);t_0,\mb{\bm$x$}_0\right)\right)\\
&=& W(t_0,\mb{\bm$x$}_0)+\int_{t_0}^{t_0+T(\tilde{b},d)}\dot{W}dt\\
&=& W(t_0,\mb{\bm$x$}_0)-\int_{t_0}^{t_0+T(\tilde{b},d)}\left(-\dot{W} \right)dt\\
&\le& \psi_2(d) - \int_{t_0}^{t_0+T(\tilde{b},d)}\left(\psi_3(\tilde{R})-2\nu\right)dt.
\end{eqnarray*}
Note that $\psi_3(\tilde{R})-2\nu > 0$ because by~(\ref{eq:rtistrbir}), $\tilde{R} > R$. We proceed with the evaluation to get
\begin{eqnarray*}
\psi_1\left(\| \mb{\bm$x$}\left(t_0+T(\tilde{b},d);t_0,\mb{\bm$x$}_0\right)\| \right) &\le& \psi_2(d)-\left(\psi_3(\tilde{R})-2\nu\right)T(\tilde{b},d)\\
&=& \psi_2(d)-\left(\psi_3(\tilde{R})-2\nu\right)\frac{\psi_2(d)-\psi_1(\tilde{R})}{\psi_3(\tilde{R})-2\nu}\\
&=&\psi_1\left(\tilde{R}\right).
\end{eqnarray*}
Thus, since $\psi_1$ is strictly increasing
\[
\| \mb{\bm$x$}\left(t_0+T(\tilde{b},d);t_0,\mb{\bm$x$}_0\right)\| \le \tilde{R}
\]
which contradicts our assumption that for all $t \in \left[t_0, t_0 + T(\tilde{b},d) \right]$ it should have been $\| \mb{\bm$x$}(t)\| > \tilde{R}$. Hence, there exists $t_1 \in \left[t_0, t_0 + T(\tilde{b},d) \right]$ such that $\| \mb{\bm$x$}(t_1)\| \le \tilde{R}$. Then using the same argument as in the previous case we can show that
\[
\| \mb{\bm$x$}(t) \| < \tilde{b}
\]
for all $t \ge t_1$ and the proof is complete.\\
\hspace*{5.5in} $\Box$\\
We have the following consequence of the above theorem.
\begin{corollary}
The ball
\[
\left\{ \mb{\bm$x$}:\|\mb{\bm$x$}\| \le \psi_1^{-1}(\psi_2(R)) \right\}
\]
is uniformly stable.
\end{corollary}
{\bf Proof}\quad Let
\[
\varepsilon > \psi_1^{-1}(\psi_2(R)),
\]
and consider $\delta=\delta(\varepsilon)$ obtained from
\[
\psi_1(\varepsilon)=\psi_2(\delta).
\]
Obviously $\delta \ge R$---see Figure~\ref{fig:ub_ball}.
\begin{figure}
\vspace{3in}
\hskip1.5in\special{bmp:c:/mybook/ub_ball.bmp x=4in y=3in}
\caption{Constructing $\delta$ given $\varepsilon$ for an uniformly stable ball.}
\label{fig:ub_ball}
\end{figure}
Hence by uniform boundedness if $\| \mb{\bm$x$}_0 \| \le \delta$, then $\| \mb{\bm$x$}(t) \| \le \varepsilon$ for all $t \ge t_0$ which means that the ball is uniformly stable.\\
\hspace*{5.5in} $\Box$
\begin{example}\rm
Consider a dynamical system model
\[
\dot{\b x} = \b A\b x+\b b(u+h(t,\b x)),
\]
where
\[
\b A=\left[\begin{array}{cc}
0 & 1\\
-1 & -1
\end{array}\right],\quad \b b=\left[\begin{array}{c}
0\\
1
\end{array}\right],\quad\mb{and}\quad |h(t,\b x)|\le 2.
\]
Let $\b P$ be the solution of the matrix Lyapunov equation $\b A^T\b P+\b P\b A=-2\b I_2$, and let
\[
u=\left\{\begin{array}{ccc}
-2\,\mb{sign}(\b b^T\b P\b x)& \mb{if}& |\b b^T\b P\b x| \ge \nu\\
-2\frac{\b b^T\b P\b x}{\nu} & \mb{if} & |\b b^T\b P\b x| < \nu,
\end{array}\right.
\]
where $\nu=1$. Suppose that the initial condition, $\b x(0)$, is such that $\|\b x(0)\|_2\le 10$. Find the radius, $b$, of a ball such that $\|\b x(t)\|_2\le b$ for all $t\ge0$.

Solving the Lyapunov equation $\b A^T\b P+\b P\b A=-2\b I_2$ yields
\[
\b P=\left[\begin{array}{cc}
3 & 1\\
1 & 2
\end{array}\right].
\]
We next evaluate $\frac{d}{dt}\left( \b x^T\b P\b x\right)$ on the trajectories of the closed-loop system. For the case when $\left| \b b^T\b P\b x\right|\ge \nu$, we get
\[
\frac{d}{dt}\left( \b x^T\b P\b x\right)\le -2\|\b x\|^2.
\]
For the case when $\left| \b b^T\b P\b x\right| < \nu$, we obtain
\begin{eqnarray*}
\frac{d}{dt}\left( \b x^T\b P\b x\right)&=&-2\|\b x\|^2+2\b x^T\b P\b b(u+h)\\
&\le& -2\|\b x\|^2-4\frac{\left| \b x^T\b P\b b\right|^2}{\nu}+4\left| \b x^T\b P\b b\right|\\
&\le& -2\|\b x\|^2 + 4\nu\\
&=& -2\|\b x\|^2+4.
\end{eqnarray*}
Hence, $\frac{d}{dt}\left( \b x^T\b P\b x\right)<0$ outside of the ball
\[
\bar{B}_R(\b 0)=\{\b x:\|\b x\| \le R=\sqrt{2} \}.
\]
To find the radius, $b$, of a ball, such that $\|\b x(t)\|_2\le b$ for all $t\ge 0$ and $\|\b x(0)\|_2\le 10=d$, we first observe that
\[
\lambda_{\min}(\b P)\|\b x\|^2 =\psi_1(\|\b x\|)\le \b x^T\b P\b x \le \psi_2(\|\b x\|)=\lambda_{\max}(\b P)\|\b x\|^2.
\]
Let $\tilde{d}=\max\{d,R\}=10$. Then, the radius $b$ is computed by solving the equation
\[
\psi_1(b)=\psi_2(\tilde{d}),
\]
that is,
\[
\lambda_{\min}(\b P)b^2=\lambda_{\max}(\b P)\tilde{d}^2.
\]
We obtain
\[
b=\tilde{d}\sqrt{\frac{\lambda_{\max}(\b P)}{\lambda_{\min}(\b P)}}=10\sqrt{\frac{(5+\sqrt{5})/2}{(5-\sqrt{5})/2}}=16.18.
\]
\end{example}\markex
\begin{example}\rm
\label{ex:ublveh}
In this example, we investigate the performance of a boundary layer tracking controller on the tracking problem considered in Example~\ref{ex:ummxveh}. Specifically, we replace the discontinuous controller given by~(\ref{eq:ummxvehtr}) 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 $\mb{sat}_{\nu}$, can be described by
\begin{equation}
\mb{sat}_{\nu}(x)=\frac{1}{2\nu}\left(|x+\nu|-|x-\nu| \right),
\label{eq:linsatu}
\end{equation}
where the parameter $\nu$ controls the width of the boundary layer. In other words,
\begin{equation}
\mb{sat}_{\nu}(x)=\left\{\begin{array}{ccl}
1&\mb{if}& x\ge \nu\\
\frac{1}{\nu}x&\mb{if}& -\nu<x<\nu\\
-1&\mb{if}& x\le \nu.
\end{array}\right.
\label{eq:worstdesu}
\end{equation}
A plot of~(\ref{eq:linsatu}) for $\nu=0.5$ is shown in Figure~\ref{fig:saturau}.
\begin{figure}
\vspace{3.5in}
\hskip1in\special{bmp:c:/mybook/saturau.bmp x=4.67in y=3.5in}
\caption{Plot of the linear saturation function.}
\label{fig:saturau}
\end{figure}
A boundary layer version of the discontinuous tracking controller~(\ref{eq:ummxvehtr}) is
\begin{eqnarray}
u &=& -\rho(\b x,r)\mb{sat}_{\nu}\left(\b b^T\b P\b e\right)\nonumber\\
&=&-(|\b k_d\b x|+|r|)\mb{sat}_{\nu}(\b b^T\b P(\b x-\b x_d))\nonumber\\
&=&-\left(\left|\left[\begin{array}{c}
0.0329\\
2.2854\\
0.2429\\
-0.1653 
\end{array}\right]^T\b x \right| + |r| \right)\mbox{sat}_{\nu}\left(\left[\begin{array}{c}
0.1571\\
26.9931\\
6.3324\\
-3.0250\end{array}\right]^T(\b x-\b x_d)\right).
\label{eq:blvodcmmx}
\end{eqnarray}
The plot of the actual output, $Y$, of the closed-loop system driven by~(\ref{eq:blvodcmmx}) for $\nu=0.5$ and zero initial conditions was indistinguishable from that shown in Figure~\ref{fig:ummxlad} when the discontinuous controller was used. The boundary layer controller action versus time is shown in Figure~\ref{fig:uveh_bla}. Chattering is basically eliminated in this case.
\begin{figure}
\vspace{3.5in}
\hskip1in\special{bmp:c:/mybook/uveh_bla.bmp x=4.67in y=3.5in}
\caption{Plot of the boundary layer controller action versus time.}
\label{fig:uveh_bla}
\end{figure}
\end{example}\markex
\section{Notes}
A landmark paper on mathematical description of linear dynamical systems from control point of view is by Kalman~\cite{kalman63}. For more examples of dynamical systems models from mechanical and electrical engineering, the reader is referred to Truxal~\cite{truxal72}, Banks~\cite{banks86}, Raven~\cite{raven87}, Mohler~\cite{mohler91}, and D{'}Azzo and Houpis~\cite{dazzo95}. Wertz~\cite{wertz97} has many excellent real-life models from space science. A well-written paper on vehicle modeling and control for broad audience is by Ackermann~\cite{ackermann97}. In addition to~\cite{wong93} and~\cite{will97}, vehicle and tire dynamics modeling are investigated in detail by Nalecz~\cite{nalecz88}, Nalecz and Bindemann~\cite{nalecz88b,nalecz88c}, and Smith and Starkey~\cite{smith94,smith95}.

Mayr writes on page~109 in~\cite{mayr70}, ``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~\cite{mayr70}.  
