%\setcounter{chapter}{6}
\chapter{Sliding Modes}
\label{chp:sliding}
\begin{flushright}
\parbox{5.5in}{As philosopher of science Karl Popper has emphasized, a good theory is characterized by the fact that it makes a number of predictions that could in principle be disproved or falsified by observation. Each time new experiments are observed to agree with the predictions the theory survives, and our confidence in it is increased; but if ever a new observation is found to disagree, we have to abandon or modify the theory. At least that is what is supposed to happen, but you can always question the competence of the person who carried out the observation.~\cite[page~130]{hawking88}}
\end{flushright}

\bigskip

\section{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 presented in the following sections.

A {\em variable structure system\/}\index{variable structure system} is a dynamical system whose structure changes in accordance with the current value of its state.\index{structure change!with current value of state} A variable structure system can be viewed as a system composed of independent structures together with a switching logic\index{switching logic} between each of the structures.\index{variable structure system!composed of independent structures} With appropriate switching logic, a variable structure system can exploit the desirable properties of each of the structures 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.
\begin{example}\rm
This example was adapted from Utkin~\cite{utkin77}. We consider a dynamical system model
\begin{equation}
\left\{\begin{array}{lll}
\dot{x}_1 &=& x_2\\
\dot{x}_2 &=& -ux_1,
\end{array}\right.
\label{eq:firstvssy}
\end{equation}
having two structures corresponding to $u=1/a$ and $u=a$, where $a$ is a positive constant greater than 1; in our simulations we take $a=5$.  The phase-plane portraits of the structures are shown in Figures~\ref{fig:vsc_str1}~(a) and~(b).
\begin{figure}
\centering
\includegraphics[scale=0.6]{vss_ex1.eps}
%\vspace{5in}
%\hskip.75in\special{eps:c:/mybook/vss_ex1.eps y=5in}
\caption{Figures~(a) and~(b) are phase-plane portraits of structures making up a variable structure system and Figure~(c) is a phase-plane portrait of the variable structure system itself, where the structure of the system is changed any time the system's trajectory crosses either of the coordinate axis. This figure was adapted from the survey paper by Utkin~\cite{utkin77}.}
\label{fig:vsc_str1}
\end{figure}
%\begin{figure}
%\vspace{3in}
%\hskip1in\special{bmp:c:/sliding/vsc_str2.bmp x=4in y=3in}
%\caption{A phase-plane portrait of the system's second structure %corresponding to {\em u}=4.}
%\label{fig:vsc_str2}
%\end{figure}
The phase-plane portraits of the individual 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,
\begin{equation}
u=\left\{\begin{array}{ccc}
1/5 & \mb{if} & x_1x_2<0\\
5& \mb{if} & x_1x_2>0.
\end{array}\right.
\label{eq:swilogofvs}
\end{equation}
A phase plane portrait of the system~(\ref{eq:firstvssy}) with the switching logic specified by~(\ref{eq:swilogofvs}) is shown in Figure~\ref{fig:vsc_clsc}. We add that the trajectories spiral clockwise towards the origin.
\begin{figure}
\centering
\includegraphics[scale=0.7]{vsc_ut1.eps}
%\vspace{3.5in}
%\hskip.75in\special{bmp:c:/mybook/vsc_ut1.bmp x=4.67in y=3.5in}
\caption{Phase-plane portrait of the variable structure system described by~(\ref{eq:firstvssy}) and~(\ref{eq:swilogofvs}).}
\label{fig:vsc_clsc}
\end{figure}
\end{example}
\begin{example}\rm
\label{ex:doub_int}
Consider the double integrator
\[
\ddot{x}=u,
\]
whose state-space model is
\begin{equation}
\left\{\begin{array}{lll}
\dot{x}_1 &=& x_2\\
\dot{x}_2 &=& u.
\end{array}\right.
\label{eq:ssdeofdouin}
\end{equation}
Let $u=\pm 2$. Typical trajectories of the above system are shown in Figure~\ref{fig:open_tr}.
\begin{figure}
\centering
\includegraphics[scale=0.5]{open_tr.eps}
%\vspace{3in}
%\centerbmp{4in}{3in}{c:/mybook/open_tr.bmp}
%\hskip1in\special{bmp:c:/sliding/open_tr.bmp x=4in y=3in}
\caption{Typical trajectories of the double integrator system.}
\label{fig:open_tr}
\end{figure}
We have two independent structures, neither of them being stable. Let us use the following switching logic\index{switching logic!between each of the structures} between each of the structures:
\begin{equation}
u=-2\,\mb{sign}\left(x_1+x_2\right).
\label{eq:colafodin}
\end{equation}
With the above switching logic, the closed-loop system changes its structure anytime the state trajectory crosses the line
\[
x_1+x_2=0.
\]
We introduce the notation,
\[
\sigma(\b x) = \sigma (x_1,x_2)= x_1+x_2.
\]
Then, we can describe the switching logic action as
\begin{equation}
u =\left\{\begin{array}{ccc}
-2 & \mbox{if} & \sigma(\b x)>0\\
2 & \mbox{if} & \sigma(\b x)<0.
\end{array}\right.
\label{eq:anodeofcl}
\end{equation}
Typical trajectories of the closed-loop system~(\ref{eq:ssdeofdouin}),~(\ref{eq:colafodin}) for the initial conditions located ``close'' to the origin of the state-plane are shown in Figure~\ref{fig:close_tr}.
\begin{figure}
\centering
\includegraphics[scale=0.5]{close_tr.eps}
%\vspace{3in}
%\centerbmp{4in}{3in}{c:/mybook/close_tr.bmp}
%\hskip1in\special{bmp:c:/sliding/close_tr.bmp x=4in y=3in}
\caption{Typical trajectories of the closed-system~(\ref{eq:ssdeofdouin}),~(\ref{eq:colafodin}) for initial conditions ``close'' to the origin of the state-plane. The applied control law is given by~(\ref{eq:colafodin}).}
\label{fig:close_tr}
\end{figure}
Close to the origin, 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 {\em chattering},\index{chattering}\index{chattering!oscillations of trajectory about switching surface} of the trajectory about the switching surface as it is explained in Figure~\ref{fig:chatter}.
\begin{figure}
\centering
\includegraphics[scale=0.5]{chatter.eps}
%\vspace{3.5in}
%\hskip1.5in\special{eps:c:/mybook/chatter.eps y=3.5in}
\caption{Chattering in a variable structure sliding mode system---a magnified view.}
\label{fig:chatter}
\end{figure}
A plot of two trajectories with an increased integration time is shown in Figure~\ref{fig:vsc_ch2}.
\begin{figure}
\centering
\includegraphics[scale=0.5]{vsc_ch2.eps}
%\vspace{3in}
%\centerbmp{4in}{3in}{c:/mybook/vsc_ch2.bmp}
%\hskip1in\special{bmp:c:/mybook/vsc_ch2.bmp x=4in y=3in}
\caption{An illustration of chattering in a variable structure sliding mode system consisting of the double integrator and the control law, $u=-\mb{sign}(x_1+x_2)$. Two trajectories starting from two different initial conditions are plotted.}
\label{fig:vsc_ch2}
\end{figure}

\end{example}

\section{Sliding Mode}
\label{sec:slidingmode}
The motion of the system while confined to the switching line or a surface is referred to as {\em sliding}.\index{sliding!motion confined to switching surface} A sliding mode will exist if in the vicinity of the switching surface the state velocity vectors are directed towards the surface. In such a case, the switching surface attracts trajectories when they are in its vicinity and once a trajectory intersects the switching surface it will stay on it thereafter.


A surface $\sigma(\b x)=0$ is attractive\index{attractive surface}\index{surface!attractive} if
\begin{enumerate}
\item any trajectory starting on the surface remains there, and
\item any trajectory starting outside the surface tends to it at least asymptotically.
\end{enumerate}
Thus, for a sliding motion\index{sliding motion!conditions to occur} to occur we need
\[
\lim_{\sigma \to 0^{+}} \dot{\sigma} < 0 \quad \mb{and} \quad \lim_{\sigma \to 0^{-}} \dot{\sigma} > 0.
\]
The above conditions are illustrated in Figure~\ref{fig:sliding}.
\begin{figure}
\centering
\includegraphics[scale=0.5]{sliding.eps}
%\vspace{3in}
%\hskip1in\special{bmp:c:/mybook/sliding.bmp x=4in y=3in}
\caption{An illustration of the sufficient condition of the attractivity of the switching surface.}
\label{fig:sliding}
\end{figure}
They ensure that the motion of the state trajectory $\b x$ on either side of the switching surface $\sigma(\b x)=0$ is towards the switching surface. The two conditions may be combined to give
\begin{equation}
\sigma \dot{\sigma} <0
\label{eq:sigmasigmadotonein}
\end{equation}
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 $\sigma(\b x)=0$. This is the reason we often refer to the switching surface as the {\em discontinuity surface}.\index{discontinuity surface|see{switching surface}}\index{switching 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 $\dot{\b x}=\b f(t, \b x)$ with a continuous right-hand side is a function $\b x(t)$ that has a derivative and satisfies the differential equation everywhere on a given interval. In the case when the right-hand side of $\dot{\b x}=\b f(t, \b x)$ is discontinuous, we have to use more general methods such as one, for example, developed by Filippov~\cite{filippov88}. We illustrate these points with a numerical example taken from Filippov~\cite[page~1]{filippov88}.
\begin{example}\rm
Consider the following differential equation with the discontinuous right-hand side,\index{differential equation!with discontinuous right-hand side}
\begin{equation}
\dot{x}=1-2\mb{sign}(x).
\label{eq:odewdrhs}
\end{equation}
For $x<0$, we have $\dot{x}=3$, while for $x>0$, $\dot{x}=-1$. In Figure~\ref{fig:filippo}, plots of $x$ versus time for different initial conditions are given.
\begin{figure}
\centering
\includegraphics[scale=0.6]{filippo.eps}
%\vspace{4in}
%\hskip1in\special{bmp:c:/mybook/filippo.bmp x=4.54in y=4in}
\caption{Plots of $x$ versus time for different initial conditions for the differential equation~(\ref{eq:odewdrhs}).}
\label{fig:filippo}
\end{figure}
As $t$ increases, the solution $x$ of~(\ref{eq:odewdrhs}) reaches the line $x=0$ and stays there. The function $x(t)$, however, does not satisfy~(\ref{eq:odewdrhs}) in the usual sense because if $\dot{x}(t)=0$ and $x(t)=0$, the right hand side of~(\ref{eq:odewdrhs}) becomes
\[
1-2\mb{sign}(0)=1\ne 0,
\]
where we used the usual definition of the sign function, where $\mb{sign}(0)=0$. To circumvent the above difficulties, the defintion of the sign function is generalized by definining its value at zero as a point-to-set mapping,\index{point-to-set mapping} that is,
\[
\mb{sign}(0) \in [-1,\, 1]
\]
and as before
\[
\mb{sign}(z)=\left\{\begin{array}{cl}
1 & \mb{if } z>0\\
-1 & \mb{if } z<0 .
\end{array}\right.
\]
\end{example}\markex


Filippov~\cite[pages~2-3]{filippov88} recommends that the generalized solution to an ordinary differential equation with a discontinuous right-hand\index{generalized solution!to differential equation with discontinuous right-hand side}\index{Filippov's generalized solution} side should meet the following criteria:
\begin{itemize}
\item[(i)] for a differential equation with a continuous right-hand side, the generalized solution should be equivalent with the usual one;
\item[(ii)] for the equation $\dot{x}=f(t)$, the generalized solution should be of the form
\[
x(t)=\int f(t)dt + \mb{constant};
\]
\item[(iii)] for any initial condition $x(t_0)=x_0$, the solution must exist at least for $t>t_0$ and should be continuable;
\item[(iv)] the generalized solution should be applicable to a broad class of equations that model dynamical systems;
\item[(v)] the limit of an uniformly convergent sequence of solutions should be a solution;
\item[(vi)] after applying to a solution a common change of variables, the result of the transformation should be a solution.
\end{itemize}

We next describe the proposed generalized solution in the sense of Filippov.\index{solution in the sense of Filippov} We begin with an example. Suppose that we are given a single-input dynamical system model, $\dot{\b x}=\b f(\b x,u)$ along with a switching surface $\sigma(\b x)=0$. Let $\b f^{-}=\b f(\b x, u^{-})$ and $\b f^{+}=\b f(\b x, u^{+})$, where $u=u^{-}$ if $\sigma<0$ and $u=u^{+}$ if $\sigma>0$. We join the velocity vectors $\b f^{-}$ and $\b f^{+}$. The resulting velocity vector $\b f^0$ is obtained by drawing the line tangent to the switching surface\index{tangent!to switching surface} that intersects the line segment joining the velocity vectors $\b f^{-}$ and $\b f^{+}$. Thus,
\[
\b f^0=\alpha \b f^{-}+(1-\alpha)\b f^{+},
\]
where $\alpha \in [0,1]$. The above construction is illustrated in Figure~\ref{fig:filip}. Observe that $\b f^0$ belongs to the smallest convex set\index{smallest convex set} containing $\b f^{-}$ and $\b f^{+}$. Recall that a set $\cal S$ is said to be convex if for every $\b x, \b y \in {\cal S}$ and every real number $\alpha$, $0<\alpha < 1$, the point $\alpha \b x+(1-\alpha)\b y \in {\cal S}$.
\begin{figure}
\centering
\includegraphics[scale=0.5]{filip.eps}
%\vspace{3in}
%\hskip1in\special{bmp:c:/mybook/filip.bmp x=4in y=3in}
\caption{Constructing a solution in the sense of Filippov in $\mb{\Bb R}^2$.}
\label{fig:filip}
\end{figure}

Following Itkis~\cite{itkis76}, we give a simple interpretation\index{Itkis'interpretation!of solution in the sense of Filippov} of the above construction. The number $\alpha$ may be interpreted as the time the system's trajectory remains below the switching surface, while $(1-\alpha)$ 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 trajectory performs one oscillation about $\sigma = 0$ in a ``small'' time interval $\Delta \tau$, then it will traverse a distance $\|\b f^{-}\|\alpha \Delta \tau$ in time $\alpha \Delta \tau$ with velocity $\b f^{-}$. During the remaining time of oscillation the trajectory will traverse a distance $\|\b f^{+}\|(1-\alpha )\Delta \tau$ in time $(1-\alpha) \Delta \tau$ with velocity $\b f^{+}$. The average displacement of the trajectory during the time $\Delta \tau$ will then be $\|\alpha \b f^{-}+(1-\alpha)\b f^{+}\|\Delta \tau$ with the average velocity $\b f^0$.


Suppose that there are $m$ discontinuity surfaces of the form
\[
\sigma_i=\{\b x:\sigma_i(\b x)=0\},\quad i=1,2,\ldots,m.
\]
Let
\[
H=\bigcup_{i=1}^{m}\sigma_i.
\]
Then, we take a solution of the differential equation $\dot{\b x}=\b f(\b x)$ with the discontinuous right-hand side to be a solution of the differential inclusion
\[
\dot{\b x}(t)\in G(\b x(t)),
\]
where for each $\b x$, the set $G(\b x)$ is the smallest closed convex set containing the cluster values, limits, of the function $\b f(\b y)$ as $\b y \to \b x$, $\b y \notin H$, that is, a solution of the dynamical system $\dot{\b x}=\b f(\b x)$ is an absolutely continuous function\index{absolutely continuous function}\index{solution!absolutely continuous function}\index{function!absolutely continuous} $\b x(t)$, in the sense of Definition~\ref{defn:abs_con}, defined on an interval, or segment, $L$ for which $\dot{\b x}(t) \in G(\b x(t))$ almost everywhere on $L$ (Filippov~\cite{filippov88}). Observe that if the function $\b f$ is continuous at a point $\b x$, that is, $\b x \notin H$, then $G(\b x)$ consists of a single point, namely $\b f(\b x)$, and the solution satisfies $\dot{\b x}=\b f(\b x)$ in the usual sense. However, if $\b x \in H$, then $\b x$ lies on one or more of the surfaces $\sigma_1, \sigma_2,\ldots,\sigma_{m}$. Suppose that $\b x$ lies on the surfaces $\sigma_{i_1},\sigma_{i_2},\ldots,\sigma_{i_k}$. It then follows that for a sufficiently small $\delta > 0$ the surfaces $\sigma_{i_1}, \sigma_{i_2},\ldots,\sigma_{i_k}$ partition the $\delta$-neighborhood of $\b x$ into $\ell \le 2^k$ regions ${\cal R}_1,{\cal R}_2,\ldots,{\cal R}_{\ell}$, in each of which the function $\b f$ is continuous. Let $\b f_j$ denote the function $\b f$ restricted to the region ${\cal R}_j$. Then, $G(\b x)$ is the smallest closed convex set containing the set of points
\begin{equation}
\left\{\lim_{{\tilde{\b x}\to \b x}\atop{\tilde{\b x} \in {\cal R}_j}} \b f_j(\tilde{\b x}) : j=1,2,\ldots,\ell \right\}.
\label{eq:cluster_f}
\end{equation}
The velocity vector, $\dot{\b x}=\b f^0$, lies then on the intersection of the tangent plane to the surfaces of discontinuity and the convex set\index{intersection!of tangent plane and convex set} described by~(\ref{eq:cluster_f}). This is illustrated in Figures~\ref{fig:filip} and~\ref{fig:filip_3d}.
\begin{figure}
\centering
\includegraphics[scale=0.5]{filip_3d.eps}
%\vspace{3.5in}
%\hskip.7in\special{bmp:c:/mybook/filip_3d.bmp x=4.67in y=3.5in}
\caption{Constructing a solution in the sense of Filippov in $\mb{\Bb R}^3$.}
\label{fig:filip_3d}
\end{figure}

\section{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:
\begin{itemize}
\item sliding (switching) surface design\index{sliding (switching) surface design}\index{switching surface design} so as to achieve the desired system behavior, like the stability to the origin, when restricted to the surface, and
\item selecting feedback gains of the controller\index{controller gains} so that the closed-system is stable to the sliding surface.
\end{itemize}
There is a number of benefits when the system is in sliding. In particular,
\begin{itemize}
\item while in sliding the system is not affected by matched uncertainties,\index{benefits of sliding!not affected by matched uncertainties}\index{matched uncertainties} and
\item while in sliding the system behavior is governed by a reduced set of differential equations. This is what we refer to as the {\em order reduction}\index{order reduction}\index{benefits of sliding!order reduction}. It is very useful in designing variable structure sliding mode controllers.
\end{itemize}

We discuss a sliding surface design\index{sliding surface design} phase for linear single-input plants. Without loss of generality, we assume that the plant model is already in its controller companion form,\index{controller companion form}
\[
\dot{\b x} = \left[\begin{array}{ccccc}
0 & 1 & 0 & \cdots & 0\\
0 & 0 & 1 & \cdots & 0\\
\vdots & {} & {} & {} & \vdots\\
0 & 0 & 0 & \cdots & 1\\
-a_0 & -a_1 & -a_2 & \cdots & -a_{n-1}
\end{array}\right]\b x + \left[\begin{array}{c}
0\\
0\\
\vdots\\
0\\
1
\end{array}\right]u.
\]
Let the sliding surface be
\begin{eqnarray*}
\sigma(\b x) &=& \b s\b x\\
&=& \left[\begin{array}{ccccc}
s_1 & s_2 &\cdots & s_{n-1} & 1
\end{array}\right]\left[\begin{array}{c}
x_1\\
\vdots\\
x_n
\end{array}\right]\\
&=&s_1x_1 +s_2x_2 + \cdots + s_{n-1}x_{n-1} + x_n\\
&=&0.
\end{eqnarray*}
Note that, without loss of generality, we assumed that $s_n=1$. If this was not the case, we could divide both sides of $\b s\b x=0$ by $s_n\ne 0$ to ensure that $s_n=1$. Using the above equation of sliding surface, we can write
\[
x_n=-s_1x_1-s_2x_2-\cdots-s_{n-1}x_{n-1}.
\]
We combine the equations of the plant model and that of the sliding surface. First note that
\[
\dot{x}_{n-1}=x_n.
\]
Hence, we can write
\begin{equation}
\left.\begin{array}{ccl}
\dot{x}_1 &=& x_2\\
\dot{x}_2 &=& x_3\\
{} & \vdots & {}\\
\dot{x}_{n-1} &=&-s_1x_1-s_2x_2-\cdots-s_{n-1}x_{n-1},
\end{array}\right\}
\label{eq:sinininslisca}
\end{equation}
or equivalently in matrix form,
\begin{equation}
\dot{\b z} = \left[\begin{array}{ccccc}
0 & 1 & 0 & \cdots & 0\\
0 & 0 & 1 & \cdots & 0\\
\vdots & {} & {} & {} & \vdots\\
0 & 0 & 0 & \cdots & 1\\
-s_1 & -s_2 & -s_3 & \cdots & -s_{n-1}
\end{array}\right]\b z,
\label{eq:sysinsliro}
\end{equation}
where $\b z=\left[\begin{array}{cccc}
x_1 & x_2 & \cdots & x_{n-1}
\end{array}\right]^{\top}$. The above equation describes the system dynamics in sliding.\index{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
\[
\lambda^{n-1}+s_{n-1}\lambda^{n-2}+\cdots+s_2\lambda+s_1=0.
\]
The response of the system in sliding is completely specified by an appropriate choice of the components $s_1, s_2, \ldots, s_{n-1}$ of the switching surface.

Consider again Example~\ref{ex:doub_int}. The sliding surface for this example can be represented as
\[
\sigma(\b x) = \left[\begin{array}{cc}
s_1 & 1
\end{array} \right]\b x =0,
\]
or equivalently
\[
s_1 x_1+x_2=0.
\]
The dynamics of the system in sliding are described by
\[
\dot{z} = -s_1 z,
\]
which is just a first-order system. Note that to ensure the reduced-system stability, we select $s_1>0$.

After designing a sliding surface, we construct a feedback controller. In this section, we discuss a controller design for linear single-input plants. The controller objective is to drive the plant state to the sliding surface and maintain it on the surface for all subsequent time. We use a generalized Lyapunov approach in constructing the controller.\index{constructing sliding controller!generalized Lyapunov approach} Specifically, we use a distance, $V=0.5 \sigma^2$, from the sliding surface $\sigma$ 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 respect to the switching surface\index{negative definite!with respect to the switching surface}, thus ensuring the motion of the state trajectory to the surface as it is illustrated in Figure~\ref{fig:sliding}. Our goal is to find $u=u(\b x)$ so that
\[
\frac{d}{dt}\left( \frac{1}{2} \sigma^2\right)=\sigma \dot{\sigma} <0.
\]
We consider the controller structure of the form:
\begin{equation}
u=k_1^{\pm}x_1 + k_2^{\pm}x_2+\cdots +k_n^{\pm}x_n,
\label{eq:costrupm}
\end{equation}
where $k_i^{\pm}$ are the feedback gains to be determined so that the condition $\sigma \dot{\sigma} <0$ is satisfied. To determine the feedback gains, we substitute $u$, given by~(\ref{eq:costrupm}), into the expression $\sigma \dot{\sigma}$ and utilize~(\ref{eq:sinininslisca}) to get
\begin{eqnarray*}
\sigma \dot{\sigma} &=& \sigma \left[\begin{array}{cccc}
s_1 & s_2 & \cdots & 1
\end{array}\right]\dot{\b x}\\
&=& \sigma \left(s_1 \dot{x}_1 + \cdots+s_{n-1}\dot{x}_{n-1}+\dot{x}_n \right)\\
&=&\sigma\left(s_1x_2 +\cdots+s_{n-1}x_n + \dot{x}_n\right)\\
&=&\sigma (s_1 x_2 +\cdots+s_{n-1}x_n- a_0x_1-\cdots-a_{n-1}x_n+u )\\
&=& \sigma(s_1x_2+\cdots+s_{n-1}x_n-a_0x_1-\cdots-a_{n-1}x_n+k_1^{\pm}x_1 + \cdots +k_n^{\pm}x_n).
\end{eqnarray*}
Rearranging the terms yields
\[
\sigma \dot{\sigma}
= \sigma x_1 \left(-a_0 + k_1^{\pm}\right)+ \sigma x_2 \left(-a_1 + s_1 + k_2^{\pm}\right)+\cdots+ \sigma x_n \left(-a_{n-1} + s_{n-1} + k_n^{\pm} \right).
\]
A possible way to ensure that $\sigma \dot{\sigma} <0$ is to select the gains to force each term in $\sigma \dot{\sigma}$ to be negative, that is,
\[
\left.\begin{array}{rll}
\sigma x_1 \left(-a_0 + k_1^{\pm}\right) &<&0\\
\sigma x_2 \left(-a_1 + s_1 + k_2^{\pm}\right) &<&0\\
{} &\vdots& {}\\
\sigma x_n \left(-a_{n-1} + s_{n-1} + k_n^{\pm} \right) &<&0.
\end{array}\right\}
\]
A possible choice of gains that leads to satisfaction of the attractivity to the surface condition, $\sigma \dot{\sigma}<0$, is:
\begin{equation}
\left.\begin{array}{lll}
k_1^{+} &<& -|a_0|\quad\mb{if}\quad \sigma x_1 >0\\
k_1^{-} &>& |a_0|\quad\mb{if}\quad \sigma x_1 <0\\
{}&\vdots& {}\\
k_n^{+} &<& -(|a_{n-1}|+|s_{n-1}|)\quad\mb{if}\quad \sigma x_n>0\\
k_n^{-} &>& |a_{n-1}|+|s_{n-1}|\quad\mb{if}\quad \sigma x_n<0.
\end{array}\right\}
\label{eq:kpmconds}
\end{equation}
We can also use the following controller gains:
\begin{equation}
\left.\begin{array}{lll}
k_1^{\pm} &=& -(|a_0|+\varepsilon)\mbox{sign}(x_1\sigma)\\
{} &\vdots&{}\\
k_n^{\pm} &=& -(|a_{n-1}|+|s_{n-1}|+\varepsilon)\mbox{sign}(x_n\sigma),
\end{array}\right\}
\label{eq:pmgsithsign}
\end{equation}
where $\varepsilon >0$ is a design parameter. Note that the controllers' gains given by~(\ref{eq:pmgsithsign}) satisfy conditions~(\ref{eq:kpmconds}). The larger the value of $\varepsilon$ the faster the trajectory converges to the sliding surface. In Figure~\ref{fig:sz_flow}, we summarize the above design method in the form of a flow chart\index{sliding controller design!flow chart}\index{sliding controller design!single-input plants}.
\begin{figure}
\centering
\includegraphics[scale=0.661]{sz_flow.eps}
\caption{Summary of a single-input sliding mode controller design. Adapted from DeCarlo et al.~\cite{decarlo88}.}
\label{fig:sz_flow}
\end{figure}

\section{Sliding in Multi-Input Systems}

In this section, we analyze sliding mode behvior in linear multi-input plants.\index{sliding controller design!multi-input plants} When the number of the plant inputs is $m$, we can use sliding surface of the form
\[
\left\{\b x:\b S\b x=\b 0,\b S\in \mb{\Bb R}^{m\times n}\right\}.
\]
We also use $\b S\b x=\b 0$ to denote the above sliding surface. In many applications nonlinear sliding surfaces arise. Here, we use linear sliding surfaces only, although one can use nonlinear sliding surfaces as well. Let $\b \sigma (\b x)=\b S\b x$. Then,
\[
\b \sigma (\b x) = \left[\begin{array}{c}
\sigma_1(\b x)\\
\sigma_2(\b x)\\
\vdots\\
\sigma_m(\b x)
\end{array}\right]
= \left[\begin{array}{c}
\b s_1\b x\\
\b s_2\b x\\
\vdots\\
\b s_m\b x
\end{array}\right],
\]
where $\b s_i\in\mb{\Bb R}^{1\times n}$. Recall, that the system is in sliding mode if $\b \sigma (\b x(t))=\b 0$ and $\dot{\b \sigma} (\b x(t))=\b 0$ for $t\ge t_0$, where $\b x$ is the state trajectory and $t_0$ is a specified time. It follows that in a sliding mode the velocity vector $\dot{\b x}(t)$ is tangent to the switching surface. Equivalently,
\[
\left(\nabla \sigma_i\right)^{\top}\dot{\b x}(t)= \b s_i\dot{\b x}(t)= 0,\quad\mb{for}\quad i=1,2,\ldots,m,
\]
that is,
\[
\dot{\b \sigma}(\b x(t)) = \b S\dot{\b x}(t)= \b 0.
\]
In summary, the system in sliding mode is described by the equations,
\begin{equation}
\left.\begin{array}{lll}
\b \sigma(\b x(t)) &=& \b 0\\
\dot{\b \sigma}(\b x(t)) &=& \b 0.
\end{array}\right\}
\label{eq:syinslisasdo}
\end{equation}

Consider now the plant model $\dot{\b x}(t)=\b A\b x(t)+\b B\b u(t)$. Combining $\b S\dot{\b x}(t) = \b 0$ and the modeling equation of the plant, we get
\[
\b S\b A\b x+\b S\b B\b u=\b 0.
\]
If the matrix $\b S\b B$ is nonsingular, then we can solve the above algebraic equation for $\b u$. The result is called the {\em equivalent control\/}\index{equivalent control!solution of algebraic equation} and it is denoted using the symbol $\b u_{eq}$. We have
\begin{equation}
\b u_{eq}=-\left(\b S\b B\right)^{-1}\b S\b A\b x.
\label{eq:equival_contro}
\end{equation}
Substituting $\b u_{eq}$ into the plant model equation yields
\begin{equation}
\dot{\b x}=\left(\b I_n-\b B(\b S\b B)^{-1}\b S \right)\b A\b x.
\label{eq:eqsyinsli}
\end{equation}
The above system is sometimes referred to as the {\em equivalent system}\index{equivalent system}. Thus, we can describe the behavior of the system in sliding alternatively as
\begin{equation}
\left.\begin{array}{lll}
\dot{\b x}&=&\left(\b I_n-\b B(\b S\b B)^{-1}\b S \right)\b A\b x\\
\b S\b x &=& \b 0.
\end{array}\right\}
\label{eq:altresisl}
\end{equation}
Note that in sliding, the system is governed by a reduced set of differential equations\index{system in sliding!governed by reduced set of differential equations} that are obtained by combining the above two sets of equations.

\section{Sliding Surface Design}
\label{sec:slidingsurf}
We give a sufficient condition for the existence of a sliding surface $\b \sigma(\b x)=\b S\b x=\b 0$ such that the plant model $\dot{\b x}=\b A\b x+\b B\b u$ restricted to it has arbitrarily prescribed poles.  In our discussion, we use the following notation. For a full column rank matrix $\b X$, its left inverse will be denoted $\b X^{g}$. Thus, $\b X^g\b X=\b I$, where $\b I$ is the identity matrix whose size is equal to the number of columns of $\b X$. The imaginary part of a complex number $z$ is denoted $\Im(z)$. The set of eigenvalues of a square matrix $\b A$, also called its spectrum, is denoted $\mb{eig}(\b A)$.\index{set of eigenvalues of a square matrix!spectrum}\index{spectrum!set of eigenvalues of a square matrix}

We now state and prove a theorem that can be found in~\cite{hui_uio93}. A general method for designing a sliding surface is contained implicitly in the proof of this theorem.
\begin{thm}\label{thm:main_slid_the}
Consider the plant model
\begin{equation}
\dot{\b x}=\b A\b x+\b B\b u,
\label{eq:plmofosude}
\end{equation}
where $\b x(t)\in \mb{\Bb R}^n$, $\b u(t)\in \mb{\Bb R}^m$, $\mb{rank}\,(\b B)=m$, and the pair $(\b A,\b B)$ is controllable. Let
\[
\{\lambda_1,\lambda_2,\ldots,\lambda_{n-m}\}\in \mb{\Bb C},
\]
where $\mb{\Bb C}$ denotes the set of complex numbers, be an arbitrary set of $n-m$ complex numbers such that any $\lambda_i$ with $\Im(\lambda_i)\ne 0$ appears in this set in conjugate pair\index{conjugate pair}. Then, there exist full rank matrices $\b W\in \mb{\Bb R}^{n\times (n-m)}$ and $\b W^{g}\in \mb{\Bb R}^{(n-m)\times n}$ so that $\b W^{g}\b W=\b I_{n-m}$, $\b W^{g}\b B=\b O$, and the spectrum of the matrix $\b W^{g}\b A\b W$ is
\[
\mb{eig}(\b W^{g}\b A\b W)=\{\lambda_1,\lambda_2,\ldots,\lambda_{n-m}\}.
\]
Furthermore, there exists a matrix $\b S\in \mb{\Bb R}^{m\times n}$ such that $\b S\b B=\b I_m$ and the system~(\ref{eq:plmofosude}) restricted to the surface $\b S\b x=\b 0$ has the poles $\lambda_1,\lambda_2,\ldots,\lambda_{n-m}$.
\end{thm}
{\bf Proof}\quad We first prove the existence of the desired matrices $\b W$ and $\b W^g$. We can find $\b T\in \mb{\Bb R}^{n\times (n-m)}$ so that $\left[\begin{array}{cc}
\b T & \b B
\end{array}\right]$ is invertible because $\b B$ has full rank. Let
\[
\left[\begin{array}{cc}
\b T & \b B
\end{array}\right]^{-1}=\left[\begin{array}{c}
\b T^{g}\\
\b B^{g}
\end{array}\right].
\]
We introduce a new coordinate $\b z$ such that $\b x=\left[\begin{array}{cc}
\b T & \b B
\end{array}\right]\b z$. Then, in the new coordinates the system $\dot{\b x}=\b A\b x+\b B\b u$ has the form
\[
\dot{\b z} = \left[\begin{array}{cc}
\b T^{g}\b A\b T & \b T^{g}\b A\b B\\
\b B^{g}\b A\b T & \b B^{g}\b A\b B
\end{array}\right]\b z+\left[\begin{array}{c}
\b O\\
\b I_m
\end{array}\right]\b u.
\]
Because the pair $(\b A,\b B)$ is controllable, we can use the PBH rank test\index{PBH rank test}, given on page~\pageref{thm:pbhtest}, to conclude that
\[
\mb{rank}\left[\begin{array}{cc}
\lambda \b I_{n-m}-\b T^{g}\b A\b T & \b T^{g}\b A\b B
\end{array}\right]=n-m \quad \mb{for all} \quad \lambda \in \mb{\Bb C},
\]
which means that the pair $(\b T^{g}\b A\b T,  \b T^{g}\b A\b B)$ is controllable. The set $\{\lambda_1,\lambda_2,\ldots,\lambda_{n-m}\}$ is symmetric with respect to the real axis, therefore we can find a matrix $\b F\in \mb{\Bb R}^{m\times (n-m)}$ such that
\[
\mb{eig}\left( \b T^{g}\b A\b T -\b T^{g}\b A\b B\b F\right)=\{\lambda_1,\lambda_2,\ldots,\lambda_{n-m}\}.
\]
Let
\begin{equation}
\b W^{g}=\b T^{g}
\label{eq:wg_equals_tg}
\end{equation}
and
\begin{equation}
\b W=\b T-\b B\b F.
\label{eq:w_eq_t_min_bf}
\end{equation}
Then, because by construction $\b T^{g}\b T=\b I_{n-m}$ and $\b T^{g}\b B=\b O$, we have $\b W^{g}\b W=\b I_{n-m}$ and $\b W^{g}\b B= \b O$. Note that
\[
\b W^{g}\b A\b W=\b T^{g}\b A\b T-\b T^{g}\b A\b B \b F.
\]
Hence,
\[
\mb{eig}\left(\b W^{g}\b A\b W\right)=\mb{eig}\left(\b T^{g}\b A\b T-\b T^{g}\b A\b B \b F\right)=\{\lambda_1,\lambda_2,\ldots,\lambda_{n-m}\}.
\]

We next prove the existence of the sliding surface with the desired properties. We first note that the matrix $\left[\begin{array}{cc}
\b W & \b B
\end{array}\right]$ is invertible because $\b W^{g}\b W=\b I_{n-m}$, $\b W^{g}\b B=\b O$, and $\b W$, $\b B$ are full rank. Hence, $\mb{Range}(\b W)\cap \mb{Range}(\b B)=\{\b 0\}$ and we write
\begin{equation}
\left[\begin{array}{cc}
\b W & \b B
\end{array}\right]^{-1}=\left[\begin{array}{c}
\b W^{g}\\
\b S
\end{array}\right]
\label{eq:inverting_wb}
\end{equation}
for some $\b S\in \mb{\Bb R}^{m \times n}$. Note that $\b S\b B=\b I_m$ and $\b S\b W=\b O$. We introduce new coordinates
\begin{equation}
\b z = \left[\begin{array}{c}
\b z_1\\
\b z_2
\end{array}\right]
= \left[\begin{array}{c}
\b W^{g}\\
\b S
\end{array}\right]\b x,
\label{eq:new_coor_in_terms_of_S}
\end{equation}
where $\b z_2=\b S\b x$. In the new coordinates, the equivalent system $\dot{\b x}=\left(\b I_n-\b B(\b S\b B)^{-1}\b S \right)\b A\b x$ takes the form
\begin{eqnarray*}
\dot{\b z} &=& \left[\begin{array}{c}
\b W^{g}\\
\b S
\end{array}\right]\left(\b I_n-\b B(\b S\b B)^{-1}\b S \right)\b A \left[\begin{array}{cc}
\b W & \b B
\end{array}\right]\b z\\
&=& \left[\begin{array}{cc}
\b W^{g}\b A\b W & \b W^{g}\b A\b B\\
\b O & \b O
\end{array}\right]\b z.
\end{eqnarray*}
The restriction to $\b z_2=\b S\b x=\b 0$ is
\[
\dot{\b z}_1=\b W^{g}\b A\b W\b z_1,
\]
and hence the system restricted to the sliding surface\index{system restricted!to the sliding surface} $\b S\b x=\b 0$ has the same eigenvalues as the matrix $\b W^{g}\b A\b W$. The proof of the theorem is now complete.\\
\hspace*{5.5in}$\Box$

Note that by~\eqref{eq:w_eq_t_min_bf},
\[
\left[\begin{array}{cc}
\b W & \b B
\end{array}\right]=\left[\begin{array}{cc}
\b T & \b B
\end{array}\right]\left[\begin{array}{cc}
\b I_{n-m} & \b O\\
-\b F & \b I_m
\end{array}\right].
\]
Therefore,
\begin{eqnarray}
\left[\begin{array}{cc}
\b W & \b B
\end{array}\right]^{-1}&=&\left[\begin{array}{cc}
\b I_{n-m} & \b O\\
-\b F & \b I_m
\end{array}\right]^{-1}\left[\begin{array}{cc}
\b T & \b B
\end{array}\right]^{-1}\nonumber\\
&=& \left[\begin{array}{cc}
\b I_{n-m} & \b O\\
\b F & \b I_m
\end{array}\right]\left[\begin{array}{c}
\b T^g\\
\b B^g
\end{array}\right].
\label{eq:exp_for_inv_wb}
\end{eqnarray}
Thus the inverse of the matrix $\left[\begin{array}{cc}
\b W & \b B
\end{array}\right]$ has the form
\[
\left[\begin{array}{cc}
\b W & \b B
\end{array}\right]^{-1}=\left[\begin{array}{c}
\b W^g\\
\b B^g+\b F\b T^g
\end{array}\right],
\]
where we invoked~\eqref{eq:wg_equals_tg}, that is, $\b W^g=\b T^g$. Taking into account~\eqref{eq:inverting_wb}, we obtain
\begin{equation}
\fbox{$\displaystyle \b S=\b B^g+\b F\b T^g $}
\label{eq:switch_surf_formu}
\end{equation}

We combine the above results into a sliding surface design algorithm.

\begin{algorithm}\label{algo:switchi_surf_design}\rm
\begin{center}
{\bf Switching Surface Construction}
\end{center}
\begin{enumerate}
\item Find a matrix $\b T\in \mb{\Bb R}^{n\times (n-m)}$ such that $\left[\begin{array}{cc}
\b T & \b B
\end{array}\right]$ is invertible.
\item Compute
\[
\left[\begin{array}{cc}
\b T & \b B
\end{array}\right]^{-1}=\left[\begin{array}{c}
\b T^{g}\\
\b B^{g}
\end{array}\right].
\]
\item Find a matrix $\b F\in \mb{\Bb R}^{m\times (n-m)}$ such that
\[
\mb{eig}\left( \b T^{g}\b A\b T -\b T^{g}\b A\b B\b F\right)=\{\lambda_1,\lambda_2,\ldots,\lambda_{n-m}\}.
\]
\item Construct the sliding surface matrix  $\b S$ using the relation,
\[
\b S=\b B^g+\b F\b T^g.
\]
\end{enumerate}
\end{algorithm}\markex

We illustrate the above algorithm with a simple numerical example.
\begin{example}\rm
Given a dynamical system model of the form~(\ref{eq:plmofosude}), where
\[
\b A=\left[\begin{array}{ccc}
0 & 1 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{array}\right],\quad \b B=\left[\begin{array}{cc}
0 & 0\\
1 & 0\\
0 & 1
\end{array}\right].
\]
We design a sliding surface
\[
\b \sigma=\b S\b x=\b 0
\]
so that the system restricted to it will behave like a linear system described by
\[
\dot{z}=-z.
\]

Following the steps of the of the design algorithm, we first find a matrix $\b T$ such that the augmented matrix $\left[\begin{array}{cc}
\b T & \b B
\end{array}\right]$ is invertible. For example, we can select
\[
\b T=\left[\begin{array}{ccc}
1 & 0 & 0
\end{array}\right]^{\top}.
\]
Note that
\[
\left[\begin{array}{cc}
\b T & \b B
\end{array}\right]=\b I_3.
\]
Hence, the inverse
\[
\left[\begin{array}{cc}
\b T & \b B
\end{array}\right]^{-1}=\left[\begin{array}{c}
\b T^g\\
\b B^g
\end{array}\right]=\b I_3.
\]
Next, we find $\b F\in \mb{\Bb R}^{m \times(n-m)}$ such that
\[
\b T^g \b A\b T-\b T^g\b A\b B\b F=-1.
\]
We have
\begin{eqnarray*}
\b T^g \b A\b T-\b T^g\b A\b B\b F &=& \left[\begin{array}{ccc}
1 & 0 & 0
\end{array}\right]\left[\begin{array}{ccc}
0 & 1 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{array}\right]\left[\begin{array}{c}
1\\
0\\
0
\end{array}\right]\\
&{}&\quad -\left[\begin{array}{ccc}
1 & 0 & 0
\end{array}\right]\left[\begin{array}{ccc}
0 & 1 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{array}\right]\left[\begin{array}{cc}
0 & 0\\
1 & 0\\
0 & 1
\end{array}\right]\left[\begin{array}{c}
f_{11}\\
f_{21}
\end{array}\right]\\
&=& - f_{11}\\
&=&-1.
\end{eqnarray*}
We select
\[
\b F=\left[\begin{array}{c}
1\\
0
\end{array}\right].
\]
We then compute
\begin{eqnarray*}
\b S &=& \b B^g+\b F\b T^g\\
&=& \left[\begin{array}{ccc}
0 & 1 & 0\\
0 & 0 & 1
\end{array}\right] + \left[\begin{array}{c}
1\\
0
\end{array}\right]\left[\begin{array}{ccc}
1 & 0 & 0
\end{array}\right].
\end{eqnarray*}
Hence,
\[
\b S=\left[\begin{array}{ccc}
1 & 1 & 0\\
0 & 0 & 1
\end{array}\right].
\]
\end{example}
\markex

\section{Sliding Mode and System Zeros}
\label{sec:sli_system_zeros}
In this section, we show that the poles of the reduced-order system in sliding\index{system in sliding!reduced-order} are the system zeros\index{system in sliding poles!system zeros}\index{invariant zeros|see{system zeros}} of the dynamical system described by
\begin{equation}
\left\{\begin{array}{lll}
\dot{\b x} &=& \b A\b x+\b B\b u\\
\b \sigma &=& \b S\b x,
\end{array}\right.
\label{eq:syzereo}
\end{equation}
where $\b S\in \mb{\Bb R}^{m \times n}$ and we assume that $\det \b S\b B \ne 0$. Recall from Section~\ref{sec:system_zeros} that the system zeros, or invariant zeros, are defined using the system matrix\index{system matrix}. The system matrix, $\b P(s)$, corresponding to the plant~(\ref{eq:syzereo}) has the form
\begin{equation}
\b P(s) = \left[\begin{array}{cc}
s\b I_n-\b A & -\b B\\
\b S & \b O
\end{array}\right].
\label{eq:sqsyma}
\end{equation}
The system matrix $\b P(s)$ corresponding to the plant~(\ref{eq:syzereo}) is a square matrix of order $n+m$. The zeros of the determinant of the system matrix~(\ref{eq:sqsyma}) are the system zeros of~(\ref{eq:syzereo}). The system zeros are invariant under\index{system zeros!invariant under set of transformations} the following set of transformations:
\begin{enumerate}
\item nonsingular state variable transformation;
\item nonsingular transformation of the inputs;
\item nonsingular transformation of the outputs;
\item state-feedback to the inputs;
\item output-feedback to the rates of change of the states.
\end{enumerate}

To proceed, we evaluate the determinant of the system matrix~(\ref{eq:sqsyma}). To facilitate this task, we premultiply  the system matrix by the matrix
\[
\left[\begin{array}{cc}
\b I_n & \b O\\
-\b S (s\b I_n-\b A)^{-1} & \b I_m
\end{array}\right].
\]
The determinant of the above matrix is unity and so the determinant of the product is the same as the determinant of the system matrix,
\begin{eqnarray}
\det \b P(s) &=&\det \left(\left[\begin{array}{cc}
\b I_n & \b O\\
-\b S (s\b I_n-\b A)^{-1} & \b I_m
\end{array}\right]\left[\begin{array}{cc}
s\b I_n-\b A & -\b B\\
\b S & \b O
\end{array}\right]\right)\nonumber\\
&=& \det \left[\begin{array}{cc}
s\b I_n-\b A & -\b B\\
\b O & \b S (s\b I_n-\b A)^{-1}\b B
\end{array}\right]\nonumber\\
&=&\det \left( s\b I_n-\b A \right)\det \left( \b S(s \b I_n - \b A)^{-1}\b B\right).
\label{eq:deofsymaps}
\end{eqnarray}
We next evaluate the characteristic polynomial of the equivalent system~(\ref{eq:eqsyinsli}). While performing manipulations, we use the result of Exercise~\ref{ex:detxyyx} to obtain
\begin{eqnarray}
{}&{}&\det \left( s\b I_n-\b A+\b B(\b S\b B)^{-1}\b S\b A\right)\nonumber\\ &{}&\quad =\det(s\b I_n-\b A) \det \left( \b I_n+(s\b I_n-\b A)^{-1}\b B(\b S\b B)^{-1}\b S\b A \right)\nonumber\\
&{}&\quad =\det(s\b I_n-\b A) \det \left( \b I_m+\b S\b A(s\b I_n-\b A)^{-1}\b B(\b S\b B)^{-1} \right).
\label{eq:prdesaisa}
\end{eqnarray}
In our further manipulations, we use the identity
\begin{equation}
\b A\left( \mb{Adj}(s\b I_n-\b A)\right)=s\, \mb{Adj}(s\b I_n-\b A)-\det(s\b I_n-\b A)\b I_n.
\label{eq:usidwadm}
\end{equation}
The above follows from the fact that
\[
(s\b I_n-\b A)(s\b I_n-\b A)^{-1}=\b I_n.
\]
which is equivalent to
\begin{equation}
(s\b I_n-\b A)\mb{Adj}(s\b I_n-\b A)=\det(s\b I_n-\b A)\b I_n.
\label{eq:eqidwadm}
\end{equation}
Then we use~(\ref{eq:usidwadm}) to get
\begin{eqnarray}
{}&{}&\det \left( \b I_m+\b S\b A(s\b I_n-\b A)^{-1}\b B(\b S\b B)^{-1} \right)\nonumber\\
&{}&=\det \left(\frac{\det(s\b I_n-\b A)\b I_m +\b S\left( s\,\mb{Adj}(s\b I_n-\b A)-\det(s\b I_n-\b A)\b I_n \right)\b B(\b S\b B)^{-1} }{\det(s\b I_n - \b A)} \right)\nonumber\\
&{}&=\det \left(\frac{\det(s\b I_n-\b A)\b I_m +s \b S\,\mb{Adj}(s\b I_n-\b A)-\det(s\b I_n-\b A)\b I_n (\b S\b B)(\b S\b B)^{-1} }{\det(s\b I_n - \b A)} \right)\nonumber\\
&{}&=\det \left(\frac{s\b S\, \mb{Adj}(s\b I_n-\b A)\b B(\b S\b B)^{-1} }{\det(s\b I_n - \b A)} \right)\nonumber\\
&{}&=s^m\det\left( \b S(s\b I_n-\b A)^{-1}\b B\right)\det (\b S\b B)^{-1}.
\label{eq:part_evalleadtosz}
\end{eqnarray}
Using~(\ref{eq:part_evalleadtosz}), we represent~(\ref{eq:prdesaisa}) as
\begin{eqnarray*}
{}&{}&\det \left( s\b I_n-\b A+\b B(\b S\b B)^{-1}\b S\b A\right)\\
&{}&\quad =s^m\det (\b S\b B)^{-1} \det(s\b I_n-\b A) \det\left( \b S(s\b I_n-\b A)^{-1}\b B\right).
\end{eqnarray*}
Taking into account~\eqref{eq:deofsymaps} gives
\begin{equation}
\fbox{$ \displaystyle \det \left( s\b I_n-\b A+\b B(\b S\b B)^{-1}\b S\b A\right)=\det (\b S\b B)^{-1}s^m \det \b P(s) $}
\label{eq:zeofabsbisa}
\end{equation}


Thus the poles of the equivalent system~\eqref{eq:eqsyinsli} contain $m$ poles at the origin and the remaining $n-m$ poles are the system zeros of the system~\eqref{eq:syzereo} characterized by the triple $(\b A, \b B, \b S)$.


We conclude that the dynamics, that is, the poles, of the system modeled by~(\ref{eq:syzereo}) in sliding along $\b \sigma =\b S\b x$ are determined by its system zeros. This fact leads to the observation that the switching surface design\index{switching surface design!system zeros locations} can be viewed as choosing an output matrix $\b S$ so that the system~(\ref{eq:syzereo}) has a desired set of system zeros locations, which in turn govern the dynamics of the system in sliding along $\b S\b x=\b 0$.





\section{Regulator Design}
In this Section, we present a regulator design algorithm for a class of nonlinear
dynamical systems using an approach that is rooted in variable structure sliding mode control.


This design methodology consists of two steps: the manifold
design and the control law design. We use Algorithm~\ref{algo:switchi_surf_design} on page~\pageref{algo:switchi_surf_design} to
construct an asymptotically stable invariant manifold so
that all trajectories confined to this manifold converge to the
origin. Once a manifold is constructed, the control strategy is obtained by solving the first-order matrix differential equation. The proposed design procedure is tested on the model of
a two-link robotic manipulator from Section~\ref{subsection:two_link_robot}.

\subsection{Outline of the Approach}
Consider a nonlinear dynamical system modeled as
\begin{equation}\label{eq:nonlinear}
\dot{\b x} = \b{f}( \b x) + \b B ( \b x ) \b u,
\end{equation}
where $\b x$ is a vector of state variables of dimension $n$,
$\b u$ is the input vector of dimension $m$, $\b{f}(\b x)$ is a
state-dependent function of dimension $n$, $\b B(\b x)$ is a
state-dependent input function matrix of dimension $n\times m$
and it has full column rank everywhere.


Let $\b{\sigma}(\b x) = \b{S}\b x:\mb{\Bb R}^n\to \mb{\Bb R}^m$ be a
vector function of dimension $m$.


The control objective is to devise a control policy that stabilizes
the system~(\ref{eq:nonlinear}) by
steering its trajectories to a
manifold,\index{manifold}
\begin{equation}
\cal{M} = \{\b x : \b{\sigma}(\b x) = \b 0\}
\label{eq:manifold}
\end{equation}
and forces the trajectories to stay on the manifold thereafter.

We further require that, (i) the system~(\ref{eq:nonlinear})
confined to the manifold $\cal{M} = \{\b x : \b{\sigma} = \b  0\}$
is asymptotically stable to the origin, (ii) $\b{\sigma}(\b x)=\b S\b x$ is
constructed so that the square $m\times m$ matrix
\[
\b{\sigma}_{\b x}
(\b x) \b B (\b x)=\b S\b B (\b x)
\]
is invertible, where $\b{ \sigma}_{\b x}$ is the
Jacobian matrix of $\b{ \sigma}(\b x)$ with respect to $\b x$. We will show that
when these two conditions are satisfied, the solution to the above
control problem is obtained by solving the associated
first-order matrix differential equation,\index{first-order matrix differential equation}\index{matrix differential equation}
\begin{equation}\label{eq:odesoc}
\b T \dot{\b{\sigma}} + \b{\sigma} = \b 0,
\end{equation}
where $\b T\in \mb{\Bb R}^{m\times m}$ is a symmetric positive
definite matrix, which is a design parameter. Substituting~\eqref{eq:nonlinear} into~\eqref{eq:odesoc} yields
\begin{eqnarray*}
\b T \dot{\b{\sigma}} + \b{\sigma} &=& \b T\b S \dot{\b x} + \b S\b x\\
&=& \b T\b S \b f(\b x) + \b T\b S \b B(\b x)\b u + \b S\b x\\
&=& \b 0.
\end{eqnarray*}
Solving the above gives
\begin{eqnarray}
\b u &=& - (\b T
\b S\b B(\b x))^{-1}\left(\b T \b S
\b{f}(\b x) +  \b S\b x\right)\nonumber\\
&=& - (\b S\b B(\b x))^{-1}\b S
\b{f}(\b x) - (\b S\b B(\b x))^{-1}\b T^{-1}\b S\b x.
\label{eq:lawsoc}
\end{eqnarray}

The use of the control law~(\ref{eq:lawsoc}) in the nonlinear
dynamical systems~(\ref{eq:nonlinear}) makes the invariant
manifold $\cal{M} = \{\b x:\b{\sigma}=\b 0\}$ attractive. The
control law~(\ref{eq:lawsoc}) drives closed-loop trajectories
of~(\ref{eq:nonlinear}) from any initial condition towards the
invariant manifold $\cal{M}$. As stated above, $\b{\sigma}$ is constructed in such a way that
all trajectories of~(\ref{eq:nonlinear}) confined on $\cal{M}$
converge to the origin. Therefore, the stability of the
closed-loop trajectories is achieved in two modes: (i) closed-loop
trajectories converging to the manifold $\cal{M}
=\{\b x:\b{\sigma}=\b 0\}$, and (ii) closed-loop trajectories
moving along the manifold $\cal{M} =\{\b x:\b{\sigma}=\b 0\}$
towards the origin.

The design method employs the following two-step design procedure:
\begin{enumerate}
    \item Construct an asymptotically stable invariant manifold\index{invariant manifold}\index{manifold!invariant} $\cal{M} =\{\b x:\b{\sigma}=\b 0\}$,
    (i.e. all trajectories of the system confined to $\cal{M}$ converge asymptotically to the origin).
    \item Construct the controller ${\b u}$ using~(\ref{eq:lawsoc}).
\end{enumerate}


This  approach shares similarity with that of the
 variable structure sliding mode control~(SMC).\index{sliding mode control (SMC)}\index{SMC|see{sliding mode control}}
This relation was discussed in~\cite{santi03,santi04,nwzakcrossleygnc07}.
%It was observed in~\cite{nwzakcrossleygnc07} that the first term of the control
%policy~(\ref{eq:lawsoc}) is the equivalent
%control, while the second term may be viewed as an
%``{approximation to the discontinuous control law}'' of the
%SMC.
In the development of the proposed regulator, we
adopt the integral tracking method  used in the sliding mode
tracking controller design described in~\cite{edwards98}. The resulting controller drives
the closed-loop dynamics so that the integral of the tracking
error\index{integral of the tracking
error} converges to zero.

\subsection{System Description}

The class of nonlinear systems that we consider is modeled by
\begin{eqnarray}
\dot{ \b x } &=& \b{f}( \b x) + \b B ( \b x ) \b u\nonumber\\
&=& \b A(\b x) \b x + \hat{\b B} \tilde{\b u},
\label{eq:nonlinear-controllable}
\end{eqnarray}
where we assume that the system~(\ref{eq:nonlinear-controllable})  is
transformable into the following ``regular form''~\cite{utkin92,edwards98},\index{regular form}
\begin{equation}\label{eq:regular}
\left[\begin{array}{c} \dot{\b x}_1 \\
\dot{\b x}_2 \end{array}\right] = \left[\begin{array}{cc}
\b A_{11} &
\b A_{12} \\
\b A_{21}(\b x) & \b A_{22}(\b x) \end{array}\right]
\left[\begin{array}{c} \b x_1
\\ \b x_2 \end{array}\right] + \left[\begin{array}{c}
\b{O} \\ \b B_u \end{array}\right] \tilde{\b u},
\end{equation}
where $\b x_1$ is in $\mb{\Bb R}^{n-m}$, $\b x_2$ is in $\mb{\Bb R}^m$, and the control
vector $\b u$ is in $\mb{\Bb R}^m$. The matrix $\b{O} \in
\mb{\Bb R}^{(n-m)\times m}$ is a matrix with zero entries. The
matrix $\b B_u \in \mb{\Bb R}^{m \times m}$ is invertible and
the matrices $\b A_{ij}$, $i=1,2$, $j=1,2$ have appropriate
dimensions. Note that $\b A_{1j}$, $j=1,2$, are constant
matrices, while $\b A_{2j}$, $j=1,2$, are state-dependent
matrices. For a constructive algorithm\index{regular form!constructive algorithm} that can be used to
transform~(\ref{eq:nonlinear-controllable}) into the regular
form given by~(\ref{eq:regular}), we refer to~\cite[p.~66]{edwards98}.

An example of physical systems that can be modeled using the above non-linear regular form are natural Lagrangian systems. A simple example of a natural Lagrangian system\index{natural Lagrangian system!planar two-link robot} whose model can be represented in the regular form is a planar two-link robot\index{planar two-link robot!natural Lagrangian system} presented in Section~\ref{subsection:two_link_robot}. Indeed, let $\b x_1=\left[\theta_1\; \theta_2\right]^\top$ be the vector of joint angles, $\b\tau=[\tau_1\;\tau_2]^\top$ be the vector of torques applied to the joints. Then the dynamics of the two-link robotic manipulator can be represented as
\begin{equation}\label{eqn:tlkoriginal}
\b H(\b x_1)\ddot{\b x}_1+\b C(\b x_1,\dot{\b x}_1)\dot{\b x}_1 + \b g(\b x_1)=\b\tau,
\end{equation}
where $\b H(\b x_1)$ is the manipulator inertia\index{manipulator inertia} matrix and $\b C(\b x_1,\dot{\b x}_1)\dot{\b x}_1$ is the vector of centripetal and Coriolis torques\index{centripetal torques}\index{Coriolis torques}, and $\b g(\b x_1)=\left[\begin{array}{cc}
g_1(\b x_1) & g_2(\b x_1)
\end{array}\right]^{\top}$ is the vector of gravitational torques. Because the inertia matrix $\b H(\b x_1)$ is positive definite over the whole workspace\index{positive definite over workspace}, we know that $\b H^{-1}(\b x_1)$ exists and is also positive definite. Let
\[
\b G(\b x_1)=\left[\begin{array}{cc}
\frac{g_1(\b x_1)}{x_1} & 0\\
0 & \frac{g_2(\b x_1)}{x_2}
\end{array}\right].
\]
Using the above we represent the term $\b g(\b x_1)$ as
\begin{equation}
\b g(\b x_1)=\b G(\b x_1)\b x_1.
\label{eqn:gterm}
\end{equation}
Taking into account~\eqref{eqn:gterm}, we rewrite~\eqref{eqn:tlkoriginal} as
\begin{equation}\label{eqn:tlk}
\ddot{\b x}_1=-\b H^{-1}(\b x_1)\b
C(\b x_1,\dot{\b x}_1)\dot{\b x}_1 - \b H^{-1}(\b x_1)\b G(\b x_1)\b x_1 + \b H^{-1}(\b x_1)\b\tau.
\end{equation}
Let $\b x_2=\dot{\b x}_1$ and $\tilde{\b u}=\b H^{-1}(\b x_1)\b\tau$.
Then, the dynamics of the planar two-link robot modeled by~\eqref{eqn:tlk} can be represented in the form
of~\eqref{eq:regular}, that is,
\begin{equation}\label{eq:two_link_regular}
\left[\begin{array}{c} \dot{\b x}_1 \\
\dot{\b x}_2 \end{array}\right] = \left[\begin{array}{cc}
\b O &
\b I_2 \\
- \b H^{-1}(\b x_1)\b G(\b x_1) & -\b H^{-1}(\b x_1)\b
C(\b x_1,\b x_2) \end{array}\right]
\left[\begin{array}{c} \b x_1
\\ \b x_2 \end{array}\right] + \left[\begin{array}{c}
\b{O} \\ \b I_2 \end{array}\right] \tilde{\b u}.
\end{equation}


\subsection{Regulator Construction}

Suppose that the output $\b y\in \mb{\Bb R}^p$ of system~(\ref{eq:regular}) has the form,
\begin{equation}
\b y = \left[\begin{array}{cc} \b C_1 & \b C_2
\end{array}\right]\left[\begin{array}{c}
\b x_1 \\ \b x_2 \end{array}\right].
\end{equation}
Our objective is to design a regulator so that the output $\b y$
tracks a constant reference vector $\b r \in
\mb{\Bb R}^p$. We introduce the vector variable $\b x_r$
defined by the following differential equation,
\begin{equation}\label{eq:errordynamic}
\dot{\b x}_r = \b r - \b y.
\end{equation}
The controller should be designed in such a way so that
\[
\lim_{t\to \infty} \dot{\b x}_r(t)=\b 0.
\]


To proceed, we define the augmented state vector,\index{augmented state vector}
\begin{equation}\label{eq:augmentedvector} \tilde{\b x}
= \left[\begin{array}{c} \b x_r \\ \b x
\end{array}\right] = \left[\begin{array}{c}
\b x_r \\ \hline \\ \underbrace{\left \{ \begin{array}{c} \b x_1 \\
\b x_2 \end{array} \right \}}_{\b x}  \end{array}\right]\in \mb{\Bb R}^N,
\end{equation}
where $N=n+p$. The dynamics of the overall augmented system\index{augmented system dynamics} are governed by the
state equation,
\begin{equation}\label{eq:augmenteddynamics}
 \left[\begin{array}{c} \dot{\b x}_r \\
\dot{\b x}_1 \\
\dot{\b x}_2 \end{array}\right] =
\underbrace{\left[\begin{array}{ccc} \b{O} &
-\b C_1 & -\b C_2 \\
\b{O} & \b A_{11} & \b A_{12} \\
\b{O} & \b A_{21}(\b x) & \b A_{22}(\b x)
\end{array}\right]}_{\tilde{\b A}} \left[\begin{array}{c} \b x_r \\
\b x_1 \\ \b x_2
\end{array}\right] + \underbrace{\left[\begin{array}{c} \b{I}_p \\ \b{O} \\ \b{O}
\end{array}\right]}_{\b B_r} \b r +
\underbrace{\left[\begin{array}{c} \b{O} \\ \b{O} \\ \b B_u
\end{array}\right]}_{\tilde{\b B}} \b u.
\end{equation}
Note that the matrices in the augmented state
equation~(\ref{eq:augmenteddynamics}) are labeled as
$\tilde{\b A}$, $\b B_r$, and $\tilde{\b B}$, respectively.
Hence, the augmented system model~(\ref{eq:augmenteddynamics})
can be represented as
\begin{equation}\label{eq:compact_augmented}
\dot{\tilde{\b x}} = \tilde{\b A} \tilde{\b x} + \b B_r \b r +
\tilde{\b B} \b u.
\end{equation}
We make the following assumption:

{\bf Assumption 1}: The pair
\[
\left(\left[\begin{array}{cc}
\b O & -\b C_1\\
\b O & \b A_{11}
\end{array}\right], \quad \left[\begin{array}{c}
-\b C_2\b B_u\\
\b A_{12}\b B_u
\end{array}\right]\right)
\]
is controllable.


An essential element of the regulator design is the construction of
an invariant manifold $\b{\sigma}=\b 0$. Let the manifold function $\b{\sigma}$ be defined as
\begin{equation}\label{eq:manifoldfunction}
\b{\sigma} = \b{\sigma}(\tilde{\b x})= \b{\sigma}(\b x_r, \b x_1, \b x_2) = -
\b S_r \b x_r + \b S_{1} \b x_1 + \b S_{2} \b x_2 =
\left[\begin{array}{ccc} -\b S_r & \b S_{1} & \b S_{2}
\end{array}\right] \tilde{\b x} = \tilde{\b S}\tilde{\b x}.
\end{equation}
The negative sign in front of $\b S_r$ is for the purpose of simplified notation.
We construct $\b{\sigma}$ so that the system modeled
by~(\ref{eq:augmenteddynamics}) restricted to the manifold\index{manifold!construction of}
$\cal{M}=\{\tilde{\b x}:\b{\sigma}(\tilde{\b x})=\b 0\}$ is asymptotically stable,
that is, the  trajectories of the system restricted to the
manifold $\cal{M}$ converge asymptotically towards the origin. We can use Algorithm~\ref{algo:switchi_surf_design} to construct such a manifold. We assume that
\begin{equation}\label{eq:detSB}
 \det\left( \tilde{\b S}\tilde{\b B}
\right) = \det\left(\b S_2 \b B_u  \right) \ne 0.
\end{equation}
Using~\eqref{eq:odesoc}, we perform manipulations to obtain,
\begin{eqnarray}
\b 0&=&\b T \dot{\b{\sigma}} + \b{\sigma} \nonumber \\
&=& \b T\tilde{\b S} \dot{\tilde{\b x}} +
\tilde{\b S} \tilde{\b x} \nonumber \nonumber \\
&=& \b T\tilde{\b S} (\tilde{\b A} \tilde{\b x} + \b B_r \b r +
\tilde{\b B}
\tilde{\b u}) + \tilde{\b S} \tilde{\b x} \nonumber \nonumber \\
&=& \b T\tilde{\b S}\tilde{\b A} \tilde{\b x} + \b T \tilde{\b S}
\b B_r \b r + \b T \tilde{\b S} \tilde{\b B}\tilde{\b u} + \tilde{\b S}
\tilde{\b x}. \label{eq:raw4control}
\end{eqnarray}
It follows from~\eqref{eq:manifoldfunction} and (\ref{eq:augmenteddynamics}) that $\tilde{\b S}\b B_r = -\b S_r$.
Hence, we can re-write~(\ref{eq:raw4control}) as
\begin{equation}\label{eq:clean4control}
 \b 0 =\b T\tilde{\b S}\tilde{\b A} \tilde{\b x} - \b T \tilde{\b S}_r \b r
 + \b T \tilde{\b S} \tilde{\b B}\tilde{\b u} + \tilde{\b S}
\tilde{\b x}.
\end{equation}
Solving~(\ref{eq:clean4control}) for $\b u$ yields
\begin{equation}\label{eq:soclawtracking}
\fbox{$\displaystyle \tilde{\b u} = -(\tilde{\b S}\tilde{\b B})^{-1}(\tilde{\b S} \tilde{\b A}
\tilde{\b x} + \b T^{-1} \tilde{\b S} \tilde{\b x} - \b S_r \b r)$}
\end{equation}


Substituting~(\ref{eq:soclawtracking}) into~(\ref{eq:augmenteddynamics}),
we obtain the closed-loop system dynamics model,
\begin{equation}\label{eq:closedlooptracking}
\dot{\tilde{\b x}} = \left(\b{I} - \tilde{\b B}
(\tilde{\b S}\tilde{\b B})^{-1} \tilde{\b S}\right)\tilde{\b A}
\tilde{\b x} - \tilde{\b B} (\tilde{\b S}\tilde{\b B})^{-1}
\b T^{-1}\tilde{\b S} \tilde{\b x} +
\tilde{\b B}(\tilde{\b S}\tilde{\b B})^{-1}\b S_r \b r+\b B_r
\b{r}.
\end{equation}

In the following subsection, we show that the control law given by~\eqref{eq:soclawtracking} forces the above closed-loop system to track the reference signal $\b r$.


\subsection{Analysis of the Closed-Loop System Behavior}

To proceed with our analysis, we need the following lemma, which can be viewed as an extension of Theorem~\ref{thm:main_slid_the} to a class of non-liner systems.
\begin{lemma}\label{lem_pre_construct}
Consider the system model given by~\eqref{eq:augmenteddynamics}, where
$\mb{rank}\;(\tilde{\b B})=m$, Assumption~1 is satisfied, and a
given set of $N-m$ complex numbers,
$\{\lambda_1,\ldots,\lambda_{N-m}\}$ is such that any $\lambda_i$
whose imaginary part is nonzero appears in the above set in a
conjugate pair. Then, there exist full-rank matrices $\b W\in
\mb{\Bb R}^{N\times (N-m)}$ and $\b W^g\in \mb{\Bb R}^{(N-m)\times
N}$ such that
\[
\b W^g\b W=\b I_{N-m},\quad \b W^g\tilde{\b B}=\b O,
\]
and the eigenvalues of $\b W^g\b A\b W$ are
\[
\mb{eig}\,(\b W^g\tilde{\b A}\b W)=\{\lambda_1,\ldots, \lambda_{N-m}\}.
\]
\end{lemma}
{\bf Proof: }  The proof of this lemma is based on the proof of Theorem~\ref{thm:main_slid_the} with an emphasis on the special structure of the non-linear plant. We begin our proof by
defining an $N\times (N-m)$ real matrix of the form,
\begin{equation}\label{eq:v11}
\b V = \left[\begin{array}{c}
\b I_{N-m}\\
\hline  \b O
\end{array}\right].
\end{equation}
Therefore,
\[
\left[\begin{array}{cc} \b V & \tilde{\b B}\end{array}\right]^{-1}
= \left[\begin{array}{cc}\b I_{N-m} & \b O \\ \b O &
\b B_u\end{array}\right]^{-1} = \left[\begin{array}{cc}
\b I_{N-m} &\b O \\ \b O & \b B_u^{-1} \end{array} \right] =
\left[\begin{array}{c} \b V^g \\ \tilde{\b B}^g \end{array}\right].
\]
Hence,
\[
\left[\begin{array}{c} \b V^g \\ \tilde{\b B}^g \end{array}\right]\left[\begin{array}{cc} \b V & \tilde{\b B}\end{array}\right]=\left[\begin{array}{cc}
\b V^g\b V & \b V^g\tilde{\b B}\\
\tilde{\b B}^g\b V & \tilde{\b B}^g\tilde{\b B}
\end{array}\right]=\left[\begin{array}{cc}
\b I_{N-m} & \b O\\
\b O & \b I_m
\end{array}\right]=
\b I_N.
\]
Note that
\begin{equation}
\b V^g\b V=\b I_{N-m}\;\;\mbox{ and }\;\; \b V^g\tilde{\b B}=\b O.
\label{eq:provgandb}
\end{equation}
By Assumptions~1, the pair
\[
\left(\b V^g \tilde{\b A} \b V,   \b V^g
\tilde{\b A} \tilde{\b B}\right)=\left(\left[\begin{array}{cc}
\b O & -\b C_1\\
\b O & \b A_{11}
\end{array}\right], \quad \left[\begin{array}{c}
-\b C_2\b B_u\\
\b A_{12}\b B_u
\end{array}\right]\right)
\]
is controllable. Therefore, for any set of complex numbers,
\[
\left\{\lambda_1,\lambda_2,\ldots,\lambda_{N-m}  \right\}
\]
symmetric with respect to the real axis, we can always find a
matrix $\b F$ such that
\[
\{\lambda_1,\lambda_2,\ldots,\lambda_{N-m}\} =
\mbox{eig} \left(\b V^g \tilde{\b A} \b V - \b V^g
\tilde{\b A} \tilde{\b B} \b{F}\right).
\]
The matrix $\b{F}$ can be determined using any available method
for pole placement.\index{pole placement} We proceed by performing the
following manipulations,
\begin{equation}
\b V^g \tilde{\b A} \b V - \b V^g \tilde{\b A} \tilde{\b B}
\b{F}= \b V^g\tilde{\b A}\left(\b V-\tilde{\b B}\b{F}\right) =
\b{W}^g \tilde{\b{A}} \b{W}, \label{eq:fromvtow}
\end{equation}
where $\b W^g=\b V^g$ and $\b W = \b V-\tilde{\b B}\b{F}$. Note
that by~(\ref{eq:provgandb}),
\begin{equation}
\b W^g\b W = \b V^g\left(\b V-\tilde{\b B}\b{F}  \right) = \b
V^g\b V = \b I_{N-m}. \label{eq:wgwisinmm}
\end{equation}
We also have, $\b V^g\tilde{\b B}=\b W^g\tilde{\b B}=\b O$, which completes the proof of the lemma.\\
\hspace*{5.5in} $\Box$
%\end{proof}

\begin{thm}\label{thm_tracking}
Suppose that the assumptions of Lemma~\ref{lem_pre_construct} are satisfied. Then there exists an invariant manifold matrix $\tilde{\b S}\in\mb{\Bb R}^{m\times N}$
such that $\tilde{\b S}\tilde{\b B}=\b I_m$ and the closed-loop system~\eqref{eq:closedlooptracking} asymptotically tracks the reference signal $\b r$, that is, $\b y(\infty)=\b r$.
\end{thm}
{\bf Proof: }  Consider the matrix, $\left[\begin{array}{cc}
\b W & \tilde{\b B}
\end{array}\right]$. Note that because $\b W = \b V-\tilde{\b B}\b{F}$, we have
\[
\left[\begin{array}{cc}
\b W & \tilde{\b B}
\end{array}\right]=\left[\begin{array}{cc}
\b V & \tilde{\b B}
\end{array}\right]\left[\begin{array}{cc}
\b I_{N-m} & \b O\\
-\b F & \b I_m
\end{array}\right].
\]
Thus the matrix $\left[\begin{array}{cc}
\b W & \tilde{\b B}
\end{array}\right]$ is invertible and its inverse has the form
\begin{eqnarray*}
\left[\begin{array}{cc}
\b W & \tilde{\b B}
\end{array}\right]^{-1} &=&\left[\begin{array}{cc}
\b I_{N-m} & \b O\\
-\b F & \b I_m
\end{array}\right]^{-1}\left[\begin{array}{cc}
\b V & \tilde{\b B}
\end{array}\right]^{-1}\\
&=& \left[\begin{array}{cc}
\b I_{N-m} & \b O\\
\b F & \b I_m
\end{array}\right]\left[\begin{array}{c}
\b V^g\\
\tilde{\b B}^g
\end{array}\right]\\
&=&\left[\begin{array}{c}
\b V^g\\
\tilde{\b B}^g+\b F\b V^g
\end{array}\right].
\end{eqnarray*}
As in~\eqref{eq:switch_surf_formu} in the proof of Theorem~\ref{thm:main_slid_the}, we take
\begin{eqnarray*}
\tilde{\b S} &=& \tilde{\b B}^g+\b F\b V^g\\
&=&\left[\begin{array}{ccc}
\b O & \b O & \b B_u
\end{array}\right]+\b F\left[\begin{array}{cc}
\b I_{N-m} & \b O
\end{array}\right]\\
&=&\left[\begin{array}{cc}
\b F & \b B_u^{-1}
\end{array}\right].
\end{eqnarray*}
Note that by~\eqref{eq:provgandb},
\[
\tilde{\b S}\tilde{\b B}=\left[\begin{array}{cc}
\b F & \b B_u^{-1}
\end{array}\right] \left[\begin{array}{c}
\b O\\
\b O\\
\b B_u
\end{array}\right]=\b I_m.
\]
Consider the following state-space transformation,
\[
\b z=\left[\begin{array}{c}
\b z_1\\
\b z_2
\end{array}\right]=\left[\begin{array}{c}
\b W^g\\
\tilde{\b S}
\end{array}\right]\tilde{\b x}=\left[\begin{array}{c}
\b V^g\\
\tilde{\b S}
\end{array}\right]\tilde{\b x}.
\]
Note that $\b z_2=\tilde{\b S}\tilde{\b x}=\b{\sigma}$. We also
have
\begin{equation}\label{eq:transformmatrix}
\left[\begin{array}{c}
\b W^g \\
\tilde{\b S}
\end{array}\right]\left[\begin{array}{cc}
\b W & \tilde{\b B}
\end{array}\right] = \left[\begin{array}{cc}
\b W^g\b W & \b W^g \tilde{\b B}\\
\tilde{\b{S}}\b W & \tilde{\b{S}}\tilde{\b B}\end{array}\right] =
\left[\begin{array}{cc}
\b{I}_{N-m} & \b{O} \\
\b{O} & \b{I}_m
\end{array}\right],
\end{equation}
where $\b{I}_{N-m}$ is the identity matrix of dimension
$(N-m)\times (N-m)$. Therefore, the matrices in~(\ref{eq:transformmatrix}) satisfy the
following relations,
\begin{equation}\label{eq:Wproperties}
\left.
\begin{array}{cc}
\begin{array}{rcl}
\b W^g \b W &=& \b{I}_{N-m},\\
\quad \tilde{\b{S}}\b W &=& \b{O} \in \mb{\Bb R}^{m\times (N-m)},\\
\end{array} &
\begin{array}{rcl}
\b W^g \tilde{\b B} &=& \b{O} \in \mb{\Bb R}^{(N-m)\times m},\\
\quad \tilde{\b{S}}\tilde{\b B}&=&\b{I}_{m}.
\end{array}
\end{array}\right\}
\end{equation}
Using matrix relations shown
in~(\ref{eq:Wproperties}), we represent the closed-loop model~\eqref{eq:closedlooptracking}
in the $\b z$ coordinates,
\begin{eqnarray}
\left[\begin{array}{c} \dot{\b z}_1 \\
\dot{\b z}_2 \end{array}\right] &=& \left[\begin{array}{c} \b W^g \\
\tilde{\b{S}} \end{array}\right] (\tilde{\b A} -
\tilde{\b B}(\tilde{\b{S}}\tilde{\b B})^{-1}\tilde{\b{S}}\tilde{\b A}
- \tilde{\b B}(\tilde{\b{S}}\tilde{\b B})^{-1}
\b T^{-1}\tilde{\b{S}}) \left[\begin{array}{cc} \b W & \tilde{\b B}
\end{array}\right] \left[\begin{array}{c} \b z_1 \\ \b z_2
\end{array}\right] \nonumber \\ && \quad
- \left[\begin{array}{c} \b W^g \\ \tilde{\b{S}} \end{array}\right]
\tilde{\b B}(\tilde{\b{S}}\tilde{\b B})^{-1} \tilde{\b{S}}\b B_r
\b r + \left[\begin{array}{c} \b W^g \\ \tilde{\b{S}}
\end{array}\right] \b B_r \b{r}
\nonumber  \\
&=& \left[\begin{array}{cc} \b W^g \tilde{\b A} \b W & \b W^g
\tilde{\b A}
\tilde{\b B} \\
\b{O} & -\b T^{-1} \end{array}\right] \left[\begin{array}{c}\b z_1 \\
\b z_2 \end{array}\right] + \left[\begin{array}{c} \b W^g \\
\tilde{\b{S}} \end{array}\right]\left(-
\tilde{\b B}(\tilde{\b{S}}\tilde{\b B})^{-1} \tilde{\b{S}} \b B_r
+\b B_r\right)\b r \nonumber \\
&=&\left[\begin{array}{cc} \b W^g \tilde{\b A} \b W & \b W^g
\tilde{\b A}
\tilde{\b B} \\
\b{O} & -\b T^{-1} \end{array}\right] \left[\begin{array}{c}\b z_1 \\
\b z_2 \end{array}\right] + \left[\begin{array}{c} \b{O} + \b{W}^g \b{B}_r \\
-\tilde{\b S} \b B_r + \tilde{\b S} \b B_r \end{array}\right]\b{r}    \nonumber \\
&=& \left[\begin{array}{cc} \b W^g \tilde{\b A} \b W & \b W^g
\tilde{\b A}
\tilde{\b B} \\
\b{O} & -\b T^{-1} \end{array}\right] \left[\begin{array}{c}\b z_1 \\
\b z_2 \end{array}\right] + \left[\begin{array}{c} \b{W}^g \b B_r
\\ \b{O} \end{array}\right] \b{r}.
\label{eq:closedloop_z}
\end{eqnarray}
Observe that $\b{W}^g\tilde{\b B}(\tilde{\b S} \tilde{\b
B})^{-1}\tilde{\b S} \b B_r = \b O$ because ${\b W^g
\tilde{\b B}=\b{O}}$. Since $\b T$ is positive definite, the eigenvalues of $-\b T^{-1}$
are all in the open left-hand complex plane. This implies
that the dynamics of $\b z_2=\b{\sigma}$ are asymptotically
stable, that is, the invariant manifold, $\{\b{\sigma}=\b 0\}$,
is asymptotically attractive. By~(\ref{eq:fromvtow}), the
matrix $\b W$ can be chosen so that the eigenvalues of the
constant matrix $\b W^g \tilde{\b A} \b W$ are in the open
left-hand complex plane. In
the steady-state the overall system trajectories converge to
the point on the manifold, $\{\b z_2=\b{\sigma}=\b 0\}$, given
by
\begin{equation}
\b{0} = \b W^g \tilde{\b A} \b W \b z_1 + \b W^g \b B_r \b r.
\label{eq:steady-state-z1-r}
\end{equation}
We partition $\b F$ as
\begin{equation}\label{eq:assume-F}
\b F = \left[\begin{array}{cc} \b F_1 & \b F_2 \end{array}\right],
\end{equation}
where $\b F_1 \in \mb{\Bb R}^{m \times p}$ and $\b F_2 \in
\mb{\Bb R}^{m \times (n-m)}$. We have,
\begin{equation}\label{eq:w-from-assume}
\b W = \b V - \tilde{\b B} \b F = \left[\begin{array}{cc}
    \b I_{p} & \b{O} \\
    \b{O} & \b I_{n-m} \\
    -\b B_u \b F_1 & - \b B_u \b F_2 \end{array}\right] \quad \mbox{and} \quad
    \b W^g \b B_r = \left[\begin{array}{l}\b I_p \\ \b{O}_{(n-m) \times p} \end{array}\right].
\end{equation}
Recall that $\b W^g=\b V^g$. Using the above and the fact that
\[
\b z_1=\b W^g\tilde{\b x}=\left[\begin{array}{c}
\b x_r\\
\b x_1
\end{array}\right],
\]
we express~(\ref{eq:steady-state-z1-r}) in the form,
\begin{eqnarray*}
\b{0} &=& \b V^g \left[\begin{array}{ccc} \b O & - \b C_1 & - \b
C_2 \\
\b O & \b A_{11} & \b A_{12} \\
\b O & \b A_{21}(\b x) & \b A_{22}(\b x) \end{array}\right] \b W
\b z_1 + \left[\begin{array}{c} \b I_p \\ \b O \end{array}\right]
\b r \\
& = & \left[\begin{array}{cc}
    \b C_2 \b B_u \b F_1 & -\b C_1 + \b C_2 \b B_u \b F_2 \\
    -\b A_{12} \b B_u \b F_1 & \b A_{11} - \b A_{12} \b B_u \b F_2
    \end{array}\right] \b W^g \tilde{\b x}+ \left[\begin{array}{c} \b I_p \\ \b O \end{array}\right]
\b r \\
& = & \left[\begin{array}{cc}
\b C_2 \b B_u \b F_1 & -\b C_1 + \b C_2 \b B_u \b F_2 \\
    -\b A_{12} \b B_u \b F_1 & \b A_{11} - \b A_{12} \b B_u \b F_2
    \end{array}\right] \left[\begin{array}{c}\b x_r \\ \b x_1
    \end{array}\right] +\left[\begin{array}{c} \b I_p \\ \b O \end{array}\right]
\b r,
\end{eqnarray*}
or, equivalently,
\begin{eqnarray}
\b{0} &=& \b C_2 \b B_u \b F_1 \b x_r -\left(\b C_1 - \b C_2 \b B_u \b F_2 \right)\b x_1 + \b r,
\label{eq:first-row} \\
\b{0} &=& -\b A_{12} \b B_u \b F_1 \b x_r + \left( \b A_{11} - \b A_{12} \b B_u \b F_2 \right) \b x_1. \label{eq:second-row}
\end{eqnarray}

In the steady-state, $\b z_2 = \b{0}$, that is,
\begin{eqnarray*}
\tilde{\b S}\tilde{\b x} &=& \left[\begin{array}{ccc}
-\b{S}_r & \b{S}_{1} & \b{S}_{2}
\end{array}\right]\left[\begin{array}{c}
\b x_r\\
\b x_1\\
\b x_2
\end{array}\right]\\
&=&\left[\begin{array}{ccc}
\b F_1 & \b F_2 & \b B_u^{-1}
\end{array}\right]\left[\begin{array}{c}
\b x_r\\
\b x_1\\
\b x_2
\end{array}\right]\\
&=&\b 0.
\end{eqnarray*}
Hence,
\begin{equation}
\b S_r \b x_r = \b S_1 \b x_1 + \b S_2 \b x_2.
\label{eq:S-steady-state}
\end{equation}
Substituting $\b S_2=\b B_u^{-1}$ and~(\ref{eq:S-steady-state})
into~(\ref{eq:first-row}), we obtain
\begin{eqnarray*}
\b{0} &=& -\b C_2 \b B_u \b S_r \b x_r - \b C_1 \b x_1 +
\b C_2 \b B_u \b S_1 \b x_1 + \b r \\
&=&\b C_2 \b B_u(- \b S_1 \b x_1 - \b S_2 \b x_2)-\b C_1 \b x_1 +
\b C_2 \b
B_u \b S_1 \b x_1 + \b r \\
&=& - \b C_2 \b x_2 - \b C_1 \b x_1 + \b r \\
&=& -\b y + \b r,
\end{eqnarray*}
that is, $\b y(\infty)=\b r$, which completes the proof of the theorem.\\
\hspace*{5.5in} $\Box$



\subsection{Example: Two-link robotic manipulator}
We synthesize a regulator\index{regulator} for the two-link robotic manipulator\index{two-link manipulator!regulator for} depicted in Figure~\ref{fig:dbl_pend} in Section~\ref{subsection:two_link_robot}.
%\begin{figure}
%\centering
%\includegraphics[scale=0.7]{DBL_PEND.eps}
%\caption{A two-link robotic manipulator.}
%\label{fig:dbl_pend}
%\end{figure}
Recall that the Lagrangian\index{Lagrangian!for two-link manipulator} has the form,
\begin{eqnarray*}
L&=&\frac{1}{2}m_1l_1^2\omega_1^2+\frac{1}{2}m_2\left( l_1^2\omega_1^2+l_2^2\omega_2^2+2l_1l_2\cos\left( \theta_2 - \theta_1\right) \omega_1\omega_2 \right)\\
&{}&\quad +m_1gl_1\cos(\theta_1)+m_2g\left(l_1\cos(\theta_1)+l_2\cos(\theta_2)\right),
\end{eqnarray*}
where $\omega_i=\dot{\theta}_i$, $i=1,2$, are the angular velocities of the links, and $m_i$ and $l_i$ are the link masses and their lengths, respectively. We assume that the input torques, $\tau_1$ and $\tau_2$ are applied at each joint. The parameter values are given in Table~\ref{tab:two_link_again_par}.
\begin{table}[h]
\caption{Parameter numerical values of the two-link robotic manipulator.}
\begin{center}
\begin{tabular}{ll}\hline
$m_1=m_2$ & 1 kg\\
$l_1=l_2$ & 0.5 m\\
%$I_1=I_2$& 1/48 kg$\cdot$m$^2$\\
$g$& 9.81 m/sec$^2$ \\ \hline
\end{tabular}
\end{center}
\label{tab:two_link_again_par}
\end{table}
The Lagrange equations\index{Lagrange equations!for two-link manipulator} of motion are,
\begin{eqnarray}
&{}&\left[\begin{array}{cc}
(m_1+m_2)l_1^2 & m_2l_1l_2\cos(\theta_2 - \theta_1)\\
m_2l_1l_2\cos(\theta_2 - \theta_1) & m_2l_2^2
\end{array}\right]\left[\begin{array}{c}
\ddot{\theta}_1\\
\ddot{\theta}_2
\end{array}\right]\nonumber\\
&{}&\qquad\qquad = \left[\begin{array}{c}
\tau_1 - \tau_2 + m_2l_1l_2\dot{\theta}_2^2 \sin(\theta_2 - \theta_1) - (m_1 + m_2) g l_1 \sin(\theta_1)\\
\tau_2-m_2l_1l_2\dot{\theta}_1^2\sin(\theta_2 - \theta_1) -m_2gl_2\sin(\theta_2)
\end{array}\right].
\label{eq:two_linkftwlags_eqns}
\end{eqnarray}
Note that the manipulator inertia matrix\index{manipulator!inertia matrix of}, $\b H$, on the left-hand side of the above equation is symmetric and positive definite, and its determinant is
\[
\det \b H = (m_1+m_2)m_2l_1^2l_2^2 - m_2^2l_1^2l_2^2 \cos^2(\theta_2 - \theta_1) > 0.
\]

We represent the above model in state-space format,\index{manipulator model!in state-space format} where the state variables are defined as, $x_1=\theta_1$, $x_2=\theta_2$, $x_3=\dot{\theta}_1$, and $x_4= \dot{\theta}_2$. The outputs are
\[
y_1=x_1\quad\mb{and}\quad y_2=x_2.
\]
Therefore, $\b C_1=\b I_2$ and $\b C_2=\b O$. Using the above notation, we represent~\eqref{eq:two_linkftwlags_eqns} as
\[
\b H(\b x_1)\ddot{\b x}_1= -\b C(\b x_1,\dot{\b x}_1)\dot{\b x}_1 - \b g(\b x_1)+\b B_u\b\tau,
\]
where
\[
\b B_u=\left[\begin{array}{cc}
1 & -1\\
0 & 1
\end{array}\right],
\]
and
\[
\b g(\b x_1)=\b G(\b x_1)\b x_1=\left[\begin{array}{cc}
-(m_1+m_2)gl_1\frac{\sin(x_1)}{x_1} & 0\\
0 & -m_2gl_2\frac{\sin(x_2)}{x_2}
\end{array}\right]\b x_1.
\]
We represent the control $\tilde{\b u}$ as in~\eqref{eq:two_link_regular}, where
\[
\tilde{\b u}=\b H^{-1}\b B_u\b{\tau}.
\]
We select
\[
\b T=0.05\b I_2,
\]
and
\[
\{\lambda_1, \lambda_2,\lambda_3,\lambda_4\}=\{-3, -3.5, -4, -4.5\}.
\]
Applying the design algorithm, we obtain
\[
\tilde{\b S} =\left[\begin{array}{cccccc}
-18. &  0  &  8.5 &  0 & 1 & 0\\
0 & -10.5 & 0 & 6.5 & 0 & 1
\end{array}\right].
\]
With the above data, we construct the regulator~\eqref{eq:soclawtracking}\index{regulator!for two-link manipulator}. In our simulations, we select the initial conditions to be,
\[
\b x_r(0)=\b 0\quad\mb{and}\quad \b x(0)=\left[\begin{array}{cccc}
-3/4\pi & \pi & 0 & 0
\end{array}\right]^{\top}.
\]
The components of the reference signal vector\index{reference!for two-link manipulator} $\b r$ are plotted in Figures~\ref{fig:theta1} and~\ref{fig:theta2}, where we show plots of link angles $\theta_1$ and $\theta_2$ versus time.
\begin{figure}
\centering
\includegraphics[scale=0.45]{theta1.eps}
\caption{A plot of $\theta_1$ and its reference versus time.}
\label{fig:theta1}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.45]{theta2.eps}
\caption{A plot of $\theta_2$ and its reference versus time.}
\label{fig:theta2}
\end{figure}
In Figures~\ref{fig:tip_x} and~\ref{fig:tip_y}, we show plots of the time history of the $x$ and $y$ coordinates, respectively, of the end effector of the robotic manipulator.
\begin{figure}
\centering
\includegraphics[scale=0.45]{tip_x.eps}
\caption{A plot of the $x$-coordinate of the end effector versus time.}
\label{fig:tip_x}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.45]{tip_y.eps}
\caption{A plot of the $y$-coordinate of the end effector versus time.}
\label{fig:tip_y}
\end{figure}
In Figure~\ref{fig:tip_xy}, we plot the path of the end effector\index{end effector} in the $(x,y)$ plane.
\begin{figure}
\centering
\includegraphics[scale=0.45]{tip_xy.eps}
\caption{A plot of the path of the end effector in  the $(x,y)$-plane.}
\label{fig:tip_xy}
\end{figure}
As can be seen from the figures the reference signal is tracked as predicted by our analysis.

\section{State Estimation of Uncertain Systems}\label{sec:statestiuncsy}
%\subsection{Sliding Mode Observer}
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\index{state vector estimate} for the inaccessible state\index{inaccessible state}, then we split a control design problem into three phases. The first phase is the design of a controller\index{controller design!three phases of} assuming the availability of the entire state vector of the plant. In the second phase, we construct an observer based on the plant input and output. Finally, in the third phase, we combine the controller and the observer into the combined controller-observer compensator,\index{combined controller-observer compensator} where the controller uses the plant state estimate instead of the actual state.

We discuss the problem of constructing a state estimator for a class of dynamical systems with matched uncertainties\index{matched uncertainties} modeled by
\begin{eqnarray*}
\dot{\b x} &=& \mb{\bm$A$}\mb{\bm$x$} + \mb{\bm$B$}\left(\mb{\bm$u$}+\mb{\bm$\xi$} \right)\\
\mb{\bm$y$} &=& \mb{\bm$C$}\mb{\bm$x$},
\end{eqnarray*}
where $\mb{\bm$C$} \in \mb{\Bb R}^{p \times n}$, and $p \ge m$, which means that the number of input channels does not exceed the number of output channels. The vector function $\mb{\bm$\xi$}$ models the lumped uncertainties\index{lumped uncertainties} or nonlinearities in the plant. We assume that there is a nonnegative real-valued function $\rho = \rho\left(\mb{\bm$y$}\right)$ such that
\[
\|\mb{\bm$\xi$}\| \le \rho.
\]
We assume that the pair $\left( \mb{\bm$A$},\mb{\bm$C$} \right)$ is observable. Our goal is to construct the plant state estimator. Let $\hat{\mb{\bm$x$}}$ be an estimate of $\mb{\bm$x$}$. Let $\b e$ denote the estimation error, that is,
\[
\mb{\bm$e$}(t)=\b x(t) - \hat{\b x}(t).
\]
The observability of $\left( \mb{\bm$A$},\mb{\bm$C$} \right)$ implies the existence of a matrix $\mb{\bm$L$} \in \mb{\Bb R}^{n \times p}$ such that $\mb{\bm$A-LC$}$ has prescribed (symmetric with respect to the real axis) eigenvalues in the open left-half plane. Because $\mb{\bm$A-LC$}$ is asymptotically stable, by the Lyapunov theorem, for any $\mb{\bm$Q$}=\mb{\bm$Q$}^{\top}>0$, there is a unique $\mb{\bm$P$}=\mb{\bm$P$}^{\top}>0$ such that
\begin{equation}
\left(\mb{\bm$A$}-\mb{\bm$L$}\mb{\bm$C$} \right) ^{\top}\mb{\bm$P$}+\mb{\bm$P$}\left(\mb{\bm$A$}-\mb{\bm$L$}\mb{\bm$C$} \right)=-\mb{\bm$Q$}.
\label{eq:sli_mode_obe_eq_one}
\end{equation}
We choose $\mb{\bm$Q$}$, if possible, so that for some $\mb{\bm$F$} \in \mb{\Bb R}^{m \times p}$,
\begin{equation}
\mb{\bm$F$}\mb{\bm$C$}=\mb{\bm$B$}^{\top}\mb{\bm$P$}.
\label{eq:sli_mode_obe_eq_two}
\end{equation}
We need this technical condition for the construction of the observer. This condition has an interesting system theoretic interpretation. Its satisfaction means that the transfer function matrix
\[
\mb{\bm$G$}(s)=\mb{\bm$FC$}\left(s\mb{\bm$I$}-\mb{\bm$A$}+\mb{\bm$LC$} \right)^{-1}\mb{\bm$B$}
\]
is positive real\index{positive real!transfer function matrix}\index{transfer function matrix!positive real}, that is, the elements of $\mb{\bm$G$}(s)$ are analytic in the open right-half plane and $\mb{\bm$G$}(s)+\mb{\bm$G$}^{*}(s) \ge 0$ in the open right-half plane. Here the superscript asterisk denotes the conjugate transpose.


An efficient method, using linear matrix inequalities~(LMIs),\index{LMIs}\index{linear matrix inequalities} for finding matrices $\b P$, $\b L$, and $\b F$ that satisfy~\eqref{eq:sli_mode_obe_eq_one} and~\eqref{eq:sli_mode_obe_eq_two} is presented in Section~\ref{sec:lmis_for_uios}.


To proceed, define the vector function
\[
\mb{\bm$E$}\left(\mb{\bm$e$}, \eta \right)=\left\{\begin{array}{lcc}
\frac{\mb{\bm$FCe$}}{\|\mb{\bm$FCe$}\|}\eta & \mb{for} & \mb{\bm$FCe$}\ne \mb{\bm$0$}\\
\b 0  & \mb{for} & \mb{\bm$FCe$}=\mb{\bm$0$},
\end{array}\right.
\]
where $\eta \ge 0$ is a design parameter. Note that if we have a single-input single-output plant, then we can write
\[
\b E(\b e,\eta)=\eta \,\mb{sign}\left(\b F\b C\b e\right).
\]

We now introduce a sliding mode observer\index{sliding mode observer} described in~\cite{walcott87}. It has the form,
\begin{equation}
\fbox{$\displaystyle \dot{\hat{\b x}}=\left(\b A -\b L\b C \right)\hat{\b x} + \b B \b E(\b e,\eta)+\b L\b y+\b B\b u $}
\label{eq:wzsmo_matched}
\end{equation}
If $\eta \ge \rho$, the function $\hat{\mb{\bm$x$}}$ is an asymptotic estimate of the state $\mb{\bm$x$}$, that is,
\[
\lim_{t \to \infty}\mb{\bm$e$}(t)=\lim_{t \to \infty} \left(\b x(t)-\hat{\b x}(t) \right)=\mb{\bm$0$}.
\]
To prove the above statement using Lyapunov's type of arguments, see Exercise~\ref{exer:conver_esti_ero}, construct the differential equation satisfied by the estimation error $\b e$,
\begin{equation}
\dot{\b e} = \dot{\mb{\bm$x$}}-\dot{\hat{\mb{\bm$x$}}}= \left(\mb{\bm$A$}-\mb{\bm$LC$} \right)\mb{\bm$e$} - \mb{\bm$BE$}(\mb{\bm$e$},\eta) + \mb{\bm$B\xi$}.
\label{eq:errodiffeqn_vsc}
\end{equation}
Then, show that
\[
\frac{d}{dt}\left(\mb{\bm$e$}^{\top}\mb{\bm$Pe$} \right)=-\mb{\bm$e$}^{\top}\mb{\bm$Qe$}<0,
\]
which implies
\[
\lim_{t \to \infty}\mb{\bm$e$}(t)=\mb{\bm$0$}.
\]
It follows from the above that the estimation error\index{estimation error!insensitive to matched uncertainty} is insensitive to matched uncertainty.


A generalization of the sliding mode observer~\eqref{eq:errodiffeqn_vsc} is presented in Chapter~\ref{chap:UIO_chapter}.

%\subsection{Boundary Layer Estimators}
%We can view the error differential equation~(\ref{eq:errodiffeqn_vsc}) as a model of a closed-loop system, with matched uncertainties, controlled by $-\mb{\bm$E$}(\mb{\bm$e$},\eta)$. This ``controller'' is discontinuous and varies its structure with respect to the switching surface $\left\{\mb{\bm$e$}:\mb{\bm$FCe$}=\mb{\bm$0$}\right\}$. In our analysis of~(\ref{eq:errodiffeqn_vsc}), we assumed that $\mb{\bm$E$}(\mb{\bm$e$},\eta)$ can switch with infinitely high frequency. In practice, the switching frequency is finite albeit very high. High frequency switching is not always desirable because it may degrade the system performance. In mechanical systems, it may lead to excessive wear and tear of the system's components. To alleviate the problem, we smooth out the discontinuous part of the estimator by introducing a boundary layer around the surface $\left\{\mb{\bm$e$}:\mb{\bm$FCe$}=\mb{\bm$0$}\right\}$. This results in replacing $\mb{\bm$E$}(\mb{\bm$e$},\eta)$ with its continuous approximation $\tilde{\mb{\bm$E$}}(\mb{\bm$e$},\eta)$. An example of such a continuous approximation is
%\[
%\tilde{\b E}\left(\b e, \eta \right)=\left\{\begin{array}{ccc}
%\frac{\b F\b C\b e}{\|\b F\b C\b e\|}\eta & \mb{for} & \|\b F\b C\b e\| \ge \nu\\
%\frac{\mb{\bm$FCe$}}{\nu}\eta & \mb{for} & \|\mb{\bm$FCe$}\| < \nu,
%\end{array}\right.
%\]
%where $\nu$ is the width of the boundary layer. However, the error equation, or more precisely $\mb{\bm$e$}=\mb{\bm$0$}$, 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.
%\begin{defn}
%The closed ball $\bar{B}_{q}(\b 0)=\left\{ \mb{\bm$e$}\,:\, \|\mb{\bm$e$}\| < q \right\}$ is uniformly stable if for any $\varepsilon > q$, there is $\delta(\varepsilon) > 0$ so that if $\|\mb{\bm$e$}(t_0) \| \le \delta(\varepsilon)$, then $\|\mb{\bm$e$}(t)\| \le \varepsilon$ for $t \ge t_0$.
%\end{defn}\markdefn
%
%\begin{defn}
%\label{def:ub}
%The equilibrium $\mb{\bm$e$}=\mb{\bm$0$}$ is uniformly bounded if for any given $r>0$, there exists $d(r)< \infty$ so that for each solution $\mb{\bm$e$}(t) = \mb{\bm$e$}(t;t_0,\mb{\bm$e$}_0)$ with $\|\mb{\bm$e$}_0\| \le r$, we have $\|\mb{\bm$e$}(t)\| \le d(r)$ for all $t\ge t_0$.
%\end{defn}\markdefn
%
%We next adapt the definition of uniform ultimate boundedness, abbreviated UUB, to the estimation error, $\b e$.
%\begin{defn}
%The error $\b e$ is uniformly ultimately bounded~(UUB) with respect to a closed ball $\bar{B}$ if for every $r>0$, there is $T(r) \ge 0$ such that for each solution $\mb{\bm$e$}(t)$ with $\|\mb{\bm$e$}(t_0)\| \le r$, we have $\b e(t) \in \bar{B}$ for $t \ge t_0+T(r)$.
%\end{defn}\markdefn
%
%We can show that the error of the boundary layer estimator is UUB. Specifically, the estimation error of the boundary layer estimator ends up in a ball whose radius depends on $\nu$. The ball radius tends to zero with $\nu$ tending to zero. To see this consider, as before, the Lyapunov function candidate $V(\b e)=\b e^{\top}\b P\b e$. For $\|\b F\b C\b e\| \ge \nu$, we have
%\begin{eqnarray}
%\frac{d}{dt} V(\b e) &=& 2\b e^{\top}\b P\dot{\b e}\nonumber\\
%&=& 2\b e^{\top}\b P\left( \left(\b A-\b L\b C \right)\b e-\b B\tilde{\b E}(\b e,\eta)-\b B\b{\xi}\right)\nonumber\\
%&=& -\b e^{\top}\b Q\b e-2\b e^{\top}\b P\b B\frac{\b F\b C\b e}{\|\b F\b C\b e\|}\eta - 2\b e^{\top}\b P \b B\b{\xi}.
%\label{eq:outsblayer}
%\end{eqnarray}
%Because by assumption, $\b F\b C=\b B^{\top}\b P$, therefore
%\[
%\b e^{\top}\b P\b B\b F\b C\b e=\|\b F\b C\b e\|^2=\|\b e^{\top}\b P\b B\|^2.
%\]
%Substituting the above into~(\ref{eq:outsblayer}) and taking into account that $\|\b e^{\top}\b P\b B\b{\xi}\|\le \|\b e^{\top}\b P\b B\|\rho$, we obtain
%\[
%\frac{d}{dt} V(\b e) \le -\b e^{\top}\b Q\b e -2\|\b F\b C\b e\|\eta +2\|\b e^{\top}\b P\b B\|\eta = -\b e^{\top}\b Q\b e.
%\]
%For $\|\b F\b C\b e\| < \nu$, we have
%\begin{eqnarray}
%\frac{d}{dt} V(\b e) &=& 2\b e^{\top}\b P\dot{\b e}\nonumber\\
%&=& -\b e^{\top}\b Q\b e-2\b e^{\top}\b P\b B\frac{\b F\b C\b e}{\nu}\eta - 2\b e^{\top}\b P \b B\b{\xi}\nonumber\\
%&\le& -\b e^{\top}\b Q\b e -2\frac{\|\b F\b C\b e\|^2}{\nu}\eta + 2\|\b e^{\top}\b P\b B\|\rho\nonumber\\
%&\le & -\b e^{\top}\b Q\b e + 2 \nu\rho\nonumber\\
%&\le & -\lambda_{\mb{min}}(\b Q)\|\b e\|^2 + 2\nu \rho.
%\label{eq:insblayer}
%\end{eqnarray}
%In summary,
%\begin{eqnarray*}
%\frac{d}{dt}V(\mb{\bm$e$}) &=& -\mb{\bm$e$}^{\top}\mb{\bm$Qe$} - 2\mb{\bm$e$}^{\top}\mb{\bm$PB$}\tilde{\mb{\bm$E$}}(\mb{\bm$e$},\eta) - 2\mb{\bm$e$}^{\top}\mb{\bm$PB$}\mb{\bm$\xi$}\\
%&\le& \left\{\begin{array}{ccc}
%-\mb{\bm$e$}^{\top}\mb{\bm$Qe$} & \mb{for} & \|\mb{\bm$FCe$}\| \ge \nu\\
%-\lambda_{\mb{min}}(\b Q)\|\b e\|^2 + 2\nu \rho &\mb{for}& \|\mb{\bm$FCe$}\| < \nu.
%\end{array}\right.
%\end{eqnarray*}
%Hence, we can conclude that $\frac{d}{dt}V(\mb{\bm$e$}) < 0$ for all $\b e\in \mb{\Bb R}^n$ such that
%\[
%\|\mb{\bm$e$}\| > \sqrt{\frac{2 \nu \rho}{\lambda_{\mb{min}}(\mb{\bm$Q$})}} \stackrel{\rm def}{=}R.
%\]
%We illustrate the regions used in the discussion of the boundary layer state estimator in Figure~\ref{fig:euub_2d}.
%\begin{figure}
%\centering
%\includegraphics[scale=0.5]{euub_2d.eps}
%%\vspace{3in}
%%\centerbmp{3.33in}{2.5in}{c:/mybook/euub_2d.bmp}
%%\hskip1.5in\special{bmp:c:/mybook/euub_2d.bmp x=4in y=3in}
%\caption{A two-dimensional illustration of an UUB error.}
%\label{fig:euub_2d}
%\end{figure}
%Referring to Figure~\ref{fig:euub_2d} and Definition~\ref{def:ub}, we have for $r\le R$,
%\[
%d(r)=\sqrt{\frac{\lambda_{\mb{max}}(\mb{\bm$P$})}{\lambda_{\mb{min}}(\mb{\bm$P$})}}\,R.
%\]
%If, on the other hand, $r>R$, then
%\[
%d(r)=\sqrt{\frac{\lambda_{\mb{max}}(\b P)}{\lambda_{\mb{min}}(\b P)}}\,r.
%\]
%We conclude that the estimation error is UUB with respect to any ball with a radius greater than
%\[
%\sqrt{\frac{\lambda_{\mb{max}}(\b P)}{\lambda_{\mb{min}}(\b P)}}\,R.
%\]
%\section{Sliding Modes in Solving Optimization Problems}
%In this section, we use sliding modes to analyze dynamics of an analog circuit, proposed by Rodr{\'{\i}}guez-V{\'{a}}zquez et al.~\cite{rodriguez90}, that can solve convex optimization problems with linear constraints. This class of networks for solving optimization problems is particularly attractive for it can be can be realized using switched-capacitor technology. The network we analyze is modeled by a set of nonlinear ordinary differential equations with discontinuous right-hand sides. In carrying out the analysis, we use concepts from the theory of differential equations with discontinuous right-hand sides and the Lyapunov stability theory. 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. Our analysis is based on the results of the paper by~Glazos, Hui, and \.{Z}ak~\cite{glazos98}
%
%
%
%\subsection{Optimization Problem Statement}
%\label{problem}
%We consider the constrained optimization problem
%\begin{equation}
%\begin{array}{ll}
%\mb{minimize} & f(\b x)\\
%\mb{subject to} & \b A\b x \le \b b,
%\end{array}
%\label{eq:opty}
%\end{equation}
%where $\b x \in \mb{\Bb R}^n$, $f:\mb{\Bb R}^n \to \mb{\Bb R}$, $\b A \in\mb{\Bb R}^{m\times n}$, and $\b b \in\mb{\Bb R}^m$. Let
%\begin{equation}
%\label{eq:feas_set}
%\Omega=\{ \b x : \b A\b x \le \b b\}.
%\end{equation}
%denote the feasible region for the above optimization problem. The feasible set $\Omega$ is the intersection of $m$ closed half-spaces and is therefore a convex polytope. Let
%\[
%h_i=\left\{ \b x:\b a_i\b x=b_i\right\},\quad i=1,2,\ldots,m,
%\]
%denote the hyperplanes associated with the above $m$ closed half-spaces, where $\b a_i$ is the $i$-th row of $\b A$ and $b_i$ is the $i$-th component of the vector $\b b$. Also, we define
%\[
%H=\bigcup_{i=1}^{m}h_i.
%\]
%We let $\Gamma$ denote the collection of all local minimizers of the optimization problem~(\ref{eq:opty}). Lastly, we introduce the index sets
%\[
%I(\b x)=\left\{ i:\b a_i\b x=b_i \right\}
%\]
%and
%\[
%J(\b x)=\left\{ i:\b a_i\b x > b_i\right\}.
%\]
%
%Recall that a function $f:\mb{\Bb R}^n \to \mb{\Bb R}$ defined on a convex set $\Omega$ is said to be convex if, for every $\b x, \b y \in \Omega$ and every $0\le \alpha \le 1$, there holds
%\[
%f(\alpha \b x + (1-\alpha)\b y) \le \alpha f(\b x)+(1-\alpha)f(\b y).
%\]
%
%Let $f\in {\cal C}^1$. Then, $f$ is convex over a convex set $\Omega$ if and only if
%\[
%f(\b y)\ge f(\b x) + \nabla f^{\top}(\b x)(\b y-\b x)
%\]
%for all $\b x,\b y \in \Omega$.
%
%We make the following assumptions:\\
%{\bf A1}. The objective function $f$ is convex over $\mb{\Bb R}^n$ and has continuous first partial derivatives on $\mb{\Bb R}^n$, that is, $f \in {\cal C}^1$;\\
%{\bf A2}. The feasible set $\Omega$ is a nonempty, bounded, and closed polyhedron, which means that $\Omega$ is a compact set;\\
%{\bf A3}. Each point in $\Omega$ is a regular point of the constraints, that is, for any $\b x \in \Omega$ the vectors $\b a_i$, $i \in I(\b x)$, are linearly independent.
%
%It follows from Assumption~{\bf A1} that the optimization problem~(\ref{eq:opty}) is a convex programming problem. Indeed, we are asked to minimize a convex objective function $f$ over a polyhedron $\Omega$.
%
%Problem~(\ref{eq:opty}) is well posed in the sense that the feasible set $\Omega$ is nonempty and the optimal value of $f$ is not $-\infty$ because by assumptions {\bf A1} and {\bf A2} the set $f(\Omega)$ is compact.
%
%An elementary overview of results concerning convex optimization problems can be found, for example, in Chong and \.{Z}ak~\cite[Chapter~21]{chong96}. We state two results related to problem~(\ref{eq:opty}) that are relevant in our analysis.
%\begin{lemma}
%\label{lemma1}
%Any local minimizer of the optimization problem~(\ref{eq:opty}) is a global minimizer.
%\end{lemma}
%Furthermore, the Karush-Kuhn-Tucker condition for optimality is both necessary and sufficient for problem~(\ref{eq:opty}). Specifically, we have the following lemma:
%\begin{lemma}
%\label{lemma2}
%Suppose that $\b x^{*} \in \Omega$. Then, $\b x^{*} \in \Gamma$ if and only if there exist non-negative constants $\mu_1,\mu_2,\ldots,\mu_m$ such that
%\[
%\nabla f(\b x^{*})+\sum_{i \in I(\b x^{*})}\mu_i\b a_i^{\top}=\b 0.
%\]
%\end{lemma}
%
%
%The network applied to the optimization problem~(\ref{eq:opty}) has the form
%\begin{eqnarray}
%\tau \dot{\b x}(t) &=& \b g(\b x(t))\nonumber \\
%&=& \left\{\begin{array}{ccc}
%-\mu \sum_{i \in J(\b x(t))}\b a_i^{\top} & \mb{if} & \b x(t) \notin \Omega\\
%-\nabla f(\b x(t)) &\mb{if} & \b x(t) \in \Omega,
%\end{array}\right.\label{eq:sys}\\
%\b x(0) &=& \b x_0,\nonumber
%\end{eqnarray}
%where $\tau$ and $\mu$ are positive design constants. The function $\b g(\b x)$ is discontinuous on the surfaces $h_1, h_2, \ldots h_{m}$ and is continuous elsewhere. A phenomenon commonly occurring in systems such as~(\ref{eq:sys}) is the so-called sliding mode that we discussed in Section~\ref{sec:slidingmode}. An analysis of the behavior of the above dynamical system using a sliding mode approach is presented in the following sections.
%
%
%We remark that we can view~(\ref{eq:sys}) as a dynamical gradient system for the unconstrained problem
%\[
%\mb{minimize}\quad \kappa_0f(\b x)+\mu\sum_{i=1}^m \kappa_i \left( \b a_i\b x-b_i\right),
%\]
%where
%\[
%\kappa_i=\left\{\begin{array}{ccc}
%0&\mb{if}& \b a_i\b x\le b_i\\
%1& \mb{if}& \b a_i\b x>b_i,
%\end{array}\right.
%\]
%and
%\[
%\kappa_0=\left\{\begin{array}{cl}
%1&\mb{if } \kappa_i=0 \mb{ for all }i=1,2,\ldots,m;  \\
%0& \mb{otherwise},
%\end{array}\right.
%\]
%that is, $\kappa_0=\prod_{i=1}^m \left( 1-\kappa_i\right)$. Using the preceding notation, we can represent system~(\ref{eq:sys}) as
%\begin{equation}
%\tau \dot{\b x}(t)=\b g(\b x(t))=-\kappa_0 \nabla f(\b x(t))-\mu \sum_{i=1}^m \kappa_i\b a_i^{\top},\quad \b x(0) = \b x_0.
%\label{eq:sys_alt}
%\end{equation}
%
%
%
%\subsection{Penalty Function Method}
%\label{background}
%An approach that can be used to construct an analog optimizer to solve a constrained optimization problem is to first convert, or approximate, 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 approximating a constrained optimization problem into an associated unconstrained optimization problem is the penalty function method. For more details on the subject of the penalty method see, for example, Luenberger~\cite{luenberger84}. The idea behind the penalty function method is to approximate the constrained optimization problem
%\begin{equation}
%\left.\begin{array}{ll}
%\mb{minimize} & f(\b x)\\
%\mb{subject to} & \b x \in \Omega,
%\end{array}\right\}
%\label{eq:consopty}
%\end{equation}
%where $\Omega \subset \mb{\Bb R}^n$ is the feasible set, with an associated unconstrained problem of the form
%\begin{equation}
%\mb{minimize}\quad f(\b x)+\zeta p(\b x),
%\label{eq:unconstropty}
%\end{equation}
%where the objective function $f:\mb{\Bb R}^n \to \mb{\Bb R}$ is continuous on $\mb{\Bb R}^n$ and $\zeta$ is a positive constant. The function $p:\mb{\Bb R}^n \to \mb{\Bb R}$ is continuous on $\mb{\Bb R}^n$ and satisfies
%\[
%\mb{(i)}\quad p(\b x) \ge 0 \quad \mb{for all}\quad \b x \in \mb{\Bb R}^n,
%\]
%and
%\[
%\mb{(ii)}\quad p(\b x)=0\quad \mb{if and only if}\quad \b x\in \Omega.
%\]
%
%If for a finite value of the penalty parameter $\zeta$ the solution of the unconstrained problem~(\ref{eq:unconstropty}) coincides with the solution of the constrained problem~(\ref{eq:consopty}), then the penalty function $p$ is said to be exact. Bertsekas~\cite{bertsekas75} has shown that except for trivial cases, an exact penalty function cannot be everywhere differentiable---see Exercise~\ref{exerc:exactnond} for an illustration of this statement. Quite often we can find an exact penalty function that results in the function $f(\b x)+\zeta p(\b x)$ 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~\cite{filippov88} and Aubin and Cellina~\cite{aubin84}. Such an approach was undertaken already in 1967 by Karpinskaya~\cite{karpinskaya67}, and later by Korovin and Utkin~\cite{korovin74}, Utkin~\cite{utkin92}, Fl{\aa}m~\cite{flam92}, Fl{\aa}m and Zowe~\cite{flam92b}, \.{Z}ak et al.~\cite{zak94}, and Chong, Hui and \.{Z}ak~\cite{chong99} among others. A differential geometric approach to the analysis of dynamical systems that solve optimization problems is reported by Faybusovich~\cite{faybusovich92} and Helmke and Moore~\cite{helmke94}.
%
%\subsection{Dynamical Gradient Circuit Analysis}
%\label{results}
%We will now show that every trajectory of system~(\ref{eq:sys}), irrespective of the initial state converges to the solution set of the optimization problem~(\ref{eq:opty}). We do this by showing that every trajectory of system~(\ref{eq:sys}) has the following properties:
%\begin{enumerate}
%\item Irrespective of the initial state $\b x_0$, the resulting trajectory reaches the feasible set $\Omega$ in finite time and is thereafter confined to $\Omega$. This portion of the trajectory is hereafter referred to as the {\em reaching phase}.
%\item While confined to the feasible set $\Omega$, the trajectory converges asymptotically to the solution set $\Gamma$. This portion of the trajectory is hereafter referred to as the {\em convergence phase}.
%\end{enumerate}
%
%\subsection*{Reaching Phase}
%
%We begin by introducing the function $V:\mb{\Bb R}^n \to \mb{\Bb R}$ of the form
%\[
%V(\b x)=\sum_{i=1}^{m}\max \left\{ \b a_i\b x-b_i,0\right\},
%\]
%or equivalently
%\begin{equation}
%V(\b x)=\left\{\begin{array}{ccc}
%\sum_{i \in J(\b x)}\left( \b a_i\b x-b_i\right) & \mb{if} & \b x \notin \Omega\\
%0 & \mb{if} & \b x \in \Omega.
%\end{array} \right.
%\label{eq:lap}
%\end{equation}
%Observe that $V(\b x) > 0$ for all $\b x \notin \Omega$, and $V(\b x)=0$ for all $\b x \in \Omega$. We can view $V(\b x)$ as a distance measure from the point $\b x$ to the feasible set $\Omega$. We shall show that $V$ is strictly decreasing when evaluated along any trajectory of system~(\ref{eq:sys}) outside $\Omega$. Specifically, we will prove the existence of a positive constant, $\eta$, with the property that
%\[
%\frac{d}{dt}V(\b x(t)) \le -\eta
%\]
%when evaluated along any trajectory of system~(\ref{eq:sys}) outside $\Omega$. In doing so we use the following result:
%\begin{lemma}
%\label{lemma3}
%Suppose $\hat{\b x} \notin \Omega$ and $\beta_1,\beta_2,\ldots,\beta_m$ are non-negative constants. If $\beta_j>0$ for $j \in J(\hat{\b x})$, then $\sum_{i \in I(\hat{\b x})\cup J(\hat{\b x})}\beta_i\b a_i \ne \b 0$.
%\end{lemma}
%{\bf Proof } Let the function $q:\mb{\Bb R}^n \to \mb{\Bb R}$ be defined as
%\[
%q(\b x)=\sum_{i\in I(\hat{\b x})\cup J(\hat{\b x})}\beta_i \left( \b a_i\b x-b_i\right).
%\]
%Clearly $q\in {\cal C}^1$ and
%\[
%\nabla q^{\top}(\b x)=\sum_{i \in I(\hat{\b x})\cup J(\hat{\b x})} \beta_i \b a_i.
%\]
%Because $\beta_1,\beta_2,\ldots,\beta_m$ are all non-negative, it follows that
%\begin{enumerate}
%\item the function $q$ is convex over $\mb{\Bb R}^n$, and
%\item $q(\b x)\le 0$ for all $\b x \in \Omega$.
%\end{enumerate}
%Furthermore, because $\beta_j>0$ for $j\in J(\hat{\b x})$, we also have $q(\hat{\b x}) >0$. Hence, we must have
%\[
%\nabla q^{\top}(\hat{\b x})=\sum_{i \in I(\hat{\b x})\cup J(\hat{\b x})} \beta_i \b a_i \ne \b 0.
%\]
%\hspace*{5.5in} $\Box$
%
%We must consider two cases when analyzing the behavior of system~(\ref{eq:sys}). The first case being when the motion of the system is not confined to any of the surfaces $h_1, h_2,\ldots,h_{m}$. The second case being when the system is in sliding mode on one or more of the surfaces $h_1, h_2,\ldots,h_{m}$. We only need to consider these two cases because every trajectory of system~(\ref{eq:sys}) is composed of these two types of motion. That is, every trajectory $\b x(t)$ of system~(\ref{eq:sys}) can be broken into a countable number of intervals, $(t_0,t_1),\ldots,(t_{l-1},t_l),\ldots$, on each of which both index sets $I(\b x(t))$ and $J(\b x(t))$ are constant. If $I(\b x(t))$ is not empty, then the system is in a sliding mode on that interval, and if $I(\b x(t))$ is empty, then the system is not in a sliding mode.
%
%\noindent{\bf Case 1 } Let $\b x(t)$ denote any particular trajectory of system~(\ref{eq:sys}), and let $t^{\prime} \in(t_{l-1},t_l)$. Suppose that on the interval $(t_{l-1},t_l)$ the trajectory $\b x(t)$ does not intersect $\Omega$ or any of the surfaces $h_1,h_2,\ldots,h_m$, that is,
%\begin{equation}
%\b x(t) \notin \Omega \cup H\quad\mb{for all}\quad t\in(t_{l-1},t_l).
%\label{eq:outside}
%\end{equation}
%Denoting the index set $J(\b x(t^{\prime}))$ by $J^{\prime}$, it follows from~(\ref{eq:outside}) and the fact that $\b x(t)$ is absolutely continuous that $I(\b x(t))=\emptyset$ and $J^{\prime} \neq \emptyset$ for all $t \in (t_{l-1},t_l)$. This in turn implies that on the interval $(t_{l-1},t_l)$ the trajectory $\b x(t)$ must satisfy~(\ref{eq:sys}) in the usual sense. Hence, we have
%\[
%\tau\dot{\b x}(t)=-\mu \sum_{i \in J^{\prime}} \b a_i^{\top}
%\]
%almost everywhere on the interval $(t_{l-1},t_l)$. Also, it follows from~(\ref{eq:lap}) that
%\begin{equation}
%V(\b x(t))=\sum_{i \in J^{\prime}} \left( \b a_i \b x(t)-b_i\right)
%\label{eq:voutside}
%\end{equation}
%for all $t \in (t_{l-1},t_l)$. Applying the chain rule to~(\ref{eq:voutside}), we obtain
%\[
%\frac{d}{dt}V(\b x(t)) = \nabla V^{\top}(\b x(t)) \dot{\b x}(t)
%= -\frac{\mu}{\tau}\left\Arrowvert  \sum_{i \in J^{\prime}} \b a_i\right\Arrowvert_2^2
%\]
%almost everywhere on the interval $(t_{l-1},t_l)$. However, Lemma~\ref{lemma3} along with the fact that there is a finite number of constraints, that is, $m$ is finite, implies the existence of a positive constant, $\eta_1$, with the property that
%\[
%\eta_1 \le \left\Arrowvert  \sum_{i \in J(\b x)} \b a_i\right\Arrowvert_2^2
%\]
%for all $\b x \notin \Omega$. Hence,
%\begin{equation}
%\frac{d}{dt}V(\b x(t)) \le -\frac{\mu \eta_1}{\tau}  < 0
%\label{eq:etaone}
%\end{equation}
%almost everywhere on the interval $(t_{l-1},t_l)$, which concludes our analysis of the first case.
%
%\noindent{\bf Case 2 } Suppose now that on an interval $(t_{l-1},t_l)$ a particular trajectory $\b x(t)$ is confined to one or more of the surfaces $h_1,h_2,\ldots,h_m$ and does not intersect $\Omega$. Specifically, suppose that there exist nonempty index sets $I^{\prime}$ and $J^{\prime}$ with the property that
%\begin{equation}
%I(\b x(t))=I^{\prime}\quad \mb{for all}\quad t\in (t_{l-1},t_l),
%\label{eq:ioutside}
%\end{equation}
%and
%\begin{equation}
%J(\b x(t))=J^{\prime}\quad \mb{for all}\quad t\in (t_{l-1},t_l).
%\label{eq:joutside}
%\end{equation}
%Let $i_1,i_2,\ldots,i_k$ denote the elements of $I^{\prime}$, and let $S=\bigcap_{i \in I^{\prime}} h_i$ denote the surface to which the trajectory $\b x(t)$ is confined on the interval $(t_{l-1},t_l)$, and let
%\[
%\tilde{\b a}=\sum_{i \in J^{\prime}} \b a_i^{\top}.
%\]
%It follows from~(\ref{eq:ioutside}) and~(\ref{eq:joutside}) that for a sufficiently small $\delta>0$ the surfaces $h_{i_1},\ldots,h_{i_k}$ partition the $\delta$-neighborhood of $\b x(t^{\prime})$ into $\ell \le 2^k$ regions, ${\cal R}_1,{\cal R}_2,\ldots,{\cal R}_{\ell}$, in each of which the function $\b g$ is continuous. Then, $G(\b x(t^{\prime}))$ is the set of all vectors $\b v$ of the form
%\begin{equation}
%\b v=-\mu \sum_{j=1}^{\ell}\alpha_j \left( \tilde{\b a}+\sum_{i \in I^{\prime}} U(i,j)\b a_i^{\top} \right),
%\label{eq:vecoutside}
%\end{equation}
%where
%\begin{equation}
%\alpha_j \ge 0,\quad j=1,2,\ldots,\ell,
%\label{eq:alphajo}
%\end{equation}
%\begin{equation}
%\sum_{j=1}^{\ell}\alpha_j=1,
%\label{eq:sumalpha}
%\end{equation}
%and
%\[
%U(i,j)=\left\{\begin{array}{cccc}
%1&\mb{if}\quad\b a_i\b x > b_i&\mb{for all}& \b x \in {\cal R}_j\\
%0&\mb{if}\quad\b a_i\b x \le b_i&\mb{for all}& \b x \in {\cal R}_j.
%\end{array}\right.
%\]
%Using~(\ref{eq:sumalpha}), we rewrite~(\ref{eq:vecoutside}) as
%\begin{equation}
%\b v=-\mu \tilde{\b a}-\mu \sum_{j=1}^{\ell}\alpha_j\left( \sum_{i \in I^{\prime}}U(i,j)\b a_i^{\top}\right).
%\label{eq:vecalt}
%\end{equation}
%For notational simplicity, we let
%\[
%\b r=\sum_{j=1}^{\ell}\alpha_j\left( \sum_{i \in I^{\prime}}U(i,j)\b a_i^{\top}\right).
%\]
%Then,~(\ref{eq:vecalt}) becomes
%\begin{equation}
%\b v=-\mu \tilde{\b a}-\mu \b r.
%\label{eq:vecfinal}
%\end{equation}
%Let $T(\b x)$ denote the tangent plane to the surface $S$ at the point $\b x$. Observe that because the trajectory $\b x(t)$ is confined to the surface $S$ on the interval $(t_{l-1},t_l)$, then $\tau\dot{\b x}(t) \in T(\b x(t))$ almost everywhere on the interval $(t_{l-1},t_l)$. Thus, $\b x(t)$ is an absolutely continuous function that satisfies
%\[
%\tau\dot{\b x}(t) \in G(\b x(t)) \cap T(\b x(t))
%\]
%almost everywhere on the interval $(t_{l-1},t_l)$. We note that because the trajectory $\b x(t)$ is confined to the surface $S$ on the interval $(t_{l-1},t_l)$, the set $G(\b x(t)) \cap T(\b x(t))$ is nonempty for all $(t_{l-1},t_l)$. In particular, $G(\b x(t^{\prime})) \cap T(\b x(t^{\prime}))$ is the set of all $\b v$ of the form~(\ref{eq:vecfinal}) that also lie on the tangent plane $T(\b x(t^{\prime}))$. However,
%\[
%\b r \in \mb{Span}\left\{\b a_{i_1}^{\top},\b a_{i_2}^{\top},\ldots,\b a_{i_k}^{\top}  \right\},
%\]
%and thus $\b r$ is orthogonal to $T(\b x(t^{\prime}))$. Therefore, the set $G(\b x(t^{\prime})) \cap T(\b x(t^{\prime}))$ consists of exactly one element, the orthogonal projection of the vector $-\mu \tilde{\b a}$ onto the tangent plane $T(\b x(t^{\prime}))$. Hence, for $(t_{l-1},t_l)$ the set $G(\b x(t)) \cap T(\b x(t))$ consists of exactly one element, and thus $\dot{\b x}(t)=-(\mu/\tau)\b P\tilde{\b a}$ is uniquely determined almost everywhere on the interval $(t_{l-1},t_l)$, where $\b P$ is the orthogonal projector onto the tangent plane $T(\b x(t))$. Note that
%\begin{equation}
%\b P=\b I_n-\b B\left( \b B^{\top}\b B\right)^{-1}\b B^{\top},
%\label{patilnozer}
%\end{equation}
%where $\b I_n$ denotes the $n\times n$ identity matrix and
%\[
%\b B=\left[\begin{array}{cccc}
%\b a_{i_1}^{\top} & \b a_{i_2}^{\top} & \cdots & \b a_{i_k}^{\top}
%\end{array}\right].
%\]
%By~(\ref{eq:lap}) and~(\ref{eq:joutside}),
%\begin{equation}
%V(\b x(t))=\sum_{i \in J^{\prime}}\left( \b a_i\b x(t)-b_i\right)
%\label{eq:lapoutside}
%\end{equation}
%for all $t \in (t_{l-1},t_l)$. Applying the chain rule to~(\ref{eq:lapoutside}) yields
%\begin{equation}
%\frac{d}{dt}V(\b x(t)) = \nabla V^{\top}(\b x(t))\dot{\b x}(t)
%= -\frac{\mu}{\tau}\tilde{\b a}^{\top}\b P\tilde{\b a}
%\label{eq:lyap_der1}
%\end{equation}
%almost everywhere on the interval $(t_{l-1},t_l)$. Observing that $\b P=\b P^{\top}=\b P^2$, we rewrite~(\ref{eq:lyap_der1}) as
%\begin{equation}
%\frac{d}{dt}V(\b x(t))=-\frac{\mu}{\tau}\left\Arrowvert \b P\tilde{\b a}\right\Arrowvert_2^2.
%\label{eq:lyap_der2}
%\end{equation}
%We can show, using Lemma~\ref{lemma3}, that $\b P\tilde{\b a} \ne \b 0$---see Exercise~\ref{ex:patildnonz}.
%%For this we will employ Theorem~21.7 from~\cite{chong96} which states that %if for a convex continuously differentiable function $f$ on the convex %constraint set $\{\b x:\b h(\b x)=\b 0\}$ there is a point $\b x^*$ such %that
%%\[
%%\nabla f(\b x^*) + \nabla \b h(\b x^*)\b{\lambda}=\b 0,
%%\]
%%then $\b x^*$ is a global minimizer of $f$ over the constraint set. In our %case the objective function is $V$ and the constraint set is
%%\begin{equation}
%%\{\b x: \b a_i\b x - b_i=0, i\in I^{\prime}\}.
%%\label{eq:constrset}
%%\end{equation}
%%Suppose now that $\b P\tilde{\b a} = \b 0$, which is true if and only if %$\tilde{\b a}=\b B\b \lambda$ for some $\b \lambda \in \mb{\Bb R}^k$. Note %that $\tilde{\b a}=\nabla V(\b x)$. Hence, $\b P\tilde{\b a} \ne \b 0$.
%Combining this with the fact that the number of constraints is finite, we conclude that there exists a positive constant, $\eta_2$, such that
%\[
%\eta_2 \le \left\Arrowvert  \b P\tilde{\b a} \right\Arrowvert_2^2 \quad \mb{for all}\quad \b x \in H \setminus \Omega.
%\]
%Hence,
%\begin{equation}
%\frac{d}{dt}V(\b x(t)) \le -\frac{\mu \eta_2}{\tau} < 0
%\label{eq:negoutside}
%\end{equation}
%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.
%\begin{thm}
%\label{thm:finitefeasible}
%Every trajectory of system~(\ref{eq:sys}) reaches $\Omega$ in finite time, and is thereafter confined to $\Omega$.
%\end{thm}
%{\bf Proof } Let $\b x(t)$ be any particular trajectory of system~(\ref{eq:sys}) and let
%\[
%\eta=\min\left\{  \frac{\mu \eta_1}{\tau},\frac{\mu \eta_2}{\tau}\right\}.
%\]
%It follows directly from~(\ref{eq:etaone}) and~(\ref{eq:negoutside}) that
%\[
%\frac{d}{dt}V(\b x(t)) \le -\eta
%\]
%for almost all $t$ when $\b x(t) \notin \Omega$. Hence, $\b x(t) \in \Omega$ for all $t \ge V(\b x(0))/\eta$.\\
%\hspace*{5.5in} $\Box$
%
%\subsection*{Convergence Phase}
%
%We now analyze dynamical behavior of system~(\ref{eq:sys}) when its motion is confined to the feasible set $\Omega$. We begin by introducing the function $F:\mb{\Bb R}^n \to \mb{\Bb R}$ such that
%\[
%F(\b x)=f(\b x)-f^{*},
%\]
%where $f^{*}$ is the optimal value of $f$ for the optimization problem~(\ref{eq:opty}). Clearly $F \in {\cal C}^1$. Furthermore, we see from Lemma~\ref{lemma1} that $F(\b x)=0$ for all $\b x \in \Gamma$ and $F(\b x)>0$ for all $\b x \in \Omega \setminus \Gamma$.
%
%Similar to the previous subsection, we consider two cases when analyzing the behavior of system~(\ref{eq:sys}) in the region $\Omega$. We first consider the case when the motion of system~(\ref{eq:sys}) is confined to the interior of $\Omega$. Then, we consider the case when the motion of system~(\ref{eq:sys}) is confined to the boundary of $\Omega$, that is, when system~(\ref{eq:sys}) is in a sliding mode on one or more of the surfaces $h_1,\ldots,h_m$.
%
%\noindent{\bf Case 1 } Let $\b x(t)$ be any particular trajectory of system~(\ref{eq:sys}). Suppose that on an interval $(t_{l-1},t_l)$ the trajectory $\b x(t)$ is confined to the interior of $\Omega$. Then, $I(\b x(t))=\emptyset$ and $J(\b x(t))=\emptyset$ for all $t \in (t_{l-1},t_l)$. This implies that on the interval $(t_{l-1},t_l)$, the trajectory $\b x(t)$ satisfies $\tau \dot{\b x}(t)=-\nabla f(\b x(t))$ almost everywhere on the interval $(t_{l-1},t_l)$ in the usual sense. Applying the chain rule to $F(\b x(t))$, we obtain
%\begin{equation}
%\frac{d}{dt}F(\b x(t)) = \nabla F^{\top}(\b x(t)) \dot{\b x}(t)
%= -\frac{1}{\tau}\left\Arrowvert \nabla f(\b x(t))\right\Arrowvert^2_2
%\label{eq:finside}
%\end{equation}
%almost everywhere on the interval $(t_{l-1},t_l)$. Using the fact that $I(\b x(t))=\emptyset$ for all $t\in (t_{l-1},t_l)$, we see from Lemma~\ref{lemma2} that on the interval $(t_{l-1},t_l)$, we have $\nabla f(\b x(t))=\b 0$ if and only if $\b x(t) \in \Gamma$. Hence, we have
%\begin{equation}
%\frac{d}{dt}F(\b x(t))=0 \quad \mb{if and only if}\quad \b x(t) \in \Gamma,
%\label{eq:xinside}
%\end{equation}
%and
%\begin{equation}
%\frac{d}{dt}F(\b x(t)) < 0\quad\mb{if and only if}\quad \b x(t) \notin \Gamma,
%\label{eq:xnoting}
%\end{equation}
%almost everywhere on the interval $(t_{l-1},t_l)$. This concludes our analysis of the first case.
%
%\noindent{\bf Case 2 } Let $\b x(t)$ denote any particular trajectory of system~(\ref{eq:sys}) and let $t \in (t_{l-1},t_l)$. Suppose that on the interval $(t_{l-1},t_l)$ the trajectory $\b x(t)$ is confined to the boundary of $\Omega$. Thus,
%\begin{enumerate}
%\item there exists a nonempty index set $I^{\prime}$ with the property that $I(\b x(t))=I^{\prime}$ for all $t \in (t_{l-1},t_l)$, and
%\item $J(\b x(t))=\emptyset$ for all $t \in (t_{l-1},t_l)$.
%\end{enumerate}
%Let $i_1,\ldots,i_k$ denote the elements of $I^{\prime}$, and let $S=\bigcap_{i \in I^{\prime}}h_i$ denote the surface to which the trajectory $\b x(t)$ is confined on the interval $(t_{l-1},t_l)$. Let $\b B \in \mb{\Bb R}^{n \times k}$ be the matrix with columns $\b a_{i_1}^{\top},\ldots,\b a_{i_k}^{\top}$. By assumption~{\bf A3} the vectors $\b a_{i_1}^{\top},\ldots,\b a_{i_k}^{\top}$ are linearly independent, and hence the matrix $\b B$ has full rank.
%
%It follows from the above that for a sufficiently small $\delta >0$ the surfaces $h_{i_1},\ldots,h_{i_k}$ partition the $\delta$-neighborhood of $\b x(t^{\prime})$ into $2^k$ regions, in each of which the function $\b g$ is continuous. Using a similar argument to the one in the previous subsection, we can show that the set $G(\b x(t^{\prime}))$ is the set of all $\b v$ of the form
%\begin{equation}
%\b v=-\alpha_0\nabla f(\b x(t^{\prime}))-\mu \sum_{j=1}^{2^k-1}\alpha_j \b B\b u_j,
%\label{eq:vecin}
%\end{equation}
%where
%\begin{equation}
%\alpha_j \ge 0,\quad j=0,1,\ldots,2^k-1,
%\label{eq:alphajin}
%\end{equation}
%\begin{equation}
%\sum_{j=0}^{2^k-1}\alpha_j=1,
%\label{eq:sumalphaj}
%\end{equation}
%and $\b u_j \in \mb{\Bb R}^{k}$, $j=1,\ldots,2^k-1$ are defined as
%\begin{eqnarray*}
%\b u_1 &=& \left[\begin{array}{ccccccc}
%1 & 0 & 0 & \cdots & 0 & 0 & 0
%\end{array}\right]^{\top},\\
%\b u_2 &=& \left[\begin{array}{ccccccc}
%0 & 1 & 0 & \cdots & 0 & 0 & 0
%\end{array}\right]^{\top},\\
%\b u_3 &=& \left[\begin{array}{ccccccc}
%1 & 1 & 0 & \cdots & 0 & 0 & 0
%\end{array}\right]^{\top},\\
%{} &{\vdots}&{}\\
%\b u_{2^k-3} &=& \left[\begin{array}{ccccccc}
%1 & 1 & 1 & \cdots & 1 & 0 & 1
%\end{array}\right]^{\top},\\
%\b u_{2^k-2} &=& \left[\begin{array}{ccccccc}
%1 & 1 & 1 & \cdots & 1 & 1 & 0
%\end{array}\right]^{\top},\\
%\b u_{2^k-1} &=& \left[\begin{array}{ccccccc}
%1 & 1 & 1 & \cdots & 1 & 1 & 1
%\end{array}\right]^{\top}.
%\end{eqnarray*}
%We illustrate this case in Figure~\ref{fig:forces} for a hypothetical situation of sliding occurring on the intersection of two hyperplanes. In the figure, the components of the vector $\b v$ as given by~(\ref{eq:vecin}) are shown.
%\begin{figure}
%\centering
%\includegraphics[scale=0.75]{forces-new.eps}
%%\centering \epsfig{file=forces.eps,height=2.2in}
%\caption{Components of the vector $\b v$ as given by~(\ref{eq:vecin}) for sliding occurring on the intersection of two hyperplanes $h_{i_1}$ and $h_{i_2}$.}
%\label{fig:forces}
%\end{figure}
%
%
%Let $T(\b x)$ denote the plane tangent to the surface $S$ at the point $\b x$. Observe that $\dot{\b x}(t) \in T(\b x(t))$ almost everywhere on the interval $(t_{l-1},t_l)$ because the trajectory $\b x(t)$ is confined to the surface $S$ on the interval $(t_{l-1},t_l)$. Thus, $\b x(t)$ is an absolutely continuous function that satisfies
%\[
%\tau \dot{\b x}(t) \in G(\b x(t)) \cap T(\b x(t))
%\]
%almost everywhere on the interval $(t_{l-1},t_l)$. We note that because the trajectory $\b x(t)$ is confined to the surface $S$ on the interval $(t_{l-1},t_l)$, the set $G(\b x(t)) \cap T(\b x(t))$ is by assumption nonempty for all $t \in (t_{l-1},t_l)$. In particular, we see that $G(\b x(t^{\prime})) \cap T(\b x(t^{\prime}))$ is the set of all $\b v$ that have the form~(\ref{eq:vecin}) and also lie on the tangent plane $T(\b x(t^{\prime}))$. Let
%\[
%\b r=\sum_{j=1}^{2^k-1}\alpha_j \b B\b u_j.
%\]
%Then, we represent~(\ref{eq:vecin}) as
%\begin{equation}
%\b v=-\alpha_0 \nabla f(\b x(t^{\prime}))-\mu \b r.
%\label{eq:insidev}
%\end{equation}
%Obviously, $\b r \in \mb{Span}\left\{ \b a_{i_1}^{\top},\ldots,\b a_{i_k}^{\top} \right\}$, and thus $\b r$ is orthogonal to $T(\b x(t^{\prime}))$. However, $\b v \in T(\b x(t^{\prime}))$ by assumption. Therefore, $\b v$ is the orthogonal projection of the vector $-\alpha_0 \nabla f(\b x(t^{\prime}))$ onto the tangent plane $T(\b x(t^{\prime}))$, that is,
%\begin{equation}
%\b v=-\alpha_0 \b P \nabla f(\b x(t^{\prime})),
%\label{eq:inprojector}
%\end{equation}
%where $\b P=\b I_n-\b B\left( \b B^{\top}\b B \right)^{-1}\b B^{\top}$.
%
%Note that by virtue of assumptions~{\bf A1} and~{\bf A2}, the set $\nabla f(\Omega)$ is compact. This implies the existence of a non-negative constant, $M$, with the property that
%\[
%M \ge \left\Arrowvert \nabla f(\b x) \right\Arrowvert_2\quad \mb{for all} \quad \b x \in \Omega.
%\]
%Let
%\[
%\gamma=\frac{\mu}{\mu + M\sqrt{k}\left\|\left(\b B^{\top}\b B \right)^{-1}\b B^{\top} \right\|_2}.
%\]
%\begin{proposition}
%\[
%\gamma \le \alpha_0 \le 1.
%\]
%\label{proposition1}
%\end{proposition}
%{\bf Proof } It follows from~(\ref{eq:alphajin}) and~(\ref{eq:sumalphaj}) that $\alpha_0 \le 1$. Thus, it remains to show that $\alpha_0 \ge \gamma$. It follows from~(\ref{eq:insidev}) and~(\ref{eq:inprojector}) that $\mu \b r$ is the orthogonal projection of the vector $-\alpha_0 \nabla f(\b x(t^{\prime}))$ onto the subspace $T(\b x(t^{\prime}))^{\perp}$, where $T(\b x(t^{\prime}))^{\perp}$ is the orthogonal complement of $T(\b x(t^{\prime}))$ in $\mb{\Bb R}^n$. Therefore,
%\begin{equation}
%-\alpha_0\left( \b I_n-\b P\right)\nabla f(\b x(t^{\prime}))=\mu \sum_{j=1}^{2^k-1}\alpha_j\b B\b u_j.
%\label{eq:comboc}
%\end{equation}
%Let $\b U\in \mb{\Bb R}^{k \times (2^k-1)}$ be the matrix with columns $\b u_1,\ldots,\b u_{2^k-1}$, and $\b \alpha \in \mb{\Bb R}^{2^k-1}$ the column vector with the components $\alpha_1,\ldots,\alpha_{2^k-1}$. Then, we can represent~(\ref{eq:comboc}) as
%\begin{equation}
%-\frac{\alpha_0}{\mu}\left( \b I_n-\b P\right)\nabla f(\b x(t^{\prime}))=\b B\b U\b \alpha.
%\label{eq:morecombo}
%\end{equation}
%Premultiplying both sides of~(\ref{eq:morecombo}) by $\left( \b B^{\top}\b B\right)^{-1}\b B^{\top}$, we obtain
%\[
%-\frac{\alpha_0}{\mu}\left( \b B^{\top}\b B\right)^{-1}\b B^{\top}\left( \b I_n-\b P\right)\nabla f(\b x(t^{\prime}))=\b U\b \alpha,
%\]
%and hence
%\begin{equation}
%\frac{\alpha_0}{\mu}\left\|\left( \b B^{\top}\b B\right)^{-1}\b B^{\top}\left( \b I_n-\b P\right)\nabla f(\b x(t^{\prime}))\right\|_2=\left\|\b U\b \alpha\right\|_2.
%\label{eq:normcombo}
%\end{equation}
%Applying the Cauchy-Schwarz inequality to~(\ref{eq:normcombo}), and using the fact that $\|\b I_n-\b P\|=1$, we have
%\[
%\frac{M\alpha_0}{\mu}\left\| \left(\b B^{\top}\b B\right)^{-1}\b B^{\top}\right\|_2 \ge \left\|\b U\b \alpha \right\|_2 \ge \frac{1}{\sqrt{k}}\left\|\b U\b \alpha\right\|_1
%\]
%since for $\b z \in \mb{\Bb R}^k$, we have $\|\b z\|_2 \ge \frac{1}{\sqrt{k}}\|\b z\|_1$. However, using~(\ref{eq:alphajin}) and~(\ref{eq:sumalphaj}), we see that
%\[
%\left\|  \b U\b \alpha\right\|_1 \ge \sum_{j=1}^{2^k-1}\alpha_j = 1-\alpha_0.
%\]
%Thus,
%\begin{equation}
%\frac{M\alpha_0 \sqrt{k}}{\mu} \left\|  \left(\b B^{\top}\b B \right)^{-1}\b B^{\top}\right\|_2 \ge 1-\alpha_0.
%\label{eq:bothside}
%\end{equation}
%It follows from~(\ref{eq:bothside}) that $\alpha_0>0$. Dividing both sides of~(\ref{eq:bothside}) by $\alpha_0$ yields
%\[
%\frac{M \sqrt{k}}{\mu} \left\|  \left(\b B^{\top}\b B \right)^{-1}\b B^{\top}\right\|_2 \ge \frac{1-\alpha_0}{\alpha_0}=\frac{1}{\alpha_0}-1.
%\]
%Hence,
%\[
%\alpha_0 \ge \frac{1}{1+\frac{M \sqrt{k}}{\mu} \left\|  \left(\b B^{\top}\b B \right)^{-1}\b B^{\top}\right\|_2}=\gamma.
%\]
%The proof of the proposition is complete.\\
%\hspace*{5.5in} $\Box$
%
%We have the following two corollaries.
%\begin{corollary}
%\label{corollary1}
%\[
%G(\b x(t^{\prime}))\cap T\left(\b x(t^{\prime})\right) \subset \left\{  -\alpha\b P\nabla f(\b x(t^{\prime})):\gamma\le \alpha \le 1\right\}.
%\]
%\end{corollary}
%{\bf Proof } This is directly from~(\ref{eq:inprojector}) and Proposition~\ref{proposition1}.\\
%\hspace*{5.5in} $\Box$
%\begin{corollary}
%\label{corollary2}
%\[
%\b P\nabla f(\b x(t^{\prime}))=\b 0 \quad \mb{if and only if}\quad \b 0 \in G(\b x(t^{\prime}))\cap T\left(\b x(t^{\prime})\right).
%\]
%\end{corollary}
%{\bf Proof } If $\b P\nabla f(\b x(t^{\prime}))=\b 0$, then $\b 0 \in G(\b x(t^{\prime}))\cap T\left(\b x(t^{\prime})\right)$ because, by Corollary~\ref{corollary1},
%\[
%\b P\nabla f(\b x(t^{\prime})) \in G(\b x(t^{\prime}))\cap T\left(\b x(t^{\prime})\right).
%\]
%Conversely, suppose that $\b 0 \in G(\b x(t^{\prime}))\cap T\left(\b x(t^{\prime})\right)$. By Corollary~\ref{corollary1}, $-\alpha\b P\nabla f(\b x(t^{\prime}))=\b 0$ for some $\alpha \in [\gamma, 1]$. Since $\gamma>0$ for all $\mu >0$, $-\alpha\b P\nabla f(\b x(t^{\prime}))=\b 0$ is possible only if $\b P\nabla f(\b x(t^{\prime}))=\b 0$, which completes the proof.\\
%\hspace*{5.5in} $\Box$
%\begin{proposition}
%\label{proposition2}
%\[
%\b P\nabla f(\b x(t^{\prime}))=\b 0 \quad\mb{if and only if}\quad \b x(t^{\prime}) \in \Gamma.
%\]
%\end{proposition}
%{\bf Proof } We first prove the sufficiency part of the proposition. Suppose $\b P\nabla f(\b x(t^{\prime}))=\b 0$. It follows from~(\ref{eq:inprojector}) and~(\ref{eq:vecin}) that
%\begin{equation}
%\label{eq:vecinzero}
%\alpha_0 \nabla f(\b x(t^{\prime})) + \mu \sum_{j=1}^{2^k-1} \alpha_j \b B\b u_j = \b 0.
%\end{equation}
%Let
%\[
%\b d=\frac{\mu}{\alpha_0} \sum_{j=1}^{2^k-1}\alpha_j\b u_j.
%\]
%Note that because $\alpha_0 \ne 0$, $\b d$ is well defined. It then follows from~(\ref{eq:vecinzero}) that
%\[
%\nabla f(\b x(t^{\prime})) + \b B\b d =\b 0.
%\]
%We see from~(\ref{eq:alphajin}) and the definition of $\b u_1,\ldots,\b u_{2^k-1}$ that each element of $\b d$ is non-negative. By Lemma~\ref{lemma2}, the point $\b x(t^{\prime})$ satisfies the Karush-Kuhn-Tucker conditions for problem~(\ref{eq:opty}) and hence $\b x(t^{\prime}) \in \Gamma$. The sufficiency condition is thus proven.
%
%We now prove the necessity condition. Suppose that $\b x(t^{\prime}) \in \Gamma$. By Lemma~\ref{lemma2}, the point $\b x(t^{\prime})$ satisfies the Karush-Kuhn-Tucker optimality conditions for problem~(\ref{eq:opty}). This implies the existence of non-negative constants $\mu_1,\ldots,\mu_k$ with the property that
%\begin{equation}
%\label{eq:kkting}
%\nabla f(\b x(t^{\prime})) + \b B\b \mu =\b 0,
%\end{equation}
%where $\b \mu=\left[\mu_1,\ldots,\mu_k\right]^{\top}$. Let
%\begin{equation}
%\label{eq:nuing}
%\nu=1+\frac{1}{\mu}\sum_{i=1}^k\mu_i,
%\end{equation}
%\begin{equation}
%\label{eq:newalpha}
%\tilde{\alpha}_0=\frac{1}{\nu},
%\end{equation}
%\begin{equation}
%\label{eq:newalphas}
%\tilde{\alpha}_{2^{i-1}}=\frac{\mu_i}{\mu \nu},\quad i=1,\ldots,k,
%\end{equation}
%\begin{equation}
%\label{eq:alphazero}
%\tilde{\alpha}_j=0,\quad j=1,\ldots,2^{k}-1,\quad j\ne 2^{i-1}\quad \mb{for}\quad i=1,\ldots,k,
%\end{equation}
%\begin{equation}
%\label{eq:vnewtilde}
%\tilde{\b v}=-\tilde{\alpha}_0\nabla f(\b x(t^{\prime}))-\mu \sum_{j=1}^{2^k-1}\tilde{\alpha}_j\b B\b u_j.
%\end{equation}
%It follows from~(\ref{eq:nuing})--(\ref{eq:alphazero}) that
%\[
%\tilde{\alpha}_j \ge 0,\quad j=0,1,\ldots,2^k-1,\quad\mb{and}\quad
%\sum_{j=0}^{2^k-1}\tilde{\alpha}_j=1.
%\]
%Therefore, $\tilde{\b v} \in G(\b x(t^{\prime}))$. Note that
%\[
%\tilde{\b v}=-\frac{\mu}{\mu+\sum_{i=1}^k\mu_i}\left(\nabla f(\b x(t^{\prime})) + \b B\b \mu \right).
%\]
%Using~(\ref{eq:kkting}), we conclude that $\tilde{\b v}=\b 0$, and thus $\tilde{\b v} \in T(\b x(t^{\prime}))$. Hence,
%\[
%\b 0 \in G(\b x(t^{\prime})) \cap T(\b x(t^{\prime})).
%\]
%Using Corollary~\ref{corollary2}, we get $\b P\nabla f(\b x(t^{\prime}))=\b 0$, which completes the proof of the necessary condition and the proof of the proposition.\\
%\hspace*{5.5in} $\Box$
%
%Applying now the chain rule to $F$, taking into account~(\ref{eq:inprojector}), Proposition~\ref{proposition1}, and the fact that $\b P=\b P^{\top}=\b P^2$, we obtain
%\begin{equation}
%\label{eq:fdecreases}
%\frac{d}{dt}F(\b x(t)) \le -\frac{\gamma}{\tau} \left\|  \b P\nabla f(\b x(t))\right\|_2^2
%\end{equation}
%almost everywhere on the interval $(t_{l-1},t_l)$. By Proposition~\ref{proposition2},
%\begin{equation}
%\label{eq:fzero}
%\frac{d}{dt}F(\b x(t)) = 0\quad\mb{if and only if}\quad \b x(t) \in \Gamma,
%\end{equation}
%and
%\begin{equation}
%\label{eq:flessthan}
%\frac{d}{dt}F(\b x(t)) < 0\quad\mb{if and only if}\quad \b x(t) \notin \Gamma,
%\end{equation}
%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~(\ref{eq:sys}) that is confined to $\Omega$ converges asymptotically to the region $\Gamma$.
%\begin{thm}
%\label{thm:asin}
%Let $\b x(t)$ be any trajectory of system~(\ref{eq:sys}). If $\b x(t) \in \Omega$ for all $t$, then the trajectory $\b x(t)$ converges asymptotically to $\Gamma$.
%\end{thm}
%{\bf Proof } Recall that $F(\b x)=0$ if $\b x \in \Gamma$ and $F(\b x)>0$ if $\b x \in \Omega \setminus \Gamma$. Therefore, to prove the theorem it is enough to show that
%\[
%\lim_{t \to \infty}F(\b x(t))=0,
%\]
%or equivalently, that for any $\varepsilon >0$ there exists a finite time, $T(\b x(0),\varepsilon)$, such that $F(\b x(t)) < \varepsilon$ for all $t\ge T(\b x(0),\varepsilon)$. We see from~(\ref{eq:xnoting}) and~(\ref{eq:flessthan}) that
%\[
%\frac{d}{dt}F(\b x(t)) < 0
%\]
%for almost all $t$ when $\b x(t) \notin \Gamma$. Consequently $F(\b x(t)) \le F(\b x(0))$ for all $t \ge 0$. Therefore, we only need to consider the case when $F(\b x(0)) \ge \varepsilon$. Let
%\[
%\theta(\varepsilon)=\left\{  \b x:\b x \in \Omega, F(\b x) \ge \frac{\varepsilon}{2}\right\}.
%\]
%Because $F(\b x(0)) \ge \varepsilon$, we have $\b x(0) \in \theta(\varepsilon)$ and $\theta(\varepsilon) \subset \Omega \setminus \Gamma$. Furthermore, because the function $F$ is continuous, the set $\theta(\varepsilon)$ is compact. We will now show that there exists a positive constant, $\rho(\varepsilon)$, such that
%\[
%\frac{d}{dt}F(\b x(t)) \le -\rho(\varepsilon)
%\]
%for almost all $t$ when $\b x(t) \in \theta(\varepsilon)$, which guarantees that $F(\b x(t)) \le \varepsilon/2$ for all
%\[
%t \ge \frac{F(\b x(0))-\varepsilon/2}{\rho(\varepsilon)}.
%\]
%
%We analyze two cases. The first case is when the trajectory $\b x(t)$ is in $\theta(\varepsilon)$ and is not in a sliding mode on any of the surfaces $h_i$. Then, we analyze the case when the trajectory is in $\theta(\varepsilon)$ in a sliding mode on a surface that is an intersection of some of the hyperplanes $h_i$.
%
%\noindent{\bf Case 1 } Because the function $f$ is convex and $f \in {\cal C}^1$, we have $\nabla f(\b x) \ne \b 0$ for all $\b x \in \Omega \setminus \Gamma$. Hence,
%\begin{equation}
%\label{eq:nablafinneg}
%-\frac{1}{\tau}\left\| \nabla f(\b x) \right\|_2^2<0 \quad \mb{for all}\quad \b x \in \theta(\varepsilon)
%\end{equation}
%because $\theta \subset \Omega \setminus \Gamma$ and $\nabla f(\b x) \ne \b 0$ in $\theta(\varepsilon)$. The fact that the function $-1/\tau\left\|\nabla f(\b x) \right\|_2^2$ is continuous on the compact set $\theta(\varepsilon)$ implies the existence of a positive constant, $\rho_1(\varepsilon)$, such that
%\[
%-\frac{1}{\tau}\left\| \nabla f(\b x) \right\|_2^2 \le -\rho_1(\varepsilon)\quad \mb{for all}\quad \b x \in \theta(\varepsilon).
%\]
%Combining the above with~(\ref{eq:finside}) yields
%\[
%\frac{d}{dt}F(\b x(t)) \le -\rho_1(\varepsilon)
%\]
%for almost all $t$ when $\b x(t) \in \theta(\varepsilon)$, which concludes our analysis of the first case.
%
%\noindent{\bf Case 2 } Let $i_1,\ldots,i_k$ denote the elements of $I^{\prime}$, and let $S=\bigcap_{i \in I^{\prime}}h_i$ denote the surface to which the trajectory $\b x(t)$ is confined on the interval $(t_{l-1},t_l)$. Suppose that $S\cap \theta(\varepsilon) \ne \emptyset$. Clearly,
%\[
%S\cap \theta(\varepsilon) \subset \Omega \setminus \Gamma.
%\]
%Using arguments similar to those of Glazos et al.~\cite[page~692]{glazos98}, we can show that
%\begin{equation}
%\label{eq:negbyprop}
%-\frac{1}{\tau}\left\|\b P \nabla f(\b x) \right\|_2^2<0 \quad \mb{for all}\quad \b x \in S \cap \theta(\varepsilon),
%\end{equation}
%where $\b P$ is the orthogonal projector onto the tangent plane to the surface $S$. By the above and the fact that the function $-\frac{1}{\tau}\left\|\b P \nabla f(\b x) \right\|_2^2$ is continuous on the compact set $S \cap \theta(\varepsilon)$, there exists a positive constant, $\hat{\rho}(\varepsilon)$, such that
%\[
%-\frac{1}{\tau}\left\|\b P \nabla f(\b x) \right\|_2^2<-\hat{\rho}(\varepsilon) \quad \mb{for all}\quad \b x \in S \cap \theta(\varepsilon).
%\]
%It then follows directly from~(\ref{eq:fdecreases}) that there exists a positive constant $\hat{\gamma}$ such that
%\[
%\frac{d}{dt}F(\b x(t)) \le -\hat{\gamma}\hat{\rho}(\varepsilon)
%\]
%for almost all $t$ when $\b x(t) \in S \cap \theta(\varepsilon)$. The above along with the fact that the number of constraints is finite implies the existence of a positive constant, $\rho_2(\varepsilon)$, such that
%\begin{equation}
%\label{eq:inrhotwo}
%\frac{d}{dt}F(\b x(t)) \le -\rho_2(\varepsilon)
%\end{equation}
%for almost all $t$ when $\b x(t) \in S \cap \theta(\varepsilon)$. This concludes our analysis for the second case.
%
%Let
%\[
%\rho(\varepsilon)=\min\left\{ \rho_1(\varepsilon),\rho_2(\varepsilon)\right\}.
%\]
%Then, using~(\ref{eq:negbyprop}) and~(\ref{eq:inrhotwo}) yields
%\[
%\frac{d}{dt}F(\b x(t)) \le -\rho(\varepsilon)
%\]
%for almost all $t$ when $\b x(t) \in \theta(\varepsilon)$. Hence, $F(\b x(t)) \le \varepsilon/2 < \varepsilon$ for all
%\[
%t \ge \frac{F(\b x(0))-\varepsilon/2}{\rho(\varepsilon)}.
%\]
%Therefore,
%\[
%T(\b x(0),\varepsilon)=\left\{\begin{array}{ccc}
%0 & \mb{if}& F(\b x(0)) < \varepsilon\\
%\frac{F(\b x(0))-\varepsilon/2}{\rho( \varepsilon)} & \mb{if}&F(\b x(0)) \ge \varepsilon.
%\end{array}\right.
%\]
%The proof of the theorem is complete.\\
%\hspace*{5.5in} $\Box$
%
%\begin{thm}
%\label{thm:main}
%Every trajectory of system~(\ref{eq:sys}), irrespective of the initial condition, $\b x(0)$, converges to a solution of problem~(\ref{eq:opty}).
%\end{thm}
%{\bf Proof } This statement follows directly from Theorems~\ref{thm:finitefeasible} and~\ref{thm:asin}.\\
%\hspace*{5.5in} $\Box$
%
%
%\begin{example}[based on~\cite{glazos98}]\label{ex:rodriguez1}\rm
%Consider the quadratic programming problem
%\[
%\begin{array}{ll}
%\mb{minimize} & \frac{1}{2}\b x^{\top}\b Q\b x+\b c^{\top}\b x,\quad \b x\in \mb{\Bb R}^2\\
%\mb{subject to} & \b A\b x\le \b b,
%\end{array}
%\]
%where
%\[
%\b Q=\left[\begin{array}{cc}
%4 & 1\\
%1 & 2
%\end{array}\right],\quad \b c=\left[\begin{array}{c}
%-12\\
%-10
%\end{array}\right],\quad \b A=\left[\begin{array}{cc}
%5 & 1\\
%-5 & 1\\
%-1 & -2
%\end{array}\right],\quad\mb{and}\quad \b b=\left[\begin{array}{c}
%0\\
%10\\
%10
%\end{array}\right].
%\]
%The above optimization problem clearly satisfies {\bf A1}--{\bf A4}, and one can easily verify that the point $\b x^{*}=\left[\begin{array}{cc}
%-19/22 & 95/22
%\end{array}\right]^{\top}$ is a unique minimizer for the problem, that is, $\Gamma=\left\{ \b x^{*}\right\}$. Note that $\b x^{*} \in h_1$. A phase-plane portrait for system~(\ref{eq:sys}) that solves the above optimization problem is shown in Figure~\ref{fig:phase1}.  The feasible region is depicted by the shaded area. Also shown are level sets of the quadratic objective function.
%\begin{figure}
%\centering
%\includegraphics[scale=0.5]{opt_sli.eps}
%%\centering \epsfig{file=opt_sli.eps,height=3.5in}
%%\vspace{4in}
%%\hskip.5in\special{bmp:c:/mike/phase1.bmp x=5.33in y=4in}
%\caption{Phase-plane portrait for the system in Example~\ref{ex:rodriguez1}.}
%\label{fig:phase1}
%\end{figure}
%\end{example}
%\markex

\section{Notes}

The idea of controlling a dynamical system through varying its structure was originated in the 1950s. Early pioneers in the field of variable structure systems\index{variable structure systems!early pioneers} were S.~V.~Emelyanov and his co-workers in the Institute of Control Sciences in Moscow in the former Soviet Union. A large body of literature was produced by these researches, unfortunately mostly published in the Russian only.
%The titles of books and papers on the subject of variable structure control systems published in the years 1957--1989 were collected by Emelyanov~\cite{emelyanov89}.
The first widely accessible books in English on variable structure sliding mode systems\index{variable structure!sliding mode systems}\index{sliding mode systems} were books by Itkis~\cite{itkis76} and Utkin~\cite{utkin78}. A survey paper by Utkin~\cite{utkin77} references many early contributions to the field; see also tutorials~\cite{decarlo88,decarlo11} and a survey by Hung, Gao, and Hung~\cite{hung93}. At about the same time, A.~F.~Filippov was developing his theory of differential equations with discontinuous right-hand sides~\cite{filippov64}\index{differential equations!with discontinuous right-hand sides}. Filippov's approach\index{Filippov's approach!analysis of sliding mode systems} is now accepted as the tool for the analysis of sliding mode systems. Sliding modes also found applications in the state estimation of dynamical uncertain systems as we show in Section~\ref{sec:statestiuncsy}. For further analysis of sliding mode observers, we refer the reader to Edwards and Spurgeon~\cite{edwards94,edwards98}. For a stability analysis of uncertain closed-loop systems with the combined variable structure controller-observer compensators\index{variable structure!controller-observer compensators} we refer to~\cite{hui92}. Output feedback sliding mode controllers\index{output feedback sliding mode!controllers} are presented in~\cite{diong92,hui93,zak93,zak93b,kwan96,edwards98p}.

Impressive applications of sliding modes to solving optimization and computational problems are reported by Bhaya and Kaszkurewicz~\cite{bhaya06}. Ignaciuk and Bartoszewicz~\cite{ignaciuk11}, on the other hand, apply discrete-time sliding-mode controllers\index{discrete-time!sliding-mode controllers} to congestion control of communication networks.

%In 1967, Karpinskaya~\cite{karpinskaya67} identified the occurrence of sliding modes in an analog circuit for solving optimization problems. The idea of using analog circuits to solve constrained optimization problems first appeared in the literature in 1956 when Pyne~\cite{pyne56} 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~\cite{rybashov61,rybashov62,rybashov65a,rybashov65b}, and Karpinskaya~\cite{karpinskaya67}, among others. More recently, many new analog architectures have been proposed for solving constrained optimization problems. In particular, Chua and Lin~\cite{chua84} developed a canonical nonlinear programming circuit for solving general nonlinear programming problems. Tank and Hopfield~\cite{tank86} 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~\cite{kennedy88}. Rodr{\'{\i}}guez-V{\'{a}}zquez et al.~\cite{rodriguez90}, and Cichocki and Unbehauen~\cite{cichocki93} proposed analog networks that use switched-capacitor integrated circuit technology for solving certain constrained optimization problems.
%We analyzed dynamical behavior of a neural network of Rodr{\'{\i}}guez-V{\'{a}}zquez et al.~\cite{rodriguez90} for solving a class of convex programming problems. In carrying out the analysis we used concepts from the theory of differential equations with discontinuous right-hand sides and Lyapunov stability theory. Our method can also be used to analyze other classes of analog dynamical optimizers whose designs are based on exact penalty functions. An open problem is to extend the results obtained to a more general class of mathematical programming problems. Preliminary results in this direction can be found in Glazos et al.~\cite{glazos98} and Chong et al.~\cite{chong99}.

\section{Exercises}
\begin{exercise}\label{exer:utlikevcssys}\rm
Consider the following variable structure system\index{variable structure system} that was analyzed in~\cite{hung93} (see also~\cite{utkin77} for a similar system):
\begin{equation}
\left.\begin{array}{lll}
\dot{x}_1 &=& x_2\\
\dot{x}_2 &=& -x_1+2x_2+u,
\end{array}\right\}
\label{eq:hungutkivsce}
\end{equation}
where
\[
u=-4x_1 \mb{sign} \sigma(x_1,x_2),\quad \sigma(x_1,x_2)=x_1\left(0.5x_1+x_2 \right).
\]
\begin{itemize}
\item[(i)] Sketch phase plane portraits of two linear subsystems that constitute the given variable structure system~(\ref{eq:hungutkivsce}).
\item[(ii)] Sketch a phase plane portrait of the variable structure system~(\ref{eq:hungutkivsce}).\index{variable structure system!phase portrait}
\end{itemize}
\end{exercise}
\begin{exercise}\rm\begin{itemize}
\item[(i)] For the dynamical system model,
\[
\dot{\b x}=\left[\begin{array}{cc}
1 & 1\\
2 & 1
\end{array}\right]\b x+\left[\begin{array}{c}
1\\
-1
\end{array}\right]u,
\]
construct the switching surface\index{switching surface!construction}
\[
\sigma(\b x)=\b s\b x=0
\]
such that the system in sliding along the surface has its pole at $-2$.
\item[(ii)] Consider the following model of a dynamical system,
\[
\dot{\b x}= \left [ \begin{array}{ccc}
2 & 0 & -1 \\
0 & -2 & 2 \\
-1 & -2 & 3
\end{array}
\right ]\b x + \left [ \begin{array}{c}
1 \\
1 \\
1
\end{array}
\right ]u.
\]
Design a switching surface $\sigma (\b x)=\b s\b x=0$ such that the system restricted to it has poles at $-4$, $-5$.
\item[(iii)] Given the following dynamical system model,
\[
\dot{\b x}=\left [ \begin{array}{ccc}
2 & 0 & -1 \\
0 & -2 & 2 \\
-1 & -2 & 3
\end{array}
\right ]\b x + \left [ \begin{array}{c}
1 \\
1 \\
1
\end{array}
\right ]u,
\]
along with the switching surface
\[
\sigma(\b x) = \b s\b x = \left [ \begin{array}{ccc}
8 & 8 & -15
\end{array}
\right ]\b x=0.
\]
Determine the reduced-order equations that describe the system behavior in a sliding mode.\index{sliding mode!reduced-order}
\end{itemize}
\end{exercise}
\begin{exercise}\rm
Consider a dynamical system modeled by the equations
\[
\left[\begin{array}{c}
\dot{x}_1\\
\dot{x}_2
\end{array}\right]=\left[\begin{array}{cc}
0 & 1\\
a_1(t) & a_2(t)
\end{array}\right]\left[\begin{array}{c}
x_1\\
x_2
\end{array}\right]+\left[\begin{array}{c}
0\\
1
\end{array}\right]u,
\]
where
\[
|a_1(t)|\le 3,\quad |a_2(t)|\le 4\quad \mb{for all}\quad t.
\]
\begin{itemize}
\item[(i)] Design a switching surface
\[
\sigma(x_1,x_2)=s_1x_1+x_2=0
\]
such that the system restricted to it has its pole at $-2$.
\item[(ii)] Design a sliding mode controller\index{sliding mode controller!design of}
\[
u=k_1^{\pm}x_1 +k_2^{\pm}x_2,
\]
so that the switching surface $\sigma=0$ is globally attractive. The gains $k_1$ and $k_2$ are functions of $x_1\sigma$ and $x_2\sigma$, respectively.
\end{itemize}
\end{exercise}
\begin{exercise}\rm
Consider the following model of an armature controlled DC motor\index{DC motor!armature controlled} controlling an inverted pendulum:
\begin{eqnarray*}
\left[\begin{array}{c}
\dot{x}_1\\
\dot{x}_2\\
\dot{x}_3
\end{array} \right]&=&\left[\begin{array}{c}
x_2\\
9.8\,\sin x_1 + x_3\\
-\frac{1}{L_a}\,x_2-\frac{1}{L_a}\,x_3
\end{array} \right]+\left[\begin{array}{c}
0\\
0\\
\frac{1}{L_a}
\end{array} \right]u\\
y&=&x_1,
\end{eqnarray*}
where $L_a$ is the armature inductance of the motor.
\begin{itemize}
\item[(i)] Derive the reduced-order model assuming $L_a$ is negligible.
\item[(ii)] For the obtained reduced-order model from the first part, design the switching surface $\sigma =0$ of the form
\[
\sigma = \sigma (x_1,\,x_2)=\left[\begin{array}{cc}
s_1 & 1
\end{array} \right] \left[ \begin{array}{c}
x_1\\
x_2
\end{array} \right]=0
\]
so that the system in sliding will have its pole at $-3$.
\item[(iii)] Use the above sliding surface to design a variable structure controller of the form
\[
u=-k_1x_1\, \mb{sign}(x_1 \sigma ) - k_2x_2 \, \mb{sign}(x_2 \sigma ).
\]
The smaller the gains $k_1$, $k_2$, the better.
\end{itemize}
\end{exercise}
\begin{exercise}\label{exer:syszeroapo}\rm
Consider a dynamical system model,
\begin{eqnarray}
\left[\begin{array}{c}
\dot{\b x}_1\\
\dot{x}_2
\end{array}\right]&=&\left[\begin{array}{cc}
\b A_1 & \b A_2\\
\b A_3 & A_4
\end{array}\right]\left[\begin{array}{c}
{\b x}_1\\
{x}_2
\end{array}\right]+\left[\begin{array}{c}
\b 0\\
b
\end{array}\right]u\label{eq:partfirst}\\
y&=& \left[\begin{array}{cc}
\b C_1 & C_2\end{array}\right]\left[\begin{array}{c}
{\b x}_1\\
{x}_2
\end{array}\right],\label{eq:partseceq}
\end{eqnarray}
where $b$ and $C_2$ are nonzero scalars. Let
\begin{equation}
\sigma=\left[\begin{array}{cc}
\b S_1 & S_2
\end{array}\right]\left[\begin{array}{c}
{\b x}_1\\
{x}_2
\end{array}\right]=F\left[\begin{array}{cc}
\b C_1 & C_2\end{array}\right]\left[\begin{array}{c}
{\b x}_1\\
{x}_2
\end{array}\right]=Fy,
\label{eq:outswitsu}
\end{equation}
where $S_2$ and $F$ are nonzero scalars.
\begin{itemize}
\item[(i)] Find the reduced-order system that models~(\ref{eq:partfirst}) when restricted to the surface $\sigma=0$.\index{reduced-order system!when restricted to the surface}\index{restricted to the surface!reduced-order system}
\item[(ii)] Find the transfer function $H(s)={\cal L}(y(t))/{\cal L}(u(t))=Y(s)/U(s)$. What is the relation between the zeros of $H(s)$ and the poles of the reduced-order system found in part~(i)?
\end{itemize}
\end{exercise}
\begin{exercise}[based on~\cite{kwan96}]\label{exer:scalbound}\rm
Prove the following
\begin{lemma}
Let
\[
\dot{\b x}=\b A\b x+\b B\b u,
\]
where the matrix $\b A$ has negative real distinct eigenvalues, that is,
\[
\mb{eig}\,(\b A)=\left\{\begin{array}{cccc}
-\lambda_1, &-\lambda_2,&\ldots, &-\lambda_n
\end{array} \right\},\quad \lambda_i> 0,\quad \lambda_i \ne \lambda_j\;\;\mb{for}\;\; i\ne j .
\]
Let $\lambda_{\min}=\min\left\{\begin{array}{cccc}
\lambda_1, &\lambda_2,&\ldots, &\lambda_n
\end{array} \right\}$ and $K$ be a positive constant such that
\[
\left\Arrowvert e^{\b A t}\right\Arrowvert  \le K e^{-\lambda_{\min} t}
\]
Then,
\[
\left\Arrowvert \b x(t)\right\Arrowvert  \le w(t),
\]
where $w(t)$ is a solution of
\[
\dot{w}= -\lambda_{\min} w +K \left\Arrowvert \b B\right\Arrowvert  \left\Arrowvert \b u\right\Arrowvert , \quad\mb{subject to}\quad w(0) \ge K\left\Arrowvert \b x(0)\right\Arrowvert .
\]
\end{lemma}
\end{exercise}
\begin{exercise}[based on~\cite{kwan96}]\label{exer:basedonkwon}\rm
Consider an $n$-th order model of an uncertain dynamical system,\index{uncertain system}
\begin{eqnarray}
\left[\begin{array}{c}
\dot{\b x}_1\\
\dot{x}_2
\end{array}\right]&=&\left[\begin{array}{cc}
\b A_1 & \b A_2\\
\b A_3 & A_4
\end{array}\right]\left[\begin{array}{c}
{\b x}_1\\
{x}_2
\end{array}\right]+\left[\begin{array}{c}
\b 0\\
b
\end{array}\right](u+\xi)\label{eq:unsertsysmodpart}\\
y&=& \left[\begin{array}{cc}
\b C_1 & C_2\end{array}\right]\left[\begin{array}{c}
{\b x}_1\\
{x}_2
\end{array}\right],\nonumber
\end{eqnarray}
where $b$ and $C_2$ are nonzero scalars and $\xi$ models system's uncertainty,\index{uncertainty} where
\[
\xi=\b D_1\b x_1+D_2 x_2+D_3 u+D_4(t).
\]
In the above, $\b D_1$ is a row vector whose size is equal to the dimension of $\b x_1\in \mb{\Bb R}^{n-1}$, $D_2$ and $D_3$ are scalars, where $|D_3|<1$ and $D_4$ is an unknown bounded function of time whose bound is known to us, that is, $|D_4(t)|\le d_4$ where $d_4$ is known to us. Let $\sigma=Fy$ be as defined in Exercise~\ref{exer:syszeroapo}. Assume that the switching surface,
\[
\sigma=Fy=F\left[\begin{array}{cc}
\b C_1 & C_2
\end{array}\right]\left[\begin{array}{c}
{\b x}_1\\
{x}_2
\end{array}\right]=0,
\]
was chosen in such a way that the nominal system~(\ref{eq:unsertsysmodpart}) restricted to it has the eigenvalues
\[
\left\{\begin{array}{cccc}
-\lambda_1, &-\lambda_2,&\ldots, &-\lambda_{n-1}
\end{array} \right\},\quad \lambda_i> 0,\quad \lambda_i \ne \lambda_j\;\;\mb{for}\;\; i\ne j .
\]
\begin{itemize}
\item[(i)] Find $\dot{\sigma}$ evaluated on the trajectories of the uncertain system model~(\ref{eq:unsertsysmodpart}).
\item[(ii)] For the controller
\[
u=\left(S_2 b\right)^{-1}\left(1-|D_3|\right)^{-1}\left(-Hw(t)-G|\sigma|-\delta \right)\mb{sign}(\sigma),
\]
where $w(t)$ is a solution of a first-order differential equation such that $\left\Arrowvert \b x_1(t)\right\Arrowvert  \le w(t)$, find lower bounds for $H$, $G$ and $\delta$ such that
\[
\sigma \dot{\sigma} \le -\eta |\sigma|
\]
for some $\eta>0$. Use the lemma of Exercise~\ref{exer:scalbound} to find the form of the differential equation whose solution satisfies the condition $\left\Arrowvert \b x_1(t)\right\Arrowvert  \le w(t)$.
\end{itemize}

\end{exercise}\rm


\begin{exercise}\rm
For the dynamical system model
\[
\left.\begin{array}{lll}
\dot{\b x} &=& \b A\b x + \b b\left(u+f(y) \right)\\
y&=&\b c\b x,
\end{array}\right\}
\]
where
\[
\b A=\left[\begin{array}{cc}
-2 & 2\\
0 & 1
\end{array}\right], \quad \b b=\left[\begin{array}{c}
0\\
1
\end{array}\right],\quad \b c=\left[\begin{array}{cc}
0 & 1
\end{array}\right], \quad\mb{and}\quad |f(y)|\le 10 y,
\]
design a sliding mode observer\index{sliding mode observer!design of}. The observer poles\index{sliding mode observer!poles} should be placed at $-2$ and $-3$. {\bf Hint:} After constructing the required observer gain\index{sliding mode observer!gain} $\b l$, find a symmetric positive definite matrix $\b Q$ such that the solution, $\b P$, to the Lyapunov matrix equation\index{sliding mode observer!Lyapunov matrix equation}, $(\b A-\b l\b c)^{\top}\b P+\b P(\b A-\b l\b c)=-\b Q$, satisfies the condition, $\b b^{\top}\b P=\b c$.
\end{exercise}

\begin{exercise}\label{exer:conver_esti_ero}\rm
Show that the estimation error $\b e$  of the sliding mode observer~\eqref{eq:wzsmo_matched} satisfies the equation
\[
\dot{\b e} = \dot{\b x}-\dot{\hat{\mb{\bm$x$}}}= \left(\mb{\bm$A$}-\mb{\bm$LC$} \right)\mb{\bm$e$} - \mb{\bm$BE$}(\mb{\bm$e$},\eta) + \mb{\bm$B\xi$}.
\]
Then show that
\[
\frac{d}{dt}\left(\mb{\bm$e$}^{\top}\mb{\bm$Pe$} \right)=-\mb{\bm$e$}^{\top}\mb{\bm$Qe$}<0,
\]
which implies that
\[
\lim_{t \to \infty}\mb{\bm$e$}(t)=\mb{\bm$0$}.
\]
\end{exercise}
%\begin{exercise}\label{exerc:exactnond}\rm
%The notion of {\em feasible direction\/}, which we present next, plays an important role in the analysis of constrained optimization problems such as that given by~(\ref{eq:consopty}). We say that a vector $\b d$ is a feasible direction at $\b x \in \Omega$ if there is an $\bar{\alpha}>0$ such that $\b x+\alpha\b d \in \Omega$ for all $\alpha \in \left[0, \bar{\alpha}\right]$. Suppose now that the minimizer $\b x^{\ast}$ of the optimization problem~(\ref{eq:consopty}) lies on the boundary of $\Omega$ and there is a feasible direction $\b d$ at $\b x^{\ast}$ such that $\b d^{\top}\nabla f\left( \b x^{\ast}\right) > 0$. Show, using contraposition, that if $p$ is an exact penalty function for the problem, then $p$ is not differentiable at $\b x^{\ast}$.
%\end{exercise}
%\begin{exercise}\label{exer:norfotrans}\rm
%Consider an uncertain, or nonlinear, dynamical system model,
%\begin{equation}
%\dot{\b x}(t)=\b A\b x(t)+\b f(t,\b x(t))+\b B\left(\b u(t)+ \b h(t,\b x(t),\b u(t))\right),
%\label{eq:uncmopl}
%\end{equation}
%where $\b x(t)\in \mb{\Bb R}^n$, $\b u(t)\in \mb{\Bb R}^m$, the constant matrices $\b A \in \mb{\Bb R}^{n \times n}$ and $\b B \in \mb{\Bb R}^{n \times m}$ are known, and the function $\b f$ and $\b h$ model uncertainties, or nonlinearities, in the system. We refer to $\b h$ as the matched uncertainty, while to $\b f$ as the unmatched uncertainty. The reason for calling $\b h$ the matched uncertainty is that it affects the system dynamics through the column space of the matrix $\b B$ in a similar fashion as the control $\b u$ does. In other words, $\b h$ matches the control action $\b u$. We assume that
%\begin{enumerate}
%\item $\|\b f(t, \b x)\|\le \alpha _f \|\b x\|+\beta _f$,
%\item $\|\b h(t, \b x, \b u)\|\le \gamma_h\|\b u\|+\alpha _h \|\b x\|+\beta _h$,
%\end{enumerate}
%where $\alpha_f$, $\beta_f$, $\gamma_h$, $\alpha_h$, and $\beta_h$ are known nonnegative constants, and $\|\cdot \|=\|\cdot \|_2$ denotes the Euclidean norm for vectors and the induced spectral norm for matrices. We also assume that
%\begin{enumerate}
%\setcounter{enumi}{2}
%\item the pair $(\b A, \b B)$ is controllable,
%\item $\mb{rank}\, \b B=m$.
%\end{enumerate}
%Let
%\[
%\tilde{\b x} = \b M \b x = \left[\begin{array}{c}
%\b W^{g}\\
%\b S
%\end{array}\right]\b x,
%\]
%where $\b W^{g}$ and $\b S$ are the matrices obtained from the Switching Surface Design Algorithm. Transform the uncertain system model~(\ref{eq:uncmopl}) into the $\tilde{\b x}$-coordinates.
%\end{exercise}
%\begin{exercise}\label{exer:evalredsys}\rm
%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 achieved in such a situation maybe UUB with some region of attraction $\Gamma \subset \mb{\Bb R}^n$ and uniformly stable set $\Sigma$ containing the origin of $\mb{\Bb R}^n$. The goal of this exercise is to estimate the ``size'' of the set $\Gamma$ and that of $\Sigma$.
%
%It follows from Exercise~\ref{exer:norfotrans} that the uncertain system~(\ref{eq:uncmopl}) confined to the switching surface $\b{\sigma}=\b S\b x=\b 0$ is described, in the new coordinates, by
%\begin{equation}
%\dot{\b z}=\b A_{11}\b z+\b W^g\tilde{\b f}(t,\b z,\b 0),
%\label{eq:redoredsy}
%\end{equation}
%where $\|\tilde{\b f}(t,\b z,\b 0)\|\le \alpha_f\|\b W\|\|\b z\|+\beta_f$. We refer to the above system as the {\it reduced-order system\/}. The matrix $\b A_{11}$ is asymptotically stable by construction. Therefore, for any symmetric and positive definite matrix $\b Q\in \mb{\Bb R}^{(n-m)\times (n-m)}$ the solution $\b P$ to the Lyapunov equation
%\[
%\b A_{11}^{\top}\b P+\b P\b A_{11}=-2\b Q
%\]
%is (symmetric) positive definite. Consider now the positive definite function:
%\[
%V_1=\left(\b z^{\top}\b P\b z \right)^{1/2}.
%\]
%The matrix $\b P$ is symmetric positive definite, hence it can be represented as
%\[
%\b P=\b P^{\frac{1}{2}}\b P^{\frac{1}{2}},
%\]
%where $\b P^{\frac{1}{2}}\in \mb{\Bb R}^{(n-m) \times (n-m)}$ is symmetric and nonsingular. We refer to $\b P^{\frac{1}{2}}$ as a square root of $\b P$---see page~\pageref{eq:squmatrpro} for a discussion of a matrix square root. Let
%\[
%\b \zeta =\b P^{\frac{1}{2}}\b z.
%\]
%Then,
%\[
%V_1 = \left(\b z^{\top}\b P\b z \right)^{1/2}= \left(\b \zeta ^{\top}\b \zeta \right)^{1/2}= \| \b \zeta \|.
%\]
%Evaluate the time derivative of $V_1$ on the trajectories of the reduced-order system~(\ref{eq:redoredsy}). Specifically, let \[
%\rho=\frac{\lambda_{\min}(\b Q)}{\lambda_{\max}(\b P)}- \alpha_f \frac{\lambda_{\max}(\b P)}{\lambda_{\min}(\b P)}\|\b W^g\|\|\b W\|,
%\]
%and
%\[
%r=\frac{\beta_f}{\rho}\frac{\lambda_{\max}(\b P)}{\lambda_{\min}^{\frac{1}{2}}(\b P)}\|\b W^g\|.
%\]
%Assume that $\rho>0$. Using the above notation, show that
%\[
%\frac{d}{dt}\| \b \zeta \|  \le -\rho(\|\b \zeta\|-r).
%\]
%\end{exercise}
%\begin{exercise}\label{ex:guubredord}\rm
%Prove the following theorem concerning stability of the motion on the switching surface.\begin{thm}
%\label{prop:reduce}
%The reduced-order system, analyzed in Exercise~\ref{exer:evalredsys},
%\[
%\dot{\b z}=\b A_{11}\b z+\b W^g\tilde{\b f}(t,\b z,\b 0)
%\]
%is globally UUB, with the uniformly stable set
%\[
%\Sigma=\left\{\b z:\|\b \zeta\|=\|\b P^{\frac{1}{2}}\b z\|\le r\right\} \subset \mb{\Bb R}^{n-m}.
%\]
%\end{thm}
%In the proof of the above theorem, use the following lemma, which we recommend that you prove before using it.
%\begin{lemma}
%\label{lemma:hit}
%Suppose $\phi(t),a,b \in \mb{\Bb R}$, and $a\ne 0$. If
%\[
%\dot{\phi}(t)-a\phi (t) \le -b,
%\]
%then, for $t\ge t_0$,
%\[
%\phi(t) \le \frac{b}{a}+\left(\phi (t_0)-\frac{b}{a} \right)e^{a(t-t_0)}.
%\]
%\end{lemma}
%\end{exercise}
%\begin{exercise}\label{exer:contpoti}\rm
%Let $\b{\sigma}=\b S\b x$, $\|\cdot\|_p$ be the $p$-norm of a vector,  and let $\nabla_{\sigma}$ denote the gradient operation with respect to $\b \sigma$. Consider the following class of controllers:
%\begin{equation}
%\fbox{$\displaystyle\b u^{(p)}=-\eta \nabla_{\sigma}\|\b \sigma\|_p,\quad 1\le p\le \infty $}
%\label{eq:pcontrb}
%\end{equation}
%where
%\[
%\nabla_{\sigma}\|\b \sigma \|_p=\left[\begin{array}{cccc}
%\frac{\partial}{\partial \sigma_1}\|\b \sigma\|_p & \frac{\partial}{\partial \sigma_2}\|\b \sigma\|_p & \cdots & \frac{\partial}{\partial \sigma_n}\|\b \sigma\|_p
%\end{array}\right]^{\top}.
%\]
%\begin{itemize}
%\item[(i)] Write down the expressions for the controllers when $p=1, 2, \infty$.
%\item[(ii)] Find the bounds of $\b u^{(p)}$ for $p=1, 2, \infty$.
%\end{itemize}
%\end{exercise}
%\begin{exercise}\label{exer:dersax}\rm
%The goal of this exercise is to obtain certain estimates that can be used to investigate stability of the closed-loop system
%\begin{equation}
%\left.\begin{array}{lll}
%\dot{\b x} &=& \b A\b x+\b f+\b B \left(\b u+\b h\right)\\
%\b u&=&\b u^{(p)} = -\eta \nabla_{\sigma}\|\b \sigma (\b x)\|_p,\; \mb{for} \quad p=1,2,\;\mb{or} \;\infty.
%\end{array}\right\}
%\label{eq:clsywpcontr}
%\end{equation}
%Let
%\[
%\left[\begin{array}{c}
%\b \zeta\\
%\b \sigma
%\end{array}\right]= \left[\begin{array}{c}
%\b P^{\frac{1}{2}}\b W^{g}\\
%\b S
%\end{array}\right]\b x.
%\]
%Note that
%\[
%\b x=\left[\begin{array}{cc}
%\b W\b P^{-\frac{1}{2}} & \b B
%\end{array}\right]\left[\begin{array}{c}
%\b \zeta\\
%\b \sigma
%\end{array}\right].
%\]
%In the new coordinates the system takes the form
%\begin{equation}
%\left[\begin{array}{c}
%\dot{\b \zeta}\\
%\dot{\b \sigma}
%\end{array}\right]\\
%= \left[\begin{array}{cc}
%\b P^{\frac{1}{2}}\b A_{11}\b P^{-\frac{1}{2}} & \b P^{\frac{1}{2}}\b A_{12}\\
%\b A_{21}\b P^{-\frac{1}{2}} & \b A_{22}
%\end{array}\right]\left[\begin{array}{c}
%\b \zeta\\
%\b \sigma
%\end{array}\right]+\left[\begin{array}{c}
%\b O\\
%\b I_m
%\end{array}\right](\b u+\tilde{\b h})+\left[\begin{array}{c}
%\b P^{\frac{1}{2}}\b W^{g}\tilde{\b f}\\
%\b S \tilde{\b f}
%\end{array}\right].
%\label{eq:sysnewcoo}
%\end{equation}
%Obtain estimates for $\frac{d}{dt}\|\b \sigma\|$ ($\b \sigma\ne \b 0$) and $\frac{d}{dt}\|\b \zeta\|$ ($\b \zeta\ne \b 0$) evaluated on any solution of the closed-loop system~(\ref{eq:sysnewcoo}). Specifically, show that, under some assumptions
%\[
%\fbox{$\displaystyle \frac{d}{dt}\|\b \sigma \| \le a_{21}\|\b \zeta\|+a_{22}\|\b \sigma \|-\mu^{(p)} $}
%\]
%and
%\[
%\fbox{$\displaystyle\frac{d}{dt}\|\b \zeta \| \le -\rho\|\b \zeta\|+a_{12}\|\b \sigma\|+\tilde{\beta}_f $}
%\]
%where
%\begin{eqnarray*}
%a_{21}&=&\|\b A_{21}\b P^{-\frac{1}{2}}\|+\alpha_h\|\b W\b P^{-\frac{1}{2}}\|+\alpha_f\|\b S\|\|\b W\b P^{-\frac{1}{2}}\|,\\
%a_{22} &=& \|\b A_{22}\|+\alpha_h\|\b B\|+\alpha_f\|\b S\|\|\b B\|,\\
%a_{12}&=&\|\b P^{\frac{1}{2}}\b A_{12}\|+\alpha_f\|\b P^{\frac{1}{2}}\b W^g\|\|\b B\|,\\
%\tilde{\beta}_f &=&\beta_f\|\b P^{\frac{1}{2}}\b W^g\|, \\
%\mu^{(1)} &=& \eta(1-\gamma_h \sqrt{m})-\beta_h-\beta_f\|\b S\|,\\
%\mu^{(2)}&=&\eta(1-\gamma_h)-\beta_h-\beta_f\|\b S\|,\\
%\mu^{(\infty)}&=&\frac{\eta}{\sqrt{m}}(1-\gamma_h \sqrt{m})-\beta_h-\beta_f\|\b S\|.
%\end{eqnarray*}
%\end{exercise}
%\begin{exercise}\label{exer:slimodo}\rm
%In the following lemma to be proved, a definition of sliding mode domain is used, which we provide below. This definition can be found, for example, in Utkin~\cite[page~218]{utkin77}.
%\begin{defn}
%A domain $\Delta$ in  the manifold $\{\b x:\b{\sigma}(\b x)=\b 0\}$ is a sliding mode domain if for each $\varepsilon>0$ there exists a $\delta>0$ such that any trajectory starting in the $n$-dimensional $\delta$-neighborhood of $\Delta$ may leave the $n$-dimensional $\varepsilon$-neighborhood of $\Delta$ only through the $n$-dimensional $\varepsilon$-neighborhood of the boundaries of $\Delta$.
%\end{defn}\markdefn
%
%The above definition is illustrated in Figure~\ref{fig:sliddom}.
%\begin{figure}
%\centering
%\includegraphics[scale=0.75]{sliddom.eps}
%%\vspace{3in}
%%\hskip1in\special{eps:c:/mybook/sliddom.eps y=3in}
%\caption{Two-dimensional illustration of a sliding mode domain.}
%\label{fig:sliddom}
%\end{figure}
%Prove the following lemma:
%\begin{lemma}
%\label{lemma:sliding}
%If $a_{21} \ne 0$, then the region
%\[
%\Delta=\left\{ (\b \zeta,\b \sigma):\|\b \zeta\|<\frac{\mu^{(p)}}{a_{21}},\b \sigma=\b 0\right\}
%\]
%is a sliding mode domain for the closed-loop system~(\ref{eq:clsywpcontr}) of Exercise~\ref{exer:dersax}.
%\end{lemma}
%The notation of the lemma is explained in Exercise~\ref{exer:dersax}.
%\end{exercise}
%\begin{exercise}\label{exer:cluubbigthm}\rm
%Prove the following result concerning UUB of the closed-loop system~(\ref{eq:clsywpcontr}) of Exercise~\ref{exer:dersax}.  Consult~\cite{glazos95} for a method of proving this result.
%\begin{thm}
%\label{thm:practical}
%Let $a_{21}\ne 0$ and
%\[
%\Gamma=\left\{ (\b \zeta,\b \sigma):\alpha\|\b \zeta \|+\beta \|\b \sigma \| < \gamma \right\},
%\]
%where
%\begin{eqnarray*}
%\alpha &=& \frac{\beta a_{21}}{\beta+\rho}\\
%\beta &=& \frac{1}{2}\left( a_{22}-\rho+\sqrt{(a_{22}+\rho)^2+4a_{12}a_{21}}\right)\\
%\gamma&=& \mu^{(p)}-\frac{\alpha \tilde{\beta}_f}{\beta}.
%\end{eqnarray*}
%Then, the closed-loop system is UUB with respect to the uncertainty region
%\[
%\Sigma \times \{\b 0\}= \left\{ (\b \zeta,\b 0):\|\b \zeta \| \le r\right\}
%\]
%with region of attraction $\Gamma$, where
%\[
%r=\frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|}{\lambda_{\min}^{\frac{1}{2}}(\b P)\rho}.
%\]
%The set $\Sigma$ is defined in Exercise~\ref{ex:guubredord}.
%\end{thm}
%For an illustration of the above mentioned region see Figure~\ref{fig:region}.
%\begin{figure}
%\centering
%\includegraphics[scale=0.55]{region.eps}
%%\vspace{3in}
%%\hskip1in\special{bmp:c:/mybook/region.bmp x=4in y=3in}
%\caption{Illustration of the region of attraction and the uncertainty region of Theorem~\ref{thm:practical}.}
%\label{fig:region}
%\end{figure}
%\end{exercise}
%\begin{exercise}\rm \label{ex:patildnonz} Show, using Lemma~\ref{lemma3}, that $\b P\tilde{\b a}\ne \b 0$, where $\b P$ and $\tilde{\b a}$ are defined  on page~\pageref{patilnozer}.
%\end{exercise} 