%\setcounter{chapter}{6}
\chapter{Sliding Modes}
\label{chp:sliding}
\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 found in the following sections.

A {\em variable structure system\/} is a dynamical system whose structure changes in accordance with the current value of its state. A variable structure system can be viewed as a system composed of independent structures together with a switching logic to switch between each of the structures. With appropriate switching logic a variable structure system can exploit the desirable properties of each of the structure the system is composed of. Even more, a variable structure system may have a property that is not a property of any of its structures. We illustrate the above ideas with two numerical examples.
\begin{example}\rm
Consider a dynamical system model
\begin{equation}
\left\{\begin{array}{lll}
\dot{x}_1 &=& x_2\\
\dot{x}_2 &=& -ax_1,
\end{array}\right.
\label{eq:firstvssy}
\end{equation}
having two structures corresponding to $a=1/4$ and $a=4$. The phase-plane portraits of the structures are shown in Figure~\ref{fig:vsc_str1} and Figure~\ref{fig:vsc_str2}.
\begin{figure}
\vspace{3in}
\hskip1in\special{bmp:c:/sliding/vsc_str1.bmp x=4in y=3in}
\caption{A phase-plane portrait of the system's first structure corresponding to {\em a}=1/4.}
\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 a}=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}
a=\left\{\begin{array}{ccc}
1/4 & \mb{if} & x_1x_2<0\\
4& \mb{if} & x_1x_2>0.
\end{array}\right.
\label{eq:swilogofvs}
\end{equation}
A typical trajectory of the system~(\ref{eq:firstvssy}) with the switching logic specified by~(\ref{eq:swilogofvs}) is shown in Figure~\ref{fig:vsc_clsc}. The starting point of the trajectory is $\b x(0)=\left[\begin{array}{cc}
0 & 10
\end{array}\right]^T$.
\begin{figure}
\vspace{3.5in}
\hskip.75in\special{bmp:c:/sliding/vsc_cls.bmp x=4.67in y=3.5in}
\caption{A trajectory of the variable structure system modeled 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}
\vspace{3in}
\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 thus have two independent structures, neither of them being stable. Let us now use the following switching logic 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}
\vspace{3in}
\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}
Note that once the trajectories intercept the switching line they are confined to it thereafter. If we increase the integration step size, then we can observe oscillations, or {\em chattering}, of the trajectory about the switching surface as can be seen in Figure~\ref{fig:vsc_ch2}.
\begin{figure}
\vspace{3in}
\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}. 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 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 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}
\vspace{3in}
\hskip1in\special{bmp:c:/papers/sliding.bmp x=4in y=3in}
\caption{An illustration of the sufficient condition of the attractivity to 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}. 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,
\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}
\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 diferential 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 since 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 defined $\mb{sign}(0)=0$.
\end{example}
\markex
We now briefly describe Filippov's approach. Filippov~\cite[pages~2-3]{filippov88} recommends that the generalized solution to an ordinary differential equation with a discontinuous right-hand 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. We begin with an example. Suppose that we are given a single-input dynmical 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 next 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 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 containing $\b f^{-}$ and $\b f^{+}$.
\begin{figure}
\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 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 now 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 $\b x(t)$, 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 described by~(\ref{eq:cluster_f}). This is illustrated in Figures~\ref{fig:filip} and~\ref{fig:filip_3d}.
\begin{figure}
\vspace{3.5in}
\hskip.7in\special{bmp:c:/mybook/filip_3d.bmp x=4.67in y=3.5in}
\caption{Construction 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 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 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, 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}. It is very useful in designing variable structure sliding mode controllers.
\end{itemize}

We now discuss a 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. Thus, the plant model we are given has the 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 now 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 x} = \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 x.
\label{eq:sysinsliro}
\end{equation}
The above equation describes the system dynamics in sliding. Observe the order reduction of the system dynamics in sliding. We now have $n-m=n-1$ first-order differential equations. The poles of the reduced-order system are the roots of the equation
\[
\lambda^{n-1}+s_{n-1}\lambda^{n-2}+\cdots+s_2\lambda+s_1=0.
\]
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
\[
s_1x_1+x_2=0.
\]
The dynamics of the system in sliding are described by
\[
\dot{x}_1 = -s_1x_1,
\]
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 goal 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. Specifically, we use a generalized distance from the sliding surface as a Lyapunov function candidate. Then, we select the controllers' gains so that the time derivative of the chosen Lyapunov function candidate evaluated on the solution of the controlled system is negative definite with resepect to the switching surface, thus ensuring the motion of the state trajectory to the surface as it is illustrated in Figure~\ref{fig:sliding}. Our goal is to find $u=u(\b x)$ so that $\sigma \dot{\sigma} <0$. 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,
\begin{eqnarray*}
\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{eqnarray*}
Using the above, we can propose a possible set of gains that lead to satisfaction of the attractivity to the surface condition. A possible choice of gains is
\begin{equation}
\left.\begin{array}{lll}
k_1^{+} &<& -|a_0|\;\mb{if}\; \sigma x_1 >0\\
k_1^{-} &>& |a_0|\;\mb{if}\; \sigma x_1 <0\\
{}&\vdots& {}\\
k_n^{+} &<& -(|a_{n-1}|+|s_{n-1}|)\;\mb{if}\; \sigma x_n>0\\
k_n^{-} &>& |a_{n-1}|+|s_{n-1}|\;\mb{if}\; \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.

\section{Sliding in Multi-Input Systems}
In this section, we describe a sliding mode in linear multi-input plants. When the number of the plant inputs is $m$, we can use sliding surfaces 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 $\b \sigma (\b x(t))=\b 0$ 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,
\[
\left(\nabla \sigma_i\right)^T\dot{\b x}(t)= \b s_i\dot{\b x}(t)= 0,\quad\mb{for}\quad i=1,2,\ldots,m,
\]
that is,
\[
\dot{\b \sigma}(\b x(t)) = \b S\dot{\b x}(t)= \b 0.
\]
We can thus characterize the system in sliding mode as
\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\/} and it is denoted using the symbol $\b u_{eq}$. We have
\[
\b u_{eq}=-\left(\b S\b B\right)^{-1}\b S\b A\b x.
\]
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}. Thus, we can describe the behavior of the system in sliding in an alternative way 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.

\section{Sliding Mode and System Zeros}
In this section, we show that the poles of the reduced-order system in sliding are the so called {\em 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 $\det \b S\b B \ne 0$. The system zeros are defined using the {\em 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 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}
We now 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. We have
\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}
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)\\
&{}&=\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)\\
&{}&=\det \left(\frac{s\left(\b S\, \mb{Adj}(s\b I_n-\b A)\b B\right)(\b S\b B)^{-1} }{\det(s\b I_n - \b A)} \right)\\
&{}&=s^m\det\left( \b S(s\b I_n-\b A)^{-1}\b B\right)\det (\b S\b B)^{-1}.
\end{eqnarray*}
We now use~(\ref{eq:deofsymaps}) to 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*}
and hence
\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}
We conclude that the dynamics 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 can be viewed as choosing an output matrix $\b S$ so that the system~(\ref{eq:syzereo}) has a desired set of system zero locations, which in turn govern the dynamics of the systems in sliding along $\b S\b x=\b 0$.

\section{Non-Ideal Sliding Mode}
We consider a nonlinear dynamical system modeled by
\begin{equation}
\dot{\b x}=\b f(t,\b x)+\b B(t,\b x)\b u,
\label{eq:nosymo}
\end{equation}
where $\b x(t) \in \mb{\Bb R}^n$, $\b f(t,\b x(t)) \in \mb{\Bb R}^n$, $\b B(t,\b x(t)) \in \mb{\Bb R}^{n\times m}$, and $\b u \in \mb{\Bb R}^m$. The functions $\b f$ and $\b B$ are continuous functions of all their arguments. The switching surface is $\{\b x:\b \sigma(\b x)=\b 0\}$, where $\b \sigma(\b x(t)) \in \mb{\Bb R}^m$. Let
\[
\b S=\frac{\partial \b \sigma}{\partial \b x},
\]
where we assume that $\b S\b B$ is invertible. Once the system enters sliding mode, an appropriately constructed controller is switched at an infinite frequency to keep the system on the surface $\{\b \sigma=\b 0\}$. Thus, once the state trajectory hits the switching surface, say at $t_0$, then for $t\ge t_0$ we have $\b \sigma=\b 0$, and $\dot{\b \sigma}(\b x)=\b 0$. Solving the algebraic equation
\[
\dot{\b \sigma}(\b x) = \b S\dot{\b x}
= \b S\left( \b f(t,\b x)+\b B(t,\b x)\b u\right) = \b 0
\]
for $\b u$ yields
\[
\b u=-\left( \b S\b B\right)^{-1}\b S\b f.
\]
Recall that the above expression for $\b u$ is called the equivalent control and is denoted $\b u_{eq}$. Substituting $\b u_{eq}$ into to the modeling equation~(\ref{eq:nosymo}), we obtain
\[
\dot{\b x}=\b f-\b B\left( \b S\b B\right)^{-1}\b S\b f= \left(\b I_n-\b B\left( \b S\b B\right)^{-1}\b S\right)\b f.
\]
The above equation together with the condition $\b \sigma=\b 0$ describes the system in ideal sliding.

We now study a non-ideal sliding mode. In non-ideal sliding mode we no longer have $\dot{\b \sigma}(\b x)=\b 0$. Hence, solving the equation
\[
\b 0 = \dot{\b \sigma}(\b x)
=\b S\left( \b f(t,\b x)+\b B(t,\b x)\b u\right)
\]
for $\b u$ that we refer now to as $\widetilde{\b u}$, yields
\[
\widetilde{\b u}=-\left( \b S\b B\right)^{-1}\b S\b f+\left( \b S\b B\right)^{-1}\dot{\b \sigma}.
\]
Substituting the above into the plant dynamics gives
\[
\dot{\widetilde{\b x}}=\b f-\b B\left( \b S\b B\right)^{-1}\b S\b f+\b B\left( \b S\b B\right)^{-1}\dot{\b \sigma}.
\]
The above describes the plant modeled by~(\ref{eq:nosymo}) in non-ideal sliding mode. The term $\b B\left( \b S\b B\right)^{-1}\dot{\b \sigma}$ can be viewed as representing non-idealities. In real sliding the state trajectory $\widetilde{\b x}$ will reside in some vicinity of the manifold $\{\b x:\b \sigma({\b x})=\b 0\}$ so that
\[
\|\b \sigma(\widetilde{\b x})\| \le \Delta,
\]
where $\Delta > 0$. We now show, following Utkin~\cite[pages~45--48]{utkin78}, that if $\Delta \to 0$, then the real sliding mode behavior approaches that of the ideal sliding mode.
\begin{thm}[Utkin, 1974]
\label{thm:real_slide}
Suppose that
\begin{enumerate}
\item \label{assump_one} a solution to $\dot{\b x}=\b f(t,\b x)+\b B(t,\b x)\widetilde{\b u}$, on the interval $[0,T]$, satisfies the condition
\[
\|\b \sigma(\widetilde{\b x})\| \le \Delta;
\]
\item \label{assump_two} there is a Lipschitz constant $L$ for the right hand side of
\[
\dot{\b x}=\b f-\b B\left( \b S\b B\right)^{-1}\b S\b f+\b B\left( \b S\b B\right)^{-1}\dot{\b \sigma};
\]
\item \label{assump_three} partial derivatives of the function
\[
\b B\left( \b S\b B\right)^{-1}
\]
exist and are bounded in any bounded domain of $\mb{\Bb R}^n$;
\item \label{assump_four} for the function
\[
\b f(t,\b x)+\b B(t,\b x)\widetilde{\b u}
\]
there exist nonnegative constants $M_1$ and $M_2$ such that
\[
\left\Arrowvert \b f(t,\b x)+\b B(t,\b x)\widetilde{\b u} \right\Arrowvert \le M_1 +M_2\|\b x\|.
\]
\end{enumerate}
Then, for any pair of state trajectories that are solutions to the ideal and non-ideal sliding models for which the initial conditions satisfy
\[
\left\Arrowvert \b x(0)-\widetilde{\b x}(0) \right\Arrowvert \le P \Delta, \quad P \ge 0,
\]
there exists a nonnegative $H$ such that
\[
\left\Arrowvert \b x(t)-\widetilde{\b x}(t) \right\Arrowvert \le H\Delta
\]
for all $t \in [0,T]$.
\end{thm}
{\bf Proof} \quad The trajectory of the system in ideal sliding mode satisfies
\begin{equation}
\b x(t)=\b x(0)+\int_0^t\left( \b f-\b B\left( \b S\b B\right)^{-1}\b S\b f\right)\b x(s)ds,
\label{eq:idslifonl}
\end{equation}
while the trajectory of the system in non-ideal sliding satisfies
\[
\widetilde{\b x}(t) =\widetilde{\b x}(0)+\int_0^t\left( \b f-\b B\left( \b S\b B\right)^{-1}\b S\b f\right)\widetilde{\b x}(s)ds + \int_0^t\b B\left( \b S\b B\right)^{-1}\dot{\b \sigma}\left(\widetilde{\b x}(s)\right)ds.
\]
Integrating the last term of the above by parts gives
\begin{eqnarray*}
\widetilde{\b x}(t)&=&\widetilde{\b x}(0)+\int_0^t\left( \b f-\b B\left( \b S\b B\right)^{-1}\b S\b f\right)\widetilde{\b x}(s)ds\\
& & +\left.\b B\left( \b S\b B\right)^{-1}\b \sigma  (\widetilde{\b x}(s))\right\arrowvert^t_0-\int_0^t\left(\frac{d}{ds}\b B\left( \b S\b B\right)^{-1}\right)\b \sigma (\widetilde{\b x}(s)) ds.
\end{eqnarray*}
Subtracting the above from~(\ref{eq:idslifonl}), taking the norms of both sides, and applying the triangle inequality yields
\begin{eqnarray*}
\|\b x-\widetilde{\b x}\| &\le& \|\b x(0)-\widetilde{\b x}(0)\|+\left\Arrowvert \left.\b B\left( \b S\b B\right)^{-1}\b \sigma  (\widetilde{\b x}(s))\right\arrowvert^t_0 \right\Arrowvert\\
& & +\int_0^t \left\Arrowvert \left(\frac{d}{ds}\b B\left( \b S\b B\right)^{-1}\right)\b \sigma (\widetilde{\b x}(s))\right\Arrowvert ds\\
& & + \int_0^t \left\Arrowvert \left( \b f-\b B\left( \b S\b B\right)^{-1}\b S\b f\right)\b x(s)-\left( \b f-\b B\left( \b S\b B\right)^{-1}\b S\b f\right)\widetilde{\b x}(s) \right\Arrowvert ds.
\end{eqnarray*}
Taking into account the hypothesis $\left\Arrowvert \b x(0)-\widetilde{\b x}(0) \right\Arrowvert \le P \Delta$ and assumption~\ref{assump_two}, we obtain
\begin{eqnarray*}
\|\b x-\widetilde{\b x}\| &\le& P\Delta +\left\Arrowvert \left.\b B\left( \b S\b B\right)^{-1}\b \sigma  (\widetilde{\b x}(s))\right\arrowvert^t_0 \right\Arrowvert\\
& & +\int_0^t \left\Arrowvert \left(\frac{d}{ds}\b B\left( \b S\b B\right)^{-1}\right)\b \sigma (\widetilde{\b x}(s))\right\Arrowvert ds+\int_0^t L\|\b x(s)-\widetilde{\b x}(s)\|ds.
\end{eqnarray*}
We next use assumption~\ref{assump_one} to get
\begin{eqnarray}
\|\b x-\widetilde{\b x}\| &\le& \left(\rule{0in}{3.5ex}P+\left\Arrowvert \b B\left( \b S\b B\right)^{-1}\widetilde{\b x}(t)-\b B\left( \b S\b B\right)^{-1}\widetilde{\b x}(0) \right\Arrowvert \right.\nonumber\\
& & +\left.\int_0^t \left\Arrowvert \left(\frac{d}{ds}\b B\left( \b S\b B\right)^{-1}\right)(\widetilde{\b x}(s))\right\Arrowvert ds\right)\Delta+\int_0^t L\|\b x(s)-\widetilde{\b x}(s)\|ds.
\label{eq:norxminutilx}
\end{eqnarray}
The trajectory of the system in non-ideal sliding is bounded on the interval $[0,T]$. Indeed, using assumption~\ref{assump_four} to the system $\dot{\widetilde{\b x}}=\b f(t,\widetilde{\b x})+\b B(t,\widetilde{\b x})\widetilde{\b u}$, we get
\begin{eqnarray*}
\|\widetilde{\b x}(t)\| &\le& \|\widetilde{\b x}(0)\|+\int_0^t \left\Arrowvert \b f(s,\widetilde{\b x})+\b B(s,\widetilde{\b x})\widetilde{\b u}\right\Arrowvert ds\\
&\le& \|\widetilde{\b x}(0)\|+\int_0^t \left( M_1+M_2 \left\Arrowvert\widetilde{\b x}\right\Arrowvert\right) ds\\
&\le& \|\widetilde{\b x}(0)\|+M_1T+\int_0^t M_2 \left\Arrowvert\widetilde{\b x}\right\Arrowvert ds.
\end{eqnarray*}
Applying now the Bellman-Gronwall lemma, from page~\pageref{lem:bellgron}, to the above integral inequality, we obtain
\begin{eqnarray*}
\|\widetilde{\b x}(t)\| &\le& \left(\|\widetilde{\b x}(0)\|+M_1T \right)e^{M_2t}\\
&\le& \left(\|\widetilde{\b x}(0)\|+M_1 T \right)e^{M_2 T},\quad t \in[0,T].
\end{eqnarray*}
The above means that $\widetilde{\b x}$ is bounded on the interval $[0,T]$ and hence the function $\b B\left( \b S\b B\right)^{-1}$ is bounded on the same interval. By assumption~\ref{assump_three}, partial derivatives of the function $\b B\left( \b S\b B\right)^{-1}$ exist and are bounded. Applying the above to~(\ref{eq:norxminutilx}), we conclude that for some $S\ge 0$ that depends on the initial conditions, time $T$, and constant $P$, we have
\[
\|\b x-\widetilde{\b x}\| \le S\Delta +L\int_0^t \left\Arrowvert \b x(s)-\widetilde{\b x}(s)\right\Arrowvert ds.
\]
Application of the Bellman-Gronwall lemma to the above inequality yields
\[
\|\b x-\widetilde{\b x}\| \le S\Delta e^{LT},\quad t \in[0,T].
\]
Let $H=Se^{LT}$. Then, we can write
\[
\|\b x-\widetilde{\b x}\| \le H\Delta ,
\]
which completes the proof of the theorem.\\
\hspace*{5.5in} $\Box$ 

\section{Sliding Surface Design}
\label{sec:slidingsurf}
We now prove 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. A general method for designing a sliding surface is contained implicitly in the proof of the following theorem that comes from Hui and \.{Z}ak~\cite{hui93}. 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 $\mb{eig}(\b A)$. 
\begin{thm}
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. 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 $\b W^{g}\b W=\b I_{n-m}$, $\b W^{g}\b B=\b 0$, 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 $[\b T, \b B]$ 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 now introduce a new coordinate $\b z$ such that $\b x=[\b T , \b B]\b z$. Then, in the new coordinates the system $\dot{\b x}=\b A\b x+\b B\b u$ has the form
\[
\dot{\b z} = \left[\begin{array}{cc}
\b T^{g}\b A\b T & \b T^{g}\b A\b B\\
\b B^{g}\b A\b T & \b B^{g}\b A\b B
\end{array}\right]\b z+\left[\begin{array}{c}
\b O\\
\b I_m
\end{array}\right]\b u.
\]
Since the pair $(\b A,\b B)$ is controllable, we can use the 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 $\b F\in \mb{\Bb R}^{m\times (n-m)}$ 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 $\b W^{g}=\b T^{g}$ and let $\b W=\b T-\b B\b F$. Then, since by construction $\b T^{g}\b T=\b I_{n-m}$ and $\b T^{g}\b B=\b 0$, we have $\b W^{g}\b W=\b I_{n-m}$ and $\b W^{g}\b B= \b 0$. 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 $[\b W,\b B]$ 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
\[
\left[\begin{array}{cc}
\b W,&\b B
\end{array}\right]^{-1}=\left[\begin{array}{c}
\b W^{g}\\
\b S
\end{array}\right]
\]
for some $\b S\in \mb{\Bb R}^{m \times n}$. Note that $\b S\b B=\b I_m$ and $\b S\b W=\b 0$. We now introduce new coordinates
\[
\b z = \left[\begin{array}{c}
\b z_1\\
\b z_2
\end{array}\right]
= \left[\begin{array}{c}
\b W^{g}\\
\b S
\end{array}\right]\b x,
\]
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 $\b S\b x=\b 0$ 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}$\Box$

We use the results of the proof of the above theorem to propose the following sliding surface design algorithm.
\begin{algorithm}\rm
Given a controllable pair $(\b A, \b B)$ with $\mb{rank}\,\b B=m$, and the set of desired poles $\{\lambda_1,\lambda_2,\ldots,\lambda_{n-m}\}$ of the system restricted to a sliding surface.
\begin{enumerate}
\item Find a matrix $\b T\in \mb{\Bb R}^{(n-m)\times n}$ such that $[\b T, \b B]$ 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 Take $\b W=\b T-\b B\b F$.
\item Compute
\[
\left[\begin{array}{cc}
\b W,&\b B
\end{array}\right]^{-1}=\left[\begin{array}{c}
\b W^{g}\\
\b S
\end{array}\right].
\]
\end{enumerate}
Form state sliding surface $\b S\b x=\b 0$ using the last $m$ rows of $\left[\begin{array}{cc}
\b W,&\b B
\end{array}\right]^{-1}$.
\end{algorithm}
\markex

We now 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 $\left[\b T, \b B\right]$ is invertible. For example, we can select
\[
\b T=\left[\begin{array}{ccc}
1 & 0 & 0
\end{array}\right]^T.
\]
Note that
\[
\left[\b T, \b B\right]=\b I_3.
\]
Hence, the inverse
\[
\left[\b T, \b B\right]^{-1}=\left[\begin{array}{c}
\b T^g\\
\b B^g
\end{array}\right]=\b I_3.
\]
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
\[
\b F=\left[\begin{array}{c}
1\\
0
\end{array}\right].
\]
We then compute $\b W=\b T-\b B\b F=\left[\begin{array}{ccc}
1& -1& 0
\end{array}\right]^T$, and find
\[
\left[\b W, \b B\right]^{-1}=\left[\begin{array}{ccc}
1 & 0 & 0\\
-1 & 1 & 0\\
0 & 0 & 1
\end{array}\right]^{-1}=\left[\begin{array}{ccc}
1 & 0 & 0\\
1 & 1 & 0\\
0 & 0 & 1
\end{array}\right]=\left[\begin{array}{c}
\b W^g\\
\b S
\end{array}\right].
\]
Hence,
\[
\b S=\left[\begin{array}{ccc}
1 & 1 & 0\\
0 & 0 & 1
\end{array}\right].
\]

\end{example}
\markex
\section{Bounded Control}
In practice, a control input signal is obtained from a physical device, the actuator, which has a limited range, that is, the actuator is capable of providing only a limited amount of energy. We say that the actuator saturates or it has {\em limited authority}. A practical control system design must account for this fact. A controller that is designed to avoid reaching the saturation limits is usually considered as overdesigned, and hence inefficient, since the actuator has the capability that is not used. Friedland~\cite[page~96]{friedland96} compares such a design to a heavily overpowered automobile. In this section, we analyze sliding mode controllers for uncertain dynamical system with actuators of limited authority. 

\subsection{Plant}
We consider a class of uncertain, or nonlinear, dynamical systems modeled by
\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),\quad \b x(t_0)=\b x_0,
\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 action of 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}
We further assume
\begin{enumerate}
\setcounter{enumi}{4}
\item {\em Existence of solutions}. For each $t_0,t_1 \in \mb{\Bb R}$, $t_0 < t_1$, $\b x_0\in \mb{\Bb R}^n$, there exists a solution $\b x(\cdot):[t_0,t_1) \to \mb{\Bb R}^n$ for the system~(\ref{eq:uncmopl}), with $\b x(t_0)=\b x_0$.
\item {\em Indefinite extension of solutions}. All solutions can be continued over $[t_0,\infty)$, that is, each solution $\b x(\cdot):[t_0,t_1) \to \mb{\Bb R}^n$, with $\b x(t_0)\in \mb{\Bb R}^n$, has an extension $\tilde{\b x}(\cdot):[t_0,\infty) \to \mb{\Bb R}^n$, where $\tilde{\b x}(t)=\b x(t)$ for all $t\in [t_0,t_1)$.
\end{enumerate}
Our goal is to develop bounded state-feedback control strategies such that the zero state of the closed-loop system is uniformly asymptotically stable in the sense of Lyapunov or, the system is at least uniformly ultimately bounded with respect to an ``acceptably'' small neighborhood of the zero state in the sense of the following definition.
\begin{defn}
\label{defn:uub}
Let the set $\Gamma \subseteq \mb{\Bb R}^n$ be a neighborhood of the origin. The system modeled by
\[
\dot{\b x}(t)=\b F(t,\b x(t)),
\]
where $t\in \mb{\Bb R}$, $\b x(t)\in \mb{\Bb R}^n$ and $\b F:\mb{\Bb R}\times \mb{\Bb R}^n \to \mb{\Bb R}^n$ is said to be uniformly ultimately bounded~(UUB) with region of attraction $\Gamma$ if there exists a compact set $\Sigma \subset \mb{\Bb R}^n$, $\Sigma \subset \Gamma$, containing the origin such that the following hold:
\begin{enumerate}
\item[(i)] {\em Uniform boundedness of solutions}. For each $\b x_0 \in \mb{\Bb R}^n$, there exists a bound $b(\b x_0)$ such that if $\b x(\cdot):[t_0,\infty) \to \mb{\Bb R}^n$ is any solution of the system $\dot{\b x}(t)=\b F(t,\b x(t))$ with $\b x(t_0)=\b x_0$, then $\|\b x(t)\| \le b(\b x_0)$ for all $t\ge t_0$.
\item[(ii)] {\em Uniform stability to within $\Sigma$}. For each neighborhood  $N$ of $\Sigma$, there exists a $\delta >0$ such that if $\b x(\cdot):[t_0,\infty) \to \mb{\Bb R}^n$ is any solution of the system $\dot{\b x}(t)=\b F(t,\b x(t))$ with $\|\b x(t_0)\| < \delta$, then $\b x(t)\in N$ for all $t\ge t_0$.
\item[(iii)] {\em Uniform attractivity of $\Sigma$}. For each neighborhood  $N$ of $\Sigma$ and each $\b x_0 \in \Gamma$, there exists $T(N,\b x_0)$ such that if $\b x(\cdot):[t_0,\infty) \to \mb{\Bb R}^n$ is any solution of the system $\dot{\b x}(t)=\b F(t,\b x(t))$ with $\b x(t_0) = \b x_0$, then $\b x(t) \in N$ for all $t \ge t_0+T(N,\b x_0)$.
\end{enumerate}
\end{defn}

Observe that if the system $\dot{\b x}(t)=\b F(t,\b x(t))$ satisfies the requirements of the above definition with $\Sigma =\left\{ \b 0 \right\}$, then it is uniformly asymptotically stable with region of attraction $\Gamma$. Also note that if a given set $\Sigma$ satisfies the requirements of the above definition, then we can say that this set is uniformly asymptotically stable.

\begin{defn}
\label{defn:guub}
The system $\dot{\b x}(t)=\b F(t,\b x(t))$ is globally UUB if it is UUB with $\mb{\Bb R}^n$ as a region of attraction.
\end{defn}

\subsection{Controllers} 
We now discuss the problem of designing controllers for the system~(\ref{eq:uncmopl}) subject to hard bounds on the control action. We analyze three bounded control laws. They are:
\[
\b u^{(1)}(\b x) = -\eta \left[\begin{array}{c}
\mb{sign} \sigma_1(\b x)\\
\mb{sign} \sigma_2(\b x)\\
\vdots\\
\mb{sign} \sigma_m(\b x)
\end{array}\right] = -\eta \mb{\bf sign} \b \sigma(\b x),
\]
\[
\b u^{(2)}(\b x)=\left\{\begin{array}{ccc}
-\eta \frac{\b \sigma (\b x)}{\|\b \sigma (\b x)\|}&\mb{if}&\b \sigma (\b x) \ne \b 0\\
\b 0 & \mb{if} & \b \sigma (\b x) =\b 0,
\end{array}\right.
\]
where $\eta > 0$ is a design parameter. The third control structure is labeled $\b u^{(\infty)}$. To describe this third controller, we need the following notation. Let
\[
\theta=\max_i|\sigma_i|,
\]
Then, the $i$-th component of $\b u^{(\infty)}$ is
\[
u_i^{(\infty)}(\b x)=\left\{\begin{array}{ccc}
-\eta \mb{sign}\sigma_i(\b x) & \mb{if} & \theta=|\sigma_i(\b x)|\\
0 & \mb{if} & \theta>|\sigma_i (\b x) |.
\end{array}\right.
\]
In other words,
\[
u_i^{(\infty)}(\b x)=\left\{\begin{array}{ccc}
-\eta \mb{sign}\sigma_i(\b x) & \mb{if} & |\sigma_i(\b x)| = \|\sigma (\b x)\|_{\infty}\\
0 & \mb{otherwise},
\end{array}\right.
\]
where $\|\cdot\|_{\infty}$ is the $\infty$ - norm of a vector. Let $\|\cdot\|_p$ denotes the $p$-norm of a vector, where in our analysis $p=1,2,\infty$, and let $\nabla_{\sigma}$ denote the gradient operation with respect to $\b \sigma$. Then, in the regions where the gradients $\nabla_{\sigma}\|\sigma \|_p$, $p=1,2,\infty$, are defined, the above controllers can be represented as
\[
\b u^{(p)}=-\eta \nabla_{\sigma}\|\b \sigma\|_p,
\]
where
\[
\nabla_{\sigma}\|\b \sigma \|_p=\left[\begin{array}{cccc}
\frac{\partial}{\partial \sigma_1}\|\b \sigma\|_p & \frac{\partial}{\partial \sigma_2}\|\b \sigma\|_p & \cdots & \frac{\partial}{\partial \sigma_n}\|\b \sigma\|_p
\end{array}\right]^T.
\]
Note that the above controllers are bounded. Indeed,
\[
\|\b u^{(1)}\|_2\le \eta \sqrt{m},\quad\|\b u^{(2)}\|_2\le \eta,\quad \mb{and}\quad \|\b u^{(\infty)}\|_2\le \eta \sqrt{m} .
\]
We can construct control laws for any $p$ such that $1\le p\le \infty$ as follows
\[
\b u^{(p)}(\b x)=-\eta \nabla_{\sigma}\|\b \sigma (\b x)\|_p,
\]
where
\[
\|\sigma\|_p=\left( \sum_{j=1}^{m}|\sigma_j|^p\right)^{1/p}.
\]
For such a generalization we refer the reader to Hui and \.{Zak}~\cite{hui93}.
\subsection{Closed-Loop System}
The closed-loop system that we analyze has the form
\begin{eqnarray*}
\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,\quad\mb{where}\; p=1,2,\;\mb{or}\;\infty.
\end{eqnarray*}
To facilitate the analysis of the above system, we use the following state variable transformation. 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 algorithm in the previous section. Note that $\b M$ is invertible with
\[
\b M^{-1}=\left[\begin{array}{cc}
\b W & \b B
\end{array}\right].
\]
Let $\b z=\b W^{g} \b x$ and $\b \sigma =\b S \b x$. Then,
\[
\tilde{\b x}=\left[\begin{array}{c}
\b z\\
\b \sigma
\end{array}\right].
\]
In the new coordinates the plant model~(\ref{eq:uncmopl}) takes the form
\begin{eqnarray*}
\dot{\tilde{\b x}}&=& \b M\b A\b M^{-1}\tilde{\b x}+\b M\b B(\b u +\b h)+\b f\\
&=& \left[\begin{array}{c}
\b W^{g}\\
\b S
\end{array}\right]\b A \left[\begin{array}{cc}
\b W & \b B
\end{array}\right]\tilde{\b x}+\left[\begin{array}{c}
\b W^{g}\\
\b S
\end{array}\right]\b B(\b u +\b h)+\b f.
\end{eqnarray*}
Let
\[
\b M\b A\b M^{-1}=\left[\begin{array}{cc}
\b A_{11} & \b A_{12}\\
\b A_{21} & \b A_{22}
\end{array}\right],
\]
where $\b A_{11}=\b W^{g}\b A\b W$, $\b A_{12}=\b W^{g}\b A\b B$, $A_{21}=\b S\b A\b W$, and $\b A_{22}=\b S\b A\b B$. Noting that $\b W^{g}\b B=\b O$ and $\b S\b B=\b I_m$, we obtain 
\[
\tilde{\b x} = \left[\begin{array}{c}
\dot{\b z}\\
\dot{\b \sigma}
\end{array}\right]
= \left[\begin{array}{cc}
\b A_{11} & \b A_{12}\\
\b A_{21} & \b A_{22}
\end{array}\right]\left[\begin{array}{c}
\b z\\
\b \sigma
\end{array}\right]+\left[\begin{array}{c}
\b O\\
\b I_m
\end{array}\right](\b u+\tilde{\b h})+\left[\begin{array}{c}
\b W^{g}\tilde{\b f}\\
\b S \tilde{\b f}
\end{array}\right],
\]
where
\[
\tilde{\b f} =\left.\b f(t,\b x)\right\arrowvert_{\b x=\b M^{-1}\tilde{\b x}}= \tilde{\b f}(t,\b z,\b \sigma)
= \b f(t,\b W\b z+\b B\b \sigma),
\]
and
\[
\tilde{\b h} =\left.\b h(t,\b x, \b u)\right\arrowvert_{\b x=\b M^{-1}\tilde{\b x}}= \tilde{\b h}(t,\b z,\b \sigma, \b u)
= \b h(t,\b W\b z+\b B\b \sigma, \b u),
\]
because $\b x=\b M^{-1}\tilde{\b x}=\b W\b z+\b B\b \sigma$. From assumptions $1$ and $2$ we have
\[
\| \tilde{\b f}\| \le \alpha_f\|\b W\|\|\b z\|+\alpha_f\|\b B\|\|\b \sigma\|+\beta_f,
\]
and
\[
\| \tilde{\b h}\| \le \gamma_h\|\b u\|+\alpha_h\|\b W\|\|\b z\|+\alpha_h\|\b B\|\|\b \sigma\|+\beta_h.
\]
In the next section, using the above background results, we estimate stability regions of the closed-loop system.

\section{Estimating Stability Regions}
The presence of unmatched uncertainties has a deteriorating effect on the stability properties of a closed-loop system. The above combined with the imposed hard bounds on the control action may make it impossible to globally asymptotically stabilize the closed-loop system. The best it can be achieved in such a situation maybe UUB with some region of attraction $\Gamma \subset \mb{\Bb R}^n$ and uniformly asymptotically stable set $\Sigma$ containing the origin of $\mb{\Bb R}^n$. Our goal here is to estimate the ``size'' of the set $\Gamma$ and that of $\Sigma$.

\subsection{Stability of Motion on the Switching Surface}

We first investigate the effect of the uncertainties  on the stability of the system restricted to the switching surface. The system confined to the switching surface is described, in the new coordinates, by
\[
\dot{\b z}=\b A_{11}\b z+\b W^g\tilde{\b f}(t,\b z,\b 0),
\]
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
\[
\b A_{11}^T\b P+\b P\b A_{11}=-2\b Q
\]
is (symmetric) positive definite. Consider now the positive definite function:
\[
V_1=\left(\b z^T\b P\b z \right)^{1/2}.
\]
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,
\[
V_1 = \left(\b z^T\b P\b z \right)^{1/2}= \left(\b \zeta ^T\b \zeta \right)^{1/2}= \| \b \zeta \|.
\]
We now evaluate the time derivative of $V_1$ on the trajectories of the reduced-order system. Since $\b P=\b P^{\frac{1}{2}}\b P^{\frac{1}{2}}$ and $\b \zeta =\b P^{\frac{1}{2}}\b z$, then for $\b z \ne \b 0$,
\begin{eqnarray*}
\frac{d}{dt}V_1 &=& \frac{d}{dt}\left(\b z^T\b P\b z \right)^{1/2}\\
&=& \frac{d}{dt}\| \b \zeta \| \\
&=& \frac{\b \zeta ^T \dot{\b \zeta}}{\|\b \zeta \|}\\
&=& \frac{\b \zeta ^T}{\| \b \zeta \|}\left(\b P^{\frac{1}{2}}\b A_{11}\b z+\b P^{\frac{1}{2}}\b W^g\tilde{\b f}(t,\b z,\b 0) \right)\\
&=& \frac{\b z^T}{\| \b P^{\frac{1}{2}}\b z \|}\left(\b P\b A_{11}\b z+\b P\b W^g\tilde{\b f}(t,\b z,\b 0) \right).
\end{eqnarray*}
Note that
\[
\b z^T\b P\b A_{11}\b z = \frac{1}{2}\b z^T\left(\b A_{11}\b P+\b P\b A_{11} \right)\b z=-\b Q.
\]
Hence,
\begin{eqnarray*}
\frac{d}{dt}\| \b \zeta \| &=& \frac{1}{\|\b P^{\frac{1}{2}}\b z\|}\left( -\b z^T\b Q\b z +\b z^T\b P\b W^g\tilde{\b f}(t,\b z,\b 0)\right)\\
&\le & \frac{-\lambda_{\min}(\b Q)\|\b z\|^2}{\|\b P^{\frac{1}{2}}\b z\|}+\frac{\lambda_{\max}(\b P)\|\b z\|}{\|\b P^{\frac{1}{2}}\b z\|}\|\b W^g\|\|\b f(t,\b z,\b 0)\|.
\end{eqnarray*}
To proceed further, note that
\[
\lambda_{\min}^{\frac{1}{2}}(\b P)\| \b z \| \le \|\b P^{\frac{1}{2}} \b z \| \le \lambda_{\max}^{\frac{1}{2}}(\b P) \| \b z \|.
\]
Taking the above into account, we get
\[
\frac{d}{dt}\| \b \zeta \|  \le  -\frac{\lambda_{\min}(\b Q)}{\lambda_{\max}^{\frac{1}{2}}(\b P)}\|\b z\|+\frac{\lambda_{\max}(\b P)}{\lambda_{\min}^{\frac{1}{2}}(\b P)}\|\b W^g\|\|\b f(t,\b z,\b 0)\|.
\]
We now eliminate $\b z$ from the right hand-side of the above inequality. For this observe that $\b z=\b P^{-\frac{1}{2}}\b \zeta$. Hence,
\[
\frac{1}{\lambda_{\max}^{\frac{1}{2}}(\b P)}\|\b \zeta\|\le \|\b z\| \le \frac{1}{\lambda_{\min}^{\frac{1}{2}}(\b P)}\|\b \zeta\|.
\]
Furthermore,
\[
\|\tilde{\b f}(t,\b z,\b 0)\| \le \alpha_f\|\b W\|\|\b z\|+\beta_f
\le \frac{\alpha_f}{\lambda_{\min}^{\frac{1}{2}}(\b P)}\|\b W\|\|\b \zeta\|+\beta_f.
\]
Therefore,
\[
\frac{d}{dt}\| \b \zeta \|  \le -\frac{\lambda_{\min}(\b Q)}{\lambda_{\max}(\b P)}\|\b \zeta\|+\alpha_f \frac{\lambda_{\max}(\b P)}{\lambda_{\min}(\b P)}\|\b W^g\|\|\b W\|\|\b \zeta\|
+\beta_f\frac{\lambda_{\max}(\b P)}{\lambda_{\min}^{\frac{1}{2}}(\b P)}\|\b W^g\|.
\]
Rearranging the above yields
\[
\frac{d}{dt}\| \b \zeta \|  \le -\left(\frac{\lambda_{\min}(\b Q)}{\lambda_{\max}(\b P)}- \alpha_f \frac{\lambda_{\max}(\b P)}{\lambda_{\min}(\b P)}\|\b W^g\|\|\b W\|\right)\|\b \zeta\|+\beta_f\frac{\lambda_{\max}(\b P)}{\lambda_{\min}^{\frac{1}{2}}(\b P)}\|\b W^g\|.
\]
To simplify our presentation 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
\[
r=\frac{\beta_f}{\rho}\frac{\lambda_{\max}(\b P)}{\lambda_{\min}^{\frac{1}{2}}(\b P)}\|\b W^g\|.
\]
We assume that
\begin{enumerate}
\setcounter{enumi}{6}
\item $\rho>0$.
\end{enumerate}
With the above notation, we can write
\[
\frac{d}{dt}\| \b \zeta \|  \le -\rho(\|\b \zeta\|-r).
\]
In our stability analysis we use the following lemma.
\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}
{\bf Proof}\quad We multiply $\dot{\phi}(t)-a\phi(t)\le -b$ by the integrating factor and represent the result in the form
\[
\frac{d}{dt}\left( e^{-at}\phi(t) \right)\le -be^{-at}.
\]
Integrating the above from $t_0$ to $t$ yields the result.\\
\hspace*{5.5in} $\Box$

Using the above results we can state and prove the following proposition concerning the stability of motion on the switching surface.
\begin{proposition}
\label{prop:reduce}
The reduced-order system
\[
\dot{\b z}=\b A_{11}\b z+\b W^g\tilde{\b f}(t,\b z,\b 0)
\]
is globally UUB, with the uniformly asymptotically 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{proposition}
{\bf Proof}\quad From the discussion before Lemma~\ref{lemma:hit}, it follows that any solution of the reduced-order system will asymptotically approach the set $\Sigma$ because for $\b z \ne \b 0$
\begin{eqnarray*}
\frac{d}{dt}|\b z^T\b P\b z|^{\frac{1}{2}}&=&\frac{d}{dt}\| \b P^{\frac{1}{2}}\b z \|\\ 
&=&\frac{d}{dt}\| \b \zeta \|\\  
&\le& -\rho(\|\b \zeta\|-r)\\
&=& -\rho \left(\|\b P^{\frac{1}{2}}\b z\|-r \right)\\
&<& 0
\end{eqnarray*}
for all $\b z$ outside the set $\Sigma=\left\{\b z:\|\b P^{\frac{1}{2}}\b z \|\le r\right\}$. Hence, all solutions of the reduced-order system are uniformly bounded, and the system is uniformly stable to within the set $\Sigma$. To prove the uniform attractivity of $\Sigma$, let $N \subset \mb{\Bb R}^{n-m}$ be any neighborhood of the set $\Sigma$, and let $R$ be a number such that $R>r$ and $\left\{ \b z:\|\b P^{\frac{1}{2}}\b z\|\le R\right\}\subset N$. By Lemma~\ref{lemma:hit}, any solution of the reduced-order system will enter the set $\left\{ \b z:\|\b P^{\frac{1}{2}}\b z\| \le R\right\}$, and hence the set $N$, and will stay there thereafter, after time $t_0+T_r(R,\b z_0)$, where
\[
T_r(R,\b z_0)=\left\{\begin{array}{ccc}
\frac{1}{\rho}\ln \left( \frac{\|\b P^{\frac{1}{2}}\b z_0\|-r}{R-r}\right) & \mb{if}& \|\b P^{\frac{1}{2}}\b z_0\|>R\\
0 & \mb{if} & \|\b P^{\frac{1}{2}}\b z_0\|\le R.
\end{array}\right.
\]
Hence, the set $\Sigma$ is uniformly attractive, and the proof is complete.\\
\hspace*{5.5in} $\Box$

We note that the size of the uniformly asymptotically stable set $\Sigma$ is determined solely by the unmatched uncertainty parameters $\alpha_f$ and $\beta_f$, and the eigenstructure of the nominal reduced-order system $\dot{\b z}=\b A_{11}\b z$ reflected by the matrices $\b W$ and $\b W^g$. The dynamics of the closed-loop system in a sliding mode are not affected by the matched uncertainty.

\subsection{Closed-Loop System Stability Analysis}

We now investigate the stability of the closed-loop system
\begin{eqnarray*}
\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{eqnarray*}
To simplify the analysis we will be working in the following coordinate system
\[
\left[\begin{array}{c}
\b \zeta\\
\b \sigma
\end{array}\right]= \left[\begin{array}{c}
\b P^{\frac{1}{2}}\b W^{g}\\
\b S
\end{array}\right]\b x.
\]
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
\[
\left[\begin{array}{c}
\dot{\b \zeta}\\
\dot{\b \sigma}
\end{array}\right]\\
= \left[\begin{array}{cc}
\b P^{\frac{1}{2}}\b A_{11}\b P^{-\frac{1}{2}} & \b P^{\frac{1}{2}}\b A_{12}\\
\b A_{21}\b P^{-\frac{1}{2}} & \b A_{22}
\end{array}\right]\left[\begin{array}{c}
\b \zeta\\
\b \sigma
\end{array}\right]+\left[\begin{array}{c}
\b O\\
\b I_m
\end{array}\right](\b u+\tilde{\b h})+\left[\begin{array}{c}
\b P^{\frac{1}{2}}\b W^{g}\tilde{\b f}\\
\b S \tilde{\b f}
\end{array}\right].
\]
\subsubsection{Some Preliminary Estimates}
We begin our analysis by obtaining estimates for $\frac{d}{dt}\|\b \sigma\|$ and $\frac{d}{dt}\|\b \zeta\|$ evaluated on any solution of the closed-loop system. For $\b \sigma \ne \b 0$, we have
\[
\frac{d}{dt}\|\b \sigma \| = \frac{\b \sigma^T\dot{\b \sigma}}{\|\b \sigma \|}
= \frac{\b \sigma^T}{\|\b \sigma \|}\left( \b A_{21}\b P^{-\frac{1}{2}}\b \zeta +\b A_{22}\b \sigma +\b u+\tilde{\b h}+\b S \tilde{\b f}\right).
\]
In further derivation, we use the fact that
\[
\mb{sign}(\sigma_i) = \frac{\sigma_i}{|\sigma_i|}
=\frac{|\sigma_i|}{\sigma_i},
\]
and hence
\[
\b \sigma^T \mb{\bf sign} (\b \sigma)=\sum_{i=1}^m |\sigma_i|.
\]
For $\b u=\b u^{(1)}$, we have
\[
\frac{d}{dt}\|\b \sigma \| \le\|\b A_{21}\b P^{-\frac{1}{2}}\|\|\b \zeta\|+\|\b A_{22}\|\|\b \sigma\|-\eta\frac{\sum_{i=1}^m|\sigma_i|}{\|\b \sigma\|}+\|\tilde{\b h}\|+\|\b S \tilde{\b f}\|.
\]
Then, since ${\sum_{i=1}^m|\sigma_i|}\ge \|\b \sigma\|$, and taking into account norm bounds on $\tilde{\b h}$ and $\tilde{\b f}$, we get
\begin{eqnarray*}
\frac{d}{dt}\|\b \sigma \| &\le&\|\b A_{21}\b P^{-\frac{1}{2}}\|\|\b \zeta\|+\|\b A_{22}\|\|\b \sigma\|-\eta\frac{\|\b \sigma \|}{\|\b \sigma\|}\\
&{}&\quad +\gamma_h\|\b u^{(1)}\|+\alpha_h\|\b W\b P^{-\frac{1}{2}}\|\|\b \zeta\|+\alpha_h\|\b B\|\|\b \sigma\|+\beta_h\\
&{}&\quad +\alpha_f\|\b S\|\|\b W\b P^{-\frac{1}{2}}\|\|\b \zeta\|+\alpha_f\|\b S\|\|\b B\|\|\b \sigma\|+\beta_f\|\b S\|\\
&=& \left(\|\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}}\| \right)\|\b \zeta\|\\
&{}&\quad +\left(\|\b A_{22}\|+\alpha_h\|\b B\|+\alpha_f\|\b S\|\|\b B\| \right)\|\b \sigma\|\\
&{}&\quad -\left(\eta(1-\gamma_h \sqrt{m})-\beta_h-\beta_f\|\b S\|\right).
\end{eqnarray*}
Let
\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\|,\\
\mu^{(1)} &=& \eta(1-\gamma_h \sqrt{m})-\beta_h-\beta_f\|\b S\|,
\end{eqnarray*}
then we can write
\[
\frac{d}{dt}\|\b \sigma \| \le a_{21}\|\b \zeta\|+a_{22}\|\b \sigma \|-\mu^{(1)}.
\]
When applying $\b u=\b u^{(1)}$, we assume that $\gamma_h < 1/\sqrt{m}$, and
\[
\eta > \frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|a_{21}}{\lambda_{\min}^{\frac{1}{2}}(\b P)\rho(1-\gamma_h\sqrt{m})}+\frac{\beta_h + \beta_f\|\b S\|}{1-\gamma_h\sqrt{m}}.
\]
We now evaluate $\frac{d}{dt}\|\b \zeta\|$ on any trajectory of the closed-loop system. For $\b \zeta \ne \b 0$, we have
\[
\frac{d}{dt}\|\b \zeta \| = \frac{\b \zeta^T\dot{\b \zeta}}{\|\b \zeta \|}
= \frac{\b \zeta^T}{\|\b \zeta \|}\left(\b P^{\frac{1}{2}}\b A_{11}\b P^{-\frac{1}{2}} \b \zeta + \b P^{\frac{1}{2}}\b A_{12}\b \sigma  + \b P^{\frac{1}{2}}\b W^{g}\tilde{\b f}\right).
\]
Recall that
\[
\| \tilde{\b f}\| \le \alpha_f\|\b W\|\|\b z\|+\alpha_f\|\b B\|\|\b \sigma\|+\beta_f.
\]
Combining the above with the derivations from the previous subsection yields
\[
\frac{d}{dt}\|\b \zeta \| \le -\rho\|\b \zeta\|+\left(\|\b P^{\frac{1}{2}}\b A_{12}\|+\alpha_f\|\b P^{\frac{1}{2}}\b W^g\|\|\b B\| \right)\|\b \sigma\|+\beta_f\|\b P^{\frac{1}{2}}\b W^g\|.
\]
Let
\begin{eqnarray*}
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\|, 
\end{eqnarray*}
then, we can write
\[
\frac{d}{dt}\|\b \zeta \| \le -\rho\|\b \zeta\|+a_{12}\|\b \sigma\|+\tilde{\beta}_f.
\]
Note that since the pair $(\b A,\b B)$ is controllable, we have
\[
\|\b P^{\frac{1}{2}}\b A_{12}\| = \|\b P^{\frac{1}{2}}\b W^g\b A\b B\|
\ne  0,
\]
and therefore $a_{12}\ne 0$. This claim can easily be verified using the PBH test.

When $\b u=\b u^{(2)}$, we obtain the same estimates except now
\[
\mu^{(2)}=\eta(1-\gamma_h)-\beta_h-\beta_f\|\b S\|
\]
because
\begin{eqnarray*}
\|\b u^{(2)}\| &=& \|\eta\frac{\b \sigma}{\|\b \sigma\|}\|\\
&=& \eta.
\end{eqnarray*}
When applying $\b u=\b u^{(2)}$, we assume that $\gamma_h < 1$, and
\[
\eta >\frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|a_{21}}{\lambda_{\min}^{\frac{1}{2}}(\b P)\rho(1-\gamma_h)}+\frac{\beta_h + \beta_f\|\b S\|}{1-\gamma_h}.
\]

While calculating estimates for $\b u=\b u^{(\infty)}$, we take into account the fact that
\[
\frac{\b \sigma^T}{\|\b \sigma \|}\nabla_{\sigma}\|\b \sigma\|_{\infty} \ge \frac{\|\b \sigma \|_{\infty}}{\|\b \sigma \|}
\ge \frac{1}{\sqrt{m}}
\]
because
\[
\|\b \sigma \| = \|\b \sigma \|_2
\le \sqrt{m}\|\b \sigma \|_{\infty}.
\]
The above implies that
\[
\mu^{(\infty)}=\frac{\eta}{\sqrt{m}}(1-\gamma_h \sqrt{m})-\beta_h-\beta_f\|\b S\|.
\]
When applying $\b u=\b u^{(\infty)}$, we assume that $\gamma_h < 1/\sqrt{m}$, and
\[
\eta > \sqrt{m}\frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|a_{21}}{\lambda_{\min}^{\frac{1}{2}}(\b P)\rho(1-\gamma_h\sqrt{m})}+\frac{\beta_h + \beta_f\|\b S\|}{1-\gamma_h\sqrt{m}}.
\]
The other parameters are the same as in the previous cases.

Note that the assumptions on the controllers' gains imply that
\[
\mu^{(p)} >\frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|a_{21}}{\lambda_{\min}^{\frac{1}{2}}(\b P)\rho},\quad p=1,2, \infty.
\]
\subsubsection{UUB Regions' Estimates}
To proceed further, we need the following definition, that 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}
The above defintion is illustrated in Figure~\ref{fig:sliddom}.
\begin{figure}
\vspace{3in}
\hskip1in\special{bmp:c:/mybook/sliddom.bmp x=4in y=3in}
\caption{Two-dimensional illustration of sliding mode domain.}
\label{fig:sliddom}
\end{figure}
\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 domain for the closed-loop system.
\end{lemma}
{\bf Proof}\quad For $\b \sigma \ne \b 0$ we have
\[
\b \sigma^T\dot{\b \sigma} \le \left( a_{21}\|\b \zeta \| + a_{22} \|\b \sigma \| -\mu^{(p)}\right)\|\b \sigma\|
\]
because
\[
\frac{d}{dt}\|\b \sigma \| = \frac{\b \sigma^T\dot{\b \sigma}}{\|\b \sigma \|}\le a_{21}\|\b \zeta\|+a_{22}\|\b \sigma \|-\mu^{(p)}.
\]
Hence, for $\b \sigma \ne \b 0$ we have $\b \sigma^T\dot{\b \sigma} \le 0$ in a neighborhood of $\Delta$, and therefore $\Delta$ is a sliding domain for the closed-loop system.\\
\hspace*{5.5in} $\Box$

We now prove a result concerning uniform practical stability of the closed-loop system.
\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 $\Gamma$.
\end{thm}
{\bf Proof}\quad Since $a_{21}>0$, the parameters $\alpha$ and $\beta$ are also positive, and hence the region $\Gamma$ is a bounded neighborhood of the origin of $\mb{\Bb R}^n$. We show that $\gamma>0$ later in the proof. For an illustration of the above mentioned region see Figure~\ref{fig:region}.
\begin{figure}
\vspace{3in}
\hskip1in\special{bmp:c:/mybook/region.bmp x=4in y=3in}
\caption{Illustration of the region of attraction and the uncertainty region.}
\label{fig:region}
\end{figure}
We first show that all solutions of the closed-loop system starting in $\Gamma$ stay in $\Gamma$, that is, the solutions are uniformly bounded. We consider the Lyapunov function candidate $V(\b \zeta,\b \sigma)=\alpha\|\b \zeta\|+\beta \|\b \sigma \|$. Note that $V$ is a positive definite function. The time derivative of $V$ evaluated on a solution of the closed-loop system, for $\b \zeta \ne \b 0$, $\b \sigma \ne \b 0$, is
\begin{eqnarray*}
\frac{d}{dt}V(\b \zeta,\b \sigma) &=& \alpha \frac{d\|\b \zeta\|}{dt}+\beta \frac{d \|\b \sigma \|}{dt}\\
&\le& \alpha\left(-\rho\|\b \zeta\|+a_{12}\|\b \sigma\|+\tilde{\beta}_f \right)+\beta \left(a_{21}\|\b \zeta\|+a_{22}\|\b \sigma \|-\mu^{(p)} \right)\\
&=& \left( -\alpha \rho+\beta a_{21}\right)\|\b \zeta\| + \left(\alpha a_{12}+\beta a_{22} \right)\|\b \sigma\|+\alpha \tilde{\beta}_f-\beta \mu^{(p)}.
\end{eqnarray*}
By assumption
\[
\gamma= \mu^{(p)}-\frac{\alpha \tilde{\beta}_f}{\beta},
\]
and hence,
\[
\beta \gamma=\beta \mu^{(p)}-\alpha \tilde{\beta}_f.
\]
Taking the above into account, we write
\[
\frac{d}{dt}V(\b \zeta,\b \sigma)\le \left( -\alpha \rho+\beta a_{21}\right)\|\b \zeta\| + \left(\alpha a_{12}+\beta a_{22} \right)\|\b \sigma\|-\beta \gamma.
\]
Substituting the expressions for $\alpha$ and $\beta$ into the above inequality yields
\begin{eqnarray*}
\frac{d}{dt}V(\b \zeta,\b \sigma) &\le& \beta \left(\alpha\|\b \zeta\|+\beta \|\b \sigma\|-\gamma \right)\\
&=& \beta \left(V(\b \zeta,\b \sigma)-\gamma \right).
\end{eqnarray*}
It follows from the above that any solution of the closed-loop system with initial condition in $\Gamma$ cannot escape $\Gamma$ through the region
\[
\left\{ (\b \zeta,\b \sigma):V(\b \zeta,\b \sigma)=\gamma,\b \zeta\ne \b 0,\b \sigma \ne \b 0\right\}.
\]
The only possibility of such a solution escaping $\Gamma$ is while moving along either of the manifolds $\left\{(\b \zeta,\b \sigma):\b \zeta = \b 0 \right\}$ or $\left\{(\b \zeta,\b \sigma):\b \sigma = \b 0 \right\}$. We now analyze the above two special cases one by one.

We first show that it is not possible for a solution of the closed-loop system to escape $\Gamma$ while moving along the manifold $\left\{(\b \zeta,\b \sigma):\b \zeta = \b 0 \right\}$. For the closed-loop system confined to $\left\{(\b \zeta,\b \sigma):\b \zeta = \b 0 \right\}$ the inequality
\[
\frac{d}{dt}\|\b \sigma\| \le a_{22}\|\b \sigma\|-\mu^{(p)}
\]
holds when evaluated on any solution of the system. We note that $\beta >a_{22}$ and $\gamma \le \mu^{(p)}$. Hence,
\[
\frac{d}{dt}\|\b \sigma\| \le \beta\|\b \sigma\|-\gamma,
\]
which means that a solution of the closed-loop system with initial condition in $\Gamma$ cannot escape $\Gamma$ while moving along the manifold $\left\{(\b \zeta,\b \sigma):\b \zeta = \b 0 \right\}$.

We now show that a solution of the closed-loop system with initial condition in $\Gamma$ cannot escape $\Gamma$ while moving along the manifold $\left\{(\b \zeta,\b \sigma):\b \sigma = \b 0 \right\}$. It follows from Proposition~\ref{prop:reduce} that a solution of the closed-loop system confined to the manifold $\left\{(\b \zeta,\b \sigma):\b \sigma = \b 0 \right\}$ asymptotically approaches the uncertainty region $\Sigma \times \{\b 0\}=\left\{(\b \zeta,\b \sigma):\|\b \zeta\|\le r,\b \sigma=\b 0 \right\}$. Thus, a solution of the closed-loop system cannot escape $\Gamma$ while moving along the manifold $\left\{(\b \zeta,\b \sigma):\b \sigma = \b 0 \right\}$ if
\[
\frac{\gamma}{\alpha} \ge r.
\]
Recall that $\mu^{(p)}>\beta_f\lambda_{\max}(\b P)\|\b W^g\|a_{21}/\left(\lambda_{\min}^{\frac{1}{2}}(\b P)\rho\right)$. Since $\alpha=\beta a_{21}/(\beta+\rho)$, we have
\begin{eqnarray*}
\mu^{(p)} &>& \frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|a_{21}}{\lambda_{\min}^{\frac{1}{2}}(\b P)\rho}\\
&=& \frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|\alpha(\beta+\rho)}{\lambda_{\min}^{\frac{1}{2}}(\b P)\rho\beta}\\
&=& \frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|\alpha}{\lambda_{\min}^{\frac{1}{2}}(\b P)\rho}+\frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|\alpha}{\lambda_{\min}^{\frac{1}{2}}(\b P)\beta}.
\end{eqnarray*}
Observe that
\begin{eqnarray*}
\tilde{\beta}_f &=& \beta_f\|\b P^{\frac{1}{2}}\b W^g\|\\
&\le& \beta_f\lambda_{\max}^{\frac{1}{2}}(\b P)\|\b W^g\|\\
&\le& \beta_f\frac{\lambda_{\max}(\b P)}{\lambda_{\min}^{\frac{1}{2}}(\b P)}\|\b W^g\|.
\end{eqnarray*}
Then, using the definition of $\gamma$, we get
\[
\gamma =\mu^{(p)}-\frac{\alpha \tilde{\beta}_f}{\beta} > \frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|\alpha}{\lambda_{\min}^{\frac{1}{2}}(\b P)\rho}.
\]
Hence,
\[
\frac{\gamma}{\alpha} > \frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|}{\lambda_{\min}^{\frac{1}{2}}(\b P)\rho}= r,
\]
which means that any solution of the closed-loop system with initial condition in $\Gamma$ cannot leave $\Gamma$ while moving along the manifold $\left\{(\b \zeta,\b \sigma):\b \sigma = \b 0 \right\}$. Thus, all solutions of the closed-loop system are uniformly bounded.

We now show that the closed-loop system is uniformly stable to within the uncertainty set $\Sigma \times \{\b 0\}= \left\{ (\b \zeta,\b 0):\|\b \zeta \| \le r\right\}$. Clearly $\Sigma \times \{\b 0\}$ is compact and contains the origin of $\mb{\Bb R}^n$. Let $N$ be any neighborhood of $\Sigma \times \{\b 0\}$, and let $\nu$ be a number such that $0<\nu \le \gamma$, and $\Gamma(\nu) \subset N$, where $\Gamma(\nu)=\left\{ (\b \zeta,\b \sigma):V(\b \zeta,\b \sigma)<\nu \right\}$. We consider now a solution of the closed-loop system with initial condition in $\Gamma(\nu)$. If $\nu/\alpha \ge r$, then it follows from the first part of this proof that such a solution cannot escape $\Gamma(\nu)$, and thus cannot escape $N$. If $\nu/\alpha < r$, then again it follows from the first part of this proof that such a solution may escape $\Gamma(\nu)$ only through the region $\Sigma \times \{\b 0\}$. However, such a solution cannot escape $\Sigma \times \{\b 0\}$, and thus cannot escape $N$. This is because $\mu^{(p)}>\beta_f\lambda_{\max}(\b P)\|\b W^g\|a_{21}/\left(\lambda_{\min}^{\frac{1}{2}}(\b P)\rho\right)$ implies
\[
\frac{\mu^{(p)}}{a_{21}} > \frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|}{\lambda_{\min}^{\frac{1}{2}}(\b P)\rho}
=r,
\]
and hence $\Sigma \times \{\b 0\}\subset \Delta$. Thus, by Proposition~\ref{prop:reduce} and Lemma~\ref{lemma:sliding} any solution of the closed-loop system that enters the region $\Sigma \times \{\b 0\}$ is thereafter confined to it. We conclude that a solution of the closed-loop system with initial condition in $\Gamma(\nu)$ cannot escape $N$, which means that the closed-loop system is uniformly stable to within $\Sigma \times \{\b 0\}$.

We now show that the region $\Sigma \times \{\b 0\}$is uniformly attractive in the sense of Definition~\ref{defn:guub}. Let $R$ be a number such that $r<R<\mu^{(p)}/a_{21}$ and $\Sigma(R) \times \{\b 0\} \subset N$, where $\Sigma(R)=\left\{ \b \zeta:\|\b \zeta \|\le R \right\}$. We will now show that a solution of the closed-loop system with initial condition in $\Gamma$ is confined to $\Sigma(R) \times \{\b 0\}$ after a finite time. Since $R<\mu^{(p)}/a_{21}$, we have $\Sigma(R) \times \{\b 0\} \subset \Delta$. By Proposition~\ref{prop:reduce} and Lemma~\ref{lemma:sliding} any solution of the closed-loop system that enters the region $\Sigma(R) \times \{\b 0\}$ is thereafter confined to it. Thus, it remains to show that such a solution, with initial condition in $\Gamma$, reaches this region in finite time. From the first part of the proof, we have
\[
\frac{d}{dt}\|\b \sigma\| \le \beta \|\b \sigma \|-\gamma
\]
on any solution that is confined to the region $\left\{ (\b \zeta,\b \sigma):\b \zeta=\b 0,\b \sigma \ne \b 0\right\}$. Hence, for such a solution
\[
\frac{d}{dt}V(\b \zeta,\b \sigma) \le \beta \left(V(\b \zeta,\b \sigma )-\gamma\right).
\]
Thus, by Lemma~\ref{lemma:hit}, for a solution that does not reside on the switching surface
\[
V(\b \zeta(t),\b \sigma(t)) \le \gamma + \left( V(\b \zeta(t_0),\b \sigma(t_0))-\gamma\right)e^{\beta(t-t_0)}.
\]
The above implies that a solution with initial condition in $\Gamma$ that never moves along the switching surface is guaranteed to reach $\Sigma(R) \times \{\b 0\}$ by time $t_0+T(R,\b \zeta(t_0),\b \sigma(t_0))$, where
\[
T(R,\b \zeta(t_0),\b \sigma(t_0))=\left\{\begin{array}{ccc}
\frac{1}{\beta}\ln\left( \frac{\gamma}{\gamma-\alpha\|\b \zeta(t_0)\|-\beta\|\b \sigma(t_0)\|}\right)& \mb{if}& (\b \zeta(t_0),\b \sigma(t_0)) \notin \Sigma(R) \times \{\b 0\}\\
0 & \mb{if}& (\b \zeta(t_0),\b \sigma(t_0)) \in \Sigma(R) \times \{\b 0\}.
\end{array}\right.
\]
By Proposition~\ref{prop:reduce} a closed-loop system solution confined to the switching surface reaches $\Sigma(R) \times \{\b 0\}$ by time $t_0+T_r(R,\b \zeta(t_0))$. Combining the above, we conclude that a solution with initial condition in $\Gamma$ is guaranteed to reach the region $\Sigma(R) \times \{\b 0\}$, and thereafter is confined to it, by time $t_0+T(R,\b \zeta(t_0),\b \sigma(t_0))+T_r(R,\b \zeta^{*})$, where $\|\b \zeta^{*}\|=\|\b \zeta(t_0)\|+\frac{\beta}{\alpha}\|\b \sigma(t_0)\|$. Hence, $\Sigma \times \{\b 0\}$ is uniformly attractive, and the proof is complete.\\
\hspace*{5.5in} $\Box$

The following corollaries are concerned with two special cases.
\begin{corollary}
\label{cor:a22n0}
If $a_{21}=0$ and $a_{22}\ne0$, then the region
\[
\left\{(\b \zeta,\b \sigma):\b \sigma=\b 0\right\}
\]
is a sliding domain of the closed-loop system, and the system is UUB with respect to the region $\Sigma \times \{\b 0\}$, and with region of attraction
\[
\Gamma=\left\{ (\b \zeta,\b \sigma):\|\b \sigma \|\le \frac{\mu^{(p)}}{a_{22}}\right\}.
\] 
\end{corollary}

\begin{corollary}
\label{cor:a220}
If $a_{21}=0$ and $a_{22}=0$, then the region
\[
\left\{(\b \zeta,\b \sigma):\b \sigma=\b 0\right\}
\]
is a sliding domain of the closed-loop system, and the system is globally UUB.
\end{corollary}

If $\beta_f=0$, then we can replace the term ``UUB'' with the term ``uniformly asymptotically stable'' in Theorem~\ref{thm:practical}, Corollary~\ref{cor:a22n0}, and Corollary~\ref{cor:a220} because $\beta_f=0$ implies that solutions of the reduced-order system satisfy
\[
\frac{d}{dt}\| \b \zeta \|  \le -\rho\|\b \zeta\|,
\]
which in turn reduces the uncertainty set $\Sigma \times \{\b 0\}$ to the origin of $\mb{\Bb R}^n$.

\section{State Estimation of Uncertain Systems}
\subsection{Discontinuous Estimators}
So far in our discussion of variable structure sliding mode control system design we assumed that the entire state vector of the plant to be controlled is available to us through measurement. If the state vector cannot be measured, then we can try to determine its suitable estimate that can be used by the controller instead of the true state vector. If we use a state vector estimate for the inaccessible state, then we split a control design problem into three phases. The first phase is the design of a controller assuming the availability of the entire state vector of the plant. In the second phase we construct a state estimator based on the plant input and output. In the final, third, phase we combine the controller and the estimator into the combined controller-estimator compensator.

We now discuss the problem of constructing a state estimator for a class of dynamical systems with matched uncertainties modeled by
\begin{eqnarray*}
\dot{\mb{\bm$x$}} &=& \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 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 the estimate of $\mb{\bm$x$}$. We denote the estimation error by $\mb{\bm$e$}$, that is,
\[
\mb{\bm$e$}(t)=\hat{\mb{\bm$x$}}(t)-\mb{\bm$x$}(t).
\]
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. Since $\mb{\bm$A-LC$}$ is asymptotically stable, by the Lyapunov theorem for any $\mb{\bm$Q$}=\mb{\bm$Q$}^{T}>0$, there is a unique $\mb{\bm$P$}=\mb{\bm$P$}^{T}>0$ such that
\[
\left(\mb{\bm$A$}-\mb{\bm$L$}\mb{\bm$C$} \right) ^{T}\mb{\bm$P$}+\mb{\bm$P$}\left(\mb{\bm$A$}-\mb{\bm$L$}\mb{\bm$C$} \right)=-\mb{\bm$Q$}.
\]
We choose $\mb{\bm$Q$}$, if possible, so that for some $\mb{\bm$F$} \in \mb{\Bb R}^{m \times p}$ we obtain
\[
\mb{\bm$F$}\mb{\bm$C$}=\mb{\bm$B$}^{T}\mb{\bm$P$}.
\]
We need this technical condition for the construction of the state estimator. This condition has a nice system theoretic interpretation. Its satisfaction means that the transfer function matrix
\[
\mb{\bm$G$}(s)=\mb{\bm$FC$}\left(s\mb{\bm$I$}-\mb{\bm$A$}+\mb{\bm$LC$} \right)^{-1}\mb{\bm$B$}
\]
is positive real, 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. To proceed, define the vector function
\[
\mb{\bm$E$}\left(\mb{\bm$e$}, \eta \right)=\left\{\begin{array}{ccc}
\frac{\mb{\bm$FCe$}}{\|\mb{\bm$FCe$}\|}\eta & \mb{for} & \mb{\bm$FCe$}\ne \mb{\bm$0$}\\
\left\{\mb{\bm$v$}:\mb{\bm$v$} \in \mb{\Bb R}^{m},\|\mb{\bm$v$}\| \le \eta \right\} & \mb{for} & \mb{\bm$FCe$}=\mb{\bm$0$},
\end{array}\right.
\]
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 the state estimator proposed by Walcott and \.{Z}ak~\cite{walcott87}. It has the form
\[
\fbox{$\displaystyle \dot{\hat{\mb{\bm$x$}}}=\left(\mb{\bm$A$}-\mb{\bm$LC$} \right)\hat{\mb{\bm$x$}}-\mb{\bm$BE$}(\mb{\bm$e$},\eta)+\mb{\bm$Ly$}+\mb{\bm$Bu$} $}
\]
If $\eta \ge \rho$, the function $\hat{\mb{\bm$x$}}$ is an asymptotic estimate of the state $\mb{\bm$x$}$, that is,
\[
\lim_{t \to \infty}\mb{\bm$e$}(t)=\lim_{t \to \infty} \left(\hat{\mb{\bm$x$}}(t)-\mb{\bm$x$}(t) \right)=\mb{\bm$0$}.
\]
We can prove the above statement using Lyapunov's type of arguments. We first construct the differential equation satisfied by the error
\begin{eqnarray*}
\dot{\mb{\bm$e$}} &=& \dot{\hat{\mb{\bm$x$}}}-\dot{\mb{\bm$x$}}\\
&=& \left(\mb{\bm$A$}-\mb{\bm$LC$} \right)\mb{\bm$e$}-\mb{\bm$BE$}(\mb{\bm$e$},\eta)-\mb{\bm$B\xi$}.
\end{eqnarray*}
We then show that
\[
\frac{d}{dt}\left(\mb{\bm$e$}^{T}\mb{\bm$Pe$} \right)=-\mb{\bm$e$}^{T}\mb{\bm$Qe$}<0,
\]
which implies
\[
\lim_{t \to \infty}\mb{\bm$e$}(t)=\mb{\bm$0$}.
\]
We conclude that the estimation error is insensitive to matched uncertainty. 

\subsection{Boundary Layer Estimators}
We can view the error differential equation $\dot{\mb{\bm$e$}} =\left(\mb{\bm$A$}-\mb{\bm$LC$} \right)\mb{\bm$e$}-\mb{\bm$BE$}(\mb{\bm$e$},\eta)-\mb{\bm$B\xi$}$ as a model of a closed-loop system, with matched uncertainties, controlled by $-\mb{\bm$E$}(\mb{\bm$e$},\eta)$. The 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, we assumed that $\mb{\bm$E$}(\mb{\bm$e$},\eta)$ can switch with infinitely high frequency. In practice, very high frequencies may excite unmodeled high frequency modes and degrade the system performance. The alleviate this problem, we smooth out the discontinuous part of the estimator by introducing a boundary layer around the surface $\left\{\mb{\bm$e$}:\mb{\bm$FCe$}=\mb{\bm$0$}\right\}$. This results in replacing $\mb{\bm$E$}(\mb{\bm$e$},\eta)$ by its continuous approximation $\tilde{\mb{\bm$E$}}(\mb{\bm$e$},\eta)$. An example of such a continuous approximation is
\[
\tilde{\mb{\bm$E$}}\left(\mb{\bm$e$}, \eta \right)=\left\{\begin{array}{ccc}
\frac{\mb{\bm$FCe$}}{\|\mb{\bm$FCe$}\|}\eta & \mb{for} & \|\mb{\bm$FCe$}\| \ge \epsilon\\
\frac{\mb{\bm$FCe$}}{\epsilon}\eta & \mb{for} & \|\mb{\bm$FCe$}\| < \epsilon,
\end{array}\right.
\]
where $\epsilon$ 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 ball $B_{\delta}(\b 0)=\left\{ \mb{\bm$e$}\,:\, \|\mb{\bm$e$}\| < \delta \right\}$ is uniformly stable if for all $\theta > \delta$, there is $r(\theta) > 0$ so that if $\|\mb{\bm$e$}(t_0) \| \le r(\theta)$, then $\|\mb{\bm$e$}(t)\| \le \theta$ for $t \ge t_0$.
\end{defn}
\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}
We next define the uniform ultimate boundedness, or for short, practical stability.
\begin{defn}
The error is UUB with respect to a closed ball $\bar{B}$ if for every $r>0$, there is $T(r) \ge t_0$ such that for each solution $\mb{\bm$e$}(t)$ with $\|\mb{\bm$e$}(t_0)\| \le r$, we have $\mb{\bm$e(t)$} \in B$ for $t \ge T(r)$.
\end{defn}

We can show that the error of the boundary layer estimator is uniformly ultimately bounded. Specifically, the estimation error of the boundary layer estimator ends in a ball whose radius depends on $\epsilon$. The ball radius tends to zero with $\epsilon$ tending to zero. To see this consider, as before, the Lyapunov function candidate $V(\mb{\bm$e$})=\mb{\bm$e$}^{T}\mb{\bm$P$}\mb{\bm$e$}$. Performing simple manipulations yields
\begin{eqnarray*}
\frac{d}{dt}V(\mb{\bm$e$}) &=& -\mb{\bm$e$}^{T}\mb{\bm$Qe$} - 2\mb{\bm$e$}^{T}\mb{\bm$PB$}\tilde{\mb{\bm$E$}}(\mb{\bm$e$},\eta) - 2\mb{\bm$e$}^{T}\mb{\bm$PB$}\mb{\bm$\xi$}\\
&\le& \left\{\begin{array}{ccc}
-\mb{\bm$e$}^{T}\mb{\bm$Qe$} & \mb{for} & \|\mb{\bm$FCe$}\| \ge \epsilon\\
-\mb{\bm$e$}^{T}\mb{\bm$Qe$}+2\epsilon \rho &\mb{for}& \|\mb{\bm$FCe$}\| < \epsilon.
\end{array}\right.
\end{eqnarray*}
To obtain the last inequality, we used the definition of $\tilde{\mb{\bm$E$}}$ and the fact that
\[
\mb{\bm$e$}^{T}\mb{\bm$PBFCe$}=\|\mb{\bm$FCe$}\|^2 \ge 0.
\]
Since
\[
-\mb{\bm$e$}^{T}\mb{\bm$Qe$}+2\epsilon \rho \le -\lambda_{\mb{min}}(\mb{\bm$Q$})\|\mb{\bm$e$}\|^2 + 2\epsilon \rho,
\]
we conclude that $\frac{d}{dt}V(\mb{\bm$e$}) < 0$ for
\[
\|\mb{\bm$e$}\| > \sqrt{\frac{2 \epsilon \rho}{\lambda_{\mb{min}}(\mb{\bm$Q$})}} \stackrel{\rm def}{=}R.
\]
We illustrate the regions used in the discussion of the boundary layer state estimator in Figure~\ref{fig:euub_2d}.
\begin{figure}
\vspace{3in}
\hskip1.5in\special{bmp:c:/mybook/euub_2d.bmp x=4in y=3in}
\caption{A two-dimensional illustration of an uniformly ultimately bounded error.}
\label{fig:euub_2d}
\end{figure}
Referring to Figure~\ref{fig:euub_2d} and Definition~\ref{def:ub} we have
\[
d(r)=\sqrt{\frac{\lambda_{\mb{max}}(\mb{\bm$P$})}{\lambda_{\mb{min}}(\mb{\bm$P$})}}\,R.
\]
The estimation error is UUB with respect to any ball with a radius greater than
\[
\sqrt{\frac{\lambda_{\mb{max}}(\mb{\bm$P$})}{\lambda_{\mb{min}}(\mb{\bm$P$})}}\,R.
\]
\section{Sliding Modes in Solving Optimization Problems}
In this section, we use sliding modes to analyze dynamics of an analog circuit that can solve convex optimization problems with linear constraints. The network is modeled by a set of nonlinear ordinary differential equations with discontinuous right-hand sides. In carrying out the analysis, concepts from the theory of differential equations with discontinuous right-hand sides and Lyapunov stability theory are used. We show that the equilibrium points of the differential equations modeling the network coincide with the optimizers of the underlying optimization problem. We then prove that irrespective of the initial state of the network, its state converges to the solution set of the optimization problem.

The idea of using analog circuits to solve constrained optimization problems first appeared in the literature in 1956 when Pyne~\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. The design of dynamical systems for solving optimization problems proposed by Rodr{\'{\i}}guez-V{\'{a}}zquez et al.~\cite{rodriguez90} is based on concepts from the penalty function method and results in analog dynamical systems modeled by differential equations with discontinuous right-hand sides. This class of optimizers is particularly attractive for two reasons. First, their design does not require the calculation of penalty parameters. Second, these networks can be realized using switched-capacitor technology, and thus are suitable for monolithic implementation.


\subsection{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 relative 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
\[
J(\b x)=\left\{ i:\b a_i\b x > b_i\right\}.
\]
We make the following assumption:\\
{\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, $f \in C^1$;\\
{\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.

\noindent{\bf Remark 1 } 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 $\Omega$.

\noindent{\bf Remark 2 } 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.

We now present two well known results related to problem~(\ref{eq:opty}). We use these results in our analysis.
\begin{lemma}
\label{lemma1}
Any relative 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
\[
\nabla f(\b x^{*})+\sum_{i \in I(\b x^{*})}\mu_i\b a_i^T=\b 0.
\]
\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^{T} & \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.

\subsection{Background Results}
\label{background}
An approach commonly used in developing analog optimizers is to first convert the constrained optimization problem into an associated unconstrained optimization problem, and then design an analog network that solves the unconstrained problem. Such a network is typically an implementation of the dynamical gradient system for minimizing the objective function of the associated unconstrained problem. One method of converting a constrained optimization problem into an associated unconstrained optimization problem is the penalty function method. The idea behind the penalty function method is to replace the constrained optimization problem
\[
\begin{array}{ll}
\mb{minimize} & f(\b x)\\
\mb{subject to} & \b x \in \Omega,
\end{array}
\]
where $\Omega \subset \mb{\Bb R}^n$ is the feasible set, with an associated unconstrained problem of the form
\[
\mb{minimize}\quad f(\b x)+\zeta p(\b x),
\]
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
\[
\mb{(ii)}\quad p(\b x)=0\quad \mb{if and only if}\quad \b x\in \Omega.
\]
For more details on the subject of the penalty method see, for example, Luenberger~\cite{luenberger84}. If for a finite value of the penalty parameter $\zeta$ the solution of the unconstrained problem coincides with the solution of the constrained problem, then the penalty function $p$ is said to be exact. Bertsekas~\cite{bertsekas75} has shown that except for trivial cases, an exact penalty function cannot be everywhere differentiable. Quite often we can find an exact penalty function that results in the function $f(\b x)+\zeta p(\b x)$ 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}, and \.{Z}ak et al.~\cite{zak94}, 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}.


System~(\ref{eq:sys}) can be viewed 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
\[
\kappa_0=\left\{\begin{array}{cl}
1&\mb{if } \kappa_i=1 \mb{ for all }i=1,2,\ldots,m;  \\
0& \mb{otherwise},
\end{array}\right.
\]
that is, $\kappa_0=\prod_{i=1}^m \left( 1-\kappa_i\right)$. Using the above 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^{T},\quad \b x(0) = \b x_0.
\label{eq:sys_alt}
\end{equation}
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}.



A detailed discussion of using $\tau \dot{\b x}(t)\in G(\b x(t))$ to analyze the behavior of the dynamical systems modeled by differential equations with discontinuous right-hand sides, as well as theorems guaranteeing the existence of solutions, is provided in Filippov~\cite[Chapter 2]{filippov88}, and in Aubin and Cellina~\cite{aubin84}. Applications of differential inclusions to the analysis of ordinary differential equations for solving convex programs are discussed in Fl{\aa}m~\cite{flam92}, and Fl{\aa}m and Zowe~\cite{flam92b}.

\subsection{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 show that $V$ is strictly decreasing when evaluated along any trajectory of system~(\ref{eq:sys}) outside $\Omega$. Specifically, we 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 $\beta_j>0$ and $j \in J(\hat{\b x})$, then $\sum_{I(\hat{\b x})\cup J(\hat{\b x})}\beta_i\b a_i \ne \b 0$.
\end{lemma}
{\bf Proof } Let the function $q:\mb{\Bb R}^n \to \mb{\Bb R}$ be defined as
\[
q(\b x)=\sum_{I(\hat{\b x})\cup J(\hat{\b x})}\beta_i \left( \b a_i\b x-b_i\right).
\]
Clearly $q\in C^1$ and
\[
\nabla^T q(\b x)=\sum_{I(\hat{\b x})\cup J(\hat{\b x})} \beta_i \b a_i.
\]
Since $\beta_1,\beta_2,\ldots,\beta_m$ 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, since $\beta_j>0$ for $j\in J(\hat{\b x})$, we also have $q(\hat{\b x}) >0$. Hence, we must have
\[
\nabla^T q(\hat{\b x})=\sum_{I(\hat{\b x})\cup J(\hat{\b x})} \beta_i \b a_i \ne \b 0.
\]
\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, since 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
\[
\tau\dot{\b x}(t)=-\mu \sum_{i \in J^{\prime}} \b a_i^T
\]
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
\begin{eqnarray*}
\frac{d}{dt}V(\b x(t)) &=& \nabla^T V(\b x(t)) \dot{\b x}(t)\\
&=& -\frac{\mu}{\tau}\left\Arrowvert  \sum_{i \in J^{\prime}} \b a_i\right\Arrowvert_2^2
\end{eqnarray*}
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
\[
\tilde{\b a}=\sum_{i \in J^{\prime}} \b a_i^T.
\]
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^T \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^T\right).
\label{eq:vecalt}
\end{equation}
For notational simplicity, we let
\[
\b r=\sum_{j=1}^{\ell}\alpha_j\left( \sum_{i \in I^{\prime}}U(i,j)\b a_i^T\right).
\]
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 since 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 since 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,
\[
\b r \in \mb{Span}\left\{\b a_{i_1}^T,\b a_{i_2}^T,\ldots,\b a_{i_k}^T  \right\},
\]
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
\[
\b P=\b I_n-\b B\left( \b B^T\b B\right)^{-1}\b B^T,
\]
where $\b I_n$ denotes the $n\times n$ identity matrix and \[
\b B=\left[\b a_{i_1}^T,\b a_{i_2}^T,\ldots,\b a_{i_k}^T \right].
\]
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{eqnarray}
\frac{d}{dt}V(\b x(t)) &=& \nabla^T V(\b x(t))\dot{\b x}(t)\nonumber\\
&=&-\frac{\mu}{\tau}\tilde{\b a}^T\b P\tilde{\b a}
\label{eq:lyap_der1}
\end{eqnarray}
almost everywhere on the interval $(t_{l-1},t_l)$. Observing that $\b P=\b P^T=\b P^2$, 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 now show that $\b P\tilde{\b a} \ne \b 0$. Suppose, on the contrary, that $\b P\tilde{\b a} = \b 0$, which is true if and only $\tilde{\b a}=\b B\b \lambda$ for some $\b \lambda \in \mb{\Bb R}^k$. Note that $\tilde{\b a}=\nabla V(\b x)$. This implies that the convex function $V$ is minimized subject to the active constraints outside of $\Omega$, which is not true. Hence, $\b P\tilde{\b a} \ne \b 0$. Combining this result with the fact that there are finitely many constraints, 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 $F \in C^1$. 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{eqnarray}
\frac{d}{dt}F(\b x(t)) &=& \nabla^T F(\b x(t)) \dot{\b x}(t) \nonumber \\
&=&-\frac{1}{\tau}\left\Arrowvert \nabla f(\b x(t))\right\Arrowvert^2_2
\label{eq:finside}
\end{eqnarray}
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 $\b a_{i_1}^T,\ldots,\b a_{i_k}^T$. By assumption~{\bf A3} the vectors $\b a_{i_1}^T,\ldots,\b a_{i_k}^T$ 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 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]^T,\\
\b u_2 &=& \left[\begin{array}{ccccccc}
0 & 1 & 0 & \cdots & 0 & 0 & 0
\end{array}\right]^T,\\
\b u_3 &=& \left[\begin{array}{ccccccc}
1 & 1 & 0 & \cdots & 0 & 0 & 0
\end{array}\right]^T,\\
{} &{\vdots}&{}\\
\b u_{2^k-3} &=& \left[\begin{array}{ccccccc}
1 & 1 & 1 & \cdots & 1 & 0 & 1
\end{array}\right]^T,\\
\b u_{2^k-2} &=& \left[\begin{array}{ccccccc}
1 & 1 & 1 & \cdots & 1 & 1 & 0
\end{array}\right]^T,\\
\b u_{2^k-1} &=& \left[\begin{array}{ccccccc}
1 & 1 & 1 & \cdots & 1 & 1 & 1
\end{array}\right]^T.
\end{eqnarray*}
Let $T(\b x)$ denote the tangent plane 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 since 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, $\b r \in \mb{Span}\left\{ \b a_{i_1}^T,\ldots,\b a_{i_k}^T \right\}$, 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 $\b P=\b I_n-\b B\left( \b B^T\b B \right)^{-1}\b B^T$.

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
\[
\gamma=\frac{\mu}{\mu + M\sqrt{k}\left\|\left(\b B^T\b B \right)^{-1}\b B^T \right\|_2}.
\]
\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 side of~(\ref{eq:morecombo}) by $\left( \b B^T\b B\right)^{-1}\b B^T$, we obtain
\[
-\frac{\alpha_0}{\mu}\left( \b B^T\b B\right)^{-1}\b B^T\left( \b I_n-\b P\right)\nabla f(\b x(t^{\prime}))=\b U\b \alpha,
\]
and hence
\begin{equation}
\frac{\alpha_0}{\mu}\left\|\left( \b B^T\b B\right)^{-1}\b B^T\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
\[
\frac{M\alpha_0}{\mu}\left\| \left(\b B^T\b B\right)^{-1}\b B^T\right\|_2 \ge \left\|\b U\b \alpha \right\|_2 \ge \frac{1}{\sqrt{k}}\left\|\b U\b \alpha\right\|_1
\]
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^T\b B \right)^{-1}\b B^T\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
\[
\frac{M \sqrt{k}}{\mu} \left\|  \left(\b B^T\b B \right)^{-1}\b B^T\right\|_2 \ge \frac{1-\alpha_0}{\alpha_0}=\frac{1}{\alpha_0}-1.
\]
Hence,
\[
\alpha_0 \ge \frac{1}{1+\frac{M \sqrt{k}}{\mu} \left\|  \left(\b B^T\b B \right)^{-1}\b B^T\right\|_2}=\gamma.
\]
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 follows 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 $\b 0 \in G(\b x(t^{\prime}))\cap T\left(\b x(t^{\prime})\right)$ since, 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 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 since $\alpha_0 \ne 0$, $\b d$ 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 $\b \mu=\left[\mu_1,\ldots,\mu_k\right]^T$. 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 $\b P=\b P^T=\b P^2$, 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
\[
\theta(\varepsilon)=\left\{  \b x:\b x \in \Omega, F(\b x) \ge \frac{\varepsilon}{2}\right\}.
\]
Since $F(\b x(0)) \ge \varepsilon$, we have $\b x(0) \in \theta(\varepsilon)$ and $\theta(\varepsilon) \subset \Omega \setminus \Gamma$. Furthermore, since 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 $\b x(t) \in \theta(\varepsilon)$, thereby guaranteeing 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 } Since the function $f$ is convex and $f \in C^1$, 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,
\[
S\cap \theta(\varepsilon) \subset \Omega \setminus \Gamma.
\]
By Proposition~\ref{proposition2},
\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 there are finitely many constraints 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}\rm
Consider the quadratic programming problem
\[
\begin{array}{ll}
\mb{minimize} & \frac{1}{2}\b x^T\b Q\b x+\b c^T\b x,\quad \b x\in \mb{\Bb R}^2\\
\mb{subject to} & \b A\b x\le \b b,
\end{array}
\]
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 $\b x^{*}=\left[\begin{array}{cc}
-19/22 & 95/22
\end{array}\right]^T$ 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}.
\begin{figure}
\vspace{4in}
\hskip.5in\special{bmp:c:/mike/phase1.bmp x=5.33in y=4in}
\caption{Phase-plane portrait for the system in the Example.}
\label{fig:phase1}
\end{figure}
Observe that each of the trajectories in the phase-plane portrait converges to the point $\b x^{*}$ while sliding along the surface $h_1$.
\end{example}





\section{Notes}
For further analysis of discontinuous state estimators, we refer the reader to Edwards and Spurgeon~\cite{edwards94}. For a stability analysis of uncertain closed-loop systems with the combined variable structure estimator-controller compensators we refer the reader to Hui and \.{Z}ak~\cite{hui92}.

We analyzed the 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. We showed that irrespective of the initial state of the system the state trajectory converges to the solution set of the convex programming problem. In carrying out the analysis we used concepts from the theory of differential equations with discontinuous right-hand sides and Lyapunov stability theory. Our analysis 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, Hui, and \.{Z}ak~\cite{glazos98}.

\section{Exercises}
\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
\[
\sigma(\b x)=\b s\b x=0
\]
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.
\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
\[
u=k_1^{\pm}x_1 +k_2^{\pm}x_2,
\]
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 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}\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 discontinuous state-estimator. The estimator poles should be placed at $-2$ and $-3$. {\bf Hint:} After constructing the required observer gain $\b l$, find a symmetric positve definite matrix $\b Q$ such that the solution, $\b P$, to the Lyapunov matrix equation, $(\b A-\b l\b c)^T\b P+\b P(\b A-\b l\b c)=-\b Q$, satisfies the condition, $\b b^T\b P=\b c$.
\end{exercise}
