%\setcounter{chapter}{6}
\chapter{Sliding Modes}
\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 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}.
\begin{figure}
\vspace{3.5in}
\hskip.75in\special{bmp:c:/sliding/vsc_clsc.bmp x=4.67in y=3.5in}
\caption{A typical trajectory of the variable structure system.}
\label{fig:vsc_clsc}
\end{figure}
\end{example}
\begin{example}\rm
\label{ex:doub_int}
Consider the double integrator
\[
\frac{d^2}{dt^2}x=u,
\]
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 bounded by 2.}
\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 of the double integrator variable structure sliding mode system for two different initial conditions. The applied control law is bounded by 1.}
\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 other words, the switching surface attracts trajectories when they are in its vicinity. 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
\[
\sigma \dot{\sigma} <0
\]
in the neighborhood of the switching surface. Note, however, that the closed-loop system is now modeled by differential equations with discontinuous right-hand sides. The modeling differential equations are discontinuous on the switching surface $\sigma(\b x)=0$. 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. We have to use more general methods that involve the theory of differential equations with discontinuous right-hand sides such as one developed by Filippov~\cite{filippov88}. We will 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 that contains $\b f^{-}$ and $\b f^{+}$.
\begin{figure}
\vspace{3in}
\hskip1in\special{bmp:c:/mybook/filip.bmp x=4in y=3in}
\caption{An illustration of Filippov's method in $\mb{\Bb R}^2$.}
\label{fig:filip}
\end{figure}
Following Itkis~\cite{itkis76} we now 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 representative point performs one oscillation about $\sigma = 0$ in a ``small'' time interval $\Delta \tau$, then the representative point 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 representative point 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 representative point 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 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 Figure~\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{An illustration of Filippov's method in $\mb{\Bb R}^3$.}
\label{fig:filip_3d}
\end{figure}

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}



\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 switching (sliding) 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}
We now discuss the switching surface design phase for  single-input plants. Without loss of generality, we can assume that the plant model is already in its controller companion form. Thus, the plant model we are given has the form
\[
\dot{\b x} = \left[\begin{array}{ccccc}
0 & 1 & 0 & \cdots & 0\\
0 & 0 & 1 & \cdots & 0\\
\vdots & {} & {} & {} & \vdots\\
0 & 0 & 0 & \cdots & 1\\
-a_0 & -a_1 & -a_2 & \cdots & -a_{n-1}
\end{array}\right]\b x + \left[\begin{array}{c}
0\\
0\\
\vdots\\
0\\
1
\end{array}\right]u.
\]
Let the switching 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]\b x\\
&=&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 $s_n=1$. 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 equation of the switching 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 switching surface. First note that
\[
\dot{x}_{n-1}=x_n.
\]
Hence, we can write
\begin{eqnarray*}
\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{eqnarray*}
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 switching 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 switching surface, we construct a feedback controller. In this section, we discuss a controller design for single-input plants. The controller goal is to drive the plant state to the switching surface and maintain it on the surface for all subsequent time. We use a generalized Lyapunov approach in constructing the controller. Specifically, we use a generalized distance from the switching surface as a Lyapunov function candidate. Then, we select the controllers gains so that the time derivative of the chosen Lyapunov function candidate evaluated on the solution of the controlled system is negative definite with resepect to the switching surface, thus ensuring the motion of the state trajectory to the surface as it is illustrated in Figure~\ref{fig:sliding}. Our goal is to find $u(\b x)$ so that $\sigma \dot{\sigma} <0$. The controller structure we consider has the form:
\[
u=k_1^{\pm}x_1 + k_2^{\pm}x_2+\cdots +k_n^{\pm}x_n,
\]
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 $u$ into the expression $\sigma \dot{\sigma}$ 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+\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{eqnarray*}
k_1^{+} &<& -|a_0|,\quad \sigma x_1 >0\\
k_1^{-} &>& |a_0|,\quad \sigma x_1 <0\\
{}&\vdots& {}\\
k_n^{+} &<& -(|a_{n-1}|+|s_{n-1}|),\quad \sigma x_n>0\\
k_n^{-} &>& |a_{n-1}|+|s_{n-1}|,\quad \sigma x_n<0.
\end{eqnarray*}
We can also use the following controller gains:
\begin{eqnarray*}
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{eqnarray*}
where $\varepsilon >0$ is a design parameter. The larger the value of $\varepsilon$ the faster the trajectory converges to the switching 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 switching 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 switching surface. In many applications nonlinear switching surfaces arise. We will use linear switching surfaces only, although one can use nonlinear switching surfaces as well. Let $\b \sigma (\b x)=\b S\b x$. 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,
\begin{eqnarray*}
\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.
\end{eqnarray*}
Hence
\begin{eqnarray*}
\frac{d}{dt}\b \sigma(\b x(t)) &=& \b S\dot{\b x}(t)\\
&=& \b 0.
\end{eqnarray*}
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 $\b u$ to obtain
\[
\b u=-\left(\b S\b B\right)^{-1}\b S\b A\b x.
\]
Substituting the above 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 $\det \b S\b B \ne 0$. 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 its determinant 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 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 now 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{Properties of Equivalent Control}
We consider a nonlinear dynamic 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 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
\begin{eqnarray*}
\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
\end{eqnarray*}
for $\b u$ yields
\[
\b u=-\left( \b S\b B\right)^{-1}\b S\b f.
\]
The above expression for $\b u$ is called the {\em equivalent control} and is denoted by $\b u_{eq}$. Substituting $\b u_{eq}$ into to the modeling equation~(\ref{eq:nosymo}), we obtain
\begin{eqnarray*}
\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.
\end{eqnarray*}
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
\begin{eqnarray*}
\b 0 &=& \dot{\b \sigma}(\b x)\\
&=&\b S\left( \b f(t,\b x)+\b B(t,\b x)\b u\right)
\end{eqnarray*}
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}
\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 $M$ and $N$ such that
\[
\left\Arrowvert \b f(t,\b x)+\b B(t,\b x)\widetilde{\b u} \right\Arrowvert \le M +N\|\b x\|.
\]
\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
\[
\widetilde{\b x}(t) =\widetilde{\b x}(0)+\int_0^t\left( \b f-\b B\left( \b S\b B\right)^{-1}\b S\b f\right)(\widetilde{\b x}(s))ds + \int_0^t\b B\left( \b S\b B\right)^{-1}\dot{\b \sigma}(\widetilde{\b x}(s))ds
\]
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.\\
& & +\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.
\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+N \left\Arrowvert\widetilde{\b x}\right\Arrowvert\right) ds\\
&\le& \|\widetilde{\b x}(0)\|+MT+\int_0^t N \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)\|+MT \right)e^{Nt}\\
&\le& \left(\|\widetilde{\b x}(0)\|+MT \right)e^{NT},\quad t \in[0,T].
\end{eqnarray*}
The above means that $\widetilde{\b x}$ is bounded on the interval $[0,T]$. By assumption~\ref{assump_three}, the function $\b B\left( \b S\b B\right)^{-1}$ is bounded. Hence, for some $S\ge 0$ that depends on the initial conditions, time $T$, and constant $P$, we have
\[
\|\b x-\widetilde{\b x}\| \le S\Delta +L\int_0^t \left\Arrowvert \b x(s)-\widetilde{\b x}(s)\right\Arrowvert ds.
\]
Applying 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{Switching Surface Design}
We now prove a sufficient condition for the existence of a switching 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 switching 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 by $\b X^{g}$. 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 switching 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
\begin{eqnarray*}
\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,
\end{eqnarray*}
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 switching 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 switching 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 $\{\lambda_1,\lambda_2,\ldots,\lambda_m\}$ of the system restricted to a switching 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 switching 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}
\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{Closed-Loop System}
We consider a class of uncertain, or nonlinear, dynamical systems modeled by
\begin{equation}
\dot{\b x}(t)=\b A\b x(t)+\b B\b u(t)+\b f(t,\b x(t))+\b B \b h(t,\b x(t),\b u(t)),\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 $\b f$ as the unmatched uncertainty, while to $\b h$ as the matched uncertainty. 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 above system, 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, or practically stable, 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 practically stable if it is practically stable with $\mb{\Bb R}^n$ as a region of attraction.
\end{defn}
 
We now discuss the problem of designing controllers for the above class of uncertain system subject to hard bounds on control action, that is, saturating actuators. We propose to analyze three bounded control laws. They are:
\begin{eqnarray*}
\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),
\end{eqnarray*}
\[
\b u^{(2)}(\b x)=\left\{\begin{array}{ccc}
-\eta \frac{\b \sigma (\b x)}{\|\b \sigma (\b x)\|}&\mb{if}&\b \sigma (\b x) \ne \b 0\\
\b 0 & \mb{if} & \b \sigma (\b x) =\b 0,
\end{array}\right.
\]
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}.

The closed-loop system that we are analyze has the form
\begin{eqnarray*}
\dot{\b x} &=& \b A\b x+\b B\b u+\b f+\b B \b h\\
\b u^{(p)} &=& -\eta \nabla_{\sigma}\|\b \sigma (\b x)\|_p,\quad p=1,2,\infty.
\end{eqnarray*}
To facilitate the analysis of the above system, we use the following state variable transformation. Let
\begin{eqnarray*}
\tilde{\b x} &=& \b M \b x\\
&=& \left[\begin{array}{c}
\b W^{g}\\
\b S
\end{array}\right]\b x,
\end{eqnarray*}
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 
\begin{eqnarray*}
\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],
\end{eqnarray*}
where
\begin{eqnarray*}
\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),
\end{eqnarray*}
and
\begin{eqnarray*}
\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),
\end{eqnarray*}
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.

\subsection{Estimating Stability Regions}
The presence of unmatched uncertainties has a deteriorating effect on the stability properties of a closed-loop system. The above combined with the imposed hard bounds on the control action may make it impossible to globally asymptotically stabilize the closed-loop system. The best it can be done in such a situation maybe uniform practical stability with some region of attraction $\Gamma \subset \mb{\Bb R}^n$ 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 $\Sigma$.

{\bf 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,
\begin{eqnarray*}
V_1 &=& \left(\b z^T\b P\b z \right)^{1/2}\\
&=& \left(\b \zeta ^T\b \zeta \right)^{1/2}\\
&=& \| \b \zeta \|.
\end{eqnarray*}
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
\begin{eqnarray*}
\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.
\end{eqnarray*}
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,
\begin{eqnarray*}
\|\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.
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
\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\|\\
&{}&\quad+\beta_f\frac{\lambda_{\max}(\b P)}{\lambda_{\min}^{\frac{1}{2}}(\b P)}\|\b W^g\|.
\end{eqnarray*}
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 practically stable, 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.

{\bf 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 B\b u+\b f+\b B \b h\\
\b u^{(p)} &=& -\eta \nabla_{\sigma}\|\b \sigma (\b x)\|_p,\quad \mb{for} \quad p=1,2,\quad\mb{or} \quad\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
\[
\left[\begin{array}{c}
\dot{\b \zeta}\\
\dot{\b \sigma}
\end{array}\right]\\
= \left[\begin{array}{cc}
\b P^{\frac{1}{2}}\b A_{11}\b P^{-\frac{1}{2}} & \b P^{\frac{1}{2}}\b A_{12}\\
\b A_{21}\b P^{-\frac{1}{2}} & \b A_{22}
\end{array}\right]\left[\begin{array}{c}
\b \zeta\\
\b \sigma
\end{array}\right]+\left[\begin{array}{c}
\b O\\
\b I_m
\end{array}\right](\b u+\tilde{\b h})+\left[\begin{array}{c}
\b P^{\frac{1}{2}}\b W^{g}\tilde{\b f}\\
\b S \tilde{\b f}
\end{array}\right].
\]
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
\begin{eqnarray*}
\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).
\end{eqnarray*}
In further derivation, we use the fact that
\begin{eqnarray*}
\mb{sign}(\sigma_i) &=& \frac{\sigma_i}{|\sigma_i|}\\
&=&\frac{|\sigma_i|}{\sigma_i},
\end{eqnarray*}
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
\begin{eqnarray*}
\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).
\end{eqnarray*}
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
\begin{eqnarray*}
\|\b P^{\frac{1}{2}}\b A_{12}\| &=& \|\b P^{\frac{1}{2}}\b W^g\b A\b B\|\\
&\ne & 0,
\end{eqnarray*}
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
\begin{eqnarray*}
\frac{\b \sigma^T}{\|\b \sigma \|}\nabla_{\sigma}\|\b \sigma\|_{\infty} &\ge& \frac{\|\b \sigma \|_{\infty}}{\|\b \sigma \|}\\
&\ge&\frac{1}{\sqrt{m}}
\end{eqnarray*}
because
\begin{eqnarray*}
\|\b \sigma \| &=& \|\b \sigma \|_2\\
&\le& \sqrt{m}\|\b \sigma \|_{\infty}.
\end{eqnarray*}
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
\[
\mu^{(p)} >\frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|a_{21}}{\lambda_{\min}^{\frac{1}{2}}(\b P)\rho},\quad p=1,2, \infty.
\]

To proceed further, we need the following lemma.
\begin{lemma}
\label{lemma:sliding}
If $a_{21} \ne 0$, then the region
\[
\Delta=\left\{ (\b \zeta,\b \sigma):\|\b \zeta\|<\frac{\mu^{(p)}}{a_{21}},\b \sigma=\b 0\right\}
\]
is a sliding 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
\begin{eqnarray*}
\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)}.
\end{eqnarray*}
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 practically stable 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
\begin{eqnarray*}
\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}.
\end{eqnarray*}
Hence,
\begin{eqnarray*}
\frac{\gamma}{\alpha} &>& \frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|}{\lambda_{\min}^{\frac{1}{2}}(\b P)\rho}\\
&=& r,
\end{eqnarray*}
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
\begin{eqnarray*}
\frac{\mu^{(p)}}{a_{21}} &>& \frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|}{\lambda_{\min}^{\frac{1}{2}}(\b P)\rho}\\
&=&r,
\end{eqnarray*}
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 practically stable 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 practically stable.
\end{corollary}

If $\beta_f=0$, then we can replace the word ``practically'' with the word ``asymptotically'' 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 uniformly ultimately bounded, or practically stable, 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 uniformly ultimately bounded, that is, practically stable, 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{Notes}