\section{Controllers}
The second stage of a sliding-mode controller design is concerned with the development of a nonlinear control strategy that will drive the closed-loop system state trajectory to the switching surface and maintain it on the surface upon interception. We propose to analyze the following three 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*}
where
\[
\mb{sign} z=\left\{\begin{array}{ccc}
-1 & \mb{if} & z<0\\
\left[-1,1\right] & \mb{if} & z=0\\
1& \mb{if}& z>0,
\end{array}\right.
\]
and $\eta > 0$ is a design parameter. The second controller, we analyze, is
\[
\b u^{(2)}(\b x)=\left\{\begin{array}{ccc}
-\eta \frac{\b \sigma (\b x)}{\|\b \sigma (\b x)\|}&\mb{if}&\b \sigma (\b x) \ne \b 0\\
\{\b v \in \mb{\Bb R}^m:\|\b v\| \le \eta \} & \mb{if} & \b \sigma (\b x) =\b 0.
\end{array}\right.
\]
The third control structure is labeled $\b u^{(\infty)}$. The reason for this label will become apparent soon. 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 $\|\cdot\|_p$ denote 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 $\nabla_{\sigma}\|\b \sigma \|_p$, $p=1,2,\infty$, 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,
\[
\|\b u^{(1)}\|_2\le \eta \sqrt{m},\quad\|\b u^{(2)}\|_2\le \eta,\quad \mb{and}\|\b u^{(\infty)}\|_2\le \eta \sqrt{m} .
\]
We can generate 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 details of such a generalization we refer the reader to Hui and \.{Zak}~\cite{hui93}.

The closed-loop system, we are analyzing, 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 system 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 M\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 M\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 $\b W^{g}\b B=\b 0$ 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 0\\
\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} &=&\b f(t,\b x)\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} &=&\b h(t,\b x, \b u)\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
\[
\| \tilde{\b f}\| \le \gamma_f\|\b u\|+\alpha_f\|\b W\|\|\b z\|+\alpha_f\|\b B\|\|\b \sigma\|+\beta_f,
\]
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 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 in this section 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 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 $\|\tilde{\b f}(t,\b z,\b 0)\|\le \alpha_f\|\b W\|\|\b z\|+\beta_f+\gamma_f\|\b u\|$. 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 $\b P$ 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 a symmetric and nonsingular. We refer to $\b P^{\frac{1}{2}}$ as a square root of $\b P$---see Luenberger~\cite[page 459]{luenberger84}. 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 trajectories of the reduced-order system. For $\b z \ne \b 0$ we have
\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*}
because $\b P=\b P^{\frac{1}{2}}\b P^{\frac{1}{2}}$ and $\b \zeta =\b P^{\frac{1}{2}}\b z$. Note that
\begin{eqnarray*}
\b z^T\b P\b A_{11}\b z &=& \frac{1}{2}\b z^T\left(\b A^T_{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, observe 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. Since $\b z=\b P^{-\frac{1}{2}}\b \zeta$, then
\[
\frac{1}{\lambda_{\max}^{\frac{1}{2}}(\b P)}\|\b \zeta\|=\left\Arrowvert \b P^{-\frac{1}{2}}\b \zeta \right\Arrowvert \le \|\b z\| \le \frac{1}{\lambda_{\min}^{\frac{1}{2}}(\b P)}\|\b \zeta\|.
\]
Furthermore,
\begin{eqnarray*}
\|\tilde{\b f}(t,\b z,\b 0)\| &\le& \alpha_f\|\b W\|\|\b z\|+\beta_f + \gamma_f \|\b u\|\\
&\le& \frac{\alpha_f}{\lambda_{\min}^{\frac{1}{2}}(\b P)}\|\b W\|\|\b \zeta\|+\beta_f + \gamma_f \|\b u\|.
\end{eqnarray*}
Hence
\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\|\\
&+&\frac{\lambda_{\max}(\b P)}{\lambda_{\min}^{\frac{1}{2}}(\b P)}\|\b W^g\|\left( \beta_f + \gamma_f \|\b u\|\right).
\end{eqnarray*}
Rearranging the above yields
\begin{eqnarray*}
\frac{d}{dt}\| \b \zeta \|  &\le&{} -\left(\frac{\lambda_{\min}(\b Q)}{\lambda_{\max}(\b P)}-  \frac{\alpha_f \lambda_{\max}(\b P)}{\lambda_{\min}(\b P)}\|\b W^g\|\|\b W\|\right)\|\b \zeta\|\\
&+&\frac{\lambda_{\max}(\b P)}{\lambda_{\min}^{\frac{1}{2}}(\b P)}\|\b W^g\|\left( \beta_f + \gamma_f \|\b u\| \right).
\end{eqnarray*}
To simplify our presentation let
\[
\rho=\frac{\lambda_{\min}(\b Q)}{\lambda_{\max}(\b P)}- \frac{\alpha_f \lambda_{\max}(\b P)}{\lambda_{\min}(\b P)}\|\b W^g\|\|\b W\|,
\]
and
\[
r=\frac{1}{\rho}\frac{\lambda_{\max}(\b P)}{\lambda_{\min}^{\frac{1}{2}}(\b P)}\|\b W^g\|\left( \beta_f + \gamma_f \|\b u\|\right).
\]
We assume that $\rho>0$. With the above notation we can write
\[
\frac{d}{dt}\| \b \zeta \|  \le -\rho(\|\b \zeta\|-r).
\]
In our stability analysis we shall use the following lemma that comes from~\cite[page 200]{madani93}.
\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}

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 uniformly 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 All solutions of the reduced-order system will asymptotically approach the set $\Sigma$ because for $\b z \ne \b 0$
\begin{eqnarray*}
\frac{d}{dt}\left(\b z^T\b P\b z\right)^{\frac{1}{2}}
&=&\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 $\Sigma$. Let $R$ be a number such that $R>r$ and
\[
\left\{ \b z:\|\b P^{\frac{1}{2}}\b z\|\le R\right\}\subset U.
\]
By Lemma~\ref{lemma:hit} any solution of the reduced-order system will enter the above set, 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 $\alpha_f$, $\beta_f$, $\gamma_f$, 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 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,\infty.
\end{eqnarray*}
To simplify the analysis we will be working in the following coordinate system:
\[
\left[\begin{array}{c}
\b \zeta\\
\b \sigma
\end{array}\right]= \left[\begin{array}{c}
\b P^{\frac{1}{2}}\b W^{g}\\
\b S
\end{array}\right]\b x.
\]
In the new coordinates the system takes the form
\[
\left[\begin{array}{c}
\dot{\b \zeta}\\
\dot{\b \sigma}
\end{array}\right]\\
= \left[\begin{array}{cc}
\b P^{\frac{1}{2}}\b A_{11}\b P^{-\frac{1}{2}} & \b P^{\frac{1}{2}}\b A_{12}\\
\b A_{21}\b P^{-\frac{1}{2}} & \b A_{22}
\end{array}\right]\left[\begin{array}{c}
\b \zeta\\
\b \sigma
\end{array}\right]+\left[\begin{array}{c}
\b 0\\
\b I_m
\end{array}\right](\b u+\tilde{\b h})+\left[\begin{array}{c}
\b P^{\frac{1}{2}}\b W^{g}\tilde{\b f}\\
\b S \tilde{\b f}
\end{array}\right].
\]
Note that
\[
\b x=\left[\begin{array}{cc}
\b W\b P^{-\frac{1}{2}} & \b B
\end{array}\right]\left[\begin{array}{c}
\b \zeta\\
\b \sigma
\end{array}\right].
\]
We begin 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\|}\\
&+&\gamma_h\|\b u^{(1)}\|+\alpha_h\|\b W\b P^{-\frac{1}{2}}\|\|\b \zeta\|+\alpha_h\|\b B\|\|\b \sigma\|+\beta_h\\
&+&\alpha_f\|\b S\|\|\b W\b P^{-\frac{1}{2}}\|\|\b \zeta\|+\alpha_f\|\b S\|\|\b B\|\|\b \sigma\|+\left(\beta_f+\gamma_f \|\b u^{(1)}\|\right)\|\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\|\\
&+&\left(\|\b A_{22}\|+\alpha_h\|\b B\|+\alpha_f\|\b S\|\|\b B\| \right)\|\b \sigma\|\\
&-&\left(\eta(1-\gamma_h \sqrt{m})-\beta_h-\left(\beta_f+\gamma_f \eta \sqrt{m}\right)\|\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}-\gamma_f \sqrt{m} \|\b S\|)-\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
\[
\eta > \frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|a_{21}}{\lambda_{\min}^{\frac{1}{2}}(\b P)\rho\left(1-\gamma_h \sqrt{m}-\gamma_f \sqrt{m} \|\b S\|\right)}+\frac{\beta_h + \beta_f\|\b S\|}{1-\gamma_h \sqrt{m}-\gamma_f \sqrt{m} \|\b S\|}>0.
\]
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 $\| \tilde{\b f}\| \le \alpha_f\|\b W\|\|\b z\|+\alpha_f\|\b B\|\|\b \sigma\|+\beta_f+\gamma_f \|\b u\|$. Combining the above with the derivations from the previous subsection yields
\[
\frac{d}{dt}\|\b \zeta \| \le -\rho\|\b \zeta\|+\left(\|\b P^{\frac{1}{2}}\b A_{12}\|+\alpha_f\|\b P^{\frac{1}{2}}W^g\|\|\b B\| \right)\|\b \sigma\|+\left(\beta_f+\gamma_f\|\b u\|\right)\|\b P^{\frac{1}{2}}\b W^g\|.
\]
Let
\begin{eqnarray*}
a_{12}&=&\|\b P^{\frac{1}{2}}\b A_{12}\|+\alpha_f\|\b P^{\frac{1}{2}}W^g\|\|\b B\|,\\
\tilde{\beta}_f &=&\left(\beta_f+\gamma_f\|\b u\|\right)\|\b P^{\frac{1}{2}}\b W^g\|, 
\end{eqnarray*}
then, we can write
\[
\frac{d}{dt}\|\b \zeta \| \le -\rho\|\b \zeta\|+a_{12}\|\b \sigma\|+\tilde{\beta}_f.
\]
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 & \b 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
\[
\mu^{(2)}=\eta(1-\gamma_h-\gamma_f\|\b S\|)-\beta_h-\beta_f\|\b S\|
\]
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
\[
\eta >\frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|a_{21}}{\lambda_{\min}^{\frac{1}{2}}(\b P)\rho(1-\gamma_h-\gamma_f\|\b S\|)}+\frac{\beta_h + \beta_f\|\b S\|}{1-\gamma_h-\gamma_f\|\b S\|}>0.
\]

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
\[
\mu^{(\infty)}=\frac{\eta}{\sqrt{m}}(1-\gamma_h m-\gamma_f\|\b S\|m)-\beta_h-\beta_f\|\b S\|.
\]
When applying $\b u=\b u^{(\infty)}$ we assume that
\[
\eta > \sqrt{m}\frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|a_{21}}{\lambda_{\min}^{\frac{1}{2}}(\b P)\rho(1-\gamma_h m -\gamma_f\|\b S\|m )}+\frac{\beta_h + \beta_f\|\b S\|}{1-\gamma_h m -\gamma_f\|\b S\|m}>0.
\]
The other parameters are the same as in the previous cases. Note that the assumptions on the controllers's 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. This result is a generalization of the main result from~\cite[page 161]{glazos95}.
\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*}
where $\gamma >0$. Then, the closed-loop system is uniformly 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 $\mb{\Bb R}^n$. 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. Similarly as in Glazos and \.{Z}ak~\cite{glazos95}, 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 $\gamma= \mu^{(p)}-\frac{\alpha \tilde{\beta}_f}{\beta}$, hence
\[
\beta \gamma=\beta \mu^{(p)}-\alpha \tilde{\beta}_f.
\]
Taking the above into account we can 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
\[
\frac{d}{dt}V(\b \zeta,\b \sigma) \le  \beta \left(V(\b \zeta,\b \sigma)-\gamma \right).
\]
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 a such 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 $\mu^{(p)}>\frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|a_{21}}{\lambda_{\min}^{\frac{1}{2}}(\b P)\rho}$. 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*}
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*}
because
\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*}
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 $\mu^{(p)}>\frac{\beta_f\lambda_{\max}(\b P)\|\b W^g\|a_{21}}{\lambda_{\min}^{\frac{1}{2}}(\b P)\rho}$ 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 $r<R<\frac{\mu^{(p)}}{a_{21}}$ 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 $R<\frac{\mu^{(p)}}{a_{21}}$, 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 $t_0+T\left(R,\b \zeta(t_0),\b \sigma(t_0)\right)$, where
\[
T\left(R,\b \zeta(t_0),\b \sigma(t_0)\right)=\left\{\begin{array}{ccc}
\frac{1}{\beta}\ln\left( \frac{\gamma}{\gamma-\alpha\|\b \zeta(t_0)\|-\beta\|\b \sigma(t_0)\|}\right)& \mb{if}& (\b \zeta(t_0),\b \sigma(t_0)) \notin \Sigma(R) \times \{\b 0\}\\
0 & \mb{if}& (\b \zeta(t_0),\b \sigma(t_0)) \notin \Sigma(R) \times \{\b 0\}.
\end{array}\right.
\]
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 $t_0+T\left(R,\b \zeta(t_0),\b \sigma(t_0)\right)+T_r(R,\b \zeta^{*})$, 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} $\Box$

We now state two corollaries that 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 uniformly 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 uniformly practically stable.
\end{corollary}

If $\beta_f=\gamma_f=0$, 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 $\beta_f=\gamma_f=0$ implies that solutions of 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 $\mb{\Bb R}^n$.