\section{Properties of Equivalent Control}
We consider a nonlinear dynamic system modeled by
\[
\dot{\b x}=\b f(t,\b x)+\b B(t,\b x)\b u,
\]
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 $\dot{\b x}=\b f(t,\b x)+\b B(t,\b x)\b u$ 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 we obtain
\[
\dot{\widetilde{\b x}}=\b f-\b B\left( \b S\b B\right)^{-1}\b S\b f+\b B\left( \b S\b B\right)^{-1}\dot{\b \sigma}.
\]
The above describes the plant modeled by $\dot{\b x}=\b f(t,\b x)+\b B(t,\b x)\b u$ 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 $\{\b x:\b \sigma(\widetilde{\b x})=\b 0\}$ so that
\[
\|\b \sigma(\widetilde{\b x})\| \le \Delta,
\]
where $\Delta > 0$. We now show 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 of $\dot{\b x}=\b f(t,\b x)+\b B(t,\b x)\widetilde{\b u}$, 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
\[
\b x(t)=\b x(0)+\int_0^t\left( \b f-\b B\left( \b S\b B\right)^{-1}\b S\b f\right)ds,
\]
while the trajectory of the system in non-ideal sliding satisfies
\[
\widetilde{\b x}(t) =\widetilde{\b x}(0)+\int_0^t\left( \b f-\b B\left( \b S\b B\right)^{-1}\b S\b f\right)(\widetilde{\b x}(t))ds + \int_0^t\b B\left( \b S\b B\right)^{-1}\dot{\b \sigma}(\widetilde{\b x}(t))ds
\]
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}(t))ds\\
& & +\left.\b B\left( \b S\b B\right)^{-1}\b \sigma  (\widetilde{\b x}(t))\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 the expression for the trajectory in ideal sliding, taking the norm of both sides, and applying the traiangle 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}(t))\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}(t))\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(t,\widetilde{\b x})+\b B(t,\widetilde{\b x})\widetilde{\b u}\right\Arrowvert dt\\
&\le& \|\widetilde{\b x}(0)\|+\int_0^T \left( M+N \left\Arrowvert\widetilde{\b x}\right\Arrowvert\right) dt\\
&\le& \|\widetilde{\b x}(0)\|+MT\int_0^T N \left\Arrowvert\widetilde{\b x}\right\Arrowvert dt.
\end{eqnarray*}
Applying now the Bellman-Gronwall lemma to the above integral inequality we obtain
\[
\|\widetilde{\b x}(t)\| \le \left(\|\widetilde{\b x}(0)\|+MT+ \right)e^{NT},\quad t \in[0,T].
\]
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
\[
\|\b x-\widetilde{\b x}\| \le S\Delta e^{LT}.
\]
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} $\Box$ 
