\section{Constructing Robust Linear Controllers}
In this section we use the Lyapunov Theorem to construct robust state-feedback controllers. One of the basic issues in the control of dynamic systems is the effect of uncertainties, or neglected nonlinearities, on the stability of the closed-loop system. A controller is viewed robust if it maintains stability for all uncertainties in an expected range. We consider a class of dynamic systems modeled by
\[
\dot{\b x}(t) = \b A\b x(t)+\b B\b u(t) +\b f(t, \b x(t))+\b B\b h(t,\b x(t),\b u(t)),
\]
where $\b x(t) \in \mb{\Bb R}^n$, $\b u(t) \in \mb{\Bb R}^m$, and the functions $\b f$ and $\b h$ model uncertainties, or nonlinearities, in the system. We refer to $\b f$ as the {\em unmatched uncertainty\/}, while to $\b h$ as the {\em matched uncertainty\/}. We assume that
\begin{enumerate}
\item $\|\b f(t, \b x)\| \le \alpha_f\|\b x\|$,
\item $\|\b h(t, \b x)\| \le \gamma_h\|\b u\|+\alpha_h\|\b x\|$,
\end{enumerate}
where $\alpha_f$, $\alpha_h$, and $\gamma_h$ are known nonnegative constants. We further assume that the matrix $\b A$ is asymptotically stable. If this is not the case, we apply a preliminary state-feedback controller $\b u=-\b K\b x$ such that $\b A-\b B\b K$ is asymptotically stable. Such a feedback exists provided the pair $(\b A,\b B)$ is stabilizable. Our goal is to construct a linear state-feedback controller that would make the closed-loop system asymptotically stable for arbitrary $\b f$ and $\b h$ that satisfy the above norm bounds.
\begin{thm}
Suppose that $\b A$ is asymptotically stable, and $\b P_c=\b P^T_c >0$ is the solution to the Lyapunov matrix equation $\b A^T\b P_c+\b P_c\b A=-2\b Q_c$ for some $\b Q_c=\b Q^T_c >0$. Suppose also that
\begin{eqnarray*}
\alpha_f &<& \frac{\lambda_{\min}(\b Q_c)}{\lambda_{\max}(\b P_c)},\\
\gamma_h &<& 1.
\end{eqnarray*}
Then, the state-feedback controller $\b u=-\gamma_c\b B^T\b P_c\b x$, where
\[
\gamma_c > \frac{\alpha_h^2}{4\left(\lambda_{\min}(\b Q_c)-\alpha_f \lambda_{\max}(\b P_c) \right)(1-\gamma_h)}
\]
stabilizes the uncertain system for arbitrary $\b f$ and $\b h$ that satisfy the norm bounds.
\end{thm}
{\bf Proof}\quad The time derivative of the positive definite function $V=\b x^T\b P_c\b x$ evaluated on any trajectory of the closed-loop system is
\begin{eqnarray*}
\dot{V} &=& 2\b x^T\b P_c\dot{\b x}\\
&=& -2\b x^T\b Q_c\b x -2\gamma_c\b x^T\b P_c\b B\b B^T\b P_c\b x +2\b x^T \b P_c\b f+2\b x^T \b P_c\b B\b h.
\end{eqnarray*}
Our goal is to determine $\tilde{\gamma}_c >0$ such that if $\gamma_c > \tilde{\gamma}_c$, then $\dot{V}<0$, which in turn implies the asymptotic stability of the closed-loop system. To proceed further we note that by Rayleigh's inequality
\[
\lambda_{\min}(\b Q_c)\|\b x\|^2 \le \b x^T\b Q_c\b x \le \lambda_{\max}(\b Q_c)\|\b x\|^2,
\]
and therefore
\[
-\b x^T\b Q_c \b x \le -\lambda_{\min}(\b Q_c)\|\b x\|^2.
\]
Furthermore,
\[
\b x^T\b P_c\b B\b B^T\b P_c\b x =\|\b x^T\b P_c\b B\|^2.
\]
Taking the above into account, and performing some manipulations yields
\begin{eqnarray}
\dot{V} &\le& -2\lambda_{\min}(\b Q_c)\|\b x\|^2-2\gamma_c\|\b x^T\b P_c\b B\|^2+2\alpha_f\lambda_{\max}(\b P_c)\|\b x\|^2\nonumber\\
&{}&+2\|\b x^T\b P_c\b B\|\left( \alpha_h\|\b x\|+\gamma_h \gamma_c\|\b x^T\b P_c\b B\|\right)
\label{eq:vdotless}
\end{eqnarray}
Let
\begin{equation}
\b Q=\left[\begin{array}{cc}
\lambda_{\min}(\b Q_c)-\alpha_f\lambda_{\max}(\b P_c) & -\frac{\alpha_h}{2}\\
-\frac{\alpha_h}{2} & \gamma_c(1-\gamma_h)
\end{array}\right]\in \mb{\Bb R}^{2 \times 2}.
\label{eq:big_matrix}
\end{equation}
Then, we can represent~(\ref{eq:vdotless}) as
\begin{equation}
\dot{V} \le -2\left[\begin{array}{cc}
\|\b x\| & \|\b x^T\b P_c\b B\|
\end{array}\right]\b Q\left[\begin{array}{c}
\|\b x\|\\
\|\b x^T\b P_c\b B\|
\end{array}\right].
\label{eq:vdotq}
\end{equation}
For $\dot{V}$ to be negative it is sufficient that the leading principal minors of the matrix $\b Q$ given by~(\ref{eq:big_matrix}) be positive. The first-order leading principal minor is positive by assumption. For the second-order leading principal minor to be positive it is sufficient that $\gamma_h < 1$ and
\[
\gamma_c > \frac{\alpha_h^2}{4\left(\lambda_{\min}(\b Q_c)-\alpha_f \lambda_{\max}(\b P_c) \right)(1-\gamma_h)}.
\]
The proof is now complete.\\
\hspace*{5.5in} $\Box$

Note that if $\gamma_h=0$, then the closed-loop system is asymptotically stable if
\[
\alpha_f < \frac{\lambda_{\min}(\b Q_c)}{\lambda_{\max}(\b P_c)}
\]
and
\[
\gamma_c > \frac{\alpha_h^2}{4\left(\lambda_{\min}(\b Q_c)-\alpha_f \lambda_{\max}(\b P_c) \right)}.
\]