Bounded Peaking in the Cheap Control Regulator

We will study the optimal linear regulator problem with a quadratic cost functional, where cost is a weighted sum of the integral-squared-output and the control energy. Reducing the weight on the control energy, we obtain increased speed of
response of the output, thus allowing higher feedback gains. (High gain systems have, for example, good disturbance rejection properties.)

Francis and Glover[1] have given conditions in the frequency domain when this can be done without loss of stability properties, i.e. that some state variables tend to peak excessively as $\epsilon \to 0$ , which we will state in this section. The purpose of our investigation is to analyse the response of the cheap control problem in the time-domain and interpret the conditions for bounded peaking geometrically.

Let $L^2$ and $L^{\infty}$ be the Lebesgue spaces of matrix-valued functions $T$ , of arbitrary but fixed dimensions, such that

(1) $\begin{equation*}\begin{array}{ccl}\Vert T\Vert_2&:=&(\int_0^{\infty}\Vert T(t)\Vert^2)^{1/2}<\infty\\\Vert T\Vert_{\infty}&:=&\sup_{t\ge 0}\Vert T(t)\Vert<\infty,\end{array}\end{equation*}$

respectively. Consider the problem of minimizing

(2) $\begin{equation*}\int_0^{\infty}(\Vert y(t)\Vert^2 +\epsilon^2\Vert u(t)\Vert^2)dt\end{equation*}$

subject to the time-invariant system

(3) $\begin{equation*}\begin{array}{ccl}\dot x&=&Ax+Bu,\quad x(0)=x_0,\\y&=&Cx.\end{array}\end{equation*}$

The control minimizing (2)-(3) is the state-feedback
$u=F_{\epsilon}x$ where

$F_{\epsilon}=-\frac 1{\epsilon^2}B^TP_{\epsilon}$

and $P_{\epsilon}$ is the unique positive semidefinite solution of the Riccati equation

$A^TP_{\epsilon}+P_{\epsilon}A-\frac 1{\epsilon^2}P_{\epsilon}BB^TP_{\epsilon}+C^TC=0.$

The resulting closed loop is described by

(4) $\begin{equation*}\dot x=(A+BF_{\epsilon})x,\quad x(0)=x^0.\end{equation*}$

The transition matrix corresponding to (4) is

$T_{\epsilon}(t):=e^{A+BF_{\epsilon}}.$

By $L^2$ -bounded peaking we mean that for each $x^0$ the trajectory of the closed-loop system is bounded in $L^2$ uniformly as $\epsilon \to 0$ , or equivalently, the set

${\Vert T_{\epsilon }\Vert_2:\epsilon_0\ge \epsilon >0}$

is bounded for some $\epsilon_0>0$ .

Let $u$ , $x$ and $z$ have dimensions $m$ , $n$ and $q$ , respectively. To make the problem well-posed we assume throughout that $(A,B)$ is stabilizable and $(D,A)$ is detectable. Furthermore, it is natural to assume that $B$ has linearly independent columns and $D$ has linearly independent rows.

Let $r:={\rm rank}~G(s)$ , where $G(s)$ is the plant transfer matrix, $\pi (s):=\det (s-A)$ , $\bar {\gamma}_r:=$ the sum of all $r$ -order principal minors of the matrix $G(s)G(-s)^T$ , and $\gamma_r:=\pi (s)\pi (-s)\bar {\gamma}_r$ . Then $\gamma_r=\gamma_r (s^2)$ is a polynomial in $s^2$ , see Wonham[2, p. 316].

Theorem 2.3: $L^2$ -bounded peaking is equivalent to the conditions that

(5) $\begin{equation*}{\rm rank}~G(s)={\rm rank}~CB\end{equation*}$

and

(6) $\begin{equation*}\gamma_r (s^2) {\rm~ has~no~zeros~in\quad Re}s=0.\end{equation*}$

Wonham[1, Theorem 13.2] has shown that if (5) holds, then, as $\epsilon \to 0$ , some of the closed-loop poles, the eigenvalues of $A+BF_{\epsilon}$ , tend to the zeros of the polynomial $\beta (s)$ , where

$\beta (s)\beta (-s)=\gamma_r (s^2)$

and $\beta (s)$ has zeros only in Re $s\le 0$ . The remaining closed-loop poles tend to infinity in Re $s<0$ . Thus, condition (6) guarantees that $\beta (s)$ has zeros only in Re $s<0$ and therefore that the closed-loop is stable in the limit. As a matter of fact, the roots of $\gamma_r (s^2)$ in Re $s\le 0$ are just the transmission zeros of $G(s)$ reflected, if necessary, in the imaginary axis.

The $L^2$ case is easier to deal with than the $L^{\infty}$ case. This is because of the boundary layer phenomenon
at $t=0$ . Consider the following decomposition. Let the state space $X$ and the input space $U$ of (3) be decomposed as follows. Let $X_2$ be an arbitrary complement of $B\cap R^{\ast}$ in $B$ :

$B=B\cap R^{\ast}\oplus X_2;$

and let $X_1$ be a complement of $X_2$ in $X$ which contains $B\cap R^{\ast}$ :

$X=X_1\oplus X_2.$

Now, let

$U_k:= B^{-1}X_k,~k=1,2$

so that

$U=U_1\oplus U_2.$

Corresponding to these decompositions we have the representations

(7) $\begin{equation*}A=\left(\begin{array}{cc}A_1 & A_2 \\A_3 & A_4 \end{array}\right),\left(\begin{array}{cc}B_1 & 0 \\0 & B_2 \\\end{array}\right),\\C=\left(\begin{array}{cc}C_1 & C_2 \end{array}\right),\end{equation*}$

where, $B_2$ is nonsingular and, since Im $B_1\subset R\subset$ Ker $C$ ,

$CB=\left(\begin{array}{cc}0& C_2B_2\end{array}\right)$

Solving the decomposed cheap control problem with singular perturbation gives rise to a singularly perturbated problem which has the reduced system

$x_1=\bar A_1x_1,\quad x_1(0)=x_1^0.$

Francis and Glover[1] have shown the following theorem.

Theorem 2.4: $L^{\infty}$ -bounded peaking is equivalent to the conditions that (5) holds and any eigenvalue of $\bar A_1$ on the imaginary axis are simple (i.e. have multiplicity one as roots of the minimal polynomial of $\bar A_1$ ).

From theorems 2.3 and 2.4 we have

Corollary: $L^2$ -bounded peaking implies $L^{\infty}$ -bounded peaking. If $\gamma_r(s^2)$ has no zeros in $\mathrm{Re}~s=0$ , then $L^2$ – and $L^{\infty}$ -bounded peaking are equivalent and are characterized by the condition rank $G(s)=\mathrm{rank}~CB$ .