Stability and \(L_2\) gain performance of linear systems subject to output saturation (Q2726008)

The authors consider the stability and \(L_2\) gain performance of linear systems subject to output saturation: NEWLINE\[NEWLINE\Sigma_1: \left\{\begin{aligned} \dot x&=Ax+Bu,\\ y&=\sigma(x),\end{aligned}\right. \qquad\Sigma_2: \left\{\begin{aligned}\dot x&=Ax+Bu+Ew,\\ z&=Cx+Du,\\ y&=\sigma(x),\end{aligned}\right.NEWLINE\]NEWLINE where signals \(x\in\mathbb R^n\), \(u\in\mathbb R^m\), \(w\in\mathbb R^r\), \(z\in\mathbb R^p\), \(y\in\mathbb R^n\) are the state, commanded control, outside disturbance, controlled output, measured output, respectively, \(A,B,C,D,E\) are matrices of appropriate dimensions, and \(\sigma(x)\) is a vector-valued saturation function defined as NEWLINE\[NEWLINE \sigma(x)=\begin{cases} x,&\|x\|\leqslant 1,\\ x/\|x\|,&\|x\|>1.\end{cases}NEWLINE\]NEWLINE It is shown that the stabilizable linear systems are semi-globally stabilizable via static feedback. In the presence of static feedback, sufficient conditions which make closed-loop systems satisfy the \(L_2\) gain performance prior given for two types of performance function respectively are derived. An illustrative numerical example is considered.