State feedback \(H_\infty\) control design problem for linear singular systems (Q2785175)

This paper discusses the state feedback \(H_\infty\) control problem for linear singular systems of the form NEWLINE\[NEWLINEE\dot x= Ax+ B_1w+ B_2u,\quad z= C_1x+ D_{11}w+ D_{12}u,NEWLINE\]NEWLINE \(E,A\in\mathbb{R}^{n\times n}\), \(\text{rank }E= r< n\), \(B_1\in \mathbb{R}^{n\times q}\), \(B_2\in \mathbb{R}^{n\times p}\), \(C_1\in\mathbb{R}^{m\times n}\), \(D_{11}\in \mathbb{R}^{m\times q}\), \(D_{12}\in \mathbb{R}^{m\times p}\).NEWLINENEWLINENEWLINEThe equivalence of this problem to the state feedback \(H_\infty\) control problem for standard state-space linear systems is presented. Using the method of linear matrix inequalities, a necessary and sufficient condition for the existence of a controller such that the closed loop system is impulse-free and stable satisfying an \(H_\infty\) performance is derived.NEWLINENEWLINENEWLINEA family of particular solutions of controllers are given.