\(H_\infty\) control of discrete-time linear systems with time-varying delays in state. (English) Zbl 0936.93019

Song, Seong-Ho; Kim, Jeom-Keun; Yim, Choong-Hyuk; Kim, Ho-Chan

Automatica 35, No. 9, 1587-1591 (1999).

This paper is concerned with the \(H_\infty\) control of discrete-time linear systems with time-varying delays. The main result shows that if the system \[ \begin{aligned} x_{k+1} & = A_1x_k+ B_1u_k+ [A_2Q^{-1/2} \gamma^{-1} B_2]w_k,\\ z_k & = \Biggl[\begin{matrix} \sqrt mQ^{-1/2}\\ C\end{matrix}\Biggr] x_k+ \Biggl[\begin{matrix} 0\\ D\end{matrix}\Biggr] u_k\end{aligned} \] is quadratically stabilizable with a unitary \(H_\infty\) norm-bound, then the system \[ \begin{aligned} x_{k+1} & = A_1 x_k+ B_1 u_k+ B_2w_k,\\ z_k & = Cx_k+ Du_k\end{aligned} \] is stabilizable with an \(H_\infty\) norm-bound \(\gamma\) by the same control law. The proof is based on fairly standard linear matrix inequalities and Lyapunov theory.

Reviewer: S.P.Banks (Sheffield)

MSC:

93B36	\(H^\infty\)-control
93C55	Discrete-time control/observation systems

Keywords:

quadratic stabilization; \(H_\infty\) control; discrete-time linear systems; time-varying delays; linear matrix inequalities