Stability, ${l}_{2}$-Gain and Asynchronous ${H}_{{\infty}}$ Control of Discrete-Time Switched Systems With Average Dwell Time

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper first investigates the stability and <formula formulatype="inline"> <tex Notation="TeX">$l_{2}$</tex></formula>-gain problems for a class of discrete-time switched systems with average dwell time (ADT) switching by allowing the Lyapunov-like functions to increase during the running time of subsystems. The obtained results then facilitate the studies on the issue of asynchronous control, where “asynchronous” means the switching of the controllers has a lag to the switching of system modes. In light of the proposed Lyapunov-like functions, the desired mode-dependent controllers can be designed since the unmatched controllers are allowed to perform in the interval of asynchronous switching before the matched ones are applied. The problem of asynchronous <formula formulatype="inline"><tex Notation="TeX">$H_{\infty}$</tex></formula> control for the underlying systems in linear cases is then formulated. The conditions of the existence of admissible asynchronous <formula formulatype="inline"> <tex Notation="TeX">$H_{\infty}$</tex></formula> controllers are derived, and a numerical example is provided to show the potential of the developed results. </para>