Time-Delay Switched Systems

Abstract

This chapter first investigates the stability problem of a class of discrete-time linear switched systems with cyclic switching and state delays, and a numerical searching algorithm is explored to compute the feasible values of dwell time of the subsystems. Then, the problem of $H_{\infty }$ output feedback control for discrete-time switched linear systems with time delays is investigated. The time delay is assumed to be time-varying and has minimum and maximum bounds, which covers the constant delay and mode-dependent constant delay as two special cases. By constructing a switched quadratic Lyapunov function for the underlying system, both static and dynamic $H_{\infty }$ output feedback controllers are designed respectively such that the corresponding closed-loop switched system under arbitrary switching signals is asymptotically stable and guarantees a prescribed $H_{\infty }$ noise attenuation level bound. Moreover, under the arbitrary switching, the problem of robust $l_{2}-l_{\infty }$ filtering is studied for discrete-time switched linear systems with polytopic uncertainties and time-varying delays. The robust switched linear filters are designed based on the mode-dependent idea and parameter-dependent stability approach, and the existence conditions of such filters, dependent on the upper and lower bound of time-varying delays, are formulated in terms of a set of linear matrix inequalities. Finally, the state estimation problem is studied for a class of discrete-time switching neural networks (NNs) with persistent dwell time (PDT) switching regularities and mode-dependent time-varying delays in $H_{\infty }$ sense. The random packet dropouts, which are governed by a Bernoulli distributed white sequence, are considered to exist together for the estimator design of underlying switching NNs. The desired mode-dependent estimators are designed such that the resulting estimation error system is exponentially mean-square stable and achieves a prescribed $H_{\infty }$ level of disturbance attenuation. The effectiveness and the superiority of the developed results are demonstrated through numerical examples.