5. Filtering

This chapter first studies the problem of robust \(H_{\infty }\) filtering for switched linear discrete-time systems with arbitrary switching and polytopic uncertainties. Based on the mode-dependent idea and parameter-dependent stability result, a robust switched linear filter is designed such that the corresponding filtering error system achieves robust asymptotic stability and guarantees a prescribed \(H_{\infty }\) performance index for all admissible uncertainties. The existence condition of such filters is derived and formulated in terms of a set of linear matrix inequalities (LMIs) by the introduction of slack variables to eliminate the cross coupling of system matrices and Lyapunov matrices among different subsystems. Then, an \(\mu \) -dependent approach proposed in Chap. 2 is used to investigate the exponential \(H_{\infty }\) filtering problem for discrete-time uncertain switched systems with average dwell time (ADT) switching, and a mode-dependent full-order filter is designed to guarantee that the resulting filtering error system is robustly exponentially stable and has an exponential \(H_{\infty }\) performance. Moreover, a class of discrete-time switched linear parameter varying (LPV) systems under ADT switching is considered to investigate the \(H_{\infty }\) filtering problem, and a mode-dependent full-order parameterised filter is then designed and the corresponding existence conditions of such filters are derived via LMIs formulation. Finally, the non-weighted \(H_{\infty }\) filtering problem is studied for a class of switched linear systems with persistent dwell-time (PDT) switching in discrete-time domain. A proper Lyapunov function suitable to the PDT switching is constructed, which is not only mode-dependent but also quasi-time-dependent (QTD). Then, a QTD filter is designed such that the resulting filtering error system is globally uniformly asymptotically stable and has a guaranteed \(H_{\infty }\) noise attenuation performance. Several examples are illustrated to show the validity of the obtained theoretical results.