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## Über dieses Buch

This book focuses on the basic control and filtering synthesis problems for discrete-time switched linear systems under time-dependent switching signals. Chapter 1, as an introduction of the book, gives the backgrounds and motivations of switched systems, the definitions of the typical time-dependent switching signals, the differences and links to other types of systems with hybrid characteristics and a literature review mainly on the control and filtering for the underlying systems. By summarizing the multiple Lyapunov-like functions (MLFs) approach in which different requirements on comparisons of Lyapunov function values at switching instants, a series of methodologies are developed for the issues on stability and stabilization, and l2-gain performance or tube-based robustness for l∞ disturbance, respectively, in Chapters 2 and 3. Chapters 4 and 5 are devoted to the control and filtering problems for the time-dependent switched linear systems with either polytopic uncertainties or measurable time-varying parameters in different sense of disturbances. The asynchronous switching problem, where there is time lag between the switching of the currently activated system mode and the controller/filter to be designed, is investigated in Chapter 6. The systems with various time delays under typical time-dependent switching signals are addressed in Chapter 7.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
This chapter introduces the backgrounds and motivations of this book. Some preliminaries, the classification of switching signals and the comparisons with other types of hybrid systems, are also provided which aims at giving readers an understanding on the results that will be presented in the book.
Lixian Zhang, Yanzheng Zhu, Peng Shi, Qiugang Lu

### Chapter 2. Stability and Stabilization

Abstract
This chapter is devoted to the stability analysis for the four types of time-dependent switched systems. We first summarize the mainly used multiple Lyapunov-like functions (MLFs) approaches, sort out several typical MLFs and review their characteristics. Then, the corresponding methodologies by different MLFs such as switched Lyapunov function, MLFs with $$\mu$$-times increase at switching instants are developed to study the stability conditions for the underlying system with arbitrary switching, dwell time (DT) switching, average dwell time (ADT) switching, persistent dwell time (PDT) switching and their mode-dependent forms, respectively. The generally analytic results in discrete-time context will be concretely formulated in terms of linear matrix equalities (LMIs). Numerical examples are provided to verify the obtained criteria and to compare the four kinds of time-dependent switched systems. The results obtained in this chapter will lay a foundation for future developments in this book.
Lixian Zhang, Yanzheng Zhu, Peng Shi, Qiugang Lu

### Chapter 3. Performance Analysis

Abstract
In this chapter, the issue of performance analysis for the time-dependent switched systems with several typical switching signals will be studied. We will first focus our attention on the switched systems with $$l_{2}$$ disturbances and present the results of weighted/non-weighted $$l_{2}$$-gain analyses for the switching signals subject to arbitrary switching, ADT switching, and PDT switching, respectively. Then, considering the switching signals are to have MPDT property, we will give the tube-based robustness analysis for switched systems with $$l_{\infty }$$ disturbances with the aid of set-theoretic method. Finally, one example is given to verify the effectiveness of developed results on the section of tube-based robustness analysis for discrete-time switched systems with MPDT switching; the verifications of the results corresponding to other switching signals will be illustrated in later chapters coping with $$H_{\infty }$$ control or filtering.
Lixian Zhang, Yanzheng Zhu, Peng Shi, Qiugang Lu

### Chapter 4. Control

Abstract
This chapter is concerned with the control problem for discrete-time switched systems with several typical switching signals. Firstly, the problem of designing $$H_{\infty }$$ state-feedback controllers is investigated for switched linear discrete-time systems with arbitrary switching and polytopic uncertainties. Two approaches on designing parameter-independent (robust) and parameter-dependent $$H_{\infty }$$ controllers are proposed and the existence conditions of the desired controllers are derived and formulated in terms of a set of linear matrix inequalities (LMIs). Then, considering the average dwell time (ADT) switching, an $$\mu$$-dependent approach is then introduced for the underlying systems to solve the $$H_{\infty }$$ controller, and the obtained conditions are dependent on the admissible increasing level $$\mu$$ of Lyapunov-like function values at switching instants. Finally, in a network-based environment, the quasi-time-dependent (QTD) $$H_{\infty }$$ control problem is investigated for a class of discrete-time switched linear systems with modal persistent dwell time (MPDT) switching. One redundant channel is introduced in the data transmission from sensor to controller to reduce the probabilities of packet dropouts occurred in the single channel case. Several examples are used to demonstrate the effectiveness of the developed theoretical results.
Lixian Zhang, Yanzheng Zhu, Peng Shi, Qiugang Lu

### Chapter 5. Filtering

Abstract
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.
Lixian Zhang, Yanzheng Zhu, Peng Shi, Qiugang Lu

### Chapter 6. Asynchronous Switched Systems: ADT Switching

Abstract
This chapter first investigates the stability and $$l_2$$-gain analysis 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 issues of asynchronous control, where “asynchronous” means the switching of the controllers has a lag to the switching of system modes. The basic asynchronous stabilization and asynchronous $$H_\infty$$ control problem are both studied and the case for the system with time-varying parameter is further addressed under the modal average dwell time (MADT). Finally, the asynchronous $$H_\infty$$ filter design problem is dealt with for the underlying switched linear systems with ADT switching. The phenomenon of “asynchronous” switching will unavoidably deteriorate the control performance such as the $$H_\infty$$ noise attenuation index. However, it can be verified that the designed controller/filter considering the synchronous switching will be not necessarily valid in the presence of asynchronous switching. Several examples are provided to show the potential of the developed results.
Lixian Zhang, Yanzheng Zhu, Peng Shi, Qiugang Lu

### Chapter 7. 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.
Lixian Zhang, Yanzheng Zhu, Peng Shi, Qiugang Lu

### Backmatter

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