Study On Some Problems Of Switched Linear Systems

As an important class of hybrid systems, switched linear systems consist of several linear subsystems and a rule named switching signal or switching law that orchestrates the switching among them. Switched linear systems have attracted increasingly more at-tention since they are of theoretical importance and have wide engineering applications. Despite the rapid progress made so far, many fundamental problems are still either unex-plored or less well understood due to the complicated behavior of switched systems.Based on previous works of others, this dissertation studies several problems of switched linear systems including static output feedback (SOF), robust H2 control and filtering, finite frequency filter design and fault detection. At present, there are few results on SOF control for switched linear systems under constrained switching. In response to this situation, this dissertation proposes a SOF control method for this class of switched systems based on Finsler’s lemma combined with average dwell time technique. The pro-posed method reduces design conservatism by introducing two sets of slack variables. For switched linear systems under arbitrary switching, the existing SOF methods are only ap-plicable to several classes of switched linear systems with special structures. By the aid of Finsler’s lemma and switched Lyapunov functions, this dissertation presents new SOF de-sign conditions which can work successfully in situations where the existing ones do not. The proposed method and the existing methods can be seen as alternative ones. Parameter-dependent Lyapunov functions have been employed to solve robust control problems of discrete-time uncertain switched linear systems. However, for continuous-time uncertain switched systems, this is not the case since it is hard to decouple system matrices with Lyapunov variables using the existing methods. In this dissertation, we overcome this difficulty by using Schur complement formula. Then a robust H2 state-feedback con-trol method and a robust H2 filtering method are proposed for continuous-time uncertain switched linear systems. The existing filters for switched linear systems are designed in full frequency domain. The full frequency approaches can increase conservatism when the frequency ranges of external disturbances are known beforehand. For this case, the con-cept of finite frequency l2 gain is first defined which is used to deal with finite frequency problems of switched linear systems. Based on it, a finite frequency filtering method is proposed for discrete-time uncertain switched linear systems. When the frequency ranges of disturbances are known, the finite frequency filters can achieve better performances than the full frequency ones. To my knowledge, fault detection for switched systems has not been fully investigated up to now. The last part of this dissertation studies the problem of fault detection for two classes of switched linear systems. The design conditions for fault detection filters and thresholds are presented for arbitrary and constrained switching, respectively.The details of the dissertation are as follows:Chapter 1 summarizes and analyzes the development and main research methods of switched systems. Preliminaries about the considered problems are also given.Chapter 2 investigates the SOF control problem of discrete-time switched linear systems. The first part studies the SOF problem of discrete-time switched linear sys-tems under constrained switching. Based on multiple Lyapunov functions combined with the average dwell time technique, sufficient conditions which guarantee switched linear systems exponentially stable with a weighted l2 gain are given. Then, combined with Finsler’s lemma, new SOF control design conditions are derived. The minimal average dwell time and the corresponding controller gains can be obtained from these conditions for a given system decay degree. The proposed method reduces design conservatism by introducing two sets of slack variables. The second part studies the SOF problem of discrete-time switched linear systems under arbitrary switching. By the aid of switched Lyapunov functions and Finslers lemma, new sufficient conditions for SOF controller synthesis are derived. The proposed approach can work successfully in situations where the existing ones fail. The simulation examples have shown the effectiveness of the pro-posed methods.Chapter 3 focuses on the problems of robust H2 control and filtering for continuous-time switched linear systems with polytopic uncertainties. The first part studies the robust H2 state-feedback control problem. By using multiple parameter-dependent Lyapunov functions and the average dwell time technique, sufficient conditions are obtained such that switched systems are exponentially stable and satisfy a prescribed H2 performance index. Then, we decouple the system matrices and Lyapunov variables by Schur com-plement formula. LMI-based conditions for robust H2 controller synthesis are derived. Additionally, another controller design method based on multiple parameter-independent Lyapunov functions is also given for comparison. The second part studies the robust H2 filtering problem. By using similar techniques, LMI-based conditions for robust H2 fil- ter design are obtained. The simulation examples have shown the effectiveness of the proposed methods.Chapter 4 studies the problem of finite frequency filter design for discrete-time un-certain switched linear systems. The concept of finite frequency l2 gain is first defined to deal with finite frequency problems of switched linear systems. Sufficient conditions are given such that discrete-time switched linear systems have a prescribed finite frequency l2 gain. Based on these conditions, low-frequency, middle-frequency and high-frequency filter design conditions are derived, respectively. When the frequency ranges of external disturbances are known beforehand, the proposed finite frequency filters can receive bet-ter results than the existing full frequency ones. The simulation examples have shown the effectiveness of the proposed methods.Chapter 5 investigates the fault detection problem of switched linear systems. The first part of this chapter studies the problem of fault detection of discrete-time switched linear systems under arbitrary switching. By the aid of switched Lyapunov functions and Finsler’s lemma, LMI-based conditions are first derived such that the filtering error system is stable with a prescribed H∞performance index. Then a fault detection filter is designed based on these conditions, and a threshold design condition is given. The second part studies the problem of continuous-time switched linear systems under constrained switching. Combined multiple Lyapunov functions with average dwell time technique, sufficient conditions are obtained which guarantee the filtering error system exponentially stable and satisfying the H∞performances. Then a fault detection filter design method is proposed. The fault signal can be estimated from the output of the filter. The simulation examples have shown the effectiveness of the proposed methods.Finally, the results of the dissertation are summarized and further research topics are pointed out in Chapter 6.