Stability and control of switched systems with impulsive effects
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Switched systems belong to a special class of hybrid systems, which consist of a collection of subsystems described by continuous dynamics together with a switching rule that specifies the switching between the subsystems. Such systems can be used to describe a wide range of practical applications, such as orbital transfer of satellites, auto-driving design, communication security, financial investment, neural networks and chaotic systems, just to name a few. For these switched systems, the occurrence of impulses and delay phenomena cannot be avoided. For example, in some circuit systems, switching speeds of amplifiers within the units’ individual circuits are finite, and hence causing delays in the transmission of signals. The abrupt changes in the voltages produced by faulty circuit elements are exemplary of impulsive phenomena. On the other hand, it is well known that stability is one of the most important issues in real applications for any dynamical system, and there is no exception for switched systems, switched systems with impulses, or delayed switched systems.With the motivations mentioned above, we present, in this thesis, new developments resulting from our work on fundamental stability theory and design methodologies for stabilizing controllers of several types of switched systems with impulses and delays. These systems and their practical motivations are first discussed in Chapter 1. Brief reviews on existing results which are directly relevant to the subject matters of the thesis are also given in the same chapter.In Chapter 2, we consider a class of impulsive switched systems with time- invariant delays and parameter uncertainties. New sufficient stability conditions are obtained for these impulsive delayed switched systems. For illustration, a numerical example is solved using the proposed approach.In Chapter 3, new asymptotic stability criteria, expressed in the form of linear matrix inequalities, are derived using the Lyapunov-Krasovskii technique for a class of impulsive switched systems with time-invariant delays. These asymptotic stability criteria are independent of time delays and impulsive switching intervals. A design methodology is then developed for the construction of a feedback controller which asymptotically stabilizes the closed-loop system. A numerical example is solved using the proposed method.In Chapter 4, new asymptotic stability criteria, expressed in the form of linear matrix inequalities, and a design procedure for the construction of a delayed stabilizing feedback controller are obtained using the receding horizon method for a class of uncertain impulsive switched systems with input delay. For illustration, a numerical example is solved using the proposed method.In Chapter 5, we consider the stabilization problem for cellular neural networks with time delays. Based on the Lyapunov stability theory, we obtain new sufficient conditions for asymptotical stability of the delayed cellular neural networks and devise a computational procedure for constructing impulsive feedback controllers which stabilize the delayed cellular neural networks. A numerical example is given, demonstrating the effectiveness of the proposed method.In Chapter 6, we consider a class of H∞ optimal control problems with systems described by uncertain impulsive differential equations. A new design method for the construction of feedback control laws which asymptotically stabilize the uncertain closed-loop systems is obtained. Furthermore, it is shown that the H∞ norm-bounded constraints on disturbance attenuation for all admissible uncertainties are satisfied. New sufficient conditions, expressed as linear matrix inequalities, for ensuring the existence of such a control law are presented. A numerical example is solved, illustrating the effectiveness of the proposed method.In Chapter 7, we conclude the thesis by making some concluding remarks and giving brief discussions on topics for further research.
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