Analysis And Synthesis For Nonlinear Jump Systems Based On Dissipative Theory

Nonlinear system control is an important branch of control theory, because an actual system essentially is nonlinear and the linear model is an approximation of the actual system to a certain extent. With the rapid development of science and technology, high accuracy is required for system control, which makes it necessary to revisit the nonlinear nature of the system. On the other hand, many practical engineering systems may suffer from unexpected changes in system parameters and structure, due to unpredictable reasons, such as the sudden component failures, and external environment mutation, etc. lead to changes in system parameters and structure, resulting in a system with two or more operating modes. Typical examples are power systems and aircraft control systems, etc. These systems can be accurately described by Markov jump systems, which has been received a great deal of attentions from the control research community. Nonlinear jump systems are a class of nonlinear systems with Markov jump parameters, which has been widely used in networked control, economics, chemical engineering and other fields. Compared with deterministic nonlinear systems, jump nonlinear systems can be more accurate, more flexible and more reliable for modeling and analyzing the actual systems. Therefore, it is of great theoretical significance and practical importance to study the nonlinear system with random jump parameters.On another research frontier, the dissipative theory has played an important role in the system analysis and synthesis since the concept of dissipative dynamical systems was introduced in 1970 s. The essence of the theory is that there exists a nonnegative energy function such that the system energy loss is always less than the supply rate of the energy. Based on the input-output description, the dissipative theory gives a framework for the analysis and design of control systems from the energy view point. Moreover, the dissipative theory is closely related to Lyapunov stability theory, which provides a new method for constructing Lyapunov function. Dissipative theory-based research for nonlinear systems has become a hot research topic in control field.Motivated by the above considerations, we investigate a class of discrete-time nonlinear jump systems, where each mode is described by a linear part and a nonlinearity verifying a sector condition. The switching between the modes is governed by a Markov chain. This type of nonlinear jump systems is also known as Markov jump Lur ’e systems. Based on the dissipative theory, the corresponding filtering, robust control and model reduction issues are investigated for the presented nonlinear jump systems respectively. The main contents and results of this work are summarized as follows.In chapter 2, the robust dissipative filtering problem is investigated for a class of Markov jump Lur’e systems with piecewise homogeneous transition probabilities(TPs) in discrete-time domain. The traditional Lur’e type of Lyapunov function combining with the sector condition for the nonlinearities is employed. The sufficient condition on the existence of desired filter are derived in terms of linear matrix inequalities(LMIs), which ensures the filtering error dynamics is stochastically stable and the dissipative performance requirement is satisfied. In addition, Chapter 2.3 is concerned with the resilient dissipative filtering problem for a special class of uncertain discrete-time nonlinear jump time-delay systems, The nonlinear functions are assumed to satisfy sector condition with arbitrary(positive or negative, and possibly asymmetric) slopes for the sector boundaries. The filter is not sensitive to the gain disturbance in the process of filter implementations, i.e. it is nonfragile. A new multiple Lyapunov-Krasovskii function is proposed, which is composed of quadratic terms with respect to the state and two cross terms between state and nonlinear function. Based on the Lyapunov direct method, a new method for designing the resilient dissipative filter is presented.Chapter 3 studies the observer-based2 ?l-l control problem for a class of nonhomogeneous nonlinear jump systems subject to sensor saturations in discrete-time domain. Sensor saturations are decomposed into a linear term and a nonlinear term satisfying a sector condition. The time-varying characteristic of TPs is considered to be in a polytopic sense. The main objective of this chapter is to design an observer-based control law resulting from both the observer states and the nonlinearity, through constant feedback gain matrices. By employing a stochastic Lyapunov function that contains three quadratic functions of the estimation error, observer states and nonlinearities, we give the sufficient conditions under which the resulting closed-loop system is stochastically stable and satisfies a given 2 ?l-l performance index. The explicit expressions of desired gain matrices for the designed observer and controller are obtained, simultaneously.Chapter 4 dedicates to the resilient dissipative dynamic output feedback control problem for a class of uncertain Markov jump Lur’e time-delays systems in discrete-time domain. The state delay is assumed to be time-varying and has minimum and maximum bounds, the parameter uncertainties are assumed to be norm-bounded, and the sector nonlinearities appear in the system states and controller states. The controller to be designed is assumed to include additive gain variations, which results from controller implementations. An effective LMI approach is proposed to design the resilient dissipative output feedback controllers such that the resulting closed-loop system is stochastically stable and strictly dissipative.In chapter 5, the dissipativity-based model reduction problem for a class of discrete-time nonlinear jump systems with piecewise homogeneous TPs is studied. Given a Markov jump nonlinear system, where the system model includes the mode-dependent sector-bounded nonlinearities. a low-order linear model is constructed such that these two models are approximate according to dissipativity performance. The sufficient condition on the existence of desired reduced-order models are given by using the Lyapunov function with the sector nonlinearities. In addition, as a first attempt, an extended dissipativity performance index is considered for the model reduction problem of Markov jump systems with sector-bounded nonlinearity. Chapter 5.3 investigates the model reduction problem for a class of discrete-time nonlinear jump systems with time-varying delays in a unified framework. By constructing a mode-dependent Lyapunov-Krasovskii functional with the sector condition assumption for the nonlinearities, the sufficient conditions in terms of LMIs are derived under which the resulting error system is stochastically stable and satisfies a unified performance index. Instead of addressing the problems of ?H,2 ?l-l, passive and dissipative, model reduction in a separate way, this part studies these model reduction problems in a unified framework by using an extended dissipativity performance index.In chapter 6, the main content of this paper is summarized, and the future research direction is prospected.