Passive Analysis And Control Problem For A Class Of Singular Systems

Singular system is a kind of dynamics of more general form compared with state-space systems. The theory of singular systems has been developed extensively from 1970s, and becomes an independent branch of modern control theory. Recently, there has been a growing interest in singular systems since they are widely applied in control theory, circuits, economics, mechanical systems and other fields. Many classical concepts and results in the normal state-space theory have been extended to singular systems.Dissipative theory plays an important role in the stability research of control systems. The implication is that there exists a nonnegative energy function (namely store function) such that the energy consumption of a control system is always less than the supply rate of the energy. The passivity is an important part of the dissipativeness. It takes the product of the input and the output as the supply rate of the energy, which embodies the attenuation property of a system under bounded exogenous input. In fact, the stabilization theory based on Lyapunov function can also be explained in view of the passivity. Therefore, the passivity is a further abstraction of the stability. When investigating the stabilization of a system, a Lyapunov function is badly in need. According to the present literatures, this process can be converted into constructing a store function that makes the system passive.By using linear matrix inequality (LMI), the paper discusses on the dissipative and passive problem for linear singular systems, the main contribution of this paper is summarized as follows:(1) The main background of the problem discussed in this dissertation is introduced. Firstly, the structure characteristics and application background for singular systems are introduced, and many practical examples are given to illustrate the extensive existence of such systems in engineering. Then the development and recent progress are presented for singular systems. Furthermore, the significance and recent development of dissipative control and passive control problem are reviewed, at the same time, the main contribution of this paper is pointed out. Some relative knowledge of observer-based control theory, non-fragile control theory, robust control and LMI is introduced, so are the mainframe and main method. Finally, the main work of this dissertation is listed.(2) The passive control problems are considered for continuous singular systems. Passive control is firstly discussed for singular systems without uncertainty, and the conditions for the closed-loop systems to be admissible and passive with dissipationηare proposed in the cases of state feedback control and observer-based feedback control. Then robust passive analysis and control are discussed for descriptor systems with norm-bounded uncertainties. The design of two types of controllers is presented for the closed-loop systems to be generalized quadratically stable and passive with dissipationη. In addition, the design procedures are given respectively to obtain the maximum dissipation. Finally, sufficient and necessary conditions are obtained for the singular systems subjected to a class of parameter uncertainties which can be expressed in a linear fraction form, to be generalized quadratically stable and strictly passive in the case of series compensation and feedforward compensation, moreover, the compensators are designed.(3) The passive control problems are discussed for continuous singular systems with time-delay. As for the time-delay singular systems without uncertainty, by using of the Lyapunov technique and introducing the Lyapuniov-Krasovskii function, the existence conditions of state feedback and observer-based feedback passive controllers with their design methods are derived in linear matrix inequalities, such that the closed-loop systems are admissible and passive. Then we investigate the problem of robust passive control for singular systems which contain structure uncertainties and time-delay. Three types of controllers are considered:state feedback controller, observer-based feedback controller and dynamic output feedback controller. The controllers are constructed such that the closed-loop systems are generalized quadratically stable and passive with dissipationη. Design procedures and algorithm are given for obtaining the maximum dissipationη.(4) The non-fragile passive control problems are considered for continuous singular systems with and without state delay. The main idea is to design non-fragile controller such that the closed-loop system is robustly stable and passive with dissipationη. The non-fragile state feedback controller is designed firstly to guarantee the relative performance of the systems, and then non-fragile observer-based controller with the perturbations in both the control gain and observer gain is discussed. The feedback gain perturbations are in both the additive form and multiplicative form. Moreover, the existence conditions and design methods are presented in terms of LMI.(5) The strictly dissipative analysis and control problems with quadratic supply rate as well as strictly passive analysis and control for discrete-time singular systems are considered. Equivalence between strict dissipativity and extended strictly positive realness is firstly established. By using the approach of LMI, necessary and sufficient conditions are given for discrete-time singular systems to be strictly dissipative, and the conditions of the strict LMIs are mainly derived. Then the state feedback dissipative control is discussed by means of non-strict LMI and strict LMI, respectively, at the same time, the design method of the controller is presented. Based on the dissipative analysis, passive control problem is studied for the discrete-time singular systems without and with uncertainty. State feedback controllers are designed respectively to guarantee the closed-loop systems satisfying relative performance. Numerical examples are proposed to illustrate the effectiveness and generality of the proposed method, and also show the advantages of the strict LMI conditions compared with non-strict LMI conditions when analyzing the dissipative or passive problem.(6) The non-fragile passive control problems are studied for discrete singular systems without and with time-delay, respectively. At first, non-fragile state feedback controller is designed for discrete systems without time-delay to be admissible, and the controller gain perturbations are in both the additive form and multiplicative form. Then the problems of the delay-dependent robustly strict passivity, asymptotical stability and correlative non-fragile control for uncertain discrete-time singular systems are considered. The delay-dependent strictly passive and asymptotically stable criteria are provided for both discrete-time singular systems and uncertain discrete-time singular systems by using LMIs and introducing free weighting matrices. Then the feedback controller with additive form perturbations is designed such that the closed-loop systems posses the given performance, at the same time, the controller is presented.(7) A summary of this paper is given. At the same time, we give an expectation for the future work.As to the conclusions obtained in the paper, simulation examples are presented to show the use and effectiveness of the proposed methods.