Study On Systems Subject To Input Saturation Based On LMI Technique

Almost all real world control systems have an associated set of constraints, for ex-ample, the opening or power of actuator has to be limited by physical constraints. The presence of saturation in control system mostly results from inherent characteristics of system components or installations (for example, the saturation of the magnetic path and certain valve stroke), and sometimes is factitiously added for safety. Once saturation in-tervenes, linear system also changes into nonlinear system. Thus, saturation has received increasing attention from research community, particularly in the control of chemical pro-cesses and in channel equalization in digital communications.This dissertation studies the linear constant and linear switched systems subject to input saturation. Firstly, for the linear constant systems with input saturation, two perfor-mance indices are considered based on the LMI technique, which include the estimation of the domain of attraction and the L2 performance. The saturation dependent lyapunov function is combined with Finsler’s lemma to add the freedom of results by adding the dimension of the LMIs. In addition, for the same systems, we combine cone comple-mentarity algorithm with the LMI technique to study the state feedback problem with the decay rate analysis or with the estimation of the domain of attraction. More slack vari-ables are introduced in the analysis conditions, which is less conservative than existing results. Secondly, for the linear switched systems with input saturation, on the one hand, by exploiting the switched saturation dependent lyapunov function, the disturbance rejec-tion and the state feedback robust control problem with the estimation of the domain of attraction are studied. On the other hand, the output feedback control problem is studied by introducing a new congruence transformation, which gives LMI conditions related to the number of system modes.The main contributions are as follows:Chapters 1-2 first summarize and analyze the development and main research meth-ods in saturation problem. Preliminaries about the considered problem are also given.Chapter 3 investigates performance analysis problem under a given feedback law for the linear constant systems subjected to input saturation, based on the linear matrix inequality(LMI). Two performance measures, the estimation of domain of attraction and L2 performance, are considered by combining the saturation-dependent Lyapunov func-tion method with Finsler’s Lemma. The method is conceptually simple. Here, difference equations are considered as constraints and these dynamical constraints are incorporated into the stability analysis conditions through the use of matrix Lagrange multipliers. New and less conservative conditions in the enlarged space containing both the state and its time difference, allowing extra degree of freedom for various performance analysis, are proposed. Furthermore, based on these results, two important lemmas and two iterative LMI-based optimization algorithms are also developed to optimize the performance in-dexes respectively. Numerical examples illustrate that the proposed methods improve recent results on the same problems.Chapter 4 studies stabilization problem with the estimation of domain of attraction(or decay rate analysis) for the linear constant systems with input saturation. The saturation-dependent Lyapunov function is exploited to propose new stability conditions by intro-ducing additional slack variables. Especially, Elimination Lemma is used to show the stable property of one slack variable. If the stable slack variable is specified a priori by a systemic and simple approach, via a cone complementarity algorithm involving convex optimization, a state feedback control law is then designed by utilizing LMI-based ap-proach which maximize the estimation of domain of attraction (or guarantees an upper bound on the decay rate) of the system. The effectiveness of the proposed methods is illustrated by a numerical example.Based on the linear matrix inequality(LMI) technique, Chapter 5 study the problem of disturbance rejection for the linear switched systems subjected to input saturation. A new analysis condition is derived by exploiting switched saturation-dependent Lyapunov functions (SSDLF) under which trajectories starting from a level set will remain inside an outer level set. In this chapter, we are interested in the maximal norm of the distur-bance that can be rejected by the system. The corresponding algorithm is developed. Furthermore, upper bound of regional L2-gain for exogenous disturbance is minimized. A numerical example is given to show the effectiveness of the proposed methods.Chapter 6 investigates the problem of robust stabilization for the linear switched sys-tems subjected to input saturation. A robust stabilizing state feedback controller is devised by exploiting switched saturation dependent Lyapunov functions (SSDLF) again. The de-sign problem of a controller which maximizes an estimation of the domain of attraction is then reduced to an optimization problem with LMI constraints. The simulation results illustrate the feasibility and effectiveness of the proposed techniques.Chapter 7 designs a stabilizing controller for the linear switched systems with in-put saturation via output feedback. A switched nonlinear output feedback controller is proposed which guarantees the closed-loop system is locally asymptotically stable. Our main contribution consists in new sufficient LMI conditions related to the number of sys-tem modes for control synthesis, which are developed by introducing a new congruence transformation. The design problem of controller (coefficient matrices) that maximizes an estimation of the domain of attraction of the considered systems is then reduced to optimization problems with LMI constraints. The effectiveness of the proposed method is illustrated by a numerical example.Finally, the results of the dissertation are summarized and further research topics are pointed out.