Stability And Stabilization Of Kinds Of Uncertain Systems

Uncertainty exists in almost all real systems. It may arise from the modeling error, the measure noise, the varying parameters and environmental disturbance. However, the design always depends on a certain system. A gap then appears. It is necessary for us to study how to make a control law which is designed based on a certain system meets the requirement of an uncertain or time-varying real system. The investigation consists of two aspects. The first one is to estimate the roust margin of a control law. The second is to find the characters of the better robust control law. This thesis will deal with the two problems.The main tool used in this thesis is the Lyapunov stability theory. Generally, there are two critical problems in the applications of the Lyapunov theory:one is how to construct a Lyapunov function or functional for the system under consideration, another is how to estimate the time derivative of the constructed Lyapunov function or functional along with the system solution. This thesis studies the robust control of kinds of uncertain systems by using the Lyapunov theory. It consists of two parts. Chapter 2 is the first part and the second part contains Chapters 3,4 and 5. The first part considers the simultaneous stabilization problem of uncertain nonlinear systems by using the control Lyapunov function (CLF). The second part considers the robust control problem of kinds of uncertain time-delay systems and polytopic-type linear discrete-time systems based on the Lyapunov-Krasovskii functional and the parameter dependent Lyapunov function, respectively. The contents and results of the thesis are as follows.Chapter I is an introduction. It firstly sums up the progress of robust control of kinds of uncertain systems. Some methods employed in simultaneous stabilization, robust control of parameter systems and uncertain time-delay systems, and their limitation are briefly introduced. Consequently, we present several system models discussed in this thesis and the used mathematical lemmas. At last, we briefly sum up the main work of this thesis.Chapter 2 studies the simultaneous stabilization problem of a collection of uncertain nonlinear systems. Firstly, we consider the single-input affine nonlinear systems. A suffi- cient condition for the simultaneous stabilization of these systems is proposed by using the CLF. The obtained results are then extended to the single-input and multi-input nonlinear systems with uncertain parameters, respectively. At the end of this chapter, the simultane-ous stabilization of unified chaotic systems is considered. Numerical examples are provided to illustrate the effectiveness of the proposed scheme.Chapter 3 studies uncertain discrete-time systems with interval time-varying delay. Un-certainties considered are polytopic-type uncertainty, linear fractal norm-bounded uncer-tainty, and quadratic nonlinear perturbations. An appreciate Lyapunov-Krasovskii func-tional is constructed, and a sum of finite inequalities is applied to estimate the time deriva-tive of the functional. Delay-range-dependent stability criteria are developed in terms of LMIs. It is shown by simulation that the proposed criteria can provide less conservatism than some existing ones. Moreover, based on the criteria, we also design the state feedback and time-delayed feedback to stabilize the system, respectively.Chapter 4 analyzes the stability of uncertain continuous-time systems which have inter-val time-varying delay and nonlinear perturbations. In the estimation of the time derivative of the constructed Lyapunov-Krassivskii functional, some useful terms are reserved such that the estimation holds less conservatism. The effectiveness of the proposed approach is demonstrated by numerical examples.Chapter 5 considers the robust stability and stabilization for a class of discrete-time polytypic linear systems. A sufficient and necessary condition for the stability of nominal system is presented by using the descriptor system transformation. This condition can be easily adapted in controller synthesis since it separates the design of Lyapunov function and the control law.In Chapter 6 the topics of this thesis are summarized and the problems for further study are presented.The main contributions of this thesis are as follows:â‘ For a collection of nonlinear uncertain systems which have the Brunovsky canonical form, a systematic algorithm is proposed to construct the common CLF. Based on the CLF, simultaneous stabilization feedback is presented for the cases of single-input and multi-input, respectively. The results simplify and generalize the corresponding works of Wu (2005,2009).â‘¡To reduce the conservatism of the reported delay-range-dependent stability criteria for uncertain interval time-delay systems, a Lyapunov-Krasovskii functional which includes the information of the range of time delay is presented, and the upper bound of the time derivative of the constructed functional is estimated by a new approach. New delay-range-dependent stability criteria are proposed. In the case of nominal discrete time-delay systems, we prove that the obtained result is less conservative than some existing criteria.â‘¢Descriptor model transformation is employed in the robust stability analysis and control design of polytopic-type discrete-time linear systems. The developed result can be viewed as a discrete-time counterpart of the continuous-time results proposed by Cao and Lin in 2004.