Research On Robust Stability And Control For Nonlinear And Time-delay Uncertain Stochastic Systems

There are some uncertain random factors in real system. When accuracy equirement isn't high or random factors can be ignored, the systems model is often simplified as deterministic systems model. The deterministic systems model is convenient for systems analysis and synthesis. When random uncertainty is modeled as stochastic processes, the stochastic systems can describe actual engineering systems more really and more accurately. The dynamic stochastic system is described by stochastic differential equation. So stochastic differential equation has been paid more attention and is widely used in the systems model and systems analysis. Meanwhile, stochastic control theory is also widely used in the economic, demographic and other social areas, as well as aviation and aerospace, navigation and control, manufacturing engineering and other engineering fields. Stochastic systems theory has become a popular research field of modern control theory.Nonlinear and time-delay of system are commonly encountered in real systems. Time-delay is frequently a source of instability or oscillation. The existence of time-delay of control systems increases the difficulty of theoretical analysis and engineering applicationl. On the other hand, the controlled systems are often impacted by parameter error, unmodeled dynamics and uncertain external disturbances. The system model has some uncertainties. The studies on the impact of uncertainty for system performance produce robust control theory. Therefore, it is of a great importance in theoretical and practical application to research into robust stability and controlof nonlinear stochastic systems and uncertain stochastic time-delay systems. By using stochastic Lyapunov stability theory, model transform and free-weighting matrix method, and by means of It(o|^) differential formula, Schur complement lemma, linear matrix inequality, some important lemmas and inequalities, we study the robust stability, robust stabilization and control of nonlinear stochastic systems and uncertain stochastic time-delay systems.The main contents and contribution of this dissertation are summarized as followings:1. The first two chapters give an introduction to the background and significance of stochastic systems, and the latest progress in the stability and control of non-linear stochastic systems and uncertain stochastic time-delay systems. Then the basic theory of stochastic systems is reviewed, including the stochastic process, Brown motion, It(o|^) stochastic stability and Lyapunov stochastic stability theorem.2. The robust exponential stability for a class of uncertain nonlinear stochastic systems with discrete and distributed delays is investigated. During the past years, distributed-delay term is often looked as a perturbation of the discussed systems. Therefore, the stability criteria given in these references may not work or may be conservative in some cases. In this paper, the distributed-delay term doesn't be treated as a perturbation. We ues descriptor model transformation with less conconservatism to obtain delay–dependent stability conditions, and different from the usual descriptor model transformation only used in the discrete-delay term, that is, the descriptor model transformation is not only used in discrete-delay term but also in distributed-delay term. Combined with a new type of Lyapunov-Krasovskii functional and integral inequality technique, delay-dependent robust exponential stability in mean square criteria are derived in terms of linear matrix inequalities (LMI).3. The robust stability problem for nonlinear time-varying delay stochastic systems with polytopic-type uncertainties is discussed. Since nonlinear term, distributed delay and discrete delay term in the uncertain systems, the model becomes more general and the upper bound of derivative of the delay term needn't less than 1. Based on parameter-dependent Lyapunov-Krasovskii functional and free-weighting matrix method, some delay-dependent and parameter-dependent stability conditions are presented in terms of linear matrix inequalities. The results in this paper improve the existing stability criteria.4. The control problem for two nonlinear stochastic systems are considered. The adaptive boundary control for a class of infinite-dimensional nonlinear stochastic (It(o|^) stochastic KdVB equations) is discussed. A nonlinear boundary control law and an adaptation law are proposed. Secondly, the state feedback control for a class of stochastic nonlinear systems with time-delay disturbs is investigated. Based on the technique of Razumikhin and backstepping method, delay-independent state feedback controller is given.5. The delay-dependent robust exponential stabilization for uncertain stochastic systems with time delay is investigated. Based on parameterized neutral model transformation and Lyapunov-Krasovskii functional approach, a sufficient condition of delay-dependent exponential stability in mean square for the closed-loop systems is derived in terms of linear matrix inequality.6. The parameter-dependent state feedback control problem for stochastic delay-varying systems with polytopic-type uncertainties is discussed. Some examples show that many systems can't be stabilization by fixed gain matrix (parameter-independent controller), but can be stabilization by parameter-dependent controller. Based on parameter-dependent Lyapunov-Krasovskii functional and free-weighting matrix method, some delay-dependent and parameter-dependent stabilization conditions are presented in terms of linear matrix inequalities. Since the number of free-weighting matrices is reduced when introducing free-weighting matrix, the given parameter-dependent controller is easier to implement.7. The non-fragile stabilization problem for stochastic delay-varying systems with polytopic-type uncertainties is discussed, and the perturbed matrix in the actual implemented controller is assumed satisfying polytopic-type uncertainties (a sort of more natural description). By using parameter-dependent Lyapunov-Krasovskii functional method and free-weighting matrix method, the product terms for Lyapunov matrix and system matrix are separated. Then a non-fragile robust exponential stabilization condition for stochastic delay-varying systems with polytopic-type uncertainties is proposed in terms of linear matrix inequalities. The non-fragile state feedback controller can be obtained by solving LMI.Finally, the main results of the dissertation are concluded, and the issues of future investigation are proposed.