Research On Stability And Control Of Nonlinear Stochastic Systems With Lévy Noise

Due to that Lévy noise can not only describe continuous Brown motion,but also be suitable to describe the random failures,abrupt change or sudden disturbances which arise in many physical systems,so the fact of engineering is in accordance with the consideration of Lévy noise.Nonlinearity is the essential characteristic of real systems,and many scholars have been concerned the research on stability and control of nonlinear stochastic systems with Lévy noise which is a research hot topic.In this dissertation,the problem of stability and control of nonlinear stochastic systems with Lévy noise is investigated.The study subjects are nonlinear stochastic systems with Lévy noise,including the almost sure stability of neutral stochastic delayed hybrid systems with Lévy noise,the input-to-state stability of switched stochastic delayed systems with Lévy noise,the moment exponential input-to-state stability of nonlinear switched stochastic systems with Lévy noise and the stability of stochastic systems with Lévy noise based on sliding mode control and adaptive control.Some analysis tools of stochastic dynamic systems are applied,such as Lyapunov stability theory,comparison principle theory,stochastic analysis,M-matrix,convergence theorem of nonnegative semimartingale,input-to-state stability,sliding mode control theory and adaptive control theory,etc.Sufficient conditions for the nonlinear stochastic systems with Lévy noise to achieve stability are established.The main contents are presented as follows:1.The research background and significance of nonlinear stochastic systems with Lévy noise are introduced.The research progress of control of nonlinear stochastic systems with Lévy noise is presented.Combining the main research content,the research status of sliding mode control and adaptive control is emphatically summarized.Then some preliminaries and related theorems,lemmas and definitions are given.Finally,the main research contents and chapters arrangement are introduced.2.The almost sure stability with general decay rate of neutral stochastic delayed hybrid systems with Lévy noise is studied.Firstly,the definitions about a kind of ?-function and almost sure stability with general decay rate are introduced.Using Lyapunov function and convergence theorem of nonnegative semimartingale,the sufficient conditions for the almost sure stability with general decay rate of considered systems can be obtained.Then according to the M-matrix theory,the upper bound of each coefficient at any mode is given.Especially,the coefficients of considered systems can be allowed to be high order nonlinear.3.The input-to-state stability of a class of switched stochastic delayed systems is investigated.By multiple Lyapunov function and average dwell time approach,the sufficient conditions of the input-to-state stability with general decay rate can be obtained if all the subsystems are input-to-state stable.Then utilizing comparison principle and the method of constant variation,the sufficient criteria of the exponential input-to-state stability of the switched stochastic delayed systems containing both input-to-state stable subsystems and non-input-to-state stable subsystems can also be derived.4.The moment exponential input-to-state stability for a class of nonlinear switched stochastic systems is studied.A continuously differentiable Lyapunov function with indefinite derivative is introduced,which generalises classic Lyapunov function method.Two situations are considered:(i)synchronous switching,i.e.candidate controllers coincide with system modes;(ii)asynchronous switching,i.e.the candidate controllers have a lag to the switching of the system modes.By employing indefinite derivative Lyapunov function method and average dwell-time approach,sufficient conditions for moment exponential input-to-state stability of the systems are derived.5.The almost sure stability of second-order nonlinear stochastic system with Lévy noise is studied by sliding mode control method.Two kinds of sliding mode surfaces and their corresponding sliding mode controllers are constructed.A conventional linear sliding mode surface is first constructed,by employing stochastic analysis technique combined with Lyapunov function method,sufficient conditions are established to ensure the almost sure stability of the system dynamics.Then a nonsingular terminal sliding mode control technique is used,corresponding controller is designed to guarantee the sufficient conditions of almost sure stability.6.The mean-square asymptotic stability of stochastic system with Markovian switching and Lévy noise is studied based on adaptive control.Firstly,a kind of general systems is considered and the information including the bounds of the nonlinear parameters and external disturbance is unavailable.Then an adaptive controller is designed to achieve the mean-square stability by utilizing Lyapunov function and M-matrix method.Next,a class of linear systems with noise possessing unknown coefficients is discussed,the corresponding adaptive controller can force the state trajectories of the systems to achieve mean-square asymptotic stability.Finally,the conclusions and some topics for future work are given.