Discrete Control Of Nonlinear Stochastic Systems Driven By Lévy Processe

Stochastic processes widely exist in natural and real dynamic systems,such as ecosystems,industrial manufacturing systems,financial systems and many other systems.However,sometimes the existence of stochastic noises will make the performance of the system worse,and even bring about the instability of the system.Therefore,it is quite necessary to study the stability of the stochastic system.Gaussian noises,as a type of common noises in stochastic systems,have obtained a large number of research results so far.However,when modeling the actual system,we should not only consider the continuous noises,but also consider the existence of discontinuous noises such as pure jump noises because in some cases,the system may have parameter jumps or large system oscillations caused by mutations.As known to all,a Lévy process can describe a continuous Brownian motion,as well as discontinuous conditions such as abrupt changes,stochastic faults or sudden disturbances in the system.In the mean time,for common continuous systems,controllers are generally designed for continuous time.However,continuous controllers are prone to fail if they work continuously for a long time.In this case,discrete control came into being.By controlling the system within the set sampling interval,a discrete control not only improves the control efficiency,but also saves resources.Through proper design,it can also achieve the same control effect as continuous controller,so it has broad prospects for development.This thesis mainly studies the discrete control and stability of nonlinear stochastic systems driven by Lévy process.By making use of Lyapunov stability theory,stochastic analysis,sliding mode control,state feedback control,adaptive control and other analysis methods,the sufficient conditions are obtained for the realization of mean square stability of nonlinear stochastic systems driven by Lévy process under discrete control and different parameters.The primary research contents of this thesis are summarized as follows:1.The discrete control problem is studied for a class of nonlinear stochastic systems driven by a Lévy process with uncertain parameters.Firstly,an integral sliding mode controller is devised to make the system reach exponential stability in mean square sense and reach the designed integral sliding mode surface in a finite time.Secondly,the continuous controller is discretized.The results show that the second moment of the difference between the system state under the action of the discrete controller and the system state under the action of the continuous controller is bounded.The stability of the closed-loop system under the action of discrete controller is further proved by derivation.Finally,the effectiveness of the results is validated by the simulation of the actual drill bit system.2.A nonlinear stochastic system driven by a Lévy process with Markov switching is considered,and the stability of the system after the discrete control is studied.Firstly,a state feedback controller is devised to keep the system exponentially stable in the sense of mean square.The continuous state feedback controller is discretized,and the upper bound of the second moment of the system state difference under the action of continuous and discrete controller is found by using the method of stochastic analysis.Secondly,the sufficient condition that the system under the action of discrete controller can still maintain mean square stability is obtained.Finally,the stability of the system and the availability of the discrete controller is validated by numerical simulation.3.For a nonlinear stochastic system driven by a Lévy process with Markov switching and unknown external disturbances,the problem of whether the discrete controller can still keep the continuous system stable is studied.Firstly,an adaptive controller is designed.Using the Mmatrix theory,it is verified that the designed adaptive controller can keep the system stable in the sense of mean square.Secondly,the adaptive controller is discretized,and the range of the second moment of the system state difference before and after the discretization of the controller is obtained.Thirdly,the mean square stability of the discrete system is proved.Finally,the validity of the results is validated by numerical simulation.