Stability Of Stochastic Switched Systems And Its Applications

Switched systems have drawn considerable attention because they can be widely used to describe lots of real-world systems.Abundant research results have been achieved on stability and analysis methods of deterministic switched systems.However,there are not many stability results for stochastic switched systems,and lots of difficult but rather important problems still remain to be solved.On the one hand,interaction between subsystems and the switching signal makes stability problem of switched systems more complex.On the other hand,for switched systems with different noises and uncertainty factors,the research on stability and control of stochastic switched systems is more difficult.Based on existing results,stability of stochastic switched systems is studied deeply in this dissertation.The main contents and results are summarized as follows:1.Exponential stability of stochastic switched systems is studied under state-dependent switching,which is motivated by active regions of subsystems.The global existence of the solution to stochastic switched systems is not given as a priori information but can be proved under some easily verified conditions.Because of switching behavior between individual subsystems,the stochastic switched system does not satisfy local Lipschitz condition which holds for individual subsystem of stochastic system family.The standard concepts of exponentially stability for It?o stochastic differential equations(SDEs)are not suitable for stochastic switched systems.Consequently,we must give new definitions of exponential stability for stochastic switched systems.Dynkin's formula is applied along the solution of stochastic switched systems to prove the moment exponential stability based on common Lyapunov function technique.To discuss the almost sure exponential stability,we should prove that the sample path of solution to stochastic switched systems starting from a non-zero initial state will also remain non-zero.The upper bound of It?o's integral of a switched function is given by the exponential martingale inequality.By the aid of It?o's formula,the criterion on almost sure exponential stability is established based on common Lyapunov function technique.The almost sure exponential stability of stochastic switched systems has not been touched before.The exponential martingale inequality is first applied to hybrid and switched systems by the aid of Dynkin's formula.Simulation examples are presented to illustrate the validity of the results.Stochastic switched systems composed by unstable subsystems are exponentially stable through designing active regions of subsystems and the state-dependent switching signal,and instability of subsystems and exponential stability of stochastic switched systems are proved respectively.2.The instability problem of stochastic switched systems is investigated under timedependent switching.The instability results of SDEs is extended to stochastic switched systems in some appropriate forms.Because switching behavior between subsystems destroys the surroundings of SDEs,definitions of instability are given in the forms of instability in probability,m-th instability,moment exponential instability and almost sure exponential instability for stochastic switched systems.By the aid of Dynkin's formula,It?o's formula,Chebyshev's inequality and strong law of large numbers,the criteria on instability of stochastic switched systems under arbitrary switching are established based on common Lyapunov function technique.To establish the criterion on instability in probability,we should prove that any solution of a stochastic switched system beginning in a bounded region,almost surely reaches the boundary of this domain in a finite time.The strong law of large numbers is first applied to hybrid and switched systems,It?o's integral of a switched function should be verified to be a higher-order infinitesimal about time using strong law of large numbers,then,we give the proof of the almost sure exponential instability.3.The problem of adaptive tracking control is considered for stochastic switched systems with unknown parameters.We firstly introduce the concept of global asymptotical practical stability in probability.A special stochastic barbalat's lemma should be given to establish the criterion on global asymptotical practical stability in probability for stochastic switched systems based on a common Lyapunov function and Chebyshev's inequality.For stochastic switched systems with unknown parameters in strict feedback form,based on adaptive backstepping tracking control with tuning functions,adaptive switched controllers are designed to guarantee that the tracking error in 4-th moment converges to an arbitrarily small neighborhood of zero,and the closed-loop system remains globally asymptotically practically stable in probability.Simulation examples on the control problem of a planar double pendulum hung from a random vibrating ceiling are given to demonstrate the efficiency of the proposed schemes.4.The research objects of three results above are stochastic switched systems described by a family of SDEs and switching signals,where the stochastic driver is a Wiener process,and its formal derivative is a gaussian white noise.For random switched systems,where the system family is composed by random differential equations(RDEs)with a non-gaussian white noise,noise-to-state stability is investigated under time-dependent switching.The global existence of the solution to random switched systems is not given,but can be verified under some conditions as in the non-switching cases.By the aid of Fubini's formula,the criteria on noise-to-state exponentially stable in the m-th moment of random switched systems are established based on a common Lyapunov function.As applications,we consider the control problems of spring pendulum hung from a random vibrating ceiling.A state feedback switching controller is designed such that this model can track a given reference signal based on backstepping techniques.Besides,nonlinear filtering is considered to ensure noise-to-state exponential stability in the m-th moment of filter error switched systems under arbitrary switching based on Riccati inequality.At the end of this dissertation,the main results are concluded and the problems to be solved in the future are presented.