Input-to-state Stability And Asynchronous Switching For Stochastic Systems

Stochastic system means that the input,output and disturbance of the system have random factors,or the system itself has some uncertainty.As is well known,stochastic systems have an extensive applications in many branches,such as science,economy,physics,control and so on.Many scholars in the world have devoted them-selves to investigating various properties of stochastic systems.Especially,stability and switched control theories have been interesting and important topics since they can describe significant characteristics of stochastic systems.Switched systems,as a class of hybrid systems,are composed by a family of subsystems and a switching signal which governs the switching between the system modes.On the other hand,it is inevitable to encounter disturbance in the practical systems,which means that stochastic models are more applicable to describe practical systems.Overall consideration results in our present models-switched stochastic sys-tems.If the controller's switching is regarded coincident with the system's switching,which is called synchronous switching.Asynchronous switching,which is opposed to the synchronous switching,is caused by the detection delay of the switching signal,which results in the mismatched period of the designed controller in each subsystem.Therefore,the investigations for switched stochastic systems are very important and challenging.Besides,we also consider another type of hybrid system-impulsive systems,which undergoes abrupt changes in the state at discrete times.Finally,some numerical examples are presented to illustrate our results.The main work of this thesis includes the following parts:1.In Chapter 2,the input-to-state stability?ISS?,stochastic-ISS and integral-ISS problems for a class of switched stochastic systems with time delays are studied.A continuously differentiable Lyapunov–Krasovskii function is employed to derive the ISS-type properties of the systems.Two cases are considered:?i?synchronous switching,i.e.candidate controllers coincide with the switching of the system mode;?ii?asynchronous switching,i.e.the candidate controllers have a lag to the switching of the system modes.For synchronous switching,we require that the coefficients of the diffusion operator of Lyapunov–Krasovskii function is indefinite,that is,the rate coefficient of the Lyapunov–Krasovskii function is time-varying,which can be pos-itive or negative along time evolution.Then,we extend it to asynchronous switching and consider two situations:1)the diffusion operator coefficient is time-varying in the synchronous interval and increasing in the asynchronous interval;2)diffusion op-erator coefficients are increasing in the synchronous interval and time-varying in the asynchronous interval.Then,by means of the average dwell-time method together with the stochastic technique,we can get the desired ISS-type results.Finally,two examples are given to show the validity of the results.2.In Chapter 3,we discuss the problems of input-to-state stability?ISS?,integral-ISS?i ISS?and e^?t-ISS for a class of switched stochastic delayed systems under asyn-chronous switching.Asynchronous switching refers to that the switching of candidate controllers does not coincident with the system modes.Different from existing work-s,we allow the coefficients of the estimated upper bound for the diffusion operator of a Lyapunov function to be time-varying and increasing during the matched time interval and unmatched time interval,respectively.Firstly,we give a function??t?,which is measurable.Then,based on measurability of the above function,further using Razumikhin theorem,comparison lemma,Fubini theorem,and methods of all sorts of inequalities technique.Especially,our results improve existing results regard-ing asynchronous switching in the literature.An example is used to demonstrate the applicability of the results.3.In Chapter 4,we deal with the stochastic input-to-state stability?SISS?for a class of switched stochastic systems under asynchronous switching.Asynchronous switching refers to that the switching of the candidate controllers does not coincide with the switching of system modes.In addition,the time delays appear in the state.Unlike the existing works,we not only allow the coefficients of the estimated upper bound for the diffusion operator of a Lyapunov function to be increasing during the matched time interval,but also in the unmatched time interval.Meanwhile,by using the switching signals which prevail over the traditional average dwell time scheme and the Razumikhin theorem,the SISS property is obtained.Especially,our results improve some existing ones regarding asynchronous switching in the literature.4.In Chapter 5,the problems of the stochastic input-to-state stability?SISS?properties are investigated for impulsive stochastic nonlinear systems with delayed impulses.By employing Lyapunov method together with average impulsive interval approach,the sufficient conditions are established to ensure properties for impulsive stochastic systems with all stable subsystems.Moreover,the ISS properties are also derived for the stochastic systems when the system is with both stable and unstable subsystems.Then,we also study impulsive systems with multiple jumps,i.e.,there exist different impulses in a system.By now,there has been no papers to consider multiple jumps for stochastic cases.Finally,two examples are provided to illustrate the effectiveness of our results.