Research On Reduction Methods For Stochastic Jump Systems

Stochastic jump systems can be widely found in practical applications due to their powerful modeling and analysis capabilities of handling stochastic jump phenomena caused by abrupt environment disturbances,random failures or component damages,or even human factors in normal operation.As a special class of switched systems,Markov jump systems(MJSs)have been widely applied in practical applications,such as eco-nomics systems,computer and communication systems,solar thermal receivers,aircraft systems,power systems,etc.However,the assumption that the sojourn-time is exponen-tially distributed and the transition probability(TP)is a constant result in conservative results,which limits the applications of MJSs in reality.Semi-Markov jump systems(S-MJSs)relax the memoryless property of TPs in MJSs to sojourn-time-dependent TPs,which makes the S-MJSs have much broader applications.On the other hand,high-order complex modeling is frequently encountered in many areas of practical applications,which makes analysis and synthesis of stochastic hybrid systems more complex.Thereby,the study of simplifying stochastic hybrid systems within a certain criterion is necessary and significant,both for theoretical researchers and engineering practitioner.Furthermore,compared with stability,the dissipativity theory is a more general and popular input-output energy-related performance index in analysis and synthesis of MJSs.This thesis investigates the exponentially mean square stability analysis,dissipa-tivity analysis,robustness control,model reduction,reduced-order controller design,and reduced-order filtering design for stochastic jump systems.The considered dynamic mod-els are delay systems,switched linear parameter-varying(LPV)systems,and S-MJSs.The main contents are concluded as follows:Chapter 1 firstly introduced the background and motivations of stochastic jump systems,especially the MJSs and S-MJSs,to show the necessity and significance of stochastic jump systems.Then,the studies of model reduction,controller design,and filter design for dynamic systems,such as time-varying delay systems,Markov jump systems,and uncertain systems were reviewed in this chapter.The analysis and synthesis of stochastic jump systems have been a hot research field and have made great progress.However,there are still many shortcomings and problems that need to be solved urgently.Chapter 2 investigated the dissipativity analysis for continuous-time S-MJSs with randomly occurring uncertainties and time-varying delays.It was assumed that the time-varying uncertainties belonged to independent Bernoulli-distributed white sequences and the TP matrix was norm-bounded.By selecting proper parameter-dependent Lyapunov-Krasovskii functional and using piecewise analysis technique,sufficient conditions to ensure the exponential mean-square stability with a strict dissipative performance were derived.The obtained results of proposed method were less conservative and more general than that of existing methods.A RCL system modeled by S-MJS was used to illustrate the effectiveness of the proposed method.Chapter 3 proposed a new mode reduction method for switched LPV systems based on time-weighted controllability and observability gramians matrix and the new time-weighted gramians matrix and time-weighted energy functions for switched LPV sys-tems were defined.The time-weighted gramians matrix can be obtained by constructing parameter-independent multiple Lyapunov-Krasovskii functional and was shown to be input-output energy bounded;The generalized balanced transformation matrix can be obtained by solving a minimization problem and the original high-order systems were transformed into a balanced form based on the abilities of controllability and observabil-ity;Then the low-order models for different weighting values were obtained by truncating or residualizing the states which were difficult to reach and simultaneously difficult to observe.The feasibility and effectiveness of the proposed model reduction method were illustrated by using two examples including a three spring-mass system.Chapter 4 was concerned with the reduced-order dynamic output feedback controller(DOFC)design for continuous-time S-MJSs with randomly occurring uncertainties and partially access polytopic TPs.It was assumed that the time-varying uncertainties obeyed independent Bernoulli-distributed white sequences and the TP matrix was polytopesis.Motivated by the conclusions of the former two chapters,the sufficient conditions for the existence of the DOFC are proposed to guarantee the stochastic stability with a strict dissi-pative performance by using model-dependent Lyapunov-Krasovskii functional.Further-more,the DOFC gains are derived by using cone complementarity linearization algorithm and the reduced-order controller model were obtained by truncating the weak states as described in last chapter.Simulations were provided to demonstrate the effectiveness and potential of the proposed reduced-order dissipative DOFC design methods.Chapter 5 considered the reduced-order dissipative filtering problem for uncertain SMJSs with an event-triggered scheme and output quantization.The parameters in polytopic TPs were assumed to be time-varying and the Lyapunov-Krasovskii functional included the sojourn-time and its change rate.The main purpose of this chapter was to estimate the output signals for a special stochastic system by designing a reduced-order filter taking the noise,time-varying delays,and limited bandwidth in the transmission network into consideration.The sufficient conditions of the required robust dissipative performance were first obtained by constructing Lyapunov-Krasovskii functional and Wirtinger-based integral inequality;Then,the required filter parameters were solved by introducing a slack matrix and matrix transform method.Furthermore,the proposed performance criterion condition was more general by covering H_?,passivity,and mixed performance.Finally,the feasibility and effectiveness of the proposed reduced-order filtering design method were illustrated by using two examples including a single-link robot arm system.