Mode Feedback Control Of Markov Jumping Linear System

Many dynamical systems have variable structures due to random abrupt changes in the structure which result from sudden environmental disturbances,faulty communi-cation links and component failures.Most of the general dynamic systems cannot deal with these random abrupt changes.It is known that Markovian jump linear systems(MJLSs)which involve time-evolving and event-driven,can be employed effectively to model the above problems.Recently,MJLSs have been receiving considerable at-tention and the research of MJLSs has important theoretical significance and practical value.MJLSs are a class of jump systems with several modes where the stochastic jump among different modes is described by a Markov chain.As a crucial factor,the oc-currence probability of each mode in MJLSs will affect the system behavior to a large extent,and the occurrence probability is determined by the mode transition rate ma-trix(MTRM)and mode transition probability matrix(MTPM)in continuous-time and discrete-time MJLSs,respectively.It is noteworthy that most of the existing results are obtained on the premise that MTRM and MTPM are fixed matrices during the operation of MJLSs.However,in many practical systems modelled as MJLSs,the MTRM and MTPM can be adjusted or controlled via technical methods.This paper investigates the mode feedback control strategy for MJLSs with controllable MTRM or MTPM.With the mode feedback control strategy applied,the occurrence probability of each mode can be adjusted,then system stability as well as better performance can be achieved because of the difference in the performance of each mode.The main work is as follows:(1)A novel control strategy is proposed for a class continuous-time MJLSs with controllable MTRM aiming at performing control on MTRM instead of system state.The control of MTRM is introduced by mode indicator equation of continuous-time MJLSs,and a quadratic stabilization cost function is given to evaluate the performance of the MJLS which is a combination of state cost and control cost.Taking into ac-count that MTRM will totally determine system stability in absence of state feedback controller,the existence of mode feedback controller is discussed where two cases are included and the restrictions of mode feedback controller are obtained to ensure the stability of the system.Based on the given cost function,a feasible solution of mode feedback control strategy is then obtained with its detailed algorithm given.(2)The mode feedback control strategy is studied for a class of discrete-time MJLSs with controllable MTPM.The mode indicator equation for discrete-time MJLSs is ob-tained from the continuous-time case with the mode feedback controller introduced.With the given quadratic stabilization cost function,a mode feedback controller is de-duced with its existence and feasible domain discussed.By adjusting the occurrence probability of system modes,the mode feedback controller reduces the stabilization cost on the basis of guaranteeing stability,which will be an effective complement to the existing control mechanism.This paper proposes the mode feedback control strategy for continuous-time and discrete-time MJLSs,respectively.Algorithms are presented to clarify the procedure of finding the mode feedback controller.Numerical examples including comparisons with current state feedback mechanism are provided to illustrate the effectiveness of the proposed strategy.