Control Of Markovian Jump Systems With Controllable Markov Chain

Switched systems with Markovian jump parameters are also called Markovian jump systems(MJSs) where system dynamics are driven by not only time but also dis-crete events. The discrete events are regard as system modes in MJSs and stochastic jump among the system modes is described by a Markov process. MJSs known as a sig-nificant branch of stochastic switched systems are widely used for modelling complex systems subjected to abrupt environmental disturbances, sudden changes in subsystem interconnection and so on. Therefore, the research of MJSs not only has theoretical significance, but also has practical engineering value.Switching law between different modes of MJSs is described by a Markov chain presented by mode transition rate matrix(MTRM). The investigation of MJSs has sug-gested that there exists a close relationship between MTRM and system stability or system performance. It is well worth pointing out that almost of the works concern-ing MJSs have supposed that MTRM is a given invariable matrix. Nevertheless, in some practical applications, an artificial control of MTRM is feasible and this has a profound meaning to research system stability and system performance. In this paper, we investigate the control problem of MJSs with controllable Markov chain. Firstly, we investigate the control problem of MTRM for continuous-time autonomous MJSs and the control of mode transition probability matrix(MTPM) for the discrete case respec-tively. And then, a joint control of MTRM and state is proposed and deeply studied. Meanwhile, other factors about MJSs’stability are considered such as nonhomogeneity, input constraints and so on. The main contributions of this dissertation are as follows:(1) For autonomous continuous-time MJSs, we take into account the control prob-lem of MTRM. In terms of mode indicator equation of MJSs, a model subject to con-trollable MTRM is established. Based on the model, we are concerned with optimal control of MTRM to minimize system performance, which is including mode cost and control expenditure. Simultaneously, for the sake of ensuring the system stability, fea-sible region of MTRM control is discussed. And the results have been extended to a discrete case with controllable mode transition probability matrix(MTPM).(2) A joint control of MTRM and system state is proposed for a class of nonho-mogeneous MJSs subject to input constraints. By assuming MTRM is piecewise ho-mogeneous, a higher-level Markov chain represented by high mode transition rate ma-trix(HMTRM) is introduced to describe the variation of MTRM and control variables are brought into HMTRM. Considering cost of MTRM control, a hybrid performance is proposed by adding it into traditional system performance. Based on this, taking in-to account the minimization of hybrid performance, a constrained controller of system state is designed and a detailed algorithm to seek optimal control of MTRM is given.This paper presents a novel method to control MTRM for MJSs with controllable Markov chain. Two issues have been concretely focused on:the sole control problem of MTRM/MTPM and the joint control problem of MTRM and system state. The cor-responding controllers have been designed theoretically and numerical examples have been simulated. Throughout the simulations, it is shown that this control method not only well ensures system stability, but also greatly improves system performance.