Control For Non-homogeneous Markov Jump Linear Systems

Markov jump linear systems are a class of stochastic hybrid systems that randomly switch between a set of linear subsystems according to a random switching signal mod-elled by a Markov process.The dynamics of such systems are driven simultaneously by continuous variables systems and discrete events that are dynamically triggered by various random factors.Compared with traditional non-jumping systems,Markov jump linear systems have great advantages in the modelling of complex dynamic systems,and are widely used in communication networks,industrial manufacturing,aerospace sys-tems,etc.Therefore,the jumping systems have successfully attracted lots of scholars'attention,and its research has important theoretical and practical significance.For a Markov jump linear system,each subsystem can be viewed as a mode,and the jump between modes represents the switching of subsystems.The switching law between modes is determined by the mode transition probability matrix(MTPM)in discrete-time systems,while it's related to mode transition rate matrix(MTRM)for continuous-time systems.As a crucial factor governing the behaviors of Markov jump linear systems,the MTPIM/MTRM is generally considered to be time-invariant,certain and often completely known in the majority of studies,which means the underlying Markov process is homogeneous.However,such consideration has problems in fully describing the practical system dynamics since MTPM/MTRM is often dependent on the environment and is hard to maintain unchanged during the whole process.Therefore non-homogeneous Markov jump linear systems with time-varying MTPM are then pro-posed.In addition,positive systems and network systems which are actually widespread are receiving increasing attention.In this dissertation,non-homogeneous Markov jump linear systems are investigated in both positive and network system scenarios.After establishing two different non-homogeneous models,we study their stability and stabi-lization problems.The specific research contents are summarized as follows:· Stability and stabilization problems for both continuous and discrete-time non-homogeneous positive Markov jump linear systems are studied.In the context of positive systems,a non-homogeneous model with two-layer Markov process is first established:the low-layer one is used to describe the jump of system modes,while the high-layer one represents the variation of MTPM/MTRM of low-layer Markov process.Based on such model,the sufficient conditions are established by designing the switched linear co-positive Lyapunov functions and the neces-sary and sufficient conditions are addressed by analyzing the time evolution of the first-order moment of state variables.It should be noted that all the proposed conditions are solvable in terms of linear programming problems.On the basis of the derived stability criterion,the state feedback controller is designed to stabilize the closed-loop system.Finally,the results of theoretical analysis are verified by numerical simulation.· Stability and stabilization problems for network system with non-homogeneous Markov chain are investigated.Firstly,considering the characteristics of pack-et loss process,a non-homogeneous Markov jump system model with arbitrary MTPM switching is established which is more general than the two-layer Markov jump non-homogeneous model and is more suitable for network systems.Then the stability of such non-homogeneous systems is analyzed and the necessary and sufficient conditions for uniformly exponentially mean square stability are estab-lished.At the same time,a feedback controller which is dependent on current mode and finite switching path of MTPM is designed to stabilize the closed-loop system.Finally,the effectiveness of such control strategy is verified by numerical simulation with network control system background.