Investigating the effects of time-delays on stochastic stability and designing l1-gain controllers for positive discrete-time Markov jump linear systems with time-delay

This paper is concerned with stochastic stability, l1-gain performance analysis and positivity-preserving l1-gain controller design for a positive discrete-time Markov jump linear system with time-delay. First, necessary and sufficient conditions for stochastic stability of the system are derived by constructing a linear co-positive stochastic Lyapunov functional and establishing a system equation whose state variables consist of the mathematical expectation of the markovianized states and whose coefficient matrices depend on time-delay and the transition probability. It is revealed that stochastic stability of the positive discrete-time Markov jump linear system with time-delay is influenced by the size of time-delay and it is demonstrated by an example that the effect of time-delay on stochastic stability can be either positive or negative. Second, exact computation on an l1-gain index of a stochastically stable positive discrete-time Markov jump linear system with time-delay is presented, and a necessary and sufficient condition for the l1-gain performance is derived in the form of linear programming. Third, an iterative algorithm is proposed to design a positivity-preserving l1-gain controller, and in single-input case, an optimal controller is obtained analytically such that the closed-loop system achieves the minimal l1-gain performance. Then, a modified pest’s structured population dynamic model is developed to illustrate the effectiveness of the designed method.