Model Predictive Control Of Continuous-time Markov Jump System

This paper is concerned with the model predictive control(MPC) problem for continuous-time Markov jump systems(MJSs). The main research motivation is to optimize the control performance, such as dynamic behavior, anti-disturbance, etc. By employing the semi-definite programming(SDP), the optimization problem of a quadratic cost function is obtained in the form of linear matrix inequalities. The set invariance principle is utilized to guarantee the feasibility of the MPC strategy. The mean square stability of the closed-loop system can be achieved based on such managements. The main contributions are summarized as follows:(1) A moving horizon control scheme is designed to ensure the mean square stability by solving an on-line optimization problem. First, we choose a mode-dependent Lyapunov function and a quadratic cost function, and then transform the optimization problem to SDP with a linear objective function and a set of matrix inequality constraints. Second, a sampled-period MPC law is introduced to deal with the continuous-time plant. Finally, the feasibility of the MPC scheme for the continuous-time MJSs and the mean square stability of the closed-loop system are discussed by using the invariant ellipsoid.(2) Aiming at a class of continuous-time singular MJSs with incomplete transition rates and norm-bounded uncertainties, we modify the cost function, which not only refers to the knowledge of system state but also considers the sampling period and the singularity. First, sufficient conditions of the stability for singular MJSs are given under the complete transition rates. Second, because the summation of each row of the transition rate matrix is zero, the unknown transition rates can be represented by the known transition rates. Therefore, we give sufficient conditions of singular MJSs with incomplete transition rates by obtaining inequality constraints under the known and unknown parts respectively. Finally, the mean square admissibility of the closed-loop system is discussed by analyzing the piecewise regularity, impulse-free and mean square stability.(3) The H∞ predictive control is proposed for MJSs with hard constraints. Off-line H∞ control often cannot lead to an ideal disturbance attenuation capability due to the unpredictability of the external disturbance. Combining with the moving horizon idea, sufficient conditions, which guarantee the H∞ performance and satisfy the physical hard constraints, are given in detail. By minimizing the γ index on-line,the closed-loop system is able to manage a good trade-off between disturbance attenuation level and hard constraints.