Model Predictive Control For Stochastic Switched Systems

Model predictive control (MPC) (which is often interchangeably called receding horizon control (RHC)) has become a systematic approach for solving complex constrained control problems in modern control theory. It has some advantages in handling hard/soft constraint and having guaranteed optimality, feasibility and stability. This work develops MPC strategy for a special class of stochastic switched system - Markov switched system (or Markov jump system (MJS)) by employing optimal control theory, set invariance principle, semi-definite programming (SDP), linear programming (LP), and receding horizon concept. The main contributions are described as follows:1. Based on the set invariance principle, the multivariable constrained control problem for MJS is studied via MPC approach. First, the contractive terminal ellipsoid sequence is introduced under the dual-mode framework to improve the normal one-step and multiple-step predictive control strategy. Outside the terminal set, a receding horizon control is employed. Once the state enters the terminal set, the control is switched to a linear state feedback control, which makes the state stay in the set forever and stabilizes the system in the mean square sense. Then, in view of enlarging the initial feasible region, an on-line algorithm of constructing maximal admissible set for MJS is developed by applying standard LP. The control gain obtained on-line is devoted to construct a series of shrinking polyhedral sets.2. The receding horizon H_?control and H_?tracking control are proposed for MJS in the case that the statistical characteristics of the measurement noise are unknown. Comparing with the linear matrix inequality (LMI) based receding horizon H_?control, the controller designed in this work is given in an iterative form, which has more design complexity but possesses lower on-line compute effort. The control target is to regulate the underlying system mean square stable or track a given reference signal when a quadratic performance index is optimized and H_?attenuation level is met.3. Aiming at MJS with partly unknown transition probabilities and persistent bounded disturbance, the state feedback and observer-based predictive control strategy are presented respectively. The polyhedral constraint is handled by disturbance invariant set. The control strategy is divided into off-line design and on-line synthesis. With the guarantee of the mean square stabilizability and a given H_?attenuation level, some part of the controller is designed off-line to reduce the on-line compute effort. This policy resolves into a H_?state feedback controller, which is designed off-line plus on-line optimized variable and the system state is regulated into a minimal constraint set over the infinite horizon.4. The finite impulse response filter (FIR) and finite memory out-put feedback controller for MJS are designed by employing receding horizon structure. The input and output knowledge over a finite horizon is fully used and processed linearly to obtain the filter and controller. The filtering and control methods based on the finite horizon concept can be applied when the knowledge of initial state and initial mode is unknown and they have a better robust property against the temporary modeling uncertainties.5. The constrained predictive control problem for nonlinear MJS, which is represented by Takagi-Sugeno (T-S) fuzzy model, is studied based on the stochastic fuzzy Lyapunov function (FLF). First, T-S fuzzy model is employed to describe the nonlinearity, and the optimized control sequence and the parallel distributed compensation (PDC) scheme are integrated to optimize the performance index when there exists additive perturbations on the state. Moreover, a decaying aggregation matrix is introduced to reduce the on-line compute effort with guaranteed optimized performance in some extent. Second, by virtue of the feedback predictive framework, a mode independent robust controller design approach is proposed to balance the contradiction between control performances and on-line compute burden.6. Specifically, the MJS under consideration involves control and state multiplicative noise and partly unknown transition probabilities. The receding horizon formulation adopts an on-line optimization paradigm that utilizes open-loop optimized control move plus linear feedback control and is solved as a SDP problem. The mean square stability, control performance and constraint satisfaction properties are guaranteed, where the terminal- weighting matrix is determined off-line and the control move is calculated on-line. Because the optimized control sequences are obtained over different finite horizon and there are no direct connections between them, an intermediate solution and its cost function are introduced to discuss the feasibility and stability of the proposed MPC strategy.