Robust Predictive Control For Constrained Discrete-Time Piecewise Affine Systems

One of the most important and challenging problem in control is the derivation of systematic tools for the computation of controllers for general constrained non-linear or hybrid system that can guarantee closed-loop stability, feasibility, and optimality. The most successful modern control strategy both in theory and in practice for this class of systems is undoubtedly Model Predictive Control (MPC), also interchangeably called Receding Horizon Control (RHC). On the other hand, the model, which is used to describe the dynamics of controlled system, always has some uncertainty. In order to guarantee the robust stability when uncertainties are present, they must be taken into account in the computation of the control law and hence, the robust perdictive control of this class of systems has garnered increasing interest in the research community.Piecewise Affine (PWA) systems are obtained by partitioning the extended state-input space into polyhedral regions and associating with each region a different affine state update equation. PWA systems represent a powerful tool for approximating non-linear systems and are (under very mild assumptions) equivalent to many other hybrid systems, such as Mixed Logical Dynamical systems, Linear Complementary systems, Hybrid Automation and so on.In this thesis, the focus lies on robust predictive contorl for a class of constrained discrete-time PWA systems with bounded disturbances. Based on the existing theoretical results on model predictive control, the thesis is devoted to the study on the on- and off-line robust predictive control with robust feasibility and stability guaranteed. To achieve this, the relevant theory and approaches, such as robust invariant set, robust contractive sequence of sets , multi-parametric programming, geometry operations on polytopes, and linear matrix inequalities (LMI), are employed in the study. Specifically, the main contributions of this thesis are as follows:1. A robust MPC based on open-loop formulation is studied and a robust dual-model control method is presented. The method is based on so-called uncertain evolution sets, which are the sets containing the predicted evolution of the uncertain system under any admissible uncertainty. By considering these sets as the sate constrain of optimization problem of MPC and choosing as terminal constrain a robust positively invariant set, the robust stability is guaranteed by the feasibility of optimization problem . This property allow us to greatly reduce the on-line computational burden.2. It is demonstrated that how multi-parametric programming can be used to simultaneously obtain robust one step reachable set and the associated PWA feedback controller. Based on robust one step set, the maximal robust positively invariant set, maximal robust control invariant set and maximal robust stabilizable set are computed by iteration.3. A robust MPC focused on the reduction of the complexity of closed-loop optimization problem is studied and a MPC scheme with stability guaranteed is proposed. Based on the robust positively invariant set of the PWA system, the robust contractive sequence of sets are computed and is incorporated as a stabilizing constraint in the optimization problem. As a result, robust feasibility and stability is guaranteed in the case of suboptimal solutions. Finally, the simplification of stability conditions is made to reduce the computational complexity of associated optimization problem.4. A new method for enlarging the domain of attraction of robust MPC for constrained PWA systems with bounded disturbances is presented. Considering a contractive sequence of robust stabilizable sets, which is computed off-line based on robust positively invariant set, as the terminal constraint of predictive state in the optimization problem, robust stability and the enlargement of domain of attraction of robust MPC are guaranteed.5. The off-line computation of robust MPC controller for constrained PWA systems is studied. Using multi-parametric programming, dynamic programming and geometry operations on polytopes, the robust time-optimal and robust receding horizon control prolems are addressed and the resulting solutions are characterised.6. In order to reduce the off-line computation complexity of robust MPC, a low complexity control scheme, referred to as robust one-step control,is proposed. The maximal robust stabilizable set is chosen as the constraint set of the first predicted state in MPC formulation, such that the resulting feasible region cover the maximal robust stabilizable set and the robust feasibility is guaranteed for all time. In the sequent stability analysis, a general formulation for searching the common quadratic Lyapunov function with LMIs is presented.