Optimal control of constrained, piecewise affine systems with bounded disturbances

The solution to the problem of optimal control of piecewise affine systems with a bounded disturbance is characterised. Results that allow one to compute the value function, its domain (robustly controllable set) and the optimal control law are presented. The tools that are employed include dynamic programming, polytopic set algebra and parametric programming. When the cost is time (robust time-optimal control problem) or the stage cost is piecewise affine (robust optimal and robust receding horizon control problems), the value function and the optimal control law are both piecewise affine and each robustly controllable set is the union of a finite set of polytopes. Conditions on the cost and constraints are also proposed in order to ensure that the optimal control laws are robustly stabilising.