Robust finite-horizon MPC using optimal worst-case closed-loop predictions

In this paper a new robust model based predictive control (MPC) algorithm for linear models with polytopic uncertainty description is presented that circumvents the need for doing explicit robust set-valued state predictions. It is known that these lead to trees of predicted states whose number of nodes grows exponentially with the number of free control moves. Instead the algorithm uses a finite horizon within which at each time step a different robustly stabilizing feedback controller is applied. This approach has the advantage of having more degrees of freedom and leading to less conservative control behaviour than by assuming a fixed controller. At the end of the horizon a terminal control law that further drives the system state to the origin is appended. In this way stability is guaranteed. By making a suitable parameterization of the different controllers that are applied, a convex LMI-based optimization problem is obtained that allows the optimal set of controllers to be calculated efficiently by minimizing the resulting upper bound of the worst-case infinite-horizon control cost. The algorithm is illustrated on a linear uncertain model of an inverted pendulum.