Polytopic Approximation of Explicit Model Predictive Controllers

A model predictive control law (MPC) is given by the solution to a parametric optimization problem that can be pre-computed offline, which provides an explicit map from state to input that can be rapidly evaluated online. However, the primary limitations of these optimal explicit solutions are that they are applicable to only a restricted set of systems and that the complexity can grow quickly with problem size. In this paper we compute approximate explicit control laws that trade-off complexity against approximation error for MPC controllers that give rise to convex parametric optimization problems. The algorithm is based on the classic double- description method and returns a polyhedral approx- imation to the optimal cost function. The proposed method has three main advantages from a control point of view: it is an incremental approach, meaning that an approximation of any specified complexity can be produced, it operates on implicitly-defined convex sets, meaning that the prohibitively complex optimal explicit solution is not required and finally it can be applied to any convex parametric optimization problem. A sub-optimal controller based on barycentric in- terpolation is then generated from this approximate polyhedral cost function that is feasible and stabiliz- ing. The resulting control law is continuous, although non-linear and defined over a non-simplical polytopic partition of the state space. The non-simplical nature of the partition generates significantly simpler approx- imate control laws, which is demonstrated on several examples

On the computation of invariant sets for constrained nonlinear systems: An interval arithmetic approach

This paper deals with the computation of control invariant sets for constrained nonlinear systems. The proposed approach is based on the computation of an inner approximation of the one step set, that is, the set of states that can be steered to a given target set by an admissible control action. Based on this procedure, control invariant sets can be computed by recursion. We present a method for the computation of the one-step set using interval arithmetic. The proposed specialized branch and bound algorithm provides an inner approximation with a given bound of the error; this makes it possible to achieve a trade off between accuracy of the computed set and computational burden. Furthermore an algorithm to approximate the one step set by an inner bounded polyhedron is also presented; this allows us to relax the complexity of the obtained set, and to make easier the recursion and storage of the sets.Ministerio de Ciencia y Tecnología DPI2004-07444-c04-01Ministerio de Ciencia y Tecnología DPI2003-04375-c03-01Ministerio de Ciencia y Tecnología DPI2003-07146-c02-0