In this thesis, it is addressed the problem of predictive control of linear systems subject to non-convex polyhedral constraints, defined as the non-convex union of several convex polyhedra. The proposed controllers are useful, first, in problems for which such constraints arise naturally and, second, as an alternative to non-linear predictive control when low sampling times don't allow the use of non-linear programming.
In the first chapters of the work, it is shown the existence of an explicit solution to the optimization problems that arise when this kind of predictive controllers is proposed. Such explicit solution is shown to be piecewise affine defined over regions described by linear and quadratic inequalities. Two different methodologies are proposed in order to obtain this explicit solution: an intersection, division and union methodology and a convex hull methodology.
The former of these methodologies is based on formulating several subproblems with convex polyhedral constraints which union forms the original constraints and obtaining the explicit solution to the original problem from the solutions to these subproblems.
The latter is based on calculating the convex hull of the non-convex constraints and obtaining the explicit solution to the convex problem defined with these new constraints. Next, tt is shown tnat a subset of the regions of the explicit solution to the original problem are equal to some of the regions of the solution to this new problem, and a procedure to identify them and obtain the rest of regions is proposed, finding this way the complete explicit solution searched.
Efficient algorithms for the online implementation of explicit control laws in the previously described form are also studied. Particularly, an algorithm based on a binary tree of the linear partition and a comparison of the cost functions when required, is proposed. The computational burden of such an algorithm is compared to that obtained by the implementation of a binary tree for each of the convex subproblems, showing that the former is always lower than the latter.
Considering the case for which a low sampling time prevents the use of the proposed algorithms because of their online computational cost, two suboptimal solutions are proposed. The first one is based on a simplification of the optimization problem, while the second one is based on the simplification of the explicit solution.
Furthermore, stability of the control systems proposed is analyzed, specifying the well-known a priori stability conditions for general receding horizon schemes to the case of systems with non-convex polyhedral constraints. To do so, a novel efficient algorithm for the computation of the closed-loop maximal admissible set for this kind of systems is proposed. This algorithm starts from the maximal admissible set of the problem with constraints defined as the convex hull of the original ones, and then eliminates specific subsets of it. Furthermore, continuity of the cost function for the proposed problem is shown which allows to prove that, if the stability conditions hold, the system is not only Lyapunov stable but also asymptotically stable.
Finally, some problems for which the techniques proposed in the thesis are useful are introduced. First, the obstacle avoidance and path planning problem is introduced, showing that, for a general case, non-convex polyhedral constraints arise. Second, the application of the developed techniques to the problem of predictive control with non-linear constraints is proposed. Particularly, predictive control of Hammerstein-Wiener systems with static non-linearities cancelation and input-output feedback linearization schemes for predictive control of non-linear systems are solved with our approach.