Determination of the domain of attraction and regions of guaranteed cost for robust model predictive controllers based on linear matrix inequalities

The robust model-based predictive control (RMPC) formulation originally proposed in [1] ensures convergence of the state trajectory to the origin and satisfaction of operational constraints, provided that a given system of LMIs is feasible at the beginning of the control task. The largest domain of attraction of the origin under the resulting closed-loop control law can be defined as the set of all state values for which the LMIs are feasible. The present paper demonstrates that such a set is convex and symmetric about the origin, which allows the determination of extreme points through the solution of a modified version of the original RMPC optimization problem. An inner approximation of the largest domain of attraction can then be generated as the convex hull of these extreme points. The convexity and symmetry properties are also demonstrated for regions of guaranteed cost, defined as the set of initial states for which the resulting cost is upper-bounded by a given value. Inner approximations of such regions can also be obtained by solving a modified version of the RMPC optimization problem. For illustration, a numerical simulation model of an angular positioning system is employed, as in [1]. In this example, the proposed approximations were found to be in agreement with the feasibility and cost results obtained in a pointwise manner for a grid of initial conditions.