A semidefinite relaxation for the quadratic minimax problem with application to H<inf>∞</inf> model predictive control

Abstract

We derive a semidefinite relaxation for a minimax problem with two players: a quadratically bounded disturbance signal and a quadratically constrained control signal, with quadratic constraints on the state and a quadratic cost function. The constraints on the state result in mixed constraints on the disturbance and control signals. We use the S-Procedure to relax the constraints on the disturbance and tighten those on the control to obtain an upper bound on the optimum value of the minimax problem. By further relaxing the constraints on disturbance, and under the assumption that the cost function and the constraint sets are convex in the control signal, we derive a second upper bound, computable using linear matrix inequality techniques. The novelty is in our procedure for separating the mixed constraints and the facts that we handle quadratic constraints and that we make no convexity assumptions concerning the disturbance. We illustrate the effectiveness of the proposed scheme through an Hscrinfin model predictive control simulation, where a finite-horizon minimax problem is solved at each time step.