On the infinite horizon constrained switched LQR problem

This paper studies the Discrete-Time Switched LQR problem over an infinite time horizon subject to polyhedral constraints on state and control input. The overall constrained, infinite-horizon problem is split into two subproblems: (i) an unconstrained, infinite-horizon problem and (ii) a constrained, finite-horizon one. We derive a stationary suboptimal policy for problem (i) with analytical bounds on its optimality, and develop a formulation of problem (ii) as a Mixed-Integer Quadratic Program. By introducing the concept of a safe set, the solutions of the two subproblems are combined to achieve the overall control objective. It is shown that, by proper choice of the design parameters, the error of the overall sub-optimal solution can be made arbitrarily small. The approach is tested through a numerical example.