On infinite horizon switched LQR problems with state and control constraints

This paper studies the Discrete-Time Switched LQR problem over an infinite time horizon, subject to polyhedral constraints on state and control inputs. Specifically, we aim to find an infinite-horizon hybrid-control sequence, i.e., a sequence of continuous and discrete (switching) control inputs, that minimizes an infinite-horizon quadratic cost function, subject to polyhedral constraints on state and (continuous) 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 novel 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. Through the connection between (i) and (ii) it is shown that, by proper choice of the design parameters, the error of the overall suboptimal solution can be made arbitrarily small. The approach is tested on a numerical example.