On infinite horizon switched LQR problems with state and control constraints

• Published in: Systems & control letters Vol. 61; no. 4; pp. 464 - 471
• Main Authors: Balandat, Maximilian, Zhang, Wei, Abate, Alessandro
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.04.2012; Elsevier
• Subjects: Applied sciences; Computer science; control theory; systems; Constraints; Control systems; Control theory. Systems; Design parameters; Discrete-Time Switched and hybrid systems; Exact sciences and technology; Horizon; Infinite horizon constrained optimal control; Joints; LQR; Mathematical analysis; Mathematical models; Optimal control; Optimization; Suboptimality; System theory; Infinite horizon constrained optimal control; LQR; Suboptimality; Discrete-Time Switched and hybrid systems; Multimodel control; Suboptimal control; Hybrid control; Mixed integer programming; Constrained optimization; Hybrid system; Polyhedron; Control constraint; Finite horizon; LQ control; Continuous control; Optimal control; Discrete time; Cost function; Quadratic cost; Infinite horizon; State constraint; Quadratic function
• ISSN: 0167-6911; 1872-7956
• DOI: 10.1016/j.sysconle.2012.01.011

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.