A Study of the Discrete-Time Switched LQR Problem

This paper studies the discrete-time switched LQR (DSLQR) problem based on a dynamic programming approach. One contribution of this paper is the analytical characterization of both the value function and the optimal hybridcontrol strategy of the DSLQR problem. Their connections to the Riccati equation and the Kalman gain of the classical LQR problem are also discussed. Several interesting properties of the value functions are derived. In particular, we show that under some mild conditions, the family of finite-horizon value functions of the DSLQR problem is homogeneous (of degree 2), uniformly bounded over the unit ball, and converges exponentially fast to the infinitehorizon value function. Based on these properties, efficient algorithms are proposed to solve the finite-horizon and infinite-horizon DSLQR problems. More importantly, we establish conditions under which the strategies generated by the algorithms are stabilizing and suboptimal. These conditions are derived explicitly in terms of subsystem matrices and are thus very easy to verify. The proposed algorithms and the analysis provide a systematic way of solving the DSLQR problem with guaranteed closed-loop stability and suboptimal performance. Simulation results indicate that the proposed algorithms can efficiently solve not only specific but also randomly generated DSLQR problems, making the NP-hard problems numerically tractable.