Linear Quadratic Regulator of Discrete-Time Switched Linear Systems

This brief studies the linear quadratic regulation problem of discrete-time switched linear systems via dynamic programming. Discrete decision variables (referred as switch inputs) are introduced to represent the subsystem selection. This formulation leads to a mixed integer quadratic programming problem, which is known to be computationally hard in general. Instead of deriving suboptimal policies with analytical bounds on its optimality, the optimal switching sequence is computed in a computationally efficient manner when the input matrix is restricted as a vector. The unique contribution of this brief is an analytical expression of both the optimal switching condition (determining the subsystem selection) and the optimal control law. Proof of optimality is presented by fractional optimization and the practical merits of our approach is validated through simulations on a second-order system in comparison with recent pruning methods.