Feedback Control Balancing Quadratic Performance and Input Sparsity

This letter focuses on the simultaneous minimization of the quadratic cost and the number of actuations in the control of discrete-time linear systems. The quadratic cost measures conventional transient performance, while penalizing the actuation frequency promotes sparse control and improves energy efficiency. The main challenge in this optimization arises from the intricate interplay between the two optimization variables; control inputs and actuation timings. To avoid this complexity, we introduce a computationally feasible approximation and divide the problem into two stages. The proposed control scheme consists of a feedback controller, whose gains are determined offline, and an online actuation scheduler that decides whether to apply the input at each time step. We also introduce two extensions to further refine the optimization process. Through numerical examples, we evaluate the effectiveness of our method and its extensions. We show that these methods allow to balance the transient performance and the sparsity of the control input.