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In this paper, we explore the discrete time sparse feedback control for a linear invariant system, where the proposed optimal feedback controller enjoys input sparsity by using a dynamic linear compensator, that is, the components of feedback control signal having the smallest possible nonzero values. The resulting augmented dynamics ensures closed-loop stability, which infers sparse feedback controller from open-loop solution to closed-loop realization. In particular, we show that the implemented sparse feedback (closed-loop) control solution is equivalent to the original sparse (open-loop) control solution under a specified basis. We then extend the dynamic compensator to a feedforward tracking control problem. Finally, numerical examples demonstrate the effectiveness of proposed control approach.

Sparse feedback controller: from open-loop solution to closed-loop realization