Design of Sparse Control With Minimax Concave Penalty

In this letter, we propose a novel computational method for sparse control, also known as maximum hands-off control, using the minimax concave penalty. The sparse control problem is formulated as an
$L^{0}$
-optimal control problem, which is known to be hard to solve. To overcome this difficulty, we propose using the minimax concave penalty as a surrogate for the
$L^{0}$
norm. We demonstrate the equivalence between the original and proposed control problems without relying on the normality assumption, which is typically required when approximating the
$L^{0}$
norm with the
$L^{1}$
norm. Furthermore, we present an effective numerical algorithm for the proposed optimal control based on the Alternating Direction Method of Multipliers (ADMM). A design example is shown to illustrate the effectiveness of the proposed method.