Sparse optimal control of cyber-physical systems via PQA approach

<p style="text-align:center;"> <span><strong>(Communicated by Lei Wang)</strong></span> <p style="text-align:justify;"> <span> In this study, we examine a linear time-invariant system with the influence of a static state feedback control mechanism, which takes the form of $Kx(t-\tau(||K||_0))$, where $K$ represents the gain matrix and $||K||_0$ denotes the number of nonzero entries of the matrix $K$. The parameter $\tau(||K||_0)$ corresponds to a varying delay, which arises due to the time required for the transmission of the system state information and the subsequent computation of the control input. Governed by the linear time-invariant system, we minimize prescribed conventional cost functions as $J^{0}(K)$ to obtain the optimal feedback matrices $K_1^{*}$. Numerous computational methods are available for the achievement of this objective. Nevertheless, it is noteworthy that the resulting $K_1^{*}$ matrix typically exhibit a dense structure. The primary objective of this paper is to minimize the $l_0$ norm of the feedback matrix $K$ under conditions that satisfy the constraint $|J^{0}(K)-J^{0}(K_1^{*})|\leq \varepsilon$. The $l_0$ norm acts as a quantifiable measure for assessing the degree of sparsity within the feedback matrix. The sparsity of the gain matrix is approximated through the application of a piecewise quadratic approximation (PQA) of the $l_0$-norm of the feedback matrix. Subsequently, we proceed to formulate an iterative algorithm designed to address the transformed problem, accompanied by a thorough analysis of its convergence properties. Finally, we undertake a numerical experiment employing the proposed algorithm, aiming to illustrate its practical utility and efficacy in solving the problem at hand.<br /> </span>