Sparse $H_\infty$ Controller for Networked Control Systems: Non-Structured and Optimal Structured Design

This paper provides a comprehensive analysis of the design of optimal structured and sparse $H_\infty$ controllers for continuous-time linear time-invariant (LTI) systems. Three problems are considered. First, designing the sparsest $H_\infty$ controller, which minimizes the sparsity of the controller while satisfying the given performance requirements. Second, designing a sparsity-promoting $H_\infty$ controller, which balances system performance and controller sparsity. Third, designing a $H_\infty$ controller subject to a structural constraint, which enhances system performance with a specified sparsity pattern. For each problem, we adopt a linearization technique that transforms the original nonconvex problem into a convex semidefinite programming (SDP) problem. Subsequently, we design an iterative linear matrix inequality (ILMI) algorithm for each problem, which ensures guaranteed convergence. We further characterize the first-order optimality using the Karush-Kuhn-Tucker (KKT) conditions and prove that any limit point of the solution sequence generated by the ILMI algorithm is a stationary point. For the first and second problems, we validate that our algorithms can reduce the number of non-zero elements and thus the communication burden through several numerical simulations. For the third problem, we refine the solutions obtained in existing literature, demonstrating that our approaches achieve significant improvements.