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

TL;DR

This paper develops a novel iterative LMI-based method for designing optimal sparse and structured $H_ abla$ controllers for LTI systems, balancing performance and sparsity with guaranteed convergence.

Contribution

It introduces a convex SDP approach and ILMI algorithm for optimal sparse and structured $H_ abla$ control, improving upon existing methods with proven convergence and optimality conditions.

Findings

Algorithms effectively reduce controller sparsity and communication load.

Proposed methods outperform existing solutions in numerical simulations.

Refined solutions demonstrate significant performance improvements.

Abstract

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.