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

Zhaohua Yang, Pengyu Wang, Haishan Zhang, Shiyue Jia, Nachuan Yang, Yuxing Zhong, Ling Shi

TL;DR

This paper provides a comprehensive analysis of the design of optimal structured and sparse controllers for continuous-time linear time-invariant (LTI) systems and designs an iterative linear matrix inequality (ILMI) algorithm for each problem, which ensures guaranteed convergence.

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.

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

TL;DR

This paper provides a comprehensive analysis of the design of optimal structured and sparse controllers for continuous-time linear time-invariant (LTI) systems and designs an iterative linear matrix inequality (ILMI) algorithm for each problem, which ensures guaranteed convergence.

Abstract

This paper provides a comprehensive analysis of the design of optimal structured and sparse controllers for continuous-time linear time-invariant (LTI) systems. Three problems are considered. First, designing the sparsest controller, which minimizes the sparsity of the controller while satisfying the given performance requirements. Second, designing a sparsity-promoting controller, which balances system performance and controller sparsity. Third, designing a 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.

Paper Structure

This paper contains 19 sections, 7 theorems, 50 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $H_{\infty}$ Performance
  4. Sparsity in NCS
  5. Sparse Controller and $l_1$ Relaxation
  6. Structured Controller
  7. A Convex Approach for Structured Control
  8. Problem Formulation
  9. Main Results
  10. Sparse Controller Design with Bounded $H_\infty$ Performance
  11. Sparsity-promoting $H_\infty$ Controller Design
  12. Structured $H_\infty$ Controller Design
  13. Simulations
  14. Sparse Controller Design with Bounded $H_\infty$ Performance
  15. Sparsity-promoting $H_\infty$ Controller Design
  16. ...and 4 more sections

Key Result

Lemma 1

The system reformulated system is asymptotically stable (i.e., for all eigenvalues $\lambda$ of $A+BK$, $\mathrm{Re}(\lambda)<0$) and $||T_{zd}(s)||_\infty \le \gamma$ if and only if there exists a symmetric positive definite matrix $P\succ0$ such that

Figures (3)

  • Figure 1: The sparsity patterns of $K_0$,$K_1$,$K_2$,$K^*_{A1}$ (from left to right, top to bottom). The nonzero elements are labeled using blue dots.
  • Figure 2: The evolution of $||K||_0$ and $||K||_1$.
  • Figure 3: The relationship between $||K||_0$, $H_\infty$ norm and $\lambda$.

Theorems & Definitions (11)

  • Lemma 1: iwasaki1994all
  • Theorem 1: pipeleers2009extended
  • Theorem 2: Theorem 2 of ferrante2020lmi
  • Remark 1
  • Theorem 3
  • Definition 1
  • Remark 2
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • ...and 1 more