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Design of Distributed Controller for Discrete-Time Systems Via the Integration of Extended LMI and Clique-Wise Decomposition

Sotaro Fushimi, Yuto Watanabe, Kazunori Sakurama

TL;DR

This work tackles distributed state-feedback synthesis for discrete-time plants with graph-induced sparsity on the controller gains $K\in\mathcal{S}$. It develops a less conservative convex relaxation by blending clique-wise decomposition with extended LMIs, avoiding the strict block-diagonal Lyapunov constraint. The main contribution is the convex set $\mathcal{K}_{\mathcal{S}, {\rm ext}}$, which satisfies $\mathcal{K}_{\mathcal{S}}\cup\mathcal{K}_{\rm ext}\subset\mathcal{K}_{\mathcal{S}, {\rm ext}}\subset\mathcal{K}_{\rm all}$ and is extending to $H_\infty$ control with a corresponding $\mathcal{K}_{\mathcal{S}, {\rm ext}}^{\infty,\gamma}$ that is less conservative than existing relaxations. Numerical experiments demonstrate improved performance over traditional diag/ext approaches, validating the approach's practical impact for large, sparse networks and scalable SDP formulations.

Abstract

This study addresses the centralized synthesis of distributed controllers using linear matrix inequalities (LMIs). Sparsity constraints on control gains of distributed controllers result in conservatism via the convexification of the existing methods such as the extended LMI method. In order to mitigate the conservatism, we introduce a novel LMI formulation for this problem, utilizing the clique-wise decomposition method from our previous work on continuous-time systems. By reformulating the sparsity constraint on the gain matrix within cliques, this method achieves a broader solution set. Also, the analytical superiority of our method is confirmed through numerical examples.

Design of Distributed Controller for Discrete-Time Systems Via the Integration of Extended LMI and Clique-Wise Decomposition

TL;DR

This work tackles distributed state-feedback synthesis for discrete-time plants with graph-induced sparsity on the controller gains . It develops a less conservative convex relaxation by blending clique-wise decomposition with extended LMIs, avoiding the strict block-diagonal Lyapunov constraint. The main contribution is the convex set , which satisfies and is extending to control with a corresponding that is less conservative than existing relaxations. Numerical experiments demonstrate improved performance over traditional diag/ext approaches, validating the approach's practical impact for large, sparse networks and scalable SDP formulations.

Abstract

This study addresses the centralized synthesis of distributed controllers using linear matrix inequalities (LMIs). Sparsity constraints on control gains of distributed controllers result in conservatism via the convexification of the existing methods such as the extended LMI method. In order to mitigate the conservatism, we introduce a novel LMI formulation for this problem, utilizing the clique-wise decomposition method from our previous work on continuous-time systems. By reformulating the sparsity constraint on the gain matrix within cliques, this method achieves a broader solution set. Also, the analytical superiority of our method is confirmed through numerical examples.
Paper Structure (11 sections, 13 theorems, 49 equations, 2 figures)

This paper contains 11 sections, 13 theorems, 49 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM STATEMENT
  3. PRELIMINARIES
  4. MAIN RESULT
  5. Application to $H_\infty$ control
  6. NUMERICAL EXAMPLES
  7. CONCLUSION
  8. Generalized Notation
  9. Proof of Lemma 5
  10. SDP variables for the proposed method
  11. Proof of Theorem 2

Key Result

Lemma 1

Let Then, $\mathcal{K}_{\rm diag} \subset\mathcal{K}_{\rm ext}\subset \mathcal{K}_{\rm all}$ holds.

Figures (2)

  • Figure 1: Solution sets
  • Figure 2: Results for $H_\infty$ control. The black, green, blue and red markers represent the value for $\mathcal{K}_{\rm diag}^{\infty,\gamma}$, $\mathcal{K}_{\rm ext}^{\infty,\gamma}$, $\mathcal{K}_\mathcal{S}^{\infty, \gamma}$ and the proposed method $\mathcal{K}_{\mathcal{S}, {\rm ext}}^{\infty, \gamma}$, respectively.

Theorems & Definitions (24)

  • Lemma 1: Distributed version of de1999new
  • proof
  • Lemma 2: watanabe2024convex
  • Lemma 3: watanabe2024convex
  • Lemma 4: Finsler's lemma de2007stability
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Theorem 1
  • ...and 14 more