Convex Reformulation of LMI-Based Distributed Controller Design with a Class of Non-Block-Diagonal Lyapunov Functions

TL;DR

This work tackles distributed static state-feedback design for continuous-time LTI systems under sparsity constraints, where traditional block-diagonal Lyapunov relaxations are conservative. It introduces a non-block-diagonal Lyapunov class that shares the controller sparsity and develops a clique-based, LMI-compatible framework using a block-diagonal factorization and Finsler's lemma, yielding conditions that are exact on chordal graphs and extendable to $H_\infty$ control. The authors present three progressively stronger convex relaxations that strictly contain the block-diagonal approach and demonstrate their effectiveness via stabilization and $H_\infty$ tests on large-scale networks, often outperforming existing methods. A state-transformation interpretation clarifies the method as operating on an expanded system while preserving distributed structure, with practical implications for scalable controller synthesis in networked settings.

Abstract

This study addresses a distributed state feedback controller design problem for continuous-time linear time-invariant systems by means of linear matrix inequalities (LMIs). As structural constraints on a control gain result in non-convexity in general, the block-diagonal relaxation of Lyapunov functions has been prevalent despite its conservatism. In this work, we target a class of non-block-diagonal Lyapunov functions with the same sparsity pattern as distributed controllers. By leveraging a block-diagonal factorization of sparse matrices and Finsler's lemma, we first present a nonlinear matrix inequality for stabilizing distributed controllers with such Lyapunov functions, which boils down to a necessary and sufficient condition for such controllers if the sparsity pattern is chordal. As its relaxation, we derive novel LMIs, one of which strictly covers the conventional relaxation, and then provide analogous results for $H_\infty$ control. Lastly, numerical examples underscore the efficacy of our results.

Figures (4)

• Figure 1: An example of systems with $N=3$. This network is chordal with maximal cliques $\mathcal{C}_1=\{1,2\}$ and $\mathcal{C}_2=\{2,3\}$.
• Figure 2: The evolution of the state vector $x(t)=[x_1(t),\ldots,x_8(t)]^\top$ in the system DIS1 leibfritz2004compleib with our proposed methods and the centralized $H_\infty$ controller against $w(t)=0.1\sin(t)$.
• Figure 3: The inclusion relationships between the block-diagonal relaxation $\mathcal{K}_{\mathcal{S},\mathrm{diag}}$, proposed method 1 $\hat{\mathcal{K}}_{\mathcal{S},\mathrm{rlx}1}$ in Theorem \ref{['theorem:LMI_1']}, extended LMI ${\mathcal{K}}_{\mathcal{S},\mathrm{ext}}$ in \ref{['def:K_ext']}, combined method $\hat{\mathcal{K}}_{\mathcal{S},\mathrm{rlx}1}^{\mathrm{ext}}$ with the proposed method and extended LMI in Theorem \ref{['theorem:K_ext_rlx']}, and the other combinations $\hat{\mathcal{K}}_{\mathcal{S},\mathrm{rlx}1}^{\mathrm{FDR}}$ and $\hat{\mathcal{K}}_{\mathcal{S},\mathrm{rlx}1}^{\mathrm{FZPK}}$.
• Figure 4: The red circles, stars, orange circles, squares, and diamonds represent the value of $|\gamma_*-\gamma_{*,\mathrm{cen}}|$ under the proposed method 1 in $\mathcal{K}_{\mathcal{S},\mathrm{rlx}1}^{\infty,\gamma}$, proposed method 2 in $\hat{\mathcal{K}}_{\mathcal{S},\mathrm{rlx}2}^{\infty,\gamma}$, proposed method 3 in $\hat{\mathcal{K}}_{\mathcal{S},\mathrm{rlx}3}^{\infty,\gamma}$, SI-based relaxation furieri2020sparsity, and block-diagonal relaxation in $\mathcal{K}_{\mathcal{S},\text{diag}}^{\infty,\gamma}$, respectively. Here, $\gamma_{*,\mathrm{cen}}$ represents the optimal value of the upper bound $\gamma$ of $\|(C+DK)(sI-(A+BK))^{-1}B_w + D_w\|_\infty$ under the centralized controller. The cases where a stabilizing controller was not generated are collected at the top as failures.

Theorems & Definitions (38)

• Proposition 1
• Example 1
• Remark 1
• Lemma 1: positive definite version of Agler's theorem
• Lemma 2
• proof
• Example 2
• Lemma 3: Finsler's lemma de2007stability
• Proposition 2
• proof