Dissipativity-Based Distributed Stability Analysis for Networks with Heterogeneous Nonlinear Agents

TL;DR

Vidyasagar's Network Dissipativity Theorem is explored, and criteria are identified for decomposable networks facilitating chordal decomposition, to develop distributed stability analysis for networks of inhomogeneous, nonlinear agents.

Abstract

Stabilizing large networks of nonlinear agents is challenging; decomposition and distributed analysis of these networks are crucial for computational tractability and information security. Vidyasagar's Network Dissipativity Theorem enables both properties concurrently in distributed network analysis. This paper explored combining it with the alternating direction methods of multipliers to develop distributed stability analysis for networks of inhomogeneous, nonlinear agents. One algorithm enhances information security by requiring agents to share only a dissipativity characterization, not a dynamical model, for stability analysis. A second algorithm further restricts this information sharing to their clique, thereby enhancing security, and can also reduce the computational burden of stability analysis if the network allows chordal decomposition. The convergence of the proposed algorithms is demonstrated, and criteria are identified for decomposable networks facilitating chordal decomposition. The effectiveness of the proposed methods is demonstrated through numerical examples involving a swarm of linearized unmanned aerial vehicles and networks beyond linear time-invariant agents.

Figures (11)

• Figure 1: Example of graph $\mathcal{H}$ and $\overline{\mathcal{Q}}$; $\mathcal{H}\subset\overline{\mathcal{Q}}$.
• Figure 2: Chordal decomposition of $\overline{\mathbf{Q}}$, corresponding to $\overline{\mathcal{Q}}$ in \ref{['fig:Augmented Graph']}, so that $\overline{\mathbf{Q}}{+}\epsilon\mathbf{I}{\preceq} 0$ if and only if $\overline{\mathbf{Q}}_i{\preceq} 0$ for $i{=}1,2$.
• Figure 3: Mapping among all block diagonal components of variables in \ref{['tab:ADMM Variables']}: The gray rectangles represent the block component indices defining each variable. The QSR parameters are grouped by agent, with blocks for each dissipativity matrix, while the chordal parameters are grouped by clique with blocks for each and a matrix equation.
• Figure 4: Hierarchical network example: The upper left and right graphs illustrate the graph of the network and $\overline{\mathbf{Q}}(\mathbf{X})$. The lower left graph is the graph of overlapped elements in $\overline{\mathbf{Q}}(\mathbf{X})$ resulting from \ref{['thm:VNDT_Chordal_Decomposition']}. The graph is disconnected and 9 components that are denoted in different colors. The lower right figure shows nodes in $\mathcal{V}(\mathbf{R})$. As in $\overline{\mathcal{Q}}_o$, $\mathcal{V}(\mathbf{R})$ can be decomposed into 9 subsets.
• Figure 5: Graph network of a

Theorems & Definitions (37)

• Theorem 1: Chordal Block-Decomposition zheng2021chordal
• Theorem 2: boyd2011distributed
• Lemma 1: boyd2011distributed
• Definition 1: QSR-Dissipativity vidyasagar1981input
• Lemma 2: Dissipativity Lemma gupta1996robust
• Definition 2: $\mathcal{L}_2$-stability vidyasagar1981input
• Theorem 3: vidyasagar1981input
• Lemma 3
• proof
• Lemma 4