Sparse Representations of Dynamical Networks: A Coprime Factorization Approach

TL;DR

This paper develops System Response-Type Realizations (SRTR) to provide sparse, structure-preserving representations for distributed control of LTI networks. It builds exact connections between SRTR, NRF, and classical left coprime factorizations, and gives numerically tractable methods to move among these representations while preserving stability and sparsity. The authors prove that SRTR pairs are coprime-factor components and show how to switch between SRTR and left coprime factorizations for Youla-like distributed controller synthesis, including a continuous-time extension. A numerical example demonstrates how SRTR-based controllers can be designed to implement ring-topology distributed control with provable stability in a continuous-time network setting.

Abstract

We study a class of dynamical networks modeled by linear and time-invariant systems which are described by state-space realizations. For these networks, we investigate the relations between various types of factorizations which preserve the structure of their component subsystems' interconnection. In doing so, we provide tractable means of shifting between different types of sparsity-preserving representations and we show how to employ these factorizations to obtain distributed implementations for stabilizing and possibly stable controllers. By formulating all these results for both discrete- and continuous-time systems, we develop specialized distributed implementations that, up to this point, were only available for networks modeled as discrete-time systems.

Figures (5)

• Figure 1: Three-hop network architecture
• Figure 1: Feedback loop of the plant $\bf G$ with the controller $\bf K$
• Figure 1: Equivalent feedback interconnection
• Figure 2: Diagram showing the means of switching between various LTI network representations
• Figure 2: Feedback loop of the plant ${\bf G}$ with the controller ${\bf K}$ in an SRTR-based implementation $\lambda\mathbf{U}(\lambda)={\bf W}(\lambda) (\mathbf{U}(\lambda)+\mathbf{\Delta_U}(\lambda))+ {\bf V}(\lambda)\mathbf{Z}(\lambda)$

Theorems & Definitions (64)

• Definition 2.1
• Remark 3.2
• Remark 3.4
• Remark 3.5
• Definition 3.6
• Remark 3.7
• Remark 3.8
• Remark 3.9
• Definition 3.10
• Remark 3.11