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Special Stable Matrices and Their Non-square Counterpart

Steven W. Su

TL;DR

This work addresses stability analysis of non-square systems by extending D-stable and Volterra-Lyapunov stable concepts to real non-square matrices, motivated by decentralized unconditional stability and controllability. The authors introduce a framework that maps a non-square matrix $A$ to square counterparts via a block-diagonal $K$ and a set of $m$-th order squared matrices $[A]^m_{s_i}$, enabling D-stability–like LMIs and Volterra-Lyapunov conditions. They show that simultaneous Volterra-Lyapunov stability implies D-stable-like behavior for non-square matrices and discuss relaxations to individual stability, with connections to common diagonal Lyapunov functions. The work provides definitions, structural constructions, and conjectures to guide future research in decentralized stability and controllability for non-square processes.

Abstract

In this note, we discuss the extension of several important stable square matrices, e.g., D-stable matrices, diagonal dominance matrices, Volterra-Lyapunov stable matrices, to their corresponding non-square matrices. The extension is motivated by some distributed control-related problems, such as decentralized unconditional stability and decentralized integral controllability for non-square processes. We will provide the connections of conditions between these special square matrices and their associated non-square counterparts. Some conjectures for these special matrices are proposed for future research.

Special Stable Matrices and Their Non-square Counterpart

TL;DR

This work addresses stability analysis of non-square systems by extending D-stable and Volterra-Lyapunov stable concepts to real non-square matrices, motivated by decentralized unconditional stability and controllability. The authors introduce a framework that maps a non-square matrix to square counterparts via a block-diagonal and a set of -th order squared matrices , enabling D-stability–like LMIs and Volterra-Lyapunov conditions. They show that simultaneous Volterra-Lyapunov stability implies D-stable-like behavior for non-square matrices and discuss relaxations to individual stability, with connections to common diagonal Lyapunov functions. The work provides definitions, structural constructions, and conjectures to guide future research in decentralized stability and controllability for non-square processes.

Abstract

In this note, we discuss the extension of several important stable square matrices, e.g., D-stable matrices, diagonal dominance matrices, Volterra-Lyapunov stable matrices, to their corresponding non-square matrices. The extension is motivated by some distributed control-related problems, such as decentralized unconditional stability and decentralized integral controllability for non-square processes. We will provide the connections of conditions between these special square matrices and their associated non-square counterparts. Some conjectures for these special matrices are proposed for future research.
Paper Structure (3 sections, 7 theorems, 25 equations)

This paper contains 3 sections, 7 theorems, 25 equations.

Table of Contents

  1. INTRODUCTION
  2. DECENTRALIZED STABILIZATION FOR NON-SQUARE SYSTEMS AND MATRICES
  3. CONCLUSION

Key Result

Theorem 1

For a proper and asymptotically stable non-square system $H(s) \in \mathbb{C}^{m \times n}$, we define its steady state transfer matrix as $A=H(0) \in \mathbb{R}^{n \times m}$. If there exists a block diagonal non-square matrix $K \in \mathbb{C}^{n \times m}$ such that for all non-negative diagonal where $\mathcal{M}$ is the index set consisting of $k$ tuples of integers in the range $1,\cdots, m

Theorems & Definitions (19)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • Definition 5
  • Definition 6
  • Definition 7
  • Lemma 2
  • ...and 9 more