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A networked small-gain theorem based on discrete-time diagonal stability

Ron Ofir, Michael Margaliot

Abstract

We present a new sufficient condition for finite-gain $L_2$ input-to-output stability of a networked system. The condition requires a matrix, that combines information on the $L_2$ gains of the sub-systems and their interconnections, to be discrete-time diagonally stable (DTDS). We show that the new result generalizes the standard small gain theorem for the negative feedback connection of two sub-systems. An important advantage of the new result is that known sufficient conditions for DTDS can be applied to derive sufficient conditions for networked input-to-output stability. We demonstrate this using several examples. We also derive a new necessary and sufficient condition for a matrix that is a rank one perturbation of a Schur diagonal matrix to be DTDS.

A networked small-gain theorem based on discrete-time diagonal stability

Abstract

We present a new sufficient condition for finite-gain input-to-output stability of a networked system. The condition requires a matrix, that combines information on the gains of the sub-systems and their interconnections, to be discrete-time diagonally stable (DTDS). We show that the new result generalizes the standard small gain theorem for the negative feedback connection of two sub-systems. An important advantage of the new result is that known sufficient conditions for DTDS can be applied to derive sufficient conditions for networked input-to-output stability. We demonstrate this using several examples. We also derive a new necessary and sufficient condition for a matrix that is a rank one perturbation of a Schur diagonal matrix to be DTDS.

Paper Structure

This paper contains 8 sections, 8 theorems, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Schur stability
  4. Discrete-time diagonal stability
  5. A condition for input-to-output stability of a networked system
  6. The case of rank one connections
  7. Applications
  8. Discussion

Key Result

Proposition 1

Let $A \in \mathbb R^{2 \times 2}$. Then $A$ is DTDS iff the following three inequalities hold:

Figures (2)

  • Figure 1: Block diagram of a feedback interconnection of two sub-systems (top), and a simplified diagram (bottom).
  • Figure 2: Comparing two sufficient conditions for netwroked stability derived in Example \ref{['exa:gene22']}.

Theorems & Definitions (16)

  • Remark 1
  • Remark 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Example 1
  • Remark 3
  • Theorem 1
  • Remark 4
  • ...and 6 more