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A Lyapunov Characterization of Robust D-Stability with Application to Decentralized Integral Control of LTI Systems

John-Paolo Casasanta, John W. Simpson-Porco

Abstract

The concept of matrix D-stability plays an important role in applications, ranging from economic and biological system models to decentralized control. Here we provide necessary and sufficient Lyapunov-type conditions for the robust (block) D-stability property. We leverage this characterization as part of a novel Lyapunov analysis of decentralized integral control for MIMO LTI systems, providing sufficient conditions guaranteeing stability under low-gain and under arbitrary connection and disconnection of individual control loops.

A Lyapunov Characterization of Robust D-Stability with Application to Decentralized Integral Control of LTI Systems

Abstract

The concept of matrix D-stability plays an important role in applications, ranging from economic and biological system models to decentralized control. Here we provide necessary and sufficient Lyapunov-type conditions for the robust (block) D-stability property. We leverage this characterization as part of a novel Lyapunov analysis of decentralized integral control for MIMO LTI systems, providing sufficient conditions guaranteeing stability under low-gain and under arbitrary connection and disconnection of individual control loops.
Paper Structure (9 sections, 6 theorems, 45 equations, 1 figure)

This paper contains 9 sections, 6 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of Block $D$-Stability Concepts
  3. Main Result: A Lyapunov Characterization of Robust Block $D$-Stability
  4. Application to Decentralized Integral Control of LTI Systems
  5. Problem Setup
  6. Main Stability Result
  7. Numerical Example
  8. Conclusion
  9. Supporting Proofs

Key Result

Proposition 1

A block partitioned square matrix $A$ is block $D$-stable if and only if for any $Q \succ 0$ and any $D \in \mathcal{D}_\mathrm{blk}$, there exists a unique $P_D \succ 0$ such that

Figures (1)

  • Figure 1: State reference tracking; vertical line indicates closure of the second control loop.

Theorems & Definitions (11)

  • Definition 1
  • Proposition 1
  • Definition 2
  • Proposition 2
  • proof
  • Proposition 3
  • Theorem 1
  • proof
  • Lemma 1
  • Theorem 2
  • ...and 1 more