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Feedback Stability Under Mixed Gain and Phase Uncertainty

Jiajin Liang, Di Zhao, Li Qiu

TL;DR

The paper tackles robust feedback stability for MIMO LTI systems subject to mixed gain and phase uncertainty, formalized as sectored-disk uncertainty. It leverages the Davis–Wielandt (DW) shell to translate the matrix sectored-disk problem into geometric DW-shell separation in $\mathbb{R}^3$, yielding new, less-conservative sufficient and necessary conditions. These matrix results are extended to frequency-domain and state-space frameworks, producing frequency-wise LMIs and KYP-based LMIs that verify robust stability against sectored-disk uncertainty, with explicit treatment of half-disk and asymmetric phase cases. The approach demonstrates improved conservatism over existing methods and provides practical computational tools (LMIs) for robust stability verification in MIMO LTI systems, with examples illustrating the advantages over small-gain/phase-based criteria.

Abstract

In this study, we investigate the robust feedback stability problem for multiple-input-multiple-output linear time-invariant systems involving sectored-disk uncertainty, namely, dynamic uncertainty subject to simultaneous gain and phase constraints. This problem is thereby called a sectored-disk problem. Employing a frequency-wise analysis approach, we derive a fundamental static matrix problem that serves as a key component in addressing the feedback stability. The study of this matrix problem heavily relies on the Davis-Wielandt (DW) shells of matrices, providing a profound insight into matrices subjected to simultaneous gain and phase constraints. This understanding is pivotal for establishing a less conservative sufficient condition for the matrix sectored-disk problem, from which we formulate several robust feedback stability conditions against sectored-disk uncertainty. Finally, several conditions based on linear matrix inequalities are developed for efficient computation and verification of feedback robust stability against sectored-disk uncertainty.

Feedback Stability Under Mixed Gain and Phase Uncertainty

TL;DR

The paper tackles robust feedback stability for MIMO LTI systems subject to mixed gain and phase uncertainty, formalized as sectored-disk uncertainty. It leverages the Davis–Wielandt (DW) shell to translate the matrix sectored-disk problem into geometric DW-shell separation in , yielding new, less-conservative sufficient and necessary conditions. These matrix results are extended to frequency-domain and state-space frameworks, producing frequency-wise LMIs and KYP-based LMIs that verify robust stability against sectored-disk uncertainty, with explicit treatment of half-disk and asymmetric phase cases. The approach demonstrates improved conservatism over existing methods and provides practical computational tools (LMIs) for robust stability verification in MIMO LTI systems, with examples illustrating the advantages over small-gain/phase-based criteria.

Abstract

In this study, we investigate the robust feedback stability problem for multiple-input-multiple-output linear time-invariant systems involving sectored-disk uncertainty, namely, dynamic uncertainty subject to simultaneous gain and phase constraints. This problem is thereby called a sectored-disk problem. Employing a frequency-wise analysis approach, we derive a fundamental static matrix problem that serves as a key component in addressing the feedback stability. The study of this matrix problem heavily relies on the Davis-Wielandt (DW) shells of matrices, providing a profound insight into matrices subjected to simultaneous gain and phase constraints. This understanding is pivotal for establishing a less conservative sufficient condition for the matrix sectored-disk problem, from which we formulate several robust feedback stability conditions against sectored-disk uncertainty. Finally, several conditions based on linear matrix inequalities are developed for efficient computation and verification of feedback robust stability against sectored-disk uncertainty.
Paper Structure (17 sections, 22 theorems, 120 equations, 16 figures)

This paper contains 17 sections, 22 theorems, 120 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Numerical range and Davis-Wielandt shell
  4. Matrix gain and phase definition
  5. Gain and phase of MIMO LTI system
  6. Matrix sectored-disk Problem
  7. Connection with matrix DW shell
  8. DW shell union of sectored-disk matrices
  9. Necessary condition and sufficient condition for matrix sectored-disk problem
  10. Solution to half-disk problem
  11. Comparison with some existing results
  12. Sectored-disk uncertainties
  13. Robust stability with sectored-disk uncertainty
  14. State-space robust stability conditions
  15. Conclusion
  16. ...and 2 more sections

Key Result

Lemma 1

The DW shell of matrix $A\in\mathbb{C}^{n \times n}$ has the following properties.

Figures (16)

  • Figure 1: Illustration of quadratic constraints on the uncertainty. The constraints are given in polar coordinates, i.e., a point on the plane is given by $c=re^{j\theta}$.
  • Figure 2: Uncertain feedback system
  • Figure 3: Bode diagram of $\Delta(s)$, and the gain and phase range at some fixed frequency.
  • Figure 4: Controlled feedback system
  • Figure 5: Negative feedback System
  • ...and 11 more figures

Theorems & Definitions (41)

  • Lemma 1: Section 2, Li2008DAVISWIELANDTSO
  • Lemma 2: Theorem 2.1,li2008eigenvalues
  • Lemma 3: Matrix small gain theorem DoyleStein
  • Lemma 4: Matrix small phase theorem, Lemma 4wang2020phases
  • Lemma 5: System small gain theorem,DoyleStein
  • Lemma 6: System small phase theorem,chen2021phase
  • Lemma 7
  • Corollary 1
  • proof
  • Proposition 1
  • ...and 31 more