Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Cyclic Small Phase Theorem

Chao Chen, Wei Chen, Di Zhao, Jianqi Chen, Li Qiu

Abstract

This paper introduces a brand-new phase definition called the segmental phase for multi-input multi-output linear time-invariant systems. The underpinning of the definition lies in the matrix segmental phase which, as its name implies, is graphically based on the smallest circular segment covering the matrix normalized numerical range in the unit disk. The matrix segmental phase has the crucial product eigen-phase bound, which makes itself stand out from several existing phase notions in the literature. The proposed bound paves the way for stability analysis of a single-loop cyclic feedback system consisting of multiple subsystems. A cyclic small phase theorem is then established as our main result, which requires the loop system phase to lie between $-π$ and $π$. The proposed theorem complements a cyclic version of the celebrated small gain theorem. In addition, a generalization of the proposed theorem is made via the use of angular scaling techniques for reducing conservatism.

A Cyclic Small Phase Theorem

Abstract

This paper introduces a brand-new phase definition called the segmental phase for multi-input multi-output linear time-invariant systems. The underpinning of the definition lies in the matrix segmental phase which, as its name implies, is graphically based on the smallest circular segment covering the matrix normalized numerical range in the unit disk. The matrix segmental phase has the crucial product eigen-phase bound, which makes itself stand out from several existing phase notions in the literature. The proposed bound paves the way for stability analysis of a single-loop cyclic feedback system consisting of multiple subsystems. A cyclic small phase theorem is then established as our main result, which requires the loop system phase to lie between and . The proposed theorem complements a cyclic version of the celebrated small gain theorem. In addition, a generalization of the proposed theorem is made via the use of angular scaling techniques for reducing conservatism.
Paper Structure (20 sections, 13 theorems, 68 equations, 8 figures)

This paper contains 20 sections, 13 theorems, 68 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. The Segmental Phase of a Matrix
  4. Definition of the Segmental Phase
  5. Comparison with Existing Phase Definitions
  6. The Segmental Phase of A MIMO LTI System
  7. The Indented Nyquist Contour and Preliminaries
  8. The Segmental Phase Response of Systems
  9. A Multi-Interval-Valued Case
  10. Main Results
  11. A Cyclic Small Phase Theorem
  12. A Mixed Small Gain/Phase Theorem
  13. Angular Scaling: The $\gamma$-Segmental Phase
  14. The Matrix $\gamma$-Segmental Phase
  15. Angular Scaling of Theorem \ref{['thm:segmental_phase_stability']}
  16. ...and 5 more sections

Key Result

Lemma 1

A cyclic feedback system in Fig. fig:feedback is stable if and only if it has no unstable pole-zero cancellation and

Figures (8)

  • Figure 1: A single-loop cyclic feedback system consisting of $m$ MIMO LTI subsystems $P_1, P_2, \ldots, P_m\in \mathcal{R}^{n\times n}$.
  • Figure 2: The region (left) from the small gain idea and the region (right) from the small phase idea for Nyquist plots.
  • Figure 3: A graphical illustration of the segmental phase of $A=\left[-3+j410{1}/(5+j)\right]$. The grey area is the normalized numerical range $\mathcal{N}(A)$ contained in the unit disk. The red-bordered smallest circular segment of the disk is adopted to cover $\mathcal{N}(A)$. The lower and upper endpoints of the arc of the segment are respectively given by $e^{j\underline{\psi}(A)}$ and $e^{j\overline{\psi}(A)}$, which defines the segmental phase interval $\Psi(A)=\interval{\underline{\psi}(A)}{\overline{\psi}(A)}=\interval[scaled]{\frac{-34.2\pi}{180}}{\frac{152.2\pi}{180}}$.
  • Figure 4: $\mathcal{N}(A)$ and $\Psi(A)$ for (a) scalar identity $A=100e^{j\frac{\pi}{4}}I$, (b) positive definite $A=\mathrm{diag}(1, 20, 100)$, (c) unitary $A=\mathrm{diag}(-j, 1, e^{j\frac{\pi}{3}}, e^{j\frac{3\pi}{5}})$ and (d) nilpotent Jordan $A=\left[0100\right]$.
  • Figure 5: An illustration of the sectorial phase of a sectorial matrix $A$. The blue-bordered smallest convex sector is adopted to bound $\mathcal{N}(A)$. The two endpoints of the sector given by $e^{j\underline{\phi}(A)}$ and $e^{j\overline{\phi}(A)}$ define the sectorial phase $\Phi(A)=\interval{\underline{\phi}(A)}{\overline{\phi}(A)}$.
  • ...and 3 more figures

Theorems & Definitions (33)

  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • Example 1
  • Definition 2
  • Proposition 1
  • proof
  • Example 2
  • Theorem 1
  • ...and 23 more