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Graphical Dominance Analysis for Linear Systems: A Frequency-Domain Approach

Chao Chen, Thomas Chaffey, Rodolphe Sepulchre

TL;DR

This work develops a frequency-domain, graphical framework for robust dominance analysis of MIMO LTI systems by introducing the matrix scaled graph (SG) and its frequency-wide extension SG(G(jω)). It proves a Graphical Dominance Theorem: if the scaled graphs of two open-loop systems P and C are separated, specifically if SG(τP(jω)) and SG†(C(jω)) do not intersect for all ω and τ∈(0,1), then the closed-loop P#C is the sum of their dominances, i.e., (p1+p2)-dominant, assuming no unstable pole-zero cancellations. The framework further extends to stable uncertainties, providing a region-based robustness test and specialized conditions that recover classical small-gain, small-phase, and passivity criteria, while offering advantages over traditional eigenloci-based approaches. By linking SG to the principal region yet highlighting its reduced conservatism, the paper provides a practical, robust tool for dominance analysis in uncertain, possibly unstable MIMO systems. The approach has potential implications for analyzing complex nonlinear networks and neuromorphic control, suggesting avenues for future extensions and applications.

Abstract

We propose a frequency-domain approach to dominance analysis for multi-input multi-output (MIMO) linear time-invariant systems. The dominance of a MIMO system is defined to be the number of its poles in the open right half-plane. Our approach is graphical: we define a frequency-wise notion of the recently-introduced scaled graph of a MIMO system plotted in a complex plane. The scaled graph provides a bound of the eigenloci of the system, which can be viewed as a robust MIMO extension of the classical Nyquist plot. Our main results characterize sufficient conditions for quantifying the dominance of a closed-loop system based upon separation of scaled graphs of two open-loop systems in a frequency-wise manner. The results reconcile existing small gain, small phase and passivity theorems for feedback dominance analysis.

Graphical Dominance Analysis for Linear Systems: A Frequency-Domain Approach

TL;DR

This work develops a frequency-domain, graphical framework for robust dominance analysis of MIMO LTI systems by introducing the matrix scaled graph (SG) and its frequency-wide extension SG(G(jω)). It proves a Graphical Dominance Theorem: if the scaled graphs of two open-loop systems P and C are separated, specifically if SG(τP(jω)) and SG†(C(jω)) do not intersect for all ω and τ∈(0,1), then the closed-loop P#C is the sum of their dominances, i.e., (p1+p2)-dominant, assuming no unstable pole-zero cancellations. The framework further extends to stable uncertainties, providing a region-based robustness test and specialized conditions that recover classical small-gain, small-phase, and passivity criteria, while offering advantages over traditional eigenloci-based approaches. By linking SG to the principal region yet highlighting its reduced conservatism, the paper provides a practical, robust tool for dominance analysis in uncertain, possibly unstable MIMO systems. The approach has potential implications for analyzing complex nonlinear networks and neuromorphic control, suggesting avenues for future extensions and applications.

Abstract

We propose a frequency-domain approach to dominance analysis for multi-input multi-output (MIMO) linear time-invariant systems. The dominance of a MIMO system is defined to be the number of its poles in the open right half-plane. Our approach is graphical: we define a frequency-wise notion of the recently-introduced scaled graph of a MIMO system plotted in a complex plane. The scaled graph provides a bound of the eigenloci of the system, which can be viewed as a robust MIMO extension of the classical Nyquist plot. Our main results characterize sufficient conditions for quantifying the dominance of a closed-loop system based upon separation of scaled graphs of two open-loop systems in a frequency-wise manner. The results reconcile existing small gain, small phase and passivity theorems for feedback dominance analysis.

Paper Structure

This paper contains 12 sections, 9 theorems, 42 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Dominance of MIMO Systems and Problem Formulation
  3. Definitions
  4. Problem Formulation
  5. The Scaled Graph of Matrices
  6. The Scaled Graph of MIMO Systems
  7. Main Results
  8. A Graphical Dominance Theorem
  9. Robust Dominance Analysis under Stable Uncertainties
  10. Specialization of the Main Result
  11. Links to the Principal Region
  12. Conclusion

Key Result

Lemma 1

For $A\in {\mathbb{C}}^{m\times m}$, it holds that for all $i=1, 2, \ldots, m$ and $\lambda_i(A)\neq 0$.

Figures (3)

  • Figure 1: A positive feedback system $\hbox{${P}\,\#\, C$}$.
  • Figure 2: A sketch of the scaled graph of $G$ over $\omega\in \interval{0}{100}~\mathrm{rad/s}$ in the upper half complex plane, within which the generalized Nyquist plot ($3$ continuous eigenloci) of $G$ lies (red).
  • Figure 3: The scaled graph of $G$ in Example \ref{['ex:1d_example']} in the upper half-plane, with the generalized Nyquist plot in red.

Theorems & Definitions (27)

  • Example 1
  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • ...and 17 more