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Exploiting Structure in MIMO Scaled Graph Analysis

Timo de Groot, Tom Oomen, Sebastiaan van den Eijnden

TL;DR

This work addresses the conservatism in scaled-graph analysis for multivariable nonlinear feedback by introducing weighted inner products through a multiplier $W$ and exploiting a commutation structure. It derives a stability framework based on weighted scaled graphs SG$^{W}(H)$ and provides an LMI-based method to compute an over-approximation for MIMO Lur'e systems, enabling tighter stability conclusions. The approach significantly reduces conservatism in a representative Lur'e example, demonstrating that appropriate weighting enlarges the class of interconnections that can be certified stable. The results offer practical insights for robustness analysis and controller design in structured, high-dimensional nonlinear feedback, with potential extensions to dynamic multipliers and broader nonlinear dynamics.

Abstract

Scaled graphs offer a graphical tool for analysis of nonlinear feedback systems. Although recently substantial progress has been made in scaled graph analysis, at present their use in multivariable feedback systems is limited by conservatism. In this paper, we aim to reduce this conservatism by introducing multipliers and exploit system structure in the analysis with scaled graphs. In particular, we use weighted inner products to arrive at a weighted scaled graph and combine this with a commutation property to formulate a stability result for multivariable feedback systems. We present a method for computing the weighted scaled graph of Lur'e systems based on solving sets of linear matrix inequalities, and demonstrate a significant reduction in conservatism through an example.

Exploiting Structure in MIMO Scaled Graph Analysis

TL;DR

This work addresses the conservatism in scaled-graph analysis for multivariable nonlinear feedback by introducing weighted inner products through a multiplier and exploiting a commutation structure. It derives a stability framework based on weighted scaled graphs SG and provides an LMI-based method to compute an over-approximation for MIMO Lur'e systems, enabling tighter stability conclusions. The approach significantly reduces conservatism in a representative Lur'e example, demonstrating that appropriate weighting enlarges the class of interconnections that can be certified stable. The results offer practical insights for robustness analysis and controller design in structured, high-dimensional nonlinear feedback, with potential extensions to dynamic multipliers and broader nonlinear dynamics.

Abstract

Scaled graphs offer a graphical tool for analysis of nonlinear feedback systems. Although recently substantial progress has been made in scaled graph analysis, at present their use in multivariable feedback systems is limited by conservatism. In this paper, we aim to reduce this conservatism by introducing multipliers and exploit system structure in the analysis with scaled graphs. In particular, we use weighted inner products to arrive at a weighted scaled graph and combine this with a commutation property to formulate a stability result for multivariable feedback systems. We present a method for computing the weighted scaled graph of Lur'e systems based on solving sets of linear matrix inequalities, and demonstrate a significant reduction in conservatism through an example.

Paper Structure

This paper contains 18 sections, 4 theorems, 34 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. System Setting
  4. Scaled Graphs
  5. Example and Problem Statement
  6. Weighting in Scaled Graphs
  7. Weighted Inner Products in Scaled Graphs
  8. Feedback Stability
  9. Computations for Lur'e Systems
  10. Lur'e System Description
  11. From LMIs to Scaled Graphs
  12. Finding a Suitable $W$
  13. Discussion
  14. Example
  15. System Parameters
  16. ...and 3 more sections

Key Result

Theorem 1

Consider a pair of finite-gain stable systems $H_1$ and $H_2$, and suppose that the negative feedback interconnection of $H_1$ and $\tau H_2$ is well-posedA feedback interconnection is well-posed if, given input signals in $\mathcal{L}_2$, there exist output signals in $\mathcal{L}_{2e}$ depending c then the feedback interconnection is finite-gain stable with an $\mathcal{L}_2$ gain bound of $1/r$

Figures (7)

  • Figure 1: General feedback interconnection.
  • Figure 2: Transformed feedback interconnection.
  • Figure 3: Transformed feedback interconnection with $H_2\in\mathcal{C}(\mathcal{X})$.
  • Figure 4: MIMO Lur'e system, where $G$ represents the LTI part of the system and $\Phi(\cdot)$ is a MIMO nonlinearity that satisfies the IQC in \ref{['eq:lure_IQC']}.
  • Figure 5: Feedback interconnection considered in the example.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Lemma 1
  • Theorem 2
  • proof
  • Theorem 3