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Data-Driven Estimation of Structured Singular Values

Margarita A. Guerrero, Braghadeesh Lakshminarayanan, Cristian R. Rojas

TL;DR

The paper tackles robust stability under structured uncertainty by aiming to estimate the structured singular value $\mu_\Delta(\boldsymbol{G}_0)$ without a plant model. It develops a fully data-driven variant of the power-iteration method that operates on input–output data and uses frequency-domain insights to concentrate experimentation on the most informative frequencies. The key contribution is an algorithm that computes a valid lower bound for $\mu_\Delta(\boldsymbol{G}_0)$ from data, with numerical results showing close alignment to the standard Mussv lower bound in many scenarios, and a detailed analysis of convergence and noise sensitivity. This data-driven approach provides a conservative robustness metric for stability certification when an explicit plant model is unavailable. Future work includes extending the framework to estimate an upper bound and improving resilience to measurement noise.

Abstract

Estimating the size of the modeling error is crucial for robust control. Over the years, numerous metrics have been developed to quantify the model error in a control relevant manner. One of the most important such metrics is the structured singular value, as it leads to necessary and sufficient conditions for ensuring stability and robustness in feedback control under structured model uncertainty. Although the computation of the structured singular value is often intractable, lower and upper bounds for it can often be obtained if a model of the system is known. In this paper, we introduce a fully data-driven method to estimate a lower bound for the structured singular value, by conducting experiments on the system and applying power iterations to the collected data. Our numerical simulations demonstrate that this method effectively lower bounds the structured singular value, yielding results comparable to the MATLAB$^©$ Robust Control Toolbox.

Data-Driven Estimation of Structured Singular Values

TL;DR

The paper tackles robust stability under structured uncertainty by aiming to estimate the structured singular value without a plant model. It develops a fully data-driven variant of the power-iteration method that operates on input–output data and uses frequency-domain insights to concentrate experimentation on the most informative frequencies. The key contribution is an algorithm that computes a valid lower bound for from data, with numerical results showing close alignment to the standard Mussv lower bound in many scenarios, and a detailed analysis of convergence and noise sensitivity. This data-driven approach provides a conservative robustness metric for stability certification when an explicit plant model is unavailable. Future work includes extending the framework to estimate an upper bound and improving resilience to measurement noise.

Abstract

Estimating the size of the modeling error is crucial for robust control. Over the years, numerous metrics have been developed to quantify the model error in a control relevant manner. One of the most important such metrics is the structured singular value, as it leads to necessary and sufficient conditions for ensuring stability and robustness in feedback control under structured model uncertainty. Although the computation of the structured singular value is often intractable, lower and upper bounds for it can often be obtained if a model of the system is known. In this paper, we introduce a fully data-driven method to estimate a lower bound for the structured singular value, by conducting experiments on the system and applying power iterations to the collected data. Our numerical simulations demonstrate that this method effectively lower bounds the structured singular value, yielding results comparable to the MATLAB Robust Control Toolbox.

Paper Structure

This paper contains 8 sections, 5 theorems, 14 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Preliminaries
  4. Proposed Approach
  5. Experiments
  6. Experimental Setting
  7. Experimental Results
  8. Conclusion

Key Result

Theorem 1

The system $\boldsymbol{G} = [\boldsymbol{I} + \boldsymbol{G}_0 \boldsymbol{\Delta}]^{-1} \boldsymbol{G}_0$ is stable for all $\boldsymbol{\Delta} \in \Delta$ such that $\| \boldsymbol{\Delta} \|_\infty \leqslant \alpha$ if and only if $\alpha\, \mu_\Delta(\boldsymbol{G}_0) < 1$.

Figures (4)

  • Figure 1: Comparison between $\mu_{\Delta}(\boldsymbol{G}_0)$ and $N$. Solid line: $\tilde{\mu}$, dashed line: $\bar{\mu}$, dotted line: mussv. From top to bottom, the plots correspond to systems with 1 block, 2 blocks, and 3 blocks, respectively.
  • Figure 2: Convergence of the lower bound over iterations $(l)$. Solid line: $\tilde{\mu}$, dashed line: $\bar{\mu}$, dotted line: last iteration.
  • Figure 3: Frequency response of the structured singular value.
  • Figure 4: Percentage of converging cases of the data-driven power method.

Theorems & Definitions (6)

  • Theorem 1: zhoudoyle
  • Definition 1: Structured singular value
  • Lemma 1: packard1988power
  • Theorem 2: doyle1982analysis
  • Theorem 3: packard1988power
  • Lemma 2: packard1988power