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Data-Driven Estimation of Vinnicombe metric

Margarita A. Guerrero, Henrik Sandberg, Cristian R. Rojas

Abstract

Quantifying model mismatch in a control-relevant manner is fundamental in robust control. A well-known metric for this purpose is the $ν$-gap, or Vinnicombe metric, which measures the discrepancy between a nominal model and the real system from a closed-loop viewpoint. However, its computation typically requires explicit knowledge of the true system. In this letter, we propose an identification-free, data-driven method to estimate the $ν$-gap between discrete-time SISO systems directly from input-output experiments. The method is complemented by a data-driven winding-number test, based on Welch-type averaging, to verify a required topological condition for the computation of the metric. Numerical simulations on heavy-duty gas-turbine models and a textbook example show that the proposed estimate closely matches MATLAB$^©$ \texttt{gapmetric}, while correctly detecting cases in which the admissibility conditions fail.

Data-Driven Estimation of Vinnicombe metric

Abstract

Quantifying model mismatch in a control-relevant manner is fundamental in robust control. A well-known metric for this purpose is the -gap, or Vinnicombe metric, which measures the discrepancy between a nominal model and the real system from a closed-loop viewpoint. However, its computation typically requires explicit knowledge of the true system. In this letter, we propose an identification-free, data-driven method to estimate the -gap between discrete-time SISO systems directly from input-output experiments. The method is complemented by a data-driven winding-number test, based on Welch-type averaging, to verify a required topological condition for the computation of the metric. Numerical simulations on heavy-duty gas-turbine models and a textbook example show that the proposed estimate closely matches MATLAB \texttt{gapmetric}, while correctly detecting cases in which the admissibility conditions fail.
Paper Structure (12 sections, 1 theorem, 12 equations, 4 figures, 2 algorithms)

This paper contains 12 sections, 1 theorem, 12 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Preliminaries
  4. The Vinnicombe Metric
  5. Power Iteration Method
  6. Proposed Approach
  7. $\nu$-gap estimation
  8. $\mathrm{wno}$ estimation
  9. Experiments
  10. Case Study: Heavy-Duty Gas Turbine
  11. Case Study: Poles in RHP
  12. Conclusion

Key Result

Proposition 1

Given a nominal plant $G_0$ and a perturbed plant $G$, the interconnection $[G,C]$ is stable for all compensators $C$ satisfying $b_{G_0,C}>\beta$ if and only if $\delta_\nu(G_0,G)\leqslant\beta$.

Figures (4)

  • Figure 1: Standard feedback configuration.
  • Figure 2: Estimate of the $\nu$-gap $\hat{\delta}_\nu$ (black, dashed), average of 100 Monte Carlo (MC) simulations (blue, solid), and actual $\delta_\nu$ (red, dashed).
  • Figure 3: Estimate of the $\nu$-gap $\hat{\delta}_\nu$ (black, dashed), average of 100 MC simulations (blue, solid), and actual $\delta_\nu$ (red, dashed).
  • Figure 4: Estimate of the $\nu$-gap $\hat{\delta}_\nu$ (black, dashed), average of 100 MC simulations (blue, solid), and actual $\delta_\nu$ (red, dashed).

Theorems & Definitions (2)

  • Definition 1
  • Proposition 1: vinnicombe_92