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Issues with Input-Space Representation in Nonlinear Data-Based Dissipativity Estimation

Ethan LoCicero, Alex Penne, Leila Bridgeman

TL;DR

A new method is proposed to quantify the robustness of machine learning-based dissipativity estimation and it is shown that this method achieves a more tractable trade-off between robustness and sample complexity.

Abstract

In data-based control, dissipativity can be a powerful tool for attaining stability guarantees for nonlinear systems if that dissipativity can be inferred from data. This work provides a tutorial on several existing methods for data-based dissipativity estimation of nonlinear systems. The interplay between the underlying assumptions of these methods and their sample complexity is investigated. It is shown that methods based on delta-covering result in an intractable trade-off between sample complexity and robustness. A new method is proposed to quantify the robustness of machine learning-based dissipativity estimation. It is shown that this method achieves a more tractable trade-off between robustness and sample complexity. Several numerical case studies demonstrate the results.

Issues with Input-Space Representation in Nonlinear Data-Based Dissipativity Estimation

TL;DR

A new method is proposed to quantify the robustness of machine learning-based dissipativity estimation and it is shown that this method achieves a more tractable trade-off between robustness and sample complexity.

Abstract

In data-based control, dissipativity can be a powerful tool for attaining stability guarantees for nonlinear systems if that dissipativity can be inferred from data. This work provides a tutorial on several existing methods for data-based dissipativity estimation of nonlinear systems. The interplay between the underlying assumptions of these methods and their sample complexity is investigated. It is shown that methods based on delta-covering result in an intractable trade-off between sample complexity and robustness. A new method is proposed to quantify the robustness of machine learning-based dissipativity estimation. It is shown that this method achieves a more tractable trade-off between robustness and sample complexity. Several numerical case studies demonstrate the results.

Paper Structure

This paper contains 9 sections, 14 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representing $\mathcal{L}_{2e}$
  4. $\delta$-covering Methods
  5. Summary and Complexity
  6. Non-robustness Demonstration
  7. Machine Learning Methods
  8. Conclusions
  9. Appendix

Figures (4)

  • Figure 2: Left: $K=625$ trajectories of Equations \ref{['eqn:example']} (LTI) and \ref{['eqn:nonlinear']} (nonlinear) generated with a $b=4$ non-sequential Fourier bases $\{\frac{1}{T},\,\frac{\sqrt{2}}{T}\sin(\frac{2\pi}{T} t),$$\frac{\sqrt{2}}{T}\sin(\frac{20\pi}{T}t),\,\frac{\sqrt{2}}{T}\sin(\frac{200\pi}{T}t)\}$ with $T=10$. Middle: the same data generated from the first $b=4$ Legendre Polynomial bases with $T=10$, and Right: with $T=1$. Each plot also shows the true conic bounds and those calculated via Appendix A with $\delta=0$, which are nearly identical up to $K=8.1e5$.
  • Figure 3: Bounds on 1000 trajectories of \ref{['eqn:example']} uniformly sampled from $\mathcal{U}_{A1234}$ with $T=20$ and different numbers of basis functions, $b$. For each case of $b$, the upper bound is identical.
  • Figure 4: 1000 trajectories of \ref{['eqn:example']} from Weiner processes with different lengths, $T$. The Weiner process is implemented as a discrete random walk with time step $0.01$ and step size from a standard normal distribution.
  • Figure 5: Data generated from a single Weiner process applied to Equations \ref{['eqn:example']} and \ref{['eqn:nonlinear']} with $T {=} 0.2$ to $4$ (left), and $T{=}0.2$ to $50$ (right) incremented by $0.2$.

Theorems & Definitions (3)

  • Definition 1
  • Definition 2
  • Definition 3