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Stability-Certified Learning of Control Systems with Quadratic Nonlinearities

Igor Pontes Duff, Pawan Goyal, Peter Benner

TL;DR

The main objective is to develop a method that facilitates the inference of quadratic control dynamical systems with inherent stability guarantees, and investigates the stability characteristics of control systems with energy-preserving nonlinearities.

Abstract

This work primarily focuses on an operator inference methodology aimed at constructing low-dimensional dynamical models based on a priori hypotheses about their structure, often informed by established physics or expert insights. Stability is a fundamental attribute of dynamical systems, yet it is not always assured in models derived through inference. Our main objective is to develop a method that facilitates the inference of quadratic control dynamical systems with inherent stability guarantees. To this aim, we investigate the stability characteristics of control systems with energy-preserving nonlinearities, thereby identifying conditions under which such systems are bounded-input bounded-state stable. These insights are subsequently applied to the learning process, yielding inferred models that are inherently stable by design. The efficacy of our proposed framework is demonstrated through a couple of numerical examples.

Stability-Certified Learning of Control Systems with Quadratic Nonlinearities

TL;DR

The main objective is to develop a method that facilitates the inference of quadratic control dynamical systems with inherent stability guarantees, and investigates the stability characteristics of control systems with energy-preserving nonlinearities.

Abstract

This work primarily focuses on an operator inference methodology aimed at constructing low-dimensional dynamical models based on a priori hypotheses about their structure, often informed by established physics or expert insights. Stability is a fundamental attribute of dynamical systems, yet it is not always assured in models derived through inference. Our main objective is to develop a method that facilitates the inference of quadratic control dynamical systems with inherent stability guarantees. To this aim, we investigate the stability characteristics of control systems with energy-preserving nonlinearities, thereby identifying conditions under which such systems are bounded-input bounded-state stable. These insights are subsequently applied to the learning process, yielding inferred models that are inherently stable by design. The efficacy of our proposed framework is demonstrated through a couple of numerical examples.
Paper Structure (13 sections, 1 theorem, 30 equations, 4 figures)

This paper contains 13 sections, 1 theorem, 30 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model Inference of quadratic control systems
  3. Stability for quadratic control systems
  4. Parametrization of a stable matrix
  5. Energy preserving quadratic nonlinearity
  6. Bounded-input bounded-state stability result
  7. Learning bounded control systems
  8. Extension to more general quadratic Lyapunov functions
  9. Numerical results
  10. Low-dimensional example I
  11. Low-dimensional example II
  12. High-dimensional Burgers' example
  13. Conclusions

Key Result

Theorem 1

Consider a quadratic control system as in eq:quad_model. Assume that the matrix $\mathbf{A}\in \mathbb{R}^{n \times n}$ is monotonically stable and can be decomposed as $\mathbf{A} = \mathbf{J}-\mathbf{R}$, where $\mathbf{J} = -\mathbf{J}^{\top}$ and $\mathbf{R} = \mathbf{R}^{\top} \succ 0$, and $\m $\sigma_{\min}(\cdot)$ is the minimum singular value of a matrix and $\|\mathbf{u}\|_{L_{\infty}} =

Figures (4)

  • Figure 1: Low-dimensional example I: A performance test for testing control inputs of the inferred models.
  • Figure 2: Low-dimensional example II: A performance test for testing control inputs of the inferred models.
  • Figure 3: Burgers' example: A performance test for a testing control input of the inferred models.
  • Figure 4: Burgers' example: A comparison of OpInfc and stable-OpInfc for $10$ test cases.

Theorems & Definitions (1)

  • Theorem 1