Learning dynamical systems using local stability priors

Arash Mehrjou; Andrea Iannelli; Bernhard Schölkopf

doi:10.3934/jcd.2022021

Article Contents

2023, Volume 10, Issue 1: 175-198. Doi: 10.3934/jcd.2022021

This issue Previous Article Low-rank kernel approximation of Lyapunov functions using neural networks Next Article Smooth complete Lyapunov functions for multifunctions

Learning dynamical systems using local stability priors

1.
Empirical Inference Department of Max Planck Institute for Intelligent Systems, Tübingen, Germany, and Department of Computer Science, ETH Zürich, Zürich, Switzerland
2.
Automatic Control Laboratory, ETH Zürich, Zürich, Switzerland

^* Corresponding author: Arash Mehrjou amehrjou@tuebingen.mpg.de
^* Corresponding author: Arash Mehrjou amehrjou@tuebingen.mpg.de

Received: July 2021

Revised: December 2021

Early access: August 2022

Published: January 2023

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Abstract

A computational approach to simultaneously learn the vector field of a dynamical system with a locally asymptotically stable equilibrium and its region of attraction from the system's trajectories is proposed. The nonlinear identification leverages the local stability information as a prior on the system, effectively endowing the estimate with this important structural property. In addition, the knowledge of the region of attraction can be used to design experiments by informing the selection of initial conditions from which trajectories are generated and by enabling the use of a Lyapunov function of the system as a regularization term. Simulation results show that the proposed method allows efficient sampling and provides an accurate estimate of the dynamics in an inner approximation of its region of attraction.

Keywords:

Mathematics Subject Classification: Primary: 93D05, 93B30; Secondary: 62J02.

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Figure 1. (a) One step of the iterative RoA estimation algorithm. The boundary of the RoA is learned as a boundary function of a classifier. The blue area is the current estimate of the RoA. The green area contains initial states that converge to the estimated RoA and eventually the equilibrium. The red area contains the initial states that diverge. Each step of the algorithm learns the boundary between stable and unstable regions. (b) For the vector field $ f $, $ f|_{\mathcal{R}_{\bar{x}}} $ denotes its restriction to the RoA, which is the objective of the ODE learning phase

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Figure 2. Pre-training the randomly initialized neural network with a quadratic function. The background color shows the values of the Lyapunov function evaluated over a fine grid of points. Lighter colors correspond to larger values. The contours show the level sets

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Figure 3. (Van der Pol Oscillator) The growth stages of the RoA estimation algorithm of Section 2.3. Color codes are as follows. Green: True RoA. Blue: Estimated RoA (the largest sublevel set of the learned Lyapunov function that satisfies the decrease condition (3)). Pink: The gap $ \mathcal{G} = \mathcal{S}_{\alpha c}(V(\cdot;\theta))\backslash \mathcal{S}_c(V(\cdot;\theta)) $ from which the initial states of the trajectories are picked

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Figure 4. Sampled trajectories from inside the estimated RoA of each growth stage

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Figure 5. (Van der Pol Oscillator) Left: True RoA. Right: The progressively learned ODE over the steps of the coupled algorithm. The darker background shows a larger mismatch between the learned and true vector fields

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Figure 6. (Van der Pol Oscillator) The progressively learned ODE using neural networks

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Figure 7. (Van der Pol Oscillator) Learned ODEs with and without the Lyapunov regularization term in the loss function

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Figure 8. (Inverted Pendulum) Sampled trajectories from inside the estimated RoA of each growth stage

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Figure 9. (Inverted Pendulum) The progressively learned ODE using neural networks

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