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Learning dynamical systems using local stability priors

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  • 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.

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

    Citation:

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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

    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

    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

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

    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

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

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

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

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

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