Learning Lyapunov Functions for Piecewise Affine Systems with Neural Network Controllers

2020/08/14 by Shaoru Chen, Chen, Shaoru, Mahyar Fazlyab +8 · 2 citations
Engineering · Mathematics · Physics and Astronomy · #Advanced Control Systems Optimization #FOS: Mathematics #Fault Detection and Control Systems #Model Reduction and Neural Networks #Optimization and Control (math.OC) #math.OC

paper · pdf · doi:10.48550/arxiv.2008.06546

openalex publication_date 2020/08/14 · openalex created_date 2020/08/21 · arxiv created 2020/11/23 · arxiv updated 2020/11/24 · openalex updated_date 2026/07/28

Abstract

We propose a learning-based method for Lyapunov stability analysis of piecewise affine dynamical systems in feedback with piecewise affine neural network controllers. The proposed method consists of an iterative interaction between a learner and a verifier, where in each iteration, the learner uses a collection of samples of the closed-loop system to propose a Lyapunov function candidate as the solution to a convex program. The learner then queries the verifier, which solves a mixed-integer program to either validate the proposed Lyapunov function candidate or reject it with a counterexample, i.e., a state where the stability condition fails. This counterexample is then added to the sample set of the learner to refine the set of Lyapunov function candidates. We design the learner and the verifier based on the analytic center cutting-plane method, in which the verifier acts as the cutting-plane oracle to refine the set of Lyapunov function candidates. We show that when the set of Lyapunov functions is full-dimensional in the parameter space, the overall procedure finds a Lyapunov function in a finite number of iterations. We demonstrate the utility of the proposed method in searching for quadratic and piecewise quadratic Lyapunov functions.

Citations

• On the number of response regions of deep feed forward networks with piece-wise linear activations
• Sequence to Sequence Learning with Neural Networks
• ImageNet classification with deep convolutional neural networks
• An approach to reachability analysis for feed-forward ReLU neural networks
• Output Range Analysis for Deep Neural Networks
• Efficient Formal Safety Analysis of Neural Networks
• Semidefinite relaxations for certifying robustness to adversarial examples
• Stability-certified reinforcement learning: A control-theoretic perspective
• Provable defenses against adversarial examples via the convex outer adversarial polytope
• Stability Analysis using Quadratic Constraints for Systems with Neural Network Controllers

Cited by

• Learning Neural Controllers with Optimality and Stability Guarantees Using Input-Output Dissipativity
• Learning Lyapunov Functions for Hybrid Systems

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