# Learning Lyapunov (Potential) Functions from Counterexamples and Demonstrations

## TL;DR

This paper introduces an iterative learning framework that synthesizes control Lyapunov functions for nonlinear systems by leveraging demonstrations, counterexamples, and convex optimization, enabling the replacement of complex MPC controllers with simpler, guaranteed controllers.

## Contribution

It presents a novel iterative learning approach combining demonstrations, counterexamples, and verification to synthesize control Lyapunov functions for nonlinear systems.

## Key findings

- Successfully synthesizes polynomial control Lyapunov functions.
- Replaces MPC controllers with simpler, guaranteed controllers.
- Proves convergence of the learning algorithm using convex optimization techniques.

## Abstract

We present a technique for learning control Lyapunov (potential) functions, which are used in turn to synthesize controllers for nonlinear dynamical systems. The learning framework uses a demonstrator that implements a black-box, untrusted strategy presumed to solve the problem of interest, a learner that poses finitely many queries to the demonstrator to infer a candidate function and a verifier that checks whether the current candidate is a valid control Lyapunov function. The overall learning framework is iterative, eliminating a set of candidates on each iteration using the counterexamples discovered by the verifier and the demonstrations over these counterexamples. We prove its convergence using ellipsoidal approximation techniques from convex optimization. We also implement this scheme using nonlinear MPC controllers to serve as demonstrators for a set of state and trajectory stabilization problems for nonlinear dynamical systems. Our approach is able to synthesize relatively simple polynomial control Lyapunov functions, and in that process replace the MPC using a guaranteed and computationally less expensive controller.