Learning Control Lyapunov Functions from Counterexamples and Demonstrations

TL;DR

This paper introduces an iterative learning framework for synthesizing control Lyapunov functions for nonlinear systems using demonstrations and counterexamples, enabling efficient controller synthesis and system stabilization.

Contribution

It proposes a novel framework combining demonstrations, counterexamples, and convex verification to learn control Lyapunov functions for nonlinear systems.

Findings

Successfully synthesizes polynomial control Lyapunov functions.

Replaces MPC controllers with simpler, guaranteed controllers.

Achieves convergence through convex optimization techniques.

Abstract

We present a technique for learning control Lyapunov-like functions, which are used in turn to synthesize controllers for nonlinear dynamical systems that can stabilize the system, or satisfy specifications such as remaining inside a safe set, or eventually reaching a target set while remaining inside a safe set. 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. We show how the verifier can be constructed efficiently using convex relaxations of the verification problem for polynomial systems to semi-definite programming (SDP) problem instances. 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.