Learning Lyapunov (Potential) Functions from Counterexamples and Demonstrations
For control engineers, this provides a method to automatically generate certified controllers from imperfect demonstrations, reducing computational cost.
The paper proposes an iterative learning framework to synthesize control Lyapunov functions from a black-box demonstrator and a verifier, proving convergence via ellipsoidal approximation. It replaces computationally expensive MPC controllers with simpler polynomial ones while maintaining guarantees.
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.