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We present a technique for learning control Lyapunov-like functions, whichare used in turn to synthesize controllers for nonlinear dynamical systems thatcan stabilize the system, or satisfy specifications such as remaining inside asafe set, or eventually reaching a target set while remaining inside a safeset. The learning framework uses a demonstrator that implements a black-box,untrusted strategy presumed to solve the problem of interest, a learner thatposes finitely many queries to the demonstrator to infer a candidate function,and a verifier that checks whether the current candidate is a valid controlLyapunov function. The overall learning framework is iterative, eliminating aset of candidates on each iteration using the counterexamples discovered by theverifier and the demonstrations over these counterexamples. We prove itsconvergence using ellipsoidal approximation techniques from convexoptimization. We also implement this scheme using nonlinear MPC controllers toserve as demonstrators for a set of state and trajectory stabilization problemsfor nonlinear dynamical systems. We show how the verifier can be constructedefficiently using convex relaxations of the verification problem for polynomialsystems to semi-definite programming (SDP) problem instances. Our approach isable to synthesize relatively simple polynomial control Lyapunov functions, andin that process replace the MPC using a guaranteed and computationally lessexpensive controller.

Learning Control Lyapunov Functions from Counterexamples and Demonstrations