Learning control lyapunov functions from counterexamples and demonstrations
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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-like 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 problem instances. Our approach is able to synthesize relatively simple polynomial control Lyapunov-like functions, and in that process replace the MPC using a guaranteed and computationally less expensive controller.
KeywordsLyapunov functions Controller synthesis Learning from demonstrations Concept learning
We are grateful to Mr. Sina Aghli, Mr. Souradeep Dutta, Prof. Christoffer Heckman and Prof. Eduardo Sontag for helpful discussions. This work was funded in part by NSF under Award Numbers SHF 1527075 and CPS 1646556. All opinions expressed are those of the authors and not necessarily of the NSF.
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