Safe Controller Synthesis for Data-Driven Differential Inclusions
We consider the problem of designing finite-horizon safe controllers for a dynamical system for which no explicit analytical model exists and limited data only along a single trajectory of the system are available. Given samples of the states and inputs of the system, and additional side information in terms of regularity of the evolution of the states, we synthesize a controller such that the evolution of the states avoid some prespecified unsafe set over a given finite horizon. Motivated by recent results on Whitney's extension theorem, we use piecewise-polynomial approximations of the trajectories based on the data along with the regularity side information to formulate a data-driven differential inclusion model that can predict the evolution of the trajectories. For these classes of data-driven differential inclusions, we propose a safety analysis theorem based on barrier certificates. As a corollary of this theorem, we demonstrate that we can design controllers ensuring safety of the solutions to the data-driven differential inclusion over a finite horizon. From a computational standpoint, our results are cast into a set of sum-of-squares programs whenever the certificates are parametrized by polynomials of fixed degree and the sets are semialgebraic.
© 2020 IEEE. Manuscript received December 24, 2018; revised June 7, 2019 and November 28, 2019; accepted January 11, 2020. Date of publication January 27, 2020; date of current version October 21, 2020. The work was supported by the AFOSR under Grant FA9550-19-1-0005 and Grant NSF 1646522. Recommended by Associate Editor Dr. F. Zhang.