Controller Synthesis for Safety of Physically-Viable Data-Driven Models

TL;DR

This paper develops a data-driven method for synthesizing finite-horizon safe controllers for systems with no explicit models, using limited trajectory data, side information, and polynomial approximations to ensure safety via barrier certificates.

Contribution

It introduces a novel approach combining Whitney's extension theorem, differential inclusions, and sum-of-squares programming for safe control synthesis without explicit system models.

Findings

Successfully synthesizes controllers ensuring safety over finite horizons.

Incorporates side information like conservation laws into the control design.

Uses sum-of-squares programs for computationally tractable safety verification.

Abstract

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 and conservation laws, we synthesize a controller such that the evolution of the states avoid some pre-specified 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 semi-algebraic. This computational method allows incorporating side information in terms of conservation laws and integral constraints using Positivstellensatz-type arguments.