Computing controlled invariant sets from data using convex optimization

TL;DR

This paper introduces a data-driven convex optimization method to approximate maximum invariant sets in nonlinear systems using only finite transition data, with provable guarantees and broad applicability.

Contribution

It develops a novel LP-based approach for invariant set approximation from data, avoiding trajectory segments and requiring minimal assumptions.

Findings

Effective approximation of invariant sets in nonlinear systems.

Provable convergence and sample complexity guarantees.

Numerical validation up to 10-dimensional systems.

Abstract

This work presents a data-driven method for approximation of the maximum positively invariant (MPI) set and the maximum controlled invariant (MCI) set for nonlinear dynamical systems. The method only requires the knowledge of a finite collection of one-step transitions of the discrete-time dynamics, without the requirement of segments of trajectories or the control inputs that effected the transitions to be available. The approach uses a novel characterization of the MPI and MCI sets as the solution to an infinite-dimensional linear programming (LP) problem in the space of continuous functions, with the optimum being attained by a (Lipschitz) continuous function under mild assumptions. The infinite-dimensional LP is then approximated by restricting the decision variable to a finite-dimensional subspace and by imposing the non-negativity constraint of this LP only on the available data samples. This leads to a single finite-dimensional LP that can be easily solved using off-the-shelf solvers. We analyze the convergence rate and sample complexity, proving probabilistic as well as hard guarantees on the volume error of the approximations. The approach is very general, requiring minimal underlying assumptions. In particular, the dynamics is not required to be polynomial or even continuous (forgoing some of the theoretical results). Detailed numerical examples up to state-space dimension ten with code available online demonstrate the method.