Precise computation of maximal controlled invariant sets for nonlinear systems

Scott Brown, Mohammad Khajenejad, and Sonia Martínez
Automatica, under review 2025

Abstract:

In this paper, we discuss the problem of computing maximal controlled invariant sets (CIS) of nonlinear systems, emphasizing the relationship between the invariant sets of a continuous-time system and those of its discretization. We demonstrate that, due to the well-known Nagumo’s theorem which is necessary and sufficient for forward invariance of a set, invariant sets of general continuous-time systems are difficult to compute exactly. This motivates the introduction of recurrent sets, which are a relaxation of the invariance condition, requiring that all solutions return to the set in a fixed, finite time. These recurrent sets are shown to be close to invariant sets in the sense of the Hausdorff distance, where the distance depends on the sampling time. We provide a method for computing an inner approximation of the maximal recurrent set contained in a given subset of the state space. This is accomplished by computing the maximal controlled invariant set for a discretization of the system. Finally, we demonstrate our method on several numerical examples, where we apply our algorithm to multiple benchmark systems.

File: main.pdf

Bib-tex entry:

@article{SB-MK-SM:25-auto,
author = {S. Brown and M. Khajenejad and and S. Mart{\'\i}nez},
title = {Precise Computation of Maximal Controlled Invariant Sets for Nonlinear Systems},
journal= {Automatica},
pages = {},
volume = {},
number = {},
note = {Under review},
year = {2025}
}