Computing robust control invariant sets of constrained nonlinear systems: A graph algorithm approach

TL;DR

This paper introduces a graph-based algorithm for computing the largest robust control invariant sets of constrained nonlinear systems, providing convergence guarantees and practical numerical examples.

Contribution

It develops a novel graph-theoretical approach to approximate RCISs for nonlinear systems, including convergence proofs and algorithms for inner and outer approximations.

Findings

Algorithm converges to the largest RCIS

Effective for input and disturbance affine systems

Numerical examples demonstrate practical efficacy

Abstract

This paper deals with the computation of the largest robust control invariant sets (RCISs) of constrained nonlinear systems. The proposed approach is based on casting the search for the invariant set as a graph theoretical problem. Specifically, a general class of discrete-time time-invariant nonlinear systems is considered. First, the dynamics of a nonlinear system is approximated with a directed graph. Subsequently, the condition for robust control invariance is derived and an algorithm for computing the robust control invariant set is presented. The algorithm combines the iterative subdivision technique with the robust control invariance condition to produce outer approximations of the largest robust control invariant set at each iteration. Following this, we prove convergence of the algorithm to the largest RCIS as the iterations proceed to infinity. Based on the developed algorithms, an algorithm to compute inner approximations of the RCIS is also presented. A special case of input affine and disturbance affine systems is also considered. Finally, two numerical examples are presented to demonstrate the efficacy of the proposed method.