Computing Robust Forward Invariant Sets of Multidimensional Non-linear Systems via Geometric Deformation of Polytopes

TL;DR

This paper introduces a geometric deformation algorithm to compute sequences of robust forward invariant sets for nonlinear systems, enabling parametric variation between minimal and maximal invariant sets.

Contribution

It presents a novel computational method using geometric deformation of polytopes to efficiently compute invariant sets for nonlinear systems.

Findings

Algorithm successfully computes invariant sets for various nonlinear systems.

Method handles arbitrary Lipschitz continuous nonlinear systems with disturbances.

Simulation results demonstrate the approach's versatility in 2D and 3D systems.

Abstract

This paper develops and implements an algorithm to compute sequences of polytopic Robust Forward Invariant Sets (RFIS) that can parametrically vary in size between the maximal and minimal RFIS of a nonlinear dynamical system. This is done through a novel computational approach that geometrically deforms a polytope into an invariant set using a sequence of homeomorphishms, based on an invariance condition that only needs to be satisfied at a finite set of test points. For achieving this, a fast computational test is developed to establish if a given polytopic set is an RFIS. The geometric nature of the proposed approach makes it applicable for arbitrary Lipschitz continuous nonlinear systems in the presence of bounded additive disturbances. The versatility of the proposed approach is presented through simulation results on a variety of nonlinear dynamical systems in two and three dimensions, for which, sequences of invariant sets are computed.