Abstract
In this paper, we show that a basic fixed point method used to enclose the greatest fixed point in a Kleene algebra will allow us to compute inner and outer approximations of invariant-based sets for continuous-time nonlinear dynamical systems. Our contribution is to provide the definitions and theorems that will allow us to make the link between the theory of invariant sets and the Kleene algebra. This link has never be done before and will allow us to compute rigorously sets that can be defined as a combination of positive invariant sets. Some illustrating examples show the nice properties of the approach.
Highlights
In this paper, we deal with a dynamical system S defined by the following state equation: Citation: Le Mézo, T.; Jaulin, L.; Massé, D.; Zerr, B
The paper proposes to compute inner and outer approximations of invariant sets in the general case where the system is continuous-time and nonlinear. It introduces for the first time an approach based on the Kleene algebra to compute sets that can be defined as operations on invariant sets
We introduce automorphism-based Kleene algebras [4], which will allow us to make a first bridge between algebraic tools and invariant sets of nonlinear dynamical systems
Summary
We deal with a dynamical system S defined by the following state equation: Citation: Le Mézo, T.; Jaulin, L.; Massé, D.; Zerr, B. We will take advantage of this algebraic structure to derive new efficient algorithms that are able to solve problems involving invariant sets that were not possible to compute with existing methods. For continuous-time nonlinear systems, computing invariant sets is much more difficult, and different types of approaches can be extracted from the literature. The paper proposes to compute inner and outer approximations of invariant sets in the general case where the system is continuous-time and nonlinear. More than that, it introduces for the first time an approach based on the Kleene algebra to compute sets that can be defined as operations on invariant sets.
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