# Invariance in Constrained Switching

## TL;DR

This paper investigates invariant multi-sets for constrained switching systems with disturbances, providing new methods for their characterization, approximation, and computation, with applications demonstrated in control benchmarks.

## Contribution

It introduces a comprehensive framework for invariant multi-sets in constrained switching systems, including new construction and approximation techniques.

## Key findings

- Established characterization of invariant multi-sets.
- Proposed methods for approximation and computation.
- Validated results on control benchmark problems.

## Abstract

We study discrete time linear constrained switching systems with additive disturbances, in which the switching may be on the system matrices, the disturbance sets, the state constraint sets or a combination of the above. In our general setting, a switching sequence is admissible if it is accepted by an automaton. For this family of systems, stability does not necessarily imply the existence of an invariant set. Nevertheless, it does imply the existence of an invariant multi-set, which is a relaxation of invariance and the object of our work. First, we establish basic results concerning the characterization, approximation and computation of the minimal and the maximal admissible invariant multi-set. Second, by exploiting the topological properties of the directed graph which defines the switching constraints, we propose invariant multi-set constructions with several benefits. We illustrate our results in benchmark problems in control.