Computation of Maximal Admissible Robust Positive Invariant Sets for Linear Systems with Parametric and Additive Uncertainties

TL;DR

This paper develops an efficient method to compute the maximal admissible robust positive invariant set for linear systems with uncertainties, ensuring safety constraints are maintained despite disturbances.

Contribution

It introduces necessary and sufficient conditions for the existence of the MARPI set and proposes a finite-time converging algorithm based on backward reachable sets.

Findings

The algorithm efficiently computes the MARPI set for uncertain linear systems.

Theoretical conditions guarantee the non-emptiness of the invariant set.

Numerical example validates the proposed method.

Abstract

In this paper, we address the problem of computing the maximal admissible robust positive invariant (MARPI) set for discrete-time linear time-varying systems with parametric uncertainties and additive disturbances. The system state and input are subjected to hard constraints, and the system parameters and the exogenous disturbance are assumed to belong to known convex polytopes. We provide necessary and sufficient conditions for the existence of the non-empty MARPI set, and explore relevant features of the set that lead to an efficient finite-time converging algorithm with a suitable stopping criterion. The analysis hinges on backward reachable sets defined using recursively computed halfspaces and the minimal RPI set. A numerical example is used to validate the theoretical development.