How scaling of the disturbance set affects robust positively invariant sets for linear systems

TL;DR

This paper investigates how the scaling of disturbance sets influences the properties of robust positively invariant (RPI) sets in linear systems, introducing a critical scaling factor that determines RPI set existence.

Contribution

It characterizes the critical disturbance scaling factor for RPI sets, provides an efficient algorithm for its computation, and analyzes its impact on RPI set properties.

Findings

The existence of RPI sets depends on a unique critical scaling factor.

An efficient algorithm for computing the critical scaling factor is proposed.

The properties of RPI sets change significantly at the critical scaling threshold.

Abstract

This paper presents new results on robust positively invariant (RPI) sets for linear discrete-time systems with additive disturbances. In particular, we study how RPI sets change with scaling of the disturbance set. More precisely, we show that many properties of RPI sets crucially depend on a unique scaling factor which determines the transition from nonempty to empty RPI sets. We characterize this critical scaling factor, present an efficient algorithm for its computation, and analyze it for a number of examples from the literature.