Explicit Bounds on the Hausdorff Distance for Truncated mRPI Sets via Norm-Dependent Contraction Rates

TL;DR

This paper presents a computable, explicit upper bound on the Hausdorff distance between truncated and infinite-horizon mRPI sets, facilitating efficient approximation and control design without iterative computations.

Contribution

It introduces a norm-dependent, analytic bound on mRPI set approximation error, enabling explicit horizon selection and improved robustness in control applications.

Findings

The bound depends only on disturbance size and system contraction rate.

Norm shaping improves the tightness of the bound and control performance.

Numerical examples demonstrate the bound's accuracy and scalability.

Abstract

We derive a computable closed-form upper bound on the Hausdorff distance between a truncated minimal robust positively invariant (mRPI) set and its infinite-horizon limit. The bound depends only on a disturbance-set size measure and an induced-norm contraction factor of the system matrix, and it yields an explicit, fully analytic horizon-selection rule that guarantees a prescribed approximation tolerance without iterative set computations. The choice of vector norm enters as a design lever: norm shaping -- through diagonal or Lyapunov-based weighting -- tightens both the contraction factor and the resulting certificate, with direct consequences for robust invariant-set approximation and tube-based model predictive control (MPC) constraint tightening. Numerical examples illustrate the accuracy, scalability, and practical impact of the proposed bound.