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

Abstract: This paper establishes the first explicit and closed-form upper bound on the Hausdorff distance between the truncated minimal robust positively invariant (mRPI) set and its infinite-horizon limit. While existing mRPI approximations guarantee asymptotic convergence through geometric or norm-based arguments, none provides a computable expression that quantifies the truncation error for a given horizon. We show that the error satisfies ( d_H(\mathcal{E}N,\mathcal{E}\infty) \le r_W\,γ^{N+1}/(1-γ), ) where $γ<1$ is the induced-norm contraction factor and $r_W$ depends only on the disturbance set. The bound is fully analytic, requires no iterative set computations, and directly characterizes the decay rate of the truncated Minkowski series. We further demonstrate that the choice of vector norm serves as a design parameter that accelerates convergence, enabling substantially tighter horizon selection for robust invariant-set computations and tube-based MPC. Numerical experiments validate the sharpness, scalability, and practical relevance of the proposed bound.