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

Jiaxun Sun

TL;DR

The paper solves a long-standing gap by providing the first explicit, closed-form upper bound on the Hausdorff distance between a truncated minimal robust positively invariant (mRPI) set and its infinite-horizon limit for discrete-time linear systems. The bound, $d_H(\mathcal{E}_N,\mathcal{E}_\infty) \le \frac{r_W\,\gamma^{N}}{1-\gamma}$, depends only on the disturbance radius $r_W$ and the induced contraction factor $\gamma<1$, and it holds without iterative set computations. It further shows that the contraction factor can be tuned via norm design (e.g., Lyapunov or diagonal-scaling norms) to accelerate convergence, enabling tighter horizon choices for robust and tube-based MPC. The results are validated numerically across moderate to high dimensions, and demonstrate practical benefits such as enlarged nominal feasible sets and reduced conservatism in MPC tubes. The framework also extends to broader contraction-based settings, offering a principled tool for explicit truncation guarantees in invariant-set computations and related control applications.

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.

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

TL;DR

The paper solves a long-standing gap by providing the first explicit, closed-form upper bound on the Hausdorff distance between a truncated minimal robust positively invariant (mRPI) set and its infinite-horizon limit for discrete-time linear systems. The bound, , depends only on the disturbance radius and the induced contraction factor , and it holds without iterative set computations. It further shows that the contraction factor can be tuned via norm design (e.g., Lyapunov or diagonal-scaling norms) to accelerate convergence, enabling tighter horizon choices for robust and tube-based MPC. The results are validated numerically across moderate to high dimensions, and demonstrate practical benefits such as enlarged nominal feasible sets and reduced conservatism in MPC tubes. The framework also extends to broader contraction-based settings, offering a principled tool for explicit truncation guarantees in invariant-set computations and related control applications.

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 is the induced-norm contraction factor and 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.

Paper Structure

This paper contains 26 sections, 2 theorems, 26 equations, 5 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Hausdorff Distance
  4. Minimal RPI and Truncated Approximations
  5. Norm-Induced Contraction Factor
  6. RELATED WORK
  7. MAIN RESULTS
  8. Explicit Hausdorff-Distance Bound
  9. Remark:Operator-Theoretic Interpretation
  10. Corollary: Minimal Truncation Index
  11. Norm-Induced Optimization of the Convergence Rate
  12. Remark: Support-Function Interpretation
  13. Remark:Generalization Potential
  14. Practical Guidelines for Using the Truncation Bound in Tube MPC
  15. Selecting the Truncation Index $N$
  16. ...and 11 more sections

Key Result

Theorem 1

For all $N\in\mathbb{N}$, the Hausdorff distance between the truncated mRPI set $\mathcal{E}_N$ and the minimal RPI set $\mathcal{E}_\infty$ satisfies

Figures (5)

  • Figure 1: Six-dimensional example: numerical Hausdorff error and explicit bound. Both exhibit geometric decay with nearly identical slope.
  • Figure 2: Experiment 2: theoretical bound under different induced norms. Aggressive scaling tightens the bound by reducing $r_W(P)$.
  • Figure 3: Experiment 3: numerical and theoretical errors for $n=10,15,20$. All curves show geometric decay with the theoretical bound consistently above the numerical error.
  • Figure 4: Experiment 4 (Part B): Tube MPC closed-loop trajectories. Real trajectories (solid) and nominal trajectories (dashed) for the baseline tube MPC c13 (green) and our method (red). Our tube, built from the truncated mRPI set plus the rigorous tail bound, is tighter yet robust, while the baseline tube is significantly more conservative.
  • Figure 5: Experiment 4 (Part A): Enlarged nominal feasible set. Baseline tightening (green), based on the classical invariant radius $r_{\text{tube}}$c13c2, is significantly more conservative than our truncation-bound-based tightening (red). The resulting feasible set $\mathcal{X}\ominus\mathcal{E}$ is therefore much larger under our method.

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • Theorem 2: Minimal truncation index