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.

翻译：我们推导了截断最小鲁棒正不变（mRPI）集合与其无限时域极限之间Hausdorff距离的可计算闭式上界。该上界仅依赖于扰动集大小度量与系统矩阵的诱导范数收缩因子，并给出一个显式、完全解析的时域选择规则，无需迭代集合计算即可保证指定的逼近容差。向量范数的选择作为设计杠杆：通过对角或李雅普诺夫加权进行的范数塑造，既收缩了收缩因子，也收紧了最终证书，对鲁棒不变集逼近和基于管道的模型预测控制（MPC）约束收紧具有直接影响。数值示例展示了所提上界的准确性、可扩展性及实际应用价值。