给出截断mRPI集与极限集的显式距离上界，可直接选控制时域。
Explicit Bounds on the Hausdorff Distance for Truncated mRPI Sets via Norm-Dependent Contraction Rates
- 基于系统矩阵范数收缩率和扰动集大小，推导闭式上界。
- 上界可保证预设精度，无需迭代计算集合。
- 适用于鲁棒约束紧化与管状MPC设计，可调范数提升精度。
我们推导出截断最小鲁棒正不变（mRPI）集与其无限时域极限之间豪斯多夫距离的可计算闭式上界。该上界仅依赖于扰动集大小度量和系统矩阵的诱导范数收缩因子，提供完全解析的时域选择规则，可保证预设逼近容差，无需迭代集合计算。向量范数的选择作为设计自由度：通过对角加权或李雅普诺夫加权进行范数塑造，可同时收紧收缩因子与所得证书，对鲁棒不变集逼近及管状模型预测控制（MPC）中的约束紧化具有直接影响。数值实验展示了所提上界的准确性、可扩展性与实际影响。
原文摘要 · Abstract (English)
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.
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