Explicit Bounds on the Hausdorff Distance for Truncated mRPI Sets via Norm-Dependent Contraction Rates
Provides a practical, analytic tool for robust invariant-set approximation and tube-based MPC, reducing computational burden for control engineers.
The paper derives a closed-form upper bound on the Hausdorff distance between a truncated minimal robust positively invariant set and its infinite-horizon limit, enabling explicit horizon selection for prescribed approximation tolerance without iterative computations. The bound depends on a disturbance-set size measure and an induced-norm contraction factor, with norm shaping improving tightness.
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.