Certified exact error-margin evaluation and input-tightening analysis for Tube MPC

Develop a certified truncation bound for evaluating the exact worst-case tracking-error margin from the convergent matrix-power series, and quantify the severity of the input-tightening reservation required for the ancillary correction in Tube MPC under bounded actuation, including its growth with the sampling period.

Background

The paper derives an exact series representation for each component of the worst-case error margin, obtained by unrolling the closed-loop error recursion and taking the absolute contribution of every admissible disturbance. Although this representation is exact, practical use requires a certified truncation bound so that the infinite series can be evaluated reliably and efficiently. The paper also notes a separate unresolved issue in the robust Tube MPC construction: the ancillary feedback correction must be reserved within the actuator limit, potentially causing substantial input tightening, particularly for larger sampling periods. These two issues are explicitly identified as future work because the report does not provide the truncation certification or analyze the resulting input-reservation severity.

References

With a certified bound on the truncation of the series, this exact value can be evaluated more cheaply than the DARE solve it depends on, which removes the practical motivation for using an approximate formula at all. The tube construction also imposes a second tightening, on the input rather than the corridor, not discussed here: the ancillary correction $-Ke_k$ must itself be reserved out of $u_{\max}$, and this reservation can be severe in practice, growing sharply with the sampling period. Both points are left for future work.

— Real-Time Model Predictive Control Algorithms for Autonomous Spacecraft Guidance  (2609.00927 - Garab, 1 Sep 2026) in Remark “Towards an exact margin,” Section 7, subsection “A Closed-Form Estimate of the Maximum Tracking Error”