Benign Geometry and Distributional Robustness of ℋ2 Synthesis
Abstract
In this paper, we study standard and distributionally robust ℋ2 synthesis problem of a stabilizing state-feedback controller for discrete-time linear time-invariant systems. Without requiring a nonsingular disturbance controllability Gramian or a positive-definite control penalty, we establish that stationarity of a stabilizing gain in standard ℋ2 synthesis is equivalent to global optimality and derive an exact Pythagorean identity for the performance difference. We then generalize the ℋ2 synthesis problem by considering i.i.d. disturbances with zero mean and uniformly bounded second moments, and extend the stationarity-optimality equivalence to the corresponding distributionally robust ℋ2 synthesis. Moreover, we show that every standard ℋ2-optimal gain is simultaneously optimal for every such distributionally robust problem. Finally, we specialize the framework to Frobenius, Kullback--Leibler, and Wasserstein-2 ambiguity sets and obtain explicit characterizations of their worst-case covariances, which, in turn, provide performance certificates for a common optimal controller.
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BibTeX
@misc{kolarijani2026benign,
title = {{Benign Geometry and Distributional Robustness of $\mathcalH_2$ Synthesis}},
author = {Arman Sharifi Kolarijani and Peyman Mohajerin Esfahani and Tamás Keviczky and Mohamad Amin Sharifi Kolarijani},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24421},
eprint = {2609.24421},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24421},
}
RIS
TY - UNPB
TI - Benign Geometry and Distributional Robustness of ℋ_2 Synthesis
AU - Arman Sharifi Kolarijani
AU - Peyman Mohajerin Esfahani
AU - Tamás Keviczky
AU - Mohamad Amin Sharifi Kolarijani
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24421
UR - https://arxiv.org/abs/2609.24421
AB - In this paper, we study standard and distributionally robust ℋ_2 synthesis problem of a stabilizing state-feedback controller for discrete-time linear time-invariant systems. Without requiring a nonsingular disturbance controllability Gramian or a positive-definite control penalty, we establish that stationarity of a stabilizing gain in standard ℋ_2 synthesis is equivalent to global optimality and derive an exact Pythagorean identity for the performance difference. We then generalize the ℋ_2 synthesis problem by considering i.i.d. disturbances with zero mean and uniformly bounded second moments, and extend the stationarity-optimality equivalence to the corresponding distributionally robust ℋ_2 synthesis. Moreover, we show that every standard ℋ_2-optimal gain is simultaneously optimal for every such distributionally robust problem. Finally, we specialize the framework to Frobenius, Kullback--Leibler, and Wasserstein-2 ambiguity sets and obtain explicit characterizations of their worst-case covariances, which, in turn, provide performance certificates for a common optimal controller.
ER -