Optimal Infinite-Horizon Mixed $\mathit{H}_2/\mathit{H}_\infty$ Control
Abstract
We study the problem of mixed control in the infinite-horizon setting. We identify the optimal causal controller that minimizes the cost of the closed-loop system subject to an constraint. Megretski proved that the optimal mixed controller is non-rational whenever the constraint is active without giving an explicit construction of the controller. In this work, we provide the first exact closed-form solution to the infinite-horizon mixed control in the frequency domain. While the optimal controller is non-rational, our formulation provides a finite-dimensional parameterization of the optimal controller. Leveraging this fact, we introduce an efficient iterative algorithm that finds the optimal causal controller in the frequency domain. We show that this algorithm is convergent when the system is scalar and present numerical evidence for exponential convergence of the proposed algorithm. Finally, we show how to find the best (in norm) fixed-order rational approximations of the optimal mixed controller and study its performance.
Keywords
Cite
@article{arxiv.2409.20020,
title = {Optimal Infinite-Horizon Mixed $\mathit{H}_2/\mathit{H}_\infty$ Control},
author = {Vikrant Malik and Taylan Kargin and Joudi Hajar and Babak Hassibi},
journal= {arXiv preprint arXiv:2409.20020},
year = {2024}
}
Comments
Accepted for presentation at the 60th Annual Allerton Conference on Communication, Control, and Computing (Allerton) 2024