Optimal Infinite-Horizon Mixed H2/H∞ Control
Abstract
We study the problem of mixed H2/H∞ control in the infinite-horizon setting. We identify the optimal causal controller that minimizes the H2 cost of the closed-loop system subject to an H∞ constraint. Megretski proved that the optimal mixed H2/H∞ 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 H2/H∞ 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 H∞ norm) fixed-order rational approximations of the optimal mixed H2/H∞ controller and study its performance.
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