Optimal Infinite-Horizon Mixed $\mathit{H}_2/\mathit{H}_\infty$ Control

TL;DR

This paper derives an exact frequency domain solution for the optimal infinite-horizon mixed H2/H∞ control problem, providing a finite-dimensional parameterization and an efficient iterative algorithm for scalar systems.

Contribution

It presents the first explicit closed-form solution and a convergent iterative method for the non-rational optimal mixed H2/H∞ controller in infinite-horizon control.

Findings

Provides a finite-dimensional parameterization of the optimal controller.

Introduces an efficient iterative algorithm with proven convergence for scalar systems.

Demonstrates how to approximate the non-rational controller with fixed-order rational controllers.

Abstract

We study the problem of mixed $\mathit{H}_2/\mathit{H}_\infty$ control in the infinite-horizon setting. We identify the optimal causal controller that minimizes the $\mathit{H}_2$ cost of the closed-loop system subject to an $\mathit{H}_\infty$ constraint. Megretski proved that the optimal mixed $\mathit{H}_2/\mathit{H}_\infty$ 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 $\mathit{H}_2/\mathit{H}_\infty$ 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 $\mathit{H}_\infty$ norm) fixed-order rational approximations of the optimal mixed $\mathit{H}_2/\mathit{H}_\infty$ controller and study its performance.