Minimum control effort state-feedback H∞-control

Optimal H_∞-controllers may exhibit large gains, resulting in large control efforts. In this paper we consider the problem of designing a minimum gain static full state-feedback controller such that the closed-loop transfer function satisfies a H_∞-constraint. The main result of the paper shows that, by minimising an upper bound for the Frobenius-norm of the feedback-gain matrix and using a parametrisation as in [6], the problem can be cast into a finite-dimensional, convex optimisation problem. Scalar cost-functions for the H_∞-bound and various other constraints allow the application of gradient-based software packages to these problems. Finally, we illustrate how to apply this theory to the mixed H₂/H_∞-control problem with minimum control effort.