$H_\infty $ Control with Transients

Abstract

In $H_\infty $ (or uniformly optimal) control problems, it is usually assumed that the system initial conditions are zero. In this paper, an $H_\infty $-like control problem that incorporates uncertainty in initial conditions is formulated. This is done by defining a worst-case performance measure. Both finite and infinite horizon problems are considered. Necessary and sufficient conditions are derived for the existence of controllers that yield a closed-loop system for which the above-mentioned performance measure is less than a prespecified value. State-space formulae for the controllers are also presented.