H_∞ optimization with time-domain constraints

Standard H_∞ optimization cannot handle specifications or constraints on the time response of a closed-loop system exactly. In this paper, the problem of H_∞ optimization subject to time-domain constraints over a finite horizon is considered. More specifically, given a set of fixed inputs wⁱ, it is required to find a controller such that a closed-loop transfer matrix has an H_∞-norm less than one, and the time responses yⁱ to the signals wⁱ belong to some prespecified sets Ωⁱ. First, the one-block constrained H_∞ optimal control problem is reduced to a finite dimensional, convex minimization problem and a standard H_∞ optimization problem. Then, the general four-block H_∞ optimal control problem is solved by reduction to the one-block case. The objective function is constructed via state-space methods, and some properties of H_∞ optimal constrained controllers are given. It is shown how satisfaction of the constraints over a finite horizon can imply good behavior overall. An efficient computational procedure based on the ellipsoid algorithm is also discussed