The general distance problem in H∞ synthesis

The general distance problem which arises in the general H_∞ optimal control problem is considered. The existence of an optimal solution is proved and the expression of the optimal norm γ_o is obtained from a somewhat abstract operator point of view. An iterative scheme, called γ-iteration, is introduced which reduces the general distance problem to a standard best approximation problem. Bounds for γ_o are also derived. The γ-iteration is viewed as a problem of finding the zero crossing of a function. This function is shown to be continuous, monotonically decreasing, convex and be bounded by some very simple functions. These properties make it possible to obtain very rapid convergence of the iterative process. The issue of model-reduction in H_∞- synthesis will also be addressed.