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We study the problem of designing the controllers that are robust with respect to the parametric uncertainty. In Part I The Rank-One Problem we consider the class of systems with restriction that the structure of uncertainty is limited to a vector. We extend the class of the allowed systems. The main result is the canonical parametrization of all destabilizing uncertainties. We also present a method of obtaining the suboptimal controller of lower order that provides the stability margin as close to the optimal one as we wish. We propose a numerical algorithm for the optimal robust control synthesis. In the special case, when the uncertainty parameter is real-valued, we show that the initial problem can be considered as finite-dimensional in the space of variables (semi-infinite convex programming). Part II Convex Duality: Matrix Case generalizes the results to the systems with matrix uncertainties.