This proposal deals with two synthesis problems 1) design a controller of a prescribed order to stabilize a given system and 2) design a controller of a prescribed order to stabilize a given system containing parameters subject to uncertainty in specified ranges. We propose to establish conditions for the existence of a controller and to obtain a useful parametrization (from the control system designer's point of view) of all controllers that solve the above problems. The problems proposed constitute two of the outstanding unsolved problems in control theory. None of the existing synthesis methods ("observed" state feedback, LQG, H , and geometric methods of synthesis) can solve them in a practical or satisfactory manner. They arise invariably and immediately in most industrial applications, and the complete lack of a synthesis theory to address this problem is a major stumbling block that control engineers encounter in attempting to apply the results of control theory to practice. Our hope that the difficult problems posed above can be solved is based on recent results which extend Kharitonov's theorem and make it applicable to control systems, and new techniques for the exact calculation of the stability ball in parameter space.
