This thesis considers the problem of designing a controller for a linear plant, which, using output measurements only, will make the plant state variables follow optimally any input signal belonging to a known class of signals. The dynamic order of the controller is determined by the observability properties of the plant and by the class of input signals. The structure of the controller is determined from the requirement that the closed loop system consisting of plant and controller be an optimum system. Two approaches to the optimization procedure are indicated: (1) Optimizing the dynamics of an augmented system consisting of plant, controller and a dynamic system generating the input signals, to minimize appropriate errors. (2) Regarding the general problem as one of optimum regulation of errors desired to be minimized. The results allow the linear servomechanism problem to be treated, for the first time, in a realistic manner.