Calculus of Variations Derivation of the Minimax Linear-Quadratic (H^) Controller

The linear-quadratic best controller for the worst bounded disturbances (LQW controller), or so-called Hoo controller, is derived as a differential game between the controller and disturber using the calculus of variations. This derivation explicitly shows that for the full-order LQW controller the worst measurement disturbance is zero (!) and the controller initial conditions must be set equal to the plant initial conditions for a well-posed differential game. As in previous derivations, the worst process disturbance is shown to be a feedback on the plant states. The derivation yields necessary conditions for reduced-order and higher order LQW controllers, useful in multiple-plan t optimization for robustness. A helicopter near hover is used to illustrate differences between LQW and linear-quadratic-Gaussian (LQG) control. This comparison suggests the relative merits of LQG and LQW control design and shows that a special case of LQW control called infinite-disturbance LQW ("optimal" H«>} control is not practical.