CALCULUS OF VARIATIONS DERIVATION OF THE MINIMAX LINEAR-QUADRATIC (H-INFINITY) CONTROLLER

The linear-quadratic best controller for the worst bounded disturbances (LQW controller), or so-called H-infinity 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 older LQW controllers, useful in multiple-plant 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-infinity) control is not practical.