Nonlinear H∞ control and the bounded real lemma

The problem of nonlinear H^∞ optimal control via measurement feedback is considered. Some necessary conditions are derived for solution of this problem in terms of Hamilton-Jacobi inequalities which may be considered to be the nonlinear generalizations of the Doyle-Glover-Khargonekar-Francis (DGKF) X and Y Riccati equations on the plant state space. Two different ways that generalize are found. First the X and Y equations each give a nonlinear first-order partial differential inequality. However, these are restrictions of a single partial differential inequality on a larger space. The authors also provide candidate equations for two out of three functions defining the feedback compensator and give plausible conditions under which the compensator must be given by these functions