Necessary and sufficient conditions for mixed H2 and H∞ optimal control

Mixed H₂ and H_∞ optimal control problems are addressed. D.S. Bernstein and W.M. Haddad (1989) considered the case of one exogenous input and two observed outputs. Using a Lagrange multiplier technique, and under the assumption that the order of the controller is specified, they derived a necessary condition for minimizing an upper bound of the H₂ norm of one transfer matrix, subject to an H_∞ norm constraint on the other. J.C. Doyle et al. (1989) later derived a sufficient condition for minimizing what may be shown to be the dual version of the upper bound defined by Bernstein and Haddad (BH) for the mixed H₂ and H_∞ optimal control problem. In contrast with the BH system, the system of Doyle et al. (DZB) has two exogenous inputs and one measured output. The conditions are derived under different algebraic frameworks. The results are presented of a study that attempts to unify the two mixed optimality conditions. The sufficient condition for the DZB mixed H₂ and H_∞ full order optimal control is shown to be the dual of the necessary condition for the BH control. Therefore, both conditions are proved to be necessary and sufficient