Simultaneous H2/H∞ optimal control with state feedback

In this paper we consider a mixed H²/H^∞-optimal control problem. It is assumed that the plant as well as the feedback controller are finite-dimensional and linear time-invariant, and that the plants state is available for feedback. More specifically, among all the state-feedback controllers that minimize the H²-norm of a closed loop transfer matrix, we give necessary and sufficient conditions for the existence of a controller that also satisfies a prescribed H^∞-norm bound on some other closed loop transfer matrix. When these conditions are satisfied, the solution to the above problem is also a global solution to the constrained optimization problem of minimizing an H²-norm performance measure subject to an H^∞-norm constraint. We also give state-space formulae for computing the solutions. An example is given in which all solutions to the constrained optimization problem are necessarily dynamic, i.e, there is no static gain solution even though plant state is available for feedback. A conclusion of this work is that a priori assumptions on the structure of the solutions to mixed H²/H^∞ problems could turn out to be conserative.