H2-optimal control with an H∞-constraint The state feedback case

A state-space approach solves the problem of finding among all state feedback controllers that minimize an HE-performance measure one that also satisfies an Ha-norm bound. Abstract-In this paper we consider a mixed HZ//-/~-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 plant state is available for feedback. More specifically, among all the state-feedback controllers that minimize the HE-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 met, the solution to the above problem is also a global solution to the contrained optimization problem of minimizing an HE-norm performance measure subject to an H®-norm constraint. We also give state-space formulae for computing the solutions. Some easily checkable sufficient conditions for the existence of solutions are given. Finally we give an example 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. 1. INTRODUCTION AND PROBLEM FORMULATION THE CONTROL problem addressed in this paper concerns the finite-dimensional linear time-invariant feedback system depicted in Fig. 1. The signals Wl and WE denote exogenous inputs while Zl and z2 denote controlled (i.e. regulated) signals. The signals u and y denote the control input and the measured output, respectively. The transfer matrices of the plant and the controller are denoted by G and K, respectively. It is also assumed that both G and K are real-rational and