Approximate solutions to H2 optimization problems with H2 and H∞ constraints

In this paper a H2 optimal control problem is considered in which the controllers are constrained to satisfy prescribed upper bounds on the H2 and H∞ norms of several closed-loop transfer functions. A sequence of H2-cost/Ή2-constraint problems is presented which aims at generating approximate solutions to the original problem — this may be viewed as a generalization of the so-called Law-son algorithm for Tchebyche approximation. Each such H2/H2 problem has a unique rational solution which can be computed through line search and spectral factorization. Under appropriate assumptions, it is shown that the sequence of optimal cost values for the H2/H2 problems is increasing and bounded from above (hence, it is convergent) and that, whenever the corresponding sequence of solutions converges, it does so to the solution of the original problem.