THE GENERALIZED H2 CONTROL PROBLEM

In this paper we consider the problem of finding an internally stabilizing controller such that the controlled or regulated signals have a guaranteed maximum peak value in response to arbitrary (but bounded) energy exogenous inputs. More specifically, we give a complete solution to the problem of finding a stabilizing controller such that the closed loop gain from L2[0, infinity) to L(infinity)[0, infinity) is below a specified level. We consider both state-feedback and output feedback problems. In the state-feedback case it is shown that if this synthesis problem is solvable, then a solution can be chosen to be a constant state-feedback gain. Necessary and sufficient conditions for the existence of solutions as well as a formula for a state-feedback gain that solves this control problem are obtained in terms of a finite dimensional convex feasibility program. After showing the separation properties of this synthesis problem, the output feedback case is reduced to a state-feedback problem. It is shown that, in the output feedback case, generalized H, controllers can be chosen to be observer based controllers. The theory is demonstrated with a numerical example.