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The generalized H 2 control problem. (English) Zbl 0772.93027
Summary: 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 L 2 [0,) to L [0,) 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 2 controllers can be chosen to be observer based controllers. The theory is demonstrated with a numerical example.
93B50Synthesis problems
93C05Linear control systems