On the distributed stable full information H∞ problem

We study the distributed suboptimal full information H^∞ problem for a stable well-posed linear system with control u. disturbance w, state x, and output y. The problem is to find all suboptimal compensators u = uw. i.e. compensators that make the norm of the closed loop input/output map from w to y less than one. Define Q(x₀, u, w) = ||y||²_{L2(R+; y)} - ||w||²_L2(r+y) We first choose u to minimize Q and w to maximize min_u Q. and show that the minimizing control and maximizing disturbance can be written in a well-posed feedback form if and only if the input/output map of the system has a (J. S)-inner-outer factorization. We then give a parameterization of all suboptimal compensators. In the generality that we allow it is possible that we have to include a feed-forward term in the compensator. This term is not present in the classical continuous time theory, but it is well-known from the discrete time theory.