Gramians of Structured Systems and an Error Bound for Structure-Preserving Model Reduction

In this paper a general framework is posed for defining the reachability and controllability gramians of structured linear dynamical systems. The novelty is that a formula for the gramian is given in the frequency domain. This formulation is surprisingly versatile and may be applied in a variety of structured problems. Moreover, this formulation enables a rather straightforward development of apriori error bounds for model reduction in the H2 norm. The bound applies to a reduced model derived from projection onto the dominant eigenspace of the appropriate gramian. The reduced models are structure preserving because they arise as direct reduction of the original system in the reduced basis. A derivation of the bound is presented and verified computationally on a second order system arising from structural analysis