Families of moment matching based, low order approximations for linear systems

The problem of finding a framework for linear moment matching based approximation of linear systems is discussed. It is shown that it is possible to define five (equivalent) notions of moments, i.e., one from a complex “s-domain” point of view, two using Krylov projections and two from a time-domain perspective. Based on these notions, classes of parameterized reduced order models that achieve moment matching are obtained. We analyze the controllability and observability properties of the models belonging to each of the classes of reduced order models. We find the subclasses of models of orders lower than the number of matched moments, i.e., we find the sets of parameters that allow for pole–zero cancellations to occur. Furthermore we compute the (subclass of) minimal order model(s). The minimal order is equal to half of the number of matched moments, i.e., generically the largest number of possible pole–zero cancellations is half of the number of matched moments. Finally, we present classes of reduced order models that match larger numbers of moments, based on interconnections between reduced order models.