Antoine Vandendorpe

Model Reduction of MIMO Systems via Tangential Interpolation

In this paper, we address the problem of constructing a reduced order system of minimal McMillan degree that satisfies a set of tangential interpolation conditions with respect to the original system under some mild conditions. The resulting reduced order transfer function appears to be generically unique and we present a simple and efficient technique to construct this interpolating reduced order system. This is a generalization of the multipoint Pade technique which is particularly suited to handle multiinput multioutput systems.

Model Reduction of Interconnected Systems

We consider a particular class of structured systems that can be modelled as a set of input/output subsystems that interconnect to each other, in the sense that outputs of some subsystems are inputs of other subsystems. Sometimes, it is important to preserve this structure in the reduced order system. Instead of reducing the entire system, it makes sense to reduce each subsystem (or a few of them) by taking into account its interconnection with the other subsystems in order to approximate the entire system in a so-called structured manner. The purpose of this paper is to present both Krylov-based and Gramian-based model reduction techniques that preserve the structure of the interconnections. Several structured model reduction techniques existing in the literature appear as special cases of our approach, permitting to unify and generalize the theory to some extent.

Model Reduction of Second-Order Systems

In this chapter, the problem of constructing a reduced order system while preserving the second order structure of the original system is discussed. After a brief introduction on second order systems and a review of first order model reduction techniques, two classes of second order structure preserving model reduction techniques Krylov subspace-based and SVD-based are presented. For the Krylov techniques, conditions on the projectors that guarantee the reduced second order system tangentially interpolates the original system at given frequencies are derived and an algorithm is described. For SVD-based techniques, a Second Order Balanced Truncation method is derived from second order gramians.

Model reduction via truncation: an interpolation point of view

In this paper, we focus our attention on linear time invariant continuous time linear systems with one input and one output (SISO LTI systems). We consider the problem of constructing a reduced order system via truncation of the original system. Given a SISO strictly proper transfer function T(s) of McMillan degree N and a strictly proper SISO transfer function (T) over cap (s) of McMillan degree n < N, we prove that (T) over cap (s) can always be constructed via truncation of the system T(s). The proof is mainly based on interpolation theory, and more precisely on multipoint Pade interpolation. Moreover, new results about Krylov subspaces are developed

On the generality of multipoint Padé approximations

Multipoint Pade interpolation methods have shown to be very efficient for model reduction of large-scale dynamical systems. The objective of this paper is to analyze the generality of this approach. We mainly focus our attention on the Single Input Single Output case. The generalization of this approach for MIMO model reduction is briefly introduced.

On the embedding of state space realizations

In this paper, the generality of the particular model reduction method, known as the projection of state space realization, is investigated. Given two transfer functions, one wants to find the necessary and sufficient conditions for the embedding of a state-space realization of the transfer function of smaller McMillan degree into a state-space realization of the transfer function of larger McMillan degree. Two approaches are considered, both in the MIMO case. First, when the difference of the McMillan degree between the transfer functions is equal to one and there is no common pole, necessary and sufficient conditions are provided. Then, the generic case is considered using a pencil approach. Finally, it is shown that the condition of embedding is related to the eigen structure of a pencil that appears in the framework of tangential interpolation.

Model reduction via projection of generalized state space systems

We show that the projection of generalized state space models of SISO systems allows to construct arbitrary lower order models and that they can be obtained via the solution of particular generalized Sylvester equations. This generalizes the results already obtained for state space systems, where both the original models and low order models were constrained to be strictly proper. We also conjecture that for MIMO systems, this approach is as general as one can hope for.