A review of proper modeling techniques

The text was written in September 2008 for the mor4ansys group.

Ersal, T., Fathy, H.K., Louca, L.S., Rideout, D.G., and Stein, J.L. (2007) “A Review of Proper Modeling Techniques.” Proc. 2007 ASME International Mechanical Engineering Congress and Exhibition, Paper No. IMECE2007-42031.

It is quite a thorough review of model reduction methods, well, actually of what the authors refer to as proper modeling. This name was puzzling to me from the first glance but right from the beginning the authors have made it clear: “A dynamic system model is proper for a particular application if it achieves the accuracy required by the application with minimal complexity”. This way the model reduction is a part of proper modeling.

The most interesting is that the methods from the control theory as well as moment matching were reviewed and this was a significant part of the review. Provided that most of authors are mechanical engineers this is a bit unusual. I have once read a book Zu-Qing Qu, “Model Order Reduction Techniques: with Applications in Finite Element Analysis”, Springer, 2004. This was a typical text for mechanical engineers – modal superposition, Gyuan reduction, component synthesis and so on. It was huge difference with the Antoulas’ book “Approximation of Large-Scale Dynamical Systems”: like the two parallel worlds. The paper in subject seems mark a transition between these two worlds.

A few comments to the review:

1) Padé approximation can produce unstable methods indeed. This is why in practice the Padé-type approximation is employed much more often. The orthogonal projection (Arnoldi) has an advantage over the oblique projection (Lanczos) that based on the congruent transform one can produce provable stable and passive reduced models (see for example the PRIMA paper). Another advantage is that it is possible to approximate the complete output. This is the main reason that in MOR for ANSYS only the Arnoldi method is used. Note that Padé-type approximation (PRIMA-like) methods have not been mentioned in the review.

2) In the paper the authors have divided model reduction methods as frequency based and projection based and made a statement:

>The frequency-based proper modeling techniques discussed
>hitherto assume, in general, that the salient dynamics of a given
>system occur over a fairly limited range in the frequency
>domain. Projection-based techniques make a conceptually
>analogous assumption in the state domain.

I would disagree here. For a linear system time domain and frequency domains are equivalent. What happens is that in the most methods one starts with an approximation of the transfer function but then just uses the low-dimensional subspace to project the original system in the time domain.

3) It is not too difficult to make a model reduction method realization-preserving provided it defines the output matrix. That is, to express the reduced model in terms of a subset of original state variables. An example for a thermal system is in

A. Augustin, T. Hauck.
Transient Thermal Compact Models for Circuit Simulation.

follow the link in http://modelreduction.com/Applications/Thermal.html

See the transition from Eq (1) to (3). The idea is to define the number of outputs equal to the number of generalized variables. It is assumed that the outputs correspond to the original state variables of your choice. Then the output matrix is square and one can easily transform the reduced model to a realization-preserving form.

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