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Interpolatory Methods for Model Reduction

A.C. Antoulas, C.A. Beattie, and S. Gügercin
Publication Date: 
Number of Pages: 
[Reviewed by
Bill Satzer
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Large computational models of dynamical systems have become very important for the analysis of complex phenomena. The push for better predictions and improved accuracy has led inevitably to models of greater complexity and substantially larger size. The picture is complicated by the need to build models that link subsystems together, and so incorporate diverse physical phenomena with different time scales and spatial resolution.
To make the computations tractable several approaches are developing; generally their aim is to concentrate on the critical subspace of the system state space where most of the action occurs. The authors of this book present an approach to model reduction that relies heavily on interpolation techniques . Model reduction means identifying a significantly smaller model whose output matches the output of a very large model as closely as possible.
This is essentially a graduate level research monograph designed to present a unified picture of the basic machinery and principal results and to give students and new researchers a way into the research literature. The authors say that theirs is the first reference work to treat interpolatory methods for model reduction in a comprehensive way. They believe that the book is accessible to anyone with a background in linear algebra, differential equations, and numerical analysis at the level of an advanced undergraduate. They also suggest that prior knowledge of the elements of system theory would be helpful. Accordingly they provide "a quick tour of most of the required fundamentals", but that's probably insufficient for anyone without the more extensive exposure to systems theory that many engineering students have.
Although the authors eventually consider a broad range of dynamical systems - linear and nonlinear, with and without parameters - the basic idea is best explained starting with a system that is linear, time invariant, and without additional parameters. Such a system consists of the pair of vector equations
\( \dot{x} (t) = A x(t) + B u(t) \)  
\(y(t) = C x(t) + D u(t)\)
\(x (0) =  0\),  
where \(E\) and \(A\)  are \(n \times n \) matrices, \( B \) is \(n \times m \), \( C \) is \(p \times n\), \( D \) is \(p \times m\), and all are constant matrices. The system state is represented by \(x\), the system input by \(u\), and the output by \(y\).  In applications \(n\) is large, often \(10^{4}\)  to \( 10^{7}\).
If the input \(u (t) \) is exponentially bounded, then \( x (t) \) and \(y (t)\) will be as well. Taking the Laplace transform of this system and solving for the Laplace transform \(\hat{y}(s)\), the Laplace transform of \(y(t)\), in terms of \(\hat{u} (s)\), the Laplace transform of \( u(t) \), we get 
\(\hat{y} (s) = H(s) \hat{u} (s)\)
where \(H(s) = C(sE - A)^{-1} B  + D\) is the matrix-valued transfer function of the complex variable \(s \) for the original system.
The object is to produce a smaller dynamical system in an r-dimensional state space with \(r \ll n\): 
\(E_r \dot{x} (t) = A_r x(t) + B_r u(t)\)
\(y_r (t) = C_r x(t) + D_r u(t)\)     
where \(E_r\) and \(A\)  are \(r \times r\) matrices, \( B \) is \(r \times m\), \( C \) is \(q \times n \), \( D \) is \(q \times m\). What is crucial here is the mapping from the input \(u(t)\) to the output \(y(t)\), not the full information on the evolution of the state \(x(t) \). In many cases the system trajectory \( x(t) \) lives in a subspace having a substantially smaller dimension than the full system dimension \( n \), so the original model behaves very nearly as if it had many fewer internal degrees of freedom.
Current approaches to model reduction include proper orthogonal decomposition, balanced truncation, and interpolatory methods. This book concentrates on interpolation methods and looks at two approaches in particular: projection and data-driven modeling. In both cases the emphasis is on finding a suitable reduced transfer function \( H_{r} \). The idea is that a carefully selected transfer function \( H_{r} \) gives a good approximation to the transfer function of the full system, and hence that the true output \( y \) is close to the reduced system output \( y_r \). Finding the reduced order transfer function is the key. The original transfer function is a high order matrix-valued function whose entries are rational functions. The goal then is to find a lower order transfer function approximation \( H_{r} \)  by rational approximation. 
The projection method uses tangential interpolation using a selection of interpolation points for the original transfer function and then imposes interpolation conditions at those points only in tangential directions. Remarkably, it is possible to interpolate the original transfer matrix without ever computing the quantities to be matched by using Petrov-Galerkin projections with carefully chosen test subspaces. This method can give optimal reduced-order models.
The alternative "data driven" interpolation method proceeds in a similar way, but it uses actual frequency response data at certain driving frequencies and works in the frequency domain to identify a reduced-order transfer function.
The core material of the book consists of a description of these two approaches to interpolation and a discussion of optimal approaches for identifying interpolation points and associated tangential directions. Within this treatment of the core concepts there are a collection of motivating examples described at a high level as well as lower-order examples that show, for example, reduction of a fourth-order model to a second-order one in some detail.
The remainder of the book provides fewer details but sketches an approach for handling systems with parameters and nonlinear systems. It concludes with short treatments of special advanced topics. This is a cogent but challenging introduction to the use of interpolation for model reduction. It offers students and researchers the opportunity to learn about an important approach to model reduction and it has a very impressive bibliography.


Bill Satzer (, now retired from 3M Company, did his PhD work in dynamical systems and celestial mechanics. He spent most of his career working on applications ranging from ceramic fibers to optical films.