Loewner Framework

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The Loewner Framework is a frequency-domain system identification, system realization and model reduction method, named after Charles Loewner, who introduced the Loewner Matrix which is essential to this method based on ideas from .

Introduction

Model order reduction is commonly used to simulate and control complex physical processes. This is done by replacing the large-scale system model described in terms of differential equations by a system of much lower dimension that has similar response characteristics. In many situations, a model (the collection of differential equations written in matrix format) for the underlying dynamical process is not explicitly available. Hence, the system matrix realization can not be directly obtained. Instead, a black box is provided that produces input-output measurements. These situations include VLSI modeling from chips or real time simulation of multi-body dynamics with constraints.

We study reduction methods based on moment matching, that is, matching of the coefficients of power series expansions of the transfer function at selected points in the complex plane. More precisely, values of the underlying transfer function, together with the values of its derivatives are interpolated. Hence, the main underlying problem is rational interpolation. Different methods based on Krylov iteration, as well as the Arnoldi or the Lanczos procedures, and multi-point (rational) versions thereof.

Interpolatory model reduction methods compute ROMs (reduced order models) by means of interpolating the response of the original system at selected points (usually the transfer function). The Loewner framework (introduced in ) falls into this category. It is a data-driven method that directly constructs a ROM using only measured data (samples of the transfer function). Additionally, by compressing the (usually large) data set, it extracts the dominant features and eliminates the inherent redundancies. For a recent survey on the Loewner framework for linear systems, see . For the nonlinear case, this method requires appropriate definition of transfer functions, but the main procedure does not differ too much from the one applied for the linear case. Some of the extensions include the bilinear systems case in , quadratic-bilinear systems and linear switched systems . The dissertation represents a comprehensive collection of these methods.

The motivation behind developing generalized versions of the Loewner framework to new systems is that such classes of systems can be viewed as a bridge between linear and nonlinear systems. For example, one can always write an approximation of a nonlinear system by means of a bilinear system (Carleman linearization ). Furthermore, for certain types of nonlinear systems, one can always find an equivalent quadratic-bilinear model without performing any approximation (McCormick relaxation ) whatsoever. Linear switched systems can be viewed as a special class of hybrid systems that are used to model coupled or distributed processes characterized by both discrete and continuous dynamics.

Loewner Framework for Linear Systems

We analyze linear time-invariant continuous-time systems described by the following equations:

$\Sigma : \begin{cases} E \dot{x}(t) = Ax(t) + Bu(t) \\ \;\;\; y(t) = Cx(t) + Du(t) \end{cases}$

where $A, E \in \mathbb{R}^{n \times n}$, $C \in \R^{p \times n}$, $B \in \mathbb{R}^{n \times m}$, $D \in \mathbb{R}^{p \times m}$ and $x(t) \in \R^n$, $u(t) \in R^m$, $y(t) \in \R^p$ are state, input and output vector respectively. The transfer function corresponding to $\Sigma:(A,B,C,D,E)$ is computed as $H(s) = C(sE-A)^{-1}B + D$.

The Loewner framework can construct a reduced-order system by means of measured data. In this context, data represent sample values of the underlying transfer function $H(s)$ at selected interpolation points (in some cases on the imaginary axis, when approximating the frequency response of the original system). A Loewner matrix pencil $\lambda\mathbb{L}-\mathbb{L}_s$ is composed of the Loewner matrix $\mathbb{L}$ and the shifted Loewner matrix $\mathbb{L}_s$. This pencil represents a surrogate of the original matrix pencil $\lambda E - A$.

The Loewner framework originates from rational approximation theory (see ) and can be used to recover or reduce the original dynamical system by means of interpolating the transfer function. For simplicity, assume that the data set contains an even number of measurements denoted with $2N$. The next step is to partition the data measurements set into two disjoint data subsets, i.e. the left subset and the right subset as described in the following

Left set: $(\mu_i,\ell_i,v_i), i = 1,\dots,N$, Right set: $(\lambda_j,r_j,w_j), j=1,\dots,N$. (2)

More precisely, the data in (2) is composed of $N$ left interpolation points: $\{\mu_i\}_{i=1}^N \subset \mathbb{C}$, $N$, left sample values: $\{v_i\}_{i=1}^N \subset \mathbb{C}^m$ and $N$ left tangential directions: $\{\ell_i\}_{i=1}^N \subset \mathbb{C}^p$. Additionally, we have $N$ distinct right interpolation points: $\{\lambda_i\}_{i=1}^N \subset \mathbb{C}$ with $N$ right sample values: $\{w_i\}_{i=1}^N \subset \mathbb{C}^p$ and right tangential directions: $\{r_i\}_{i=1}^N \subset \mathbb{C}^m$. We seek a linear system $\widehat{Sigma}$ such that the associated transfer function $\widehat{H}(s)$ is a tangential interpolant to the original one, $H(s)$. More precisely, the following interpolation conditions should then hold:

$\begin{array}{rcl} \ell_i^* \widehat{H}(\mu_i) &= \ell_i^*H(\mu_i) = v_i^* \quad \text{for} \;\; i = 1,\dots,N, \\ \widehat{H}(\lambda_j)r_j &= H(\lambda_j) r_j = w_j \quad \text{for} \;\; j = 1,\dots,N. \end{array}$

Next, arrange the measured data into matrix format as follows

Summary

In this note we have addressed the Loewner framework for model reduction of some classes of (nonlinear) dynamical systems. The underlying philosophy valid for each of the extensions is as follows: One needs to collect data, i.e. values of high-order transfer functions (either by performing measurements or by direct numerical simulations) and then extract the desired information directly from the model constructed. The strength of the proposed approaches consists in the fact that, given input-output data, one can construct with basically no computational cost, a model of the data in generalized state-space format.