An Efficient, Memory-Saving Approach for the Loewner Framework

The Loewner framework is one of the most successful data-driven model order reduction techniques. If N is the cardinality of a given data set, the so-called Loewner and shifted Loewner matrices [Formula: see text] and [Formula: see text] can be defined by solely relying on information encoded in the considered data set and they play a crucial role in the computation of the sought rational model approximation.In particular, the singular value decomposition of a linear combination of [Formula: see text] and [Formula: see text] provides the tools needed to construct accurate models which fulfill important approximation properties with respect to the original data set. However, for highly-sampled data sets, the dense nature of [Formula: see text] and [Formula: see text] leads to numerical difficulties, namely the failure to allocate these matrices in certain memory-limited environments or excessive computational costs. Even though they do not possess any sparsity pattern, the Loewner and shifted Loewner matrices are extremely structured and, in this paper, we show how to fully exploit their Cauchy-like structure to reduce the cost of computing accurate rational models while avoiding the explicit allocation of [Formula: see text] and [Formula: see text] . In particular, the use of the hierarchically semiseparable format allows us to remarkably lower both the computational cost and the memory requirements of the Loewner framework obtaining a novel scheme whose costs scale with [Formula: see text]

Interpolatory methods for H∞\mathcal{H}_\infty model reduction of multi-input/multi-output systems

We develop here a computationally effective approach for producing high-quality $\mathcal{H}_\infty$-approximations to large scale linear dynamical systems having multiple inputs and multiple outputs (MIMO). We extend an approach for $\mathcal{H}_\infty$ model reduction introduced by Flagg, Beattie, and Gugercin for the single-input/single-output (SISO) setting, which combined ideas originating in interpolatory $\mathcal{H}_2$-optimal model reduction with complex Chebyshev approximation. Retaining this framework, our approach to the MIMO problem has its principal computational cost dominated by (sparse) linear solves, and so it can remain an effective strategy in many large-scale settings. We are able to avoid computationally demanding $\mathcal{H}_\infty$ norm calculations that are normally required to monitor progress within each optimization cycle through the use of "data-driven" rational approximations that are built upon previously computed function samples. Numerical examples are included that illustrate our approach. We produce high fidelity reduced models having consistently better $\mathcal{H}_\infty$ performance than models produced via balanced truncation; these models often are as good as (and occasionally better than) models produced using optimal Hankel norm approximation as well. In all cases considered, the method described here produces reduced models at far lower cost than is possible with either balanced truncation or optimal Hankel norm approximation

Generating parametric models from tabulated data

This paper presents an approach for generating parametric systems from frequency response measurements performed with respect to the frequency, and also with respect to one or more design parameters (geometry or material properties). These allow for fast construction of models for other parameter values, so it is suited for optimizing a certain performance measure over the design variables. We validate the proposed approach on an example with two parameters

A new approach to modeling multiport systems from frequency-domain data

This paper addresses the problem of modeling systems from measurements of their frequency response. For multiport devices, currently available techniques are expensive. We propose a new approach which is based on a system-theoretic tool, the Loewner matrix pencil constructed in the context of tangential interpolation. Several implementations are presented. They are fast, accurate, they build low order models and are especially designed for a large number of terminals. Moreover, they identify the underlying system, rather than merely fitting the measurements. The numerical results show that our algorithms yield smaller models in less time, when compared to vector fitting

Predicting the air temperature of a building zone by detecting different configurations using a switched system identification technique

Considerable efforts have been made to find a reliable model able to accurately describe and predict the thermal behavior of the indoor environment of a building. Such a model is essential, in particular, for designing climate control strategies for optimizing both the comfort level and the energy consumption. However, a building is a complex system characterized by a nonlinear thermal behavior, so the task to find such a reliable model is rather difficult. This paper aims at overcoming some of these difficulties by presenting a data driven approach based on a switched system identification to detect and model the thermal behavior of a building zone during normal usage. The proposed technique relies on a PieceWise AutoRegressive model with eXogeneous inputs (PWARX) consisting of a set of sub-models with each one of them describing a certain configuration/state of the dwelling, e.g. turning the heating ON/OFF or opening/closing windows, doors and shutters. The approach is data-driven, easy to implement and its computational time is inferior to creating a detailed model under a specialized software. Therefore, it is particularly suitable for providing a quick description of the thermal behavior of existing buildings for which it is possible to install sensors and perform measurements. Using the available measurements, the algorithm is able to detect various configurations as will be shown by two numerical examples. Such a collection of sub-models provides a better temperature estimate than using ARX models, so it will eventually allow to select better strategies for improving energy efficiency. © 2019 Elsevier Lt

Modeling systems based on noisy frequency and time domain measurements

The Loewner matrix framework can identify the underlying system from given noise-free measurements either in the frequency, or in the time domain[1, 2]. This paper provides an analysis of the effects of noise on the performance of the SVD implementation of the Loewner matrix framework for different noise levels and proposes an improved approach which is able to identify an approximation of the original system even for high levels of noise. Moreover, for frequency domain measurements, our framework can handle systems with a large number of inputs and outputs while requiring small computational time and storage

System identification of microwave filters from multiplexers by rational interpolation

Microwave multiplexers are multi-port structures composed of several two-port filters connected to a common junction. This paper addresses the de-embedding problem, in which the goal is to determine the filtering components given the measured scattering parameters of the overall multiplexer at several frequencies. Due to structural properties, the transmission zeros of the filters play a crucial role in this problem, and, consequently, in our approach. We propose a system identification algorithm for deriving a rational model of the filters’ scattering matrix. The approach is based on rational interpolation with derivative constraints, with the interpolation conditions being located precisely at the filters’ transmission zeros

De-embedding multiplexers by Schur reduction

This paper proposes a method to extract the characteristics of the filters composing a multiplexer. Scattering measurements of the multiplexer are given at some frequencies, together with information regarding the location of the transmission zeros of each filter. From these measurements we first compute a stable rational approximation of the scattering matrix (transfer function). Then we show that the chain matrices of the filters can be recovered, up to a constant matrix, by unchaining suitable components at each port of the multiplexer. The unchaining operation relies on a Schur reduction for an interpolation problem formulated for the structure thereby obtained, with the interpolation points being precisely the transmission zeros of the filter. Since transmission zeros are often located on the imaginary axis, we are dealing with a boundary interpolation problem. © 2013 IEEE