Loewner Framework Research Articles

Noise-robust modal parameter identification and damage assessment for aero-structures

PurposeGround vibration testing is critical for aircraft design and certification. Fast relaxed vector fitting (FRVF) and Loewner framework (LF), recently extended to modal parameter extraction in mechanical systems to address the computational challenges of time and frequency domain techniques, are applied for damage detection on aeronautically relevant structures.Design/methodology/approachFRVF and LF are applied to numerical datasets to assess noise robustness and performance for damage detection. Computational efficiency is also evaluated. In addition, they are applied to a novel damage detection benchmark of a high aspect ratio wing, comparing their performance with the state-of-the-art method N4SID.FindingsFRVF and LF detect structural changes effectively; LF exhibits better noise robustness, while FRVF is more computationally efficient.Practical implicationsLF is recommended for noisy measurements.Originality/valueTo the best of the authors’ knowledge, this is the first study in which the LF and FRVF are applied for the extraction of the modal parameters in aeronautically relevant structures. In addition, a novel damage detection benchmark of a high-aspect-ratio wing is introduced.

Distribution of Relaxation Times Analysis of PEMFC through Loewner Framework: A Direct and Regularization Free Approach

The analysis and interpretation of electrochemical impedance spectroscopy (EIS) data are often complex and challenging, despite its popularity within the field of electrochemistry. Hence, recently, we have introduced novel data-driven approach for extracting the distribution of relaxation times (DRT) from impedance data in order to improve further analysis [1, 2]. This approach is based on the Loewner framework, which is a versatile and straightforward data-driven modeling method that computes reduced-order models in a state-space representation directly from frequency response data. Thus, it allows estimation of DRT without need of regularization procedures or iterative optimization algorithms.The effectiveness of this approach is successfully demonstrated by analysing the experimental EIS data of a polymer electrolyte membrane fuel cell (PEMFC). The results show that this method allows clear identification of the individual polarization processes based on their characteristic time constants (see Figure). Furthermore, the peaks observed in the DRT plot are correctly associated with different processes by examining spectra under various operating conditions. This is further validated by employing a physics-based model [3]. In this way, unambiguous fault identification and monitoring of the degradation state of the cell is possible and it can be the foundation for an on-board diagnostic tool.

Construction of Nonlinear Interpolants With Second-Order Equation Structure in the Loewner Framework

This paper studies the construction of nonlinear interpolants consisting of second-order equations in the Loewner framework. First, conditions are given for which a system of second-order equations interpolates sets of right and left tangential data mappings, which results in second-order tangential generalized controllability and observability mappings allowing for the direct treatment of systems in second-order form. Following this, a family of interpolants with second-order equation structure is given. The results are demonstrated by constructing a reduced order model of a two link robotic manipulator in second-order equation form.

Identification of Low Order Systems in a Loewner Framework

This paper considers the problem of non-parametric identification of low-order models from time-domain experimental data using a combination of Caratheodory Fejer and Loewner-based interpolation, followed by a Loewner matrix Balanced Reduction (LBR) step. As we show in the paper, the Loewner matrix is an estimator for the trace norm of a system, playing a role similar to the one played by the Hankel matrix. However, utilizing Zolotarev numbers to establish decay rate bounds for singular values reveals that the decay of singular values in the Loewner matrix is considerably faster than that in the Hankel matrix. Thus, Loewner-based methods yield lower order systems, with the same error bound, than comparable ones based on Hankel matrices. The effectiveness of our method is demonstrated through a numerical example.

Loewner functions for bilinear systems

This work brings together the moment matching approach based on Loewner functions and the classical Loewner framework based on the Loewner pencil in the case of bilinear systems. New Loewner functions are defined based on the bilinear Loewner framework, and a Loewner equivalent model is produced using these functions. This model is composed of infinite series that needs to be truncated in order to be implemented in practice. In this context, a new notion of approximate Loewner equivalence is introduced. In the end, it is shown that the moment matching procedure based on the proposed Loewner functions and the classical interpolatory bilinear Loewner framework both result in κ-Loewner equivalent models, the main difference being that the latter preserves bilinearity at the expense of a higher order.

(Invited) Beyond Electrochemical Impedance Spectroscopy: Advanced Methods for Studying the Dynamics of Electrochemical Processes

The most popular technique for characterizing various electrochemical devices such as fuel cells, water electrolysis or batteries is electrochemical impedance spectroscopy (EIS). However, due to the coupling of dynamic phenomena with similar time constants, EIS often fails to separate the contributions of individual processes to overall performance losses. In this case, obtaining useful information from the EIS spectra can bebe difficult and the interpretation of the observed patterns is unclear. In this talk, I show that new dynamical methods based on the use of nonlinearities in the system response (nonlinear frequency response analysis (NFRA) [1]), non-electrical inputs (concentration-alternating frequency response analysis (cFRA) [2]), or data-driven analyses (Loewner framework [3]) can provide additional information for understanding electrochemical conversion processes. Our current examples are related to the study of polymer electrolyte membrane electrolysis and polymer electrolyte membrane fuel cell.

Toward a certified greedy Loewner framework with minimal sampling

We propose a strategy for greedy sampling in the context of non-intrusive interpolation-based surrogate modeling for frequency-domain problems. We rely on a non-intrusive and cheap error indicator to drive the adaptive selection of the high-fidelity samples on which the surrogate is based. We develop a theoretical framework to support our proposed indicator. We also present several practical approaches for the termination criterion that is used to end the greedy sampling iterations. To showcase our greedy strategy, we numerically test it in combination with the well-known Loewner framework. To this effect, we consider several benchmarks, highlighting the effectiveness of our adaptive approach in approximating the transfer function of complex systems from a few samples.

On the construction and parameterization of interpolants in the Loewner framework

We develop a general method for the construction of interpolants in the Loewner framework for nonlinear differential–algebraic systems. The approach involves building a family of systems preserving the properties of Loewner equivalence and matching of tangential data functions. It is shown that under mild conditions this family of systems parameterizes all interpolants of sufficiently large dimension matching the tangential data while possessing tangential generalized controllability and observability functions with full column and row rank Jacobians. As a result, this family of systems provides all possible degrees of freedom that an interpolant can have. The results are also discussed in the linear setting and, when taken in combination with existing results, provide a broader framework for the construction of linear interpolating systems.

Determination of the distribution of relaxation times through Loewner framework: A direct and versatile approach

In this work, a direct data-driven approach is employed to extract the distribution of relaxation times (DRT) from impedance data. The proposed procedure is based on the Loewner framework, that is a data driven methodology with theoretical basis in the linear system theory. One of the main advantages is that the DRT can be computed without need of regularization procedures or iterative optimization algorithms. We test the Loewner framework based algorithm on synthetic impedance data generated through equivalent circuit models widely used for modeling electrochemical systems. Impedance data set including noise are also processed. In addition, experimental electrochemical impedance spectroscopy (EIS) data are analyzed. The study shows a significant improvement in the accuracy and a high versatility of the methodology. However, the presence of significant amount of noise compromises the reliability of the obtained DRT due to the occurrence of artifact peaks. Some recommendations how to deal with noisy data are discussed.

A tangential interpolation framework for the AAA algorithm

AbstractInterpolation‐based data‐driven methods, such as the Loewner framework or the Adaptive Antoulas‐Anderson (AAA) algorithm, are established and effective approaches to find a realization of a dynamical system from frequency response data (measurements of the system's transfer function). If a system‐theoretic representation of the original model is not available or unfeasible to evaluate efficiently, such reduced realizations enable effective analysis and simulation. This is especially relevant for models of interconnected dynamical systems, which typically have a high number of inputs and outputs to model their coupling conditions correctly.Tangential interpolation is an established strategy to construct accurate reduced‐order models while ensuring a reasonably small size even if many inputs and/or outputs have to be considered. In this contribution, we evaluate the applicability and effectiveness of data‐driven interpolation methods to compute reduced‐order models of dynamical systems with many inputs and outputs. Additionally, we extend AAA to a tangential interpolation setting and thus enable the use of AAA‐like methods in the context of transfer function interpolation for systems with many inputs and outputs.

The Accuracy and Computational Efficiency of the Loewner Framework for the System Identification of Mechanical Systems

The Loewner framework has recently been proposed for the system identification of mechanical systems, mitigating the limitations of current frequency domain fitting processes for the extraction of modal parameters. In this work, the Loewner framework computational performance, in terms of the elapsed time till identification, is assessed. This is investigated on a hybrid, numerical and experimental dataset against two well-established system identification methods (least-squares complex exponential, LSCE, and subspace state space system identification, N4SID). Good results are achieved, in terms of better accuracy than LSCE and better computational performance than N4SID.

A Loewner-Based System Identification and Structural Health Monitoring Approach for Mechanical Systems

Data-driven structural health monitoring (SHM) requires precise estimates of the target system behaviour. In this sense, SHM by means of modal parameters is strictly linked to system identification (SI). However, existing frequency-domain SI techniques have several theoretical and practical drawbacks. This paper proposes using an input-output system identification technique based on rational interpolation, known as the Loewner framework (LF), to estimate the modal properties of mechanical systems. Pioneeringly, the Loewner framework mode shapes and natural frequencies estimated by LF are then applied as damage-sensitive features for damage detection. To assess its capability, the Loewner framework is validated on both numerical and experimental datasets and compared to established system identification techniques. Promising results are achieved in terms of accuracy and reliability.

A data-driven nonlinear frequency response approach based on the Loewner framework: preliminary analysis

We propose a hybrid method based on the combination of computed-aided nonlinear frequency response analysis with the Loewner framework, for the characterization of nonlinear dynamical processes with application in electrochemistry. The method is data-driven, i.e., requiring only samples of the first two generalized transfer functions of the underlying system, given as values of the sampled nonlinear frequency response. Then, the established system fitting and complexity reduction approach (in the frequency domain), known as the Loewner framework, is used to extract the system's invariant quantities. In this analysis, we have used a nonlinear electrical circuit model as a test case, for which the new method provides good results.

Model Reduction for Linear Port-Hamiltonian Systems in the Loewner Framework

The problem of model order reduction with assignment and preservation of port-Hamiltonian structure in the reduced order model is tackled in the Loewner framework. Given a set of right-tangential interpolation data, the (subset of) left-tangential interpolation data that allow for the construction of an interpolant possessing port-Hamiltonian structure is characterized. Conditions under which an interpolant retains the underlying port-Hamiltonian structure of the system generating the data are given by requiring a particular structure of the generalized observability matrix.

Interpolants With Second-Order Structure in the Loewner Framework

We consider the problem of assigning a particular structure to interpolants constructed in the Loewner framework. Specifically, a dynamically extended family of interpolants matching sets of right and left tangential data is used to parameterize a family of systems matching the tangential data while also possessing a specific second-order structure. At the cost of adding states to the interpolant, the approach does not require the addition of any strict conditions on the structure of the tangential data. Conditions are stated under which, if satisfied, the interpolant corresponds to a system with physical meaning, and an illustrative example is provided.

Enforcing Stability of Linear Interpolants in the Loewner Framework

Enforcing Stability of Linear Interpolants in the Loewner Framework

Black box stability preserving reduction techniques in the Loewner framework for the efficient time domain simulation of dynamical systems with damping treatments

New applications, such as material characterization, auralization and virtual sensing, have renewed the need and interest in efficient transient time-domain simulations of dynamical systems. However, the typical required numerical simulations for these applications are cumbersome to compute due to the high computational burden involved in the calculation of mechanical systems. Model order reduction techniques are important enablers to reduce this computation time. However, many reduction techniques fail when viscous or thermal damping is present in the system. This work elaborates on two non-intrusive reduction techniques using the Loewner framework which can serve as a tool for the efficient time domain simulation of second order dynamical systems with damping treatments. The reduced models are constructed using frequency domain data, however, for time domain simulations, the reduced models need to be real-valued and stable. One methodology exploits the well-known inverse fast Fourier transform, while the second methodology employs a reduced order model which serves as an input for a time marching technique. Although the Loewner reduction technique does not preserve the stability of the original system, both proposed methodologies show to be able to return an accurate and sufficiently stable time domain response.

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 {mathbb {L}}in {mathbb {C}}^{Ntimes N} and {mathbb {S}}in {mathbb {C}}^{Ntimes N} 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 {mathbb {S}} and {mathbb {L}} 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 {mathbb {L}} and {mathbb {S}} 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 {mathbb {L}} and {mathbb {S}}. 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 N log N.

Exact and Inexact Lifting Transformations of Nonlinear Dynamical Systems: Transfer Functions, Equivalence, and Complexity Reduction

In this work, we deal with the problem of approximating and equivalently formulating generic nonlinear systems by means of specific classes thereof. Bilinear and quadratic-bilinear systems accomplish precisely this goal. Hence, by means of exact and inexact lifting transformations, we are able to reformulate the original nonlinear dynamics into a different, more simplified format. Additionally, we study the problem of complexity/model reduction of large-scale lifted models of nonlinear systems from data. The method under consideration is the Loewner framework, an established data-driven approach that requires samples of input–output mappings. The latter are known as generalized transfer functions, which are appropriately defined for both bilinear and quadratic-bilinear systems. We show connections between these mappings as well as between the matrices of reduced-order models. Finally, we illustrate the theoretical discussion with two numerical examples.

Data-Driven Balancing of Linear Dynamical Systems

We present a novel reformulation of balanced truncation, a classical model reduction method. The principal innovation that we introduce comes through the use of system response data that has been either measured or computed, without reference to any prescribed realization of the original model. Data are represented by sampled values of the transfer function or the impulse response corresponding to the original model. We discuss parallels that our approach bears with the Loewner framework, another popular data-driven method. We illustrate our approach numerically in both continuous-time and discrete-time cases.