# Development of an advanced time-varying and nonlinear nonparametric estimation framework for industrial measurements

## Preliminaries

Siemens Industry Software NV (hereinafter referred to as SISW) deals with the measurement and modeling problems of its industrial partners. From one side, SISW develops an advanced hardware and software framework to support the design and verification of mechanical structures such as air and road vehicles, and from the other side SISW provides companies with engineering services including technical advice, design and execution of (on the spot) measurements, and advanced analysis based on experimental data.In this project, we focus on two different issues originated from SISW and its industrial partners.

The first one deals with modal analysis of time-varying (TV) systems.

The second goal focuses on the experimental analysis of nonlinear structural dynamics systems, which is of rising concern in industry.

At the first glance, these two issues may seem not to be interrelated but there is a strong link between them: the application areas are common – vibrational structures – and they are seen from different aspects, dealing with different issues as detailed below.

## Nonparametric time-varying Operational Modal Analysis

### Introduction

The first subproject deals with the time variations of (vibrational) mechanical structures described by the nonparametric Operational Modal Analysis (OMA). OMA is a special identification technique for estimating the modal properties (e.g. resonance frequencies, damping) of structures based on vibration data collected when the structures are under real operating conditions without having access to the excitation signals. This technique can provide the engineers with useful information to understand the dynamic behavior of the underlying structure in real-life usage scenario and it can be used to validate and update the numerical models developed in the design phase [1][2]. OMA is a very important tool because in the case of vibrating structures it is common that the real operating conditions differ significantly from dynamic measurements performed in laboratory conditions. Methods for OMA have hence been developed and now are widely used also in the industrial environment.### Problem formulation

The main issue is that the dynamics of underlying systems may vary significantly when operating in real-life conditions. In this case, advanced modeling is needed taking into account the time-varying (TV) behavior because the unmodeled time variations might lead to instability and structural failures. Contrary to the classical identification frameworks, a further challenge with the OMA framework is that the excitation signal is not known exactly, but it is assumed to be white noise [3].A good example of a time-varying mechanical structure can be, for instance, an airplane. The TV behavior originates from the decreasing weight due to the fuel consumption, and from different surface configurations during take-off, cruise and landing [4]. Moreover, the resonance frequency and damping of most vibrating parts (for instance the wings) of a plane vary as a function of the flight speed and height [5]. A road vehicle example can be, for instance, the use of active suspensions in cars, the pitting corrosion of elements.

It is already shown – [6] – that (linear) TV systems can be nonparametrically described in the time domain with a two dimensional (2D) impulse response function (IRF) or equivalently with a two dimensional frequency response function (FRF) [7].

Similarly to these cases, using the OMA framework, the underlying time-varying systems can be uniquely represented by their scaled 2D FRF The scaling in this case originates from the fact that the input excitation is not known exactly but it is assumed to be white. In this particular case, the FRF is scaled by an unknown factor.

The problem lies in the fact that due to the high number of parameters and the underdetermined system of linear equations, the estimation procedure is not trivial. Using nonparametric modeling, these equations will have very high degrees of freedom [6].

In practice this means that we have infinitely many solutions, which are equally possible, when only the data are considered. As a consequence, time-varying systems cannot be uniquely determined from a single set of (input and) output signals – unlike in the general case of linear time invariant systems. Additional user selected properties, e.g. smoothness, will be imposed to select a unique model.

The main challenge is to build accurate models which can track the varying dynamics of these systems, while using as few experiments as possible.

### The-state-of-the art

The identification literature is rich on different – typically parametric – time-varying identification techniques (e.g. [13]-[17]). Apart from the below-mentioned technique, each of the existing techniques requires the knowledge of either the excitation or the reference signal(s), respectively.At SISW, at the present state, a special short-time Fourier transform technique [9] is used: models of time-varying systems are built by using auto-correlation techniques on short data records [10][11][12]. Using a shifted time window results in a series of scaled FRFs without imposing a connection between the adjacent FRFs. However, as it has been shown by [9], this technique can provide an accurate model only if the time-variations are very slow. Unfortunately, this cannot be guaranteed as we do not have direct influence on the type and rate of the time-variations.

## Dealing with nonlinearities of industrial measurements

### Introduction

The second subproject deals with the linearity measures of vibration testing (structural dynamic testing) techniques, when a physical prototype is available. The goal of the vibration testing is to obtain experimental data of the whole vibrating structure such as road and air vehicles. Using these data it is possible to validate and to improve the dynamic models of systems under test. In aircraft engineering, these models are for instance used to predict the flutter behaviour, and to take into account safety measures.### Problem formulation and the state-of-the-art

Vibration testings are typically at the end of the development process. There is a very high time pressure on the available time for this test, due to limited availability of the fully assembled physical prototypes, and due to the tight deadline of the production date – since the competition on the market is very high. On the other hand, the vibration testing methods are very important because they help to improve the product quality and to avoid safety and comfort issues.The increasing needs for higher accuracy and faster testing techniques inspirited a lot of international researches ([18]– [21]).

For more than three decades, the use of the phase resonance method (known as normal mode testing) ([22]–[25]) has been almost exclusively required for vibration testing. The main disadvantage is that it is a very time-consuming testing procedure. Therefore, the phase resonance method is complemented and nowadays almost fully substituted by so-called phase separation techniques that find the modes by evaluating FRFs.

Since the development of advanced digital signal processing algorithms and the increased computational capability, it became possible to use complex input signals such that a large variety of shaker excitation signals can be used to experimentally determine the broadband FRFs, which are required in phase separation testing.

One of the (best) possibilities is the usage of special multisines because they can avoid spectral leakage, inconsistency, non-persistency, and they provide a handy, robust solution to build linear models (FRFs) and to detect the level and type of nonlinearities [26][27][28].

The state-of-the-art knowledge is already available for single-input, single-output (SISO) systems at the VUB [26][27].

### Proposal

The primary goal is to extend the nonlinear nonparametric SISO analysis techniques to MIMO case with a typical number of input channels between 2 and 10. The secondary goal is to develop a user-friendly software framework which provides the user with most relevant information needed for the nonlinear classification. Where the use of multiple uncorrelated pseudorandom (i.e. special case of a multisine) excitation signals for MIMO FRF estimation is well-established in industry, the use of special multisines and special averaging techniques is not yet a proven technology. The extension to MIMO FRF estimation of the underlying linear system may be straightforward, but the use of multiple uncorrelated special multisines for non-linearity characterization still needs to be elaborated and tuned towards mechanical testing (e.g. need for taking shaker-structure interaction into account)## References

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# A system identification framework for multi-fidelity modelling

## Preliminaries

The goal of this project is to develop simple models of complex systems using data-driven modelling techniques.In practice, everything in thermodynamics and fluid mechanics is to some extent unsteady, turbulent, and/or nonlinear. FLOW has developed a strong expertise in using high-fidelity numerical and experimental techniques to study these complex phenomena. However, high-fidelity techniques are often too computationally expensive for crucial engineering applications such as control or optimisation. Presently, low-fidelity alternatives exist, but these are often too strongly simplified to include the relevant unsteadiness, nonlinearity or turbulent nature of thermal-fluid systems.

Therefore, we need simplified models that include the complexity (nonlinearity, turbulence, unsteadiness) that is required for the application, but that are still computationally fast enough.

The main strategic objective of the applicants is to develop a methodology that allows the construction of such simplified models. This methodology will first be developed specifically for the thermal-fluid applications that are currently in the area of expertise of the applicants. This will strengthen the collaborations and interactions within the FLOW team and create new interdisciplinary research opportunities in the strategically-important area of sustainable energy production.