Emmanuel Ramasso

Benoît Verdin

Gaël Chevallier

MONITORING A BOLTED VIBRATING STRUCTURE USING MULTIPLE ACOUSTIC EMISSION SENSORS: A BENCHMARK

Keywords: Acoustic emission, Bolted joints, Health monitoring, Laser vibrometry, Benchmarking

The data set presented in this work, called ORION-AE, is made of raw AE data streams collected by three different AE sensors and a laser vibrometer during five campaigns of measurements by varying the tightening conditions of two bolted plates submitted to harmonic vibration tests. With seven different operating conditions, this data set was designed to challenge supervised and unsupervised machine/deep learning as well as signal processing methods which are developed for material characterization or Structural Health Monitoring (SHM). One motivation of this work was to create a common benchmark for comparing data-driven methods dedicated to AE data interpretation. The data set is made of time-series collected during an experiment designed to reproduce the loosening phenomenon observed in aeronautics, automotive or civil engineering structures where parts are assembled together by means of bolted joints. Monitoring loosening in jointed structures during operation remains challenging because contact and friction in bolted joints induce a nonlinear stochastic behavior.

Summary

Acoustic emission (AE) is one of the most promising non-destructive techniques for its ability to detect changes in the integrity of materials at the microscale. When a damage occurs, an elastic wave is released which propagates onto the surface creating displacements ranging from picometers to nanometers. The displacements are detected by highly sensitive piezoelectric sensors permanently attached onto a material. Data are continuously collected at sampling rate around 5 MHz. The obtained AE data stream (time-series) is then processed by algorithms to detect AE signals related to damages. AE is used in many laboratories for materials characterization [START_REF] Sendrowicz | Challenges and accomplishments in mechanical testing instrumented by in situ techniques: Infrared thermography, digital image correlation, and acoustic emission[END_REF] and in industrial applications for real time monitoring of manufacturing process [START_REF] Sergey | Deep learning for in situ and real-time quality monitoring in additive manufacturing using acoustic emission[END_REF] or storage facilities [START_REF] Ai | Source localization on large-scale canisters for used nuclear fuel storage using optimal number of acoustic emission sensors[END_REF]. Despite many publications on the topic, the precise identification of the source of AE signals remains a challenge. Indeed, AE source identification is an inverse problem which is difficult to solve due to the sensitivity of the sensors which provide many AE signals corrupted by noise, as well as due to the effects of wave propagation in damaged materials which creates wave scattering [START_REF] Schubert | Numerical modeling of acoustic emission sources and wave propagation in concrete[END_REF].

AE signals clustering:

The signals obtained in the previous step fed a clustering method which aims at creating groups (clusters) of AE signals [START_REF] Zhou | Cluster analysis of acoustic emission signals and deformation measurement for delaminated glass fiber epoxy composites[END_REF].

3. Clustering validation: The validation of the results obtained in the previous step is performed in a subjective manner because of the lack of prior knowledge of AE sources. To circumvent this problem, AE users apply different clustering strategies in the previous step and then compute clustering validity indices [START_REF] Vendramin | On the combination of relative clustering validity criteria[END_REF][START_REF] Vendramin | Relative clustering validity criteria: A comparative overview[END_REF] which provide a scalar value that allows end-users to compare different results. The results can also be visually analyzed, like in interactive clustering [START_REF] Bae | Interactive clustering: a comprehensive review[END_REF], to evaluate whether the clusters can be related to AE sources with a physical meaning [START_REF] Sendrowicz | Challenges and accomplishments in mechanical testing instrumented by in situ techniques: Infrared thermography, digital image correlation, and acoustic emission[END_REF].

An alternative to these three-steps strategy is based on anomaly detection [START_REF] Campo | Online feature learning for condition monitoring of rotating machinery[END_REF][START_REF] Nasir | A review on deep learning in machining and tool monitoring: methods, opportunities, and challenges[END_REF]. A baseline is considered to train a one-class supervised learning algorithm which can then be used to generate a health indicator for online monitoring [START_REF] Farrar | Structural Health Monitoring: A Machine Learning Perspective[END_REF]. Anomaly detection was also exploited for clustering in [START_REF] Ramasso | Learning the representation of raw acoustic emission signals by direct generative modelling and its use in chronology-based clusters identification[END_REF].

In real applications or for complex materials, AE signals detection (step 1), clustering (step 2) or validation (step 3), can be difficult. In that case, specific algorithms are developed. However, there is no common AE data sets that can serve as references (benchmark) to compare the methods of the literature. This work presents a benchmark data set that can be used for this purpose.

The experiment was designed to reproduce the loosening phenomenon observed in aeronautics, automotive or civil engineering structures where parts are assembled together by means of bolted joints. During the service life of a bolted structures, the torque can evolve according to operational conditions such as loading, in particular when these structures are submitted to vibrations. Monitoring the loosening condition in bolted structures during operation remains challenging because contact and friction in bolted joints induce a nonlinear stochastic behavior [START_REF] Chen | Quantitative monitoring of bolt looseness using multichannel piezoelectric active sensing and cbam-based convolutional neural network[END_REF][START_REF] Li | Monitoring of bolt looseness using piezoelectric transducers: Threedimensional numerical modeling with experimental verification[END_REF][START_REF] Majid | A review paper on looseness detection methods in bolted structures[END_REF][START_REF] Zhang | A quantitative investigation on vibration durability of viscoelastic relaxation in bolted composite joints[END_REF][START_REF] Asamene | Analysis of experimentally generated friction related acoustic emission signals[END_REF][START_REF] Amerini | Structural health monitoring of bolted joints using linear and nonlinear acoustic/ultrasound methods[END_REF]. A common solution consists in measuring the pretightening force using a strain gauge bonded on the top surface of a bolt head [START_REF] Wang | Bolt pretightening force measurement based on strain distribution of bolt head surface[END_REF]. This is a local approach since we need to instrument every bolt. On the other hand, global techniques make use of only a few sensors for bolt condition monitoring. There are two categories of global techniques: active and passive approaches. Active approaches such as lamb waves were used in the past and consists in using a transmitter of high harmonics waveforms tuned to be sensitive to the changes in the contacts. The receiver is used to collect the waveforms after their propagation in the bolted structure. Machine learning can then be used to build an anomaly detector like in [START_REF] Farrar | Structural Health Monitoring: A Machine Learning Perspective[END_REF] and [START_REF] Nasir | A review on deep learning in machining and tool monitoring: methods, opportunities, and challenges[END_REF]. Conversely, passive approaches do not use an external source, but the transmitter is actually the structure itself (if a damage occurs). Therefore, a continuous reading of the piezoelectric sensors is necessary in order to not miss any damage or event. The use of AE sensors for condition monitoring of bolts is not frequent. For example, in recent works [START_REF] Zhang | Continuous monitoring of tightening condition of single-lap bolted composite joints using intrinsic mode functions of acoustic emission signals: a proof-of-concept study[END_REF][START_REF] Chevallier | Detection and analysis of loosening in jointed structures using acoustic emission sensors and smart bolts[END_REF], it was shown that high sensitivity piezoelectric sensors used in this technique are able to detect AE signals despite low signal-to-noise ratio in bolted joints under vibration.

In the context of stationary and harmonic vibration tests, the bolts can be subject to self-loosening under vibrations, eventually leading to a critical failure of the assembly. Therefore, it is of paramount importance to develop sensing strategies and algorithms for early loosening estimation. The main characteristics of this data set are the following:

• During the tests, the tightening level of one of the bolts was controlled to reach seven different values, from highly tightened to loosened. Therefore, the end-user knows that the tightening level has been set to a specific value for some periods of the data stream. This reference can be used either to train supervised (deep/transfer) learning methods (with possibly 7 classes) and for unsupervised or semi supervised learning. This reference can thus be helpful to validate steps 2 and 3 in the aforementioned methodology.

• The data set is made of raw acoustic emission data streams acquired by three different sensors and a laser vibrometer. Multisensor data fusion can thus be performed and evaluated using the reference. This can be helpful to improve algorithms in steps 2 and 3.

• The vibrometer data also allows use users to evaluate the performance of wave-picking algorithms with low signal-to-noise ratio. This can be helpful to validate step 1 in the aforementioned methodology.

Data Description

The data set was obtained on a test rig called ORION [START_REF] Chevallier | Detection and analysis of loosening in jointed structures using acoustic emission sensors and smart bolts[END_REF][START_REF] Rafael De O Teloli | Good practices for designing and experimental testing of dynamically excited jointed structures: The orion beam[END_REF]. It is constituted of a jointed structure, dynamically loaded with a vibration shaker and monitored with acoustic emission, force, and velocity sensors.

For each series of measurements, four sensors were used simultaneously: a laser vibrometer and three different AE sensors (micro80, F50A, micro200HF), each with a given frequency band, all sampled at 5 MHz, yielding approximately between 1.4 and 1.9 GB of data. A total of five series of measurements were performed denoted as measurementSeries_B, measurementSeries_C, measurementSeries_D, measurementSeries_E and measurementSeries_F.

The organization of repositories and files is depicted in Figure 1 for series of measurements B. There is one folder for each series of measurements. The ORION-AE data set is thus composed of five folders. For each series of measurements, there are seven subfolders corresponding to seven tightening levels: 60, 50, 40, 30, 20, 10, 05 cNm, except for measurementSeries_C for which 20 cNm is missing.

Note:

The order is from 60 cNm to 5 cNm, not the reverse Note: Timestamps must be used to sort the files (by name) to read them in the good order, but must not be used to assign the timestamps to AE signals within files (can be done by a Matlab script provided)

Each file is composed of four variables: « A » for micro80 sensor « B » for F50A sensor « C » for micro200HF « D » for laser vibrometer Note: Name of variables not to be confused with campaigns It is important to read the subfolders in this precise order, otherwise the physical meaning of the data becomes wrong. Th upper bolt have been untightened with a precise sequence. For example, reading from 5 cNm to 60 cNm (reverse order) does not correspond to the "tightening" of bolts since the process is not reversible in terms of acoustic emission characteristics.

Each subfolder is made of .mat files generated using MATLAB 2016b. There is about one file per second. The files in a subfolder are named according to the timestamps (time of recording) and must be read in the correct order. Each file has the following format: • Variable "A" corresponding to micro80 sensor data (sampling frequency: 5MHz).

• Variable "B" corresponding to F50A sensor data (sampling frequency: 5MHz).

• Variable "C" corresponding to micro200HF sensor data (sampling frequency: 5MHz).

• Variable "D" corresponding to laser velocimeter data (sampling frequency: 5MHz).

The number of samples is about 5 × 4 millions per file (each file corresponds to about 1 second), with all sensors data sampled at 5 MHz. The labels corresponding to the sensors must not be confused with the labels corresponding to the test campaigns. Illustrations of the data are given in Figure 4 for sensor "F50A" (variable "B" in a file) in measurementSeries_B. About 6 periods are depicted (using the red curve representing the displacement of the upper beam).

Figure 5 depicts the acoustic emission and laser vibrometer data, superimposed with the labels (tightening levels) evolving from 1 to 7, for measurement B. This figure (red curve) shows that the control of the displacement of the beam by a feedback from the laser vibrometer (as explained in the "Method" Section) leads to a similar displacement during a whole test (about 70 s). In this figure, the green curve representing the acoustic emission shows that the amplitude can not be used as a feature to discriminate the levels.

It is important to note that the real timestamps in filenames must not be taken as the timestamps of the AE signals in a data stream. The real timestamps must be calculated from the exact number of samples in each file, starting from t = 0 s for 60 cNm. There are about 10 s of continuous recording of data per level and for the four sensors. The exact duration of a period can be found according to the number of files in each subfolder, the number of points per file and the sampling frequency. The precise values are given in Table Figure 2: Sample of the data in measurements series "B" with sensor "micro80" for each tightening level (to be continued). Blue curve: AE signal; red curve: laser vibrometer data (harmonic vibration signal at 100 Hz with amplitude control). Figure 5: Tightening levels, acoustic emission and laser vibrometer data superimposed for measurements "B" and sensor micro-200-HF (variable C). Due to the amount of data involved, a subsampling by a factor of 10 was applied for each file. The x-axis is here represented using the time of test (about 70 s), starting from 60 cNm from the left (around between t ∈ [0, 10] s) to 5 cNm on the right (from t > 60 s).

Methods

The ORION test rig (Figure 6) has been designed for the study of vibration damping and makes the vibration tests highly reproducible as shown in [START_REF] Chevallier | Detection and analysis of loosening in jointed structures using acoustic emission sensors and smart bolts[END_REF][START_REF] Rafael De O Teloli | Good practices for designing and experimental testing of dynamically excited jointed structures: The orion beam[END_REF]. It is a jointed structure made of two plates manufactured with 2024 aluminium alloy, linked together by three M4 bolts (Figure 6). Contact patches are machined overlays which have been added at each bolt connection to retain the contact between both beams in a small area, minimizing uncertainties on the structure's response and enhancing the repeatability between measurements. The contact patches have an area of 12 × 12 mm 2 and are 2 mm thick. The central bolt is dedicated to "static" functions, i.e. to ensure structural integrity and provide resistance to dynamic loads without substantial stiffness changes. The external bolts, in turn, perform "damping" functions, i.e. to increase energy dissipation due to frictional contact. It is worth noticing that these two functions have to be monitored in order to ensure the integrity (central bolt) and the damping (lateral bolts). The harmonic excitation load was applied using a Tira TV51120, electromagnetic shaker which can deliver a 200N force. The structure was submitted to a 100 Hz harmonic excitation force during about 10 s. The force was measured using a PCB 208C02 piezoelectric force sensor.

The vibration magnitude was measured using a Polytec PSV500Xtra, laser vibrometer. The obtained laser data allowed us to make sure that the amplitude of the displacement of the top of the upper plate remains constant for all tightening levels.

Seven tightening levels, corresponding to seven operating conditions of the structure under vibration, were applied on the upper bolt. The tightening level was first set to 60 cNm with a torque screwdriver. After about a 10 s vibration test, the shaker was stopped and this vibration test was repeated after a torque modification at 50 cNm. Then torque modifications at 40, 30, 20, 10 and 5 cNm were applied. The change in the tightening level was made using a CDI torque screwdrivers. The torque was checked three times at each change.

The acoustic emission sensors used were a "micro-200-HF", a "micro-80" and a "F50A", made by Euro-Physical Acoustics. They were connected to a preamplifer set to 60 dB (model 2/4/6 preamplifier made by Europhysical acoustics). Their detailed characteristics, such as dimensions and frequency bands, are provided in the datasheets enclosed in the data set, with a summary of the main ones given in Table 2. The sensors were attached onto the lower plate (5 cm above the end of the plate) using a silicone grease. All data were sampled at 5 MHz using a Picoscope 4824 [START_REF] Schubert | Numerical modeling of acoustic emission sources and wave propagation in concrete[END_REF] In ORION-AE data set, we know that, during each period of 10 s, the tightening level has been set to a specific value. The tightening level can be considered as constant during each period, and therefore can be used as a reference. However, during each cycle of the vibration test for a given tightening level, different AE sources can generate AE signals and those sources may be activated or not, depending on the tribological conditions which are not known. The tightening level may thus slightly change during the period of 10 seconds. The tribological conditions may also be different from a campaign to another. Despite this variability, preliminary results shown in [START_REF] Ramasso | Clustering acoustic emission data streams with sequentially appearing clusters using mixture models[END_REF] demonstrated that the content of AE signals in those series of measurements is sufficient to discriminate between the periods, which seem to corroborate that, first, the AE sources have indeed a signature depending on the tightening level, and second, that the tightening level and the amplitude of the vibrations (controlled by a closed-loop using the laser vibrometer) remains constant during the period of 10 s.

ORION-AE for AE signal detection

In addition to machine learning tasks, this data set can be used for signal processing and in particular wave-picking (step 1, presented in introduction). As for seismic wave detection [START_REF] Baillard | An automatic kurtosis-based p-and s-phase picker designed for local seismic networks[END_REF], this step aims at finding relevant signals in a data stream. Wave-picking is one the most important task in acoustic emission analysis because it paves the way for feature extraction, classification, localization of acoustic emission sources and interpretation by machine learning.

In ORION-AE, the AE sources are related to tribological phenomenon characterized by low amplitude and with low signal-to-noise ratio. Denoising methods are thus generally used as a preprocessing step. Figure 7 depicts the result of wavelet denoising with 14 levels of decomposition and using a Daubechies "db45" wavelet (the same settings as [START_REF] Kharrat | A signal processing approach for enhanced acoustic emission data analysis in high activity systems: Application to organic matrix composites[END_REF]). This was done for two sensors in the same chunk of data as shown in Fig. 7(a) and 7(b). It can be observed that the responses of the sensors differ since they have different characteristics (in particular the sensitivity and frequency band). We can also observe the presence of AE signals that occur at each cycle with some regularity.

We took the starting time, called onsets, of each signal detected after wavelet denoising (from Figure 7 using [START_REF] Kharrat | A signal processing approach for enhanced acoustic emission data analysis in high activity systems: Application to organic matrix composites[END_REF]). Onsets were then reported onto the vibrometer data in Figure 8. Each blue circle corresponds to one AE signal detected. The vertical position represents the amplitude of the displacement (measured by the vibrometer) when the signal was detected. We can observe that the amplitude is very similar between cycles (there is about 1 second of test represented on this figure), which means that the activation of the AE sources is highly reproducible between cycles. Similar results have been observed on other levels and considering measurements B to F. However, we also observed that several AE sources can occur in some levels, and that the average amplitude of displacement can change. It means that the instant of activation of the AE sources (related to the signals) seems to depend on the tightening levels. These observations can be dependent on the wave-picking algorithm used (here [START_REF] Kharrat | A signal processing approach for enhanced acoustic emission data analysis in high activity systems: Application to organic matrix composites[END_REF]) and should be confirmed in the future.

Possible challenges

ORION-AE data set can be used for different tasks.

Train supervised learning methods The classification task can consider up to 7 classes in order to discriminate between the levels using various machine and deep learning methods such as support vector machines or neural networks [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF][START_REF] Frank Hesser | Identification of acoustic emission sources for structural health monitoring applications based on convolutional neural networks and deep transfer learning[END_REF]. By considering only the tightened levels, such as 60 cNm and 50 cNm, it can also be used to evaluate anomaly detectors based on autoencoders, self-organizing maps or one-class classifiers [START_REF] Farrar | Structural Health Monitoring: A Machine Learning Perspective[END_REF][START_REF] Nasir | A review on deep learning in machining and tool monitoring: methods, opportunities, and challenges[END_REF] precisely obtained by using a zero-crossing detector on the vibrometer data. Illustration of such CWT images are depicted in Figure 9 using two measurements series (B, D).

Data normalization aims at compensating time-varying operational variations which can occur in a system. According to our experience on this data set, it seems that, for a given tightening level, the signature of the signals accross campaigns seems to change. This can be due to the tribological conditions in the contacts that change between levels and campaigns. The classification algorithms (using deep learning and autoencoders) that we tested seem to work well when we mix levels and campaigns. However, the classification performance drastically drops when we distinguish the campaigns (for example training on B, C, D, E and testing on F supposed unknown as in a SHM task). A data normalization is thus needed (starting from data in the first levels, for example 60 cNm or {60, 50} cNm). This is an important topic in SHM as presented in [9, Chap. 12]. Validate unsupervised learning (clustering) using the tightening levels as a reference to be compared with, using, for example, the Adjusted Rand Index [START_REF] Vinh | Information theoretic measures for clusterings comparison: Is a correction for chance necessary?[END_REF]. Initial results on this data set can be found in [START_REF] Ramasso | Clustering acoustic emission data streams with sequentially appearing clusters using mixture models[END_REF] using various algorithms.

Challenge wave-picking algorithms by trying to find either one set of signals per cycle or a set of several signals per cycle which should occur at similar displacement level along cycles for a given tightening level. Some results have been presented in the previous paragraphs using the method [START_REF] Kharrat | A signal processing approach for enhanced acoustic emission data analysis in high activity systems: Application to organic matrix composites[END_REF].

The generalization capabilities of the methods can be evaluated by considering different campaigns for training and testing. This is a critical point for structural health monitoring tasks.

Potential sources of error or variability

According to our experience on this data set, the low signal-to-noise ratio represents a real challenge to solve the three aforementioned tasks. Some preliminary results have been shown in [START_REF] Ramasso | Clustering acoustic emission data streams with sequentially appearing clusters using mixture models[END_REF] with a good performance of various clustering methods. These results were obtained by considering campaigns independently. Therefore, this study shown that the information hold by the data are relevant for estimating the tightening levels.

The variability accross campaigns can represent a difficulty for simple classification methods, as discussed in the previous section. To tackle this issue, a proper data normalization is needed.

In terms of error in the data set, we are aware that one level is missing: 20 cNm in campaign C. It means that this campaign is made of six classes.

This data set was already used by Master students in 2021 and 2022 for machine learning illustration and a SHM course, at both the University of Franche-Comté and National Engineering School of Mechanics and Microtechnics of Besançon, who reported only one issue concerning the presence of a directory "5cNm" (made of one file) in addition to "05cNm" (made of 9 files). This folder appears after the downloading but not on the online directory. We recommend to delete the former after having transfered the file in the latter. This "phantom" directory is a small issue reported to the support of Harvard Dataverse and still under consideration at the time of writing the present work.

Finally, we acknowledge that this data set is not representative of all possible data sets that can be collected using the AE technique. The purpose of this data set is simply to provide a common reference to researchers in the AE community interested in developing new machine / deep learning or pattern recognition methods or new algorithms dedicated for AE data interpretation. With the expansion of deep learning, we think that this data set can be useful for benchmarking, which can give a hand in discovering new and better ways of processing and interpreting acoustic emission data.

Data availability

Directory name: Harvard Dataverse.

Figure 1 :

 1 Figure 1: Tree structure of the repositories and files exemplified for series of measurements "B". The root directory on Dataverse also contains two additional repositories: One with images illustrating the content of the data in each level and campaign, and one with the sensors datasheet. Finally a Matlab script is provided to read the files and correctly extract the timestamps.

  salves_out_XcNm_Y_Fs5MHz_Tf1s_2019-04-02-HH-MM-SS_1.mat where X ∈ {5, 10, 20, 30, 40, 50, 60} representing the tightening level, Y ∈ {B, C, D, E, F } for the series of measurements. The production date is "2019-04-02" and the timestamp began at HH-MM-SS. For example, salves_out_05cNm_B_Fs5MHz_Tf1s_2019-04-02-16-22-09_1.mat represents AE data for 5 cNm in series of measurements B taken at 16:22:09. Each .mat file is composed of vectors of data with the following names:

Figure 3 :Figure 4 :

 34 Figure 3: Sample of the data in measurements series "B" with sensor "micro80" for each tightening level. Blue curve: AE signal; red curve: laser vibrometer data (harmonic vibration signal at 100 Hz with amplitude control). (Figure to be continued) 6

Figure 6 :

 6 Figure 6: Setup description: part dimensions, sensors positions, bolts positions

  After wavelet denoising: Measurement B, first file in 60 cNm, sensor "micro-200-HF" (sensor 3).

Figure 7 :

 7 Figure 7: Presence of signals in each cycle after wavelet denoising using two sensors (on the same period of time). For a given tightening level, the displacement measured by the vibrometer is quite close for each signal found at different cycles showing the reproducibility of the test rig.

Figure 8 :

 8 Figure 8: Acoustic emission onsets found by [11] and positioned onto vibrometer data for measurement B, first file in 60 cNm, sensor "micro-200-HF". The vertical position of the blue circle corresponds to the amount of displacement required to activate the AE sources. A similar amount is found in each cycle which illustrate the reproducibility of the test rig.

Figure 9 :

 9 Figure 9: Continuous wavelet transform (CWT) images extracted for some cycles in sensor "F50A" (vector B) in measurement B (top figures) and in measurement D (bottom figures). Extraction of cycles can be automatically made by detecting zero-crossing in the vibrometer data (vector D).

Table 1 :

 1 [START_REF] Ai | Source localization on large-scale canisters for used nuclear fuel storage using optimal number of acoustic emission sensors[END_REF]. A Matlab code (ORION_AE_sample_read_files.m) is provided in the data directory to read the files and reproduce the duration of each period. Duration in seconds of each period for a given tightening level. The exact duration of a period can be found according to the number of files in each subfolder, the number of points per file and the sampling frequency. This table can be reproduced using a A Matlab code (ORION_AE_sample_read_files.m) provided in the data directory.

	Meas. series 60 cNm 50 cNm 40 cNm 30 cNm 20 cNm 10 cNm	5 cNm
	B	10.0000 10.0000 10.0000 9.0170 10.0000 10.7725 9.2275
	C	10.0000 10.0000 10.0000 10.0000	N.A.	10.0000 10.0000
	D	10.0000 10.0000 10.8119 9.1881 10.0000 10.0000 10.0000
	E	10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000
	F	10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000

Table 2 :

 2 MHz bandwidth, low noise, 12-bit resolution, 256 MS buffer memory, USB 3) connected to a linux PC with MATLAB ® 2016b and the Data Acquisition ® toolbox. The datasheet of the Picoscope is provided in the data set. Frequency and sensitivity of the acoustic emission sensors used. The data sheets are provided in the data set.

		Units	F50A	micro80 micro200HF
	Peak Sensitivity, Ref V/(m/s)	dB	65	57	62
	Operating Frequency Range (kHz)	kHz 200-800 kHz 200-900	500-4500
	Resonant Frequency, Ref V/(m/s)	kHz	100 kHz	250	2500

  . In both cases, it is possible to use convolutionnal neural networks (CNN) taking as inputs the RGB images computed by the continuous wavelet transform (CWT) of the raw signals in each cycle. Cycles can be

		30
										8000
		20								6000
	stream after "denoising" (mV)	-20 -10 0 10								-4000 -2000 0 4000 2000	Vibrometer data (mV)
	AE	-30								-6000
										-8000
		0.3 -40	0.31	0.32	0.33	0.34	0.35	0.36	0.37	0.38
						Temps (s)
		(a) After wavelet denoising: Measurement B, first file in 60 cNm, sensor "F50A" (sensor 2).
		0.3	0.31	0.32	0.33	0.34	0.35	0.36	0.37	0.38
						Temps (s)

Acknowledgement

This work was partly carried out in the framework of: EIPHI Graduate school (contract ANR-17-EURE-0002), project RESEM-COALESCENCE funded by "Institut de Recherche Technologique" Matériaux Métallurgie Procédés (IRT M2P) and "Agence Nationale de la Recherche" (ANR); project CLIMA funded by the "Fond Unique Interministériel" (FUI 19). The authors are also thankful to MIFHySTO and AMETISTE platforms.

Data identification number: 10.7910/DVN/FBRDU0. Direct URL to data: https://doi.org/10.7910/DVN/FBRDU0, comprising a Matlab code to read the files. Codes and main results will be updated on https://github.com/emmanuelramasso/ORION_AE_acoustic_emission_ multisensor_datasets_bolts_loosening.


A Pister

C Prieur

JD Fekete

Visual Queries on Bipartite Multivariate Dynamic Social Networks

Figure 1: Proposed interface, used to search for people who are mutually guarantors to each other in construction contracts [CR18], retrieved from dynamic visual queries. (A) shows the visual query the user created. (B) shows the bipartite node-link view and (C) the document map view. (D) shows the query results panel where one attribute is selected, showing its distribution.

Introduction

Historians and sociologists aim at characterizing the social relationships between people of a period and region of interest. For this, they rely on a corpus of documents-which can be birth certificates, marriage acts, work contracts, or migration forms-which mention people and their social ties. From this corpus, they construct social networks on which they can follow a Social Network Analysis (SNA) [Fre04]. This type of dataset is best modeled using Bipartite Multivariate Dynamic Social Networks, where both documents and persons are modeled as nodes, containing additional information extracted from the original sources, such as the document time and location [Bor09]. A link represent a mention of a person in a document and encode the role of the person in the document (husband, wife, or witness in a marriage act for example). However, most visual analytics tools for social network either enforce a simple person-to-person model [START_REF] Bastian | Gephi: An Open Source Software for Exploring and Manipulating Networks[END_REF] or no model at all [START_REF] Stasko | Jigsaw: supporting investigative analysis[END_REF], and lack complex interactions needed by social sci-entists. We thus present a new visual analytics system to explore Bipartite Multivariate Dynamic Social Networks, based on a visual query system that allows social scientists to easily filter and characterize specific groups of persons and patterns of interests.

Visual Analytic System

Our proposed tool allows exploring the data with two coordinated views shown in Figure 1: the node-link view and the map view. The node-link view allows users to explore the topology of the graph, while the map allows seeing the geolocation of the documents. Two documents are linked if they share at least one person in common. This way, the map allows to rapidly see in which areas people are interacting. The two views are coordinated and selecting a person or a document in the bipartite view highlights her/him in the map view and vice versa. Moreover, every attribute in the data can be selected to show a plot of its distribution. When selected, the nodes are colored in the two views according their attribute value.

Visual Query

Our visual query system is similar to VERTIGo [CSIP21] and [SW13] for topological queries, but enforces the bipartite network model and allows more complex attribute constraints. It is based on the Neo4j database [neo] and the Cypher language. In contrast to other visual query systems that hide the raw script request, we show both representations (visual and textual) and enforce a one-to-one synchronisation between the two. This way, users can start to build a query visually, and modify it later textually for very complex constraints. Users can build two types of constraints:

Topological: An interactive node-link diagram let users construct patterns visually. They can build documents or person nodes by clicking on the canvas, and links by clicking and dragging between nodes. They can choose any of the roles of the dataset when creating a link, along an ANY link type, representing a link which can be any of the existing role. Attributes: Widgets let create attribute constraints on the created nodes, using the dynamic queries principle [START_REF] Shneiderman | Dynamic Queries for Visual Information Seeking[END_REF]. The widgets designs differ according to the attribute type: text input for nominal attributes, checkboxes for categorical and double sliders for numerical ones. These simple designs allow social scientists to easily make attribute constraints.

When modifying the query visually or textually, the results are directly highlighted in the two views, and summary results are shown along an occurrence table and attribute distribution plots.

Use Cases

We elaborated our system in collaboration with historians and sociologists who have a variety of datasets: marriage acts, birth certificates, construction contracts, work contracts and migration forms. The questions they had on their data often consisted in extracting groups of people or finding document and persons with specific patterns, and studying their characteristics in respect of the rest of the network, or to compare two groups of nodes between them (see supplementary material).

We describe here a use case in collaboration with one historian Here are some of the questions he had that we collected from him, which can be answer with queries:

1. Find the pairs or persons who are mutually guarantor to each other in two separate construction documents. 2. Find the persons who are collaborating with the Zo family.

What are the characteristics of the construction documents in

Torino, and who are the people working there. 4. Who are the persons with 3 different roles: Guarantor, Associate and Approbator.

Figure 2 shows the visual representations of the constructed queries to find the patterns one of our collaborator is interested in. After he constructed the queries, he was able to explore the results in the two views, as shown in Figure 1. Figure 3 shows the results of the four queries with an occurrences table (left) or attribute distributions (right). It shows that the constructions in Torino are mostly military (mil) and civil (civ). The Zo family collaborates with other persons mostly in Torino (but not only). Finally, the persons who have the 3 different roles only come from Biela (BI).

Conclusion

We present a new visual analytics for Bipartite Multivariate Dynamic Social Networks constructed from historical sources. It allows to explore the topology of the network and the localization of the documents, along node attribute distributions, including time. Moreover, the interface includes a new visual query system. Using it, social scientists can isolate specific patterns or groups of entities and look at their positions in the network, along with the values of their attributes compared to the global distributions. We demonstrate the power of the query system with one real-world use case, showing that it can be used to answer sociological questions.

Figure 2 :Figure 3 :

 23 Figure 2: Queries to answer our collaborator's four questions.

© 2022 The Author(s) Eurographics Proceedings © 2022 The Eurographics Association.


Julien Huillery
email: julien.huillery@lis.inpg.fr

Nadine Martin
email: nadine.martin@lis.inpg.fr

Detection of Time-Frequency Components of Signals: Bayes and focus SNR

The Bayesian time-frequency detector operating on spectrogram of non stationary signals is studied. As direct evaluation of the likelihood ratio is impossible, an a priori user parameter called focus is introduced. It is defined as a local time-frequency signal to noise ratio at which the detection is tuned to be optimal. Leading to a unique detection threshold, the parameter introduced is equivalent to the probability of false alarm used in the Neyman-Pearson detection strategy. However, we expect the formulation in terms of local signal to noise ratio to be of intuitive and practical interest.

Introduction

Detection procedures are used to reveal wether a signal is present in a given observation or not. The detection may be proceeded in the temporal domain [START_REF] Van Trees | Detection, Estimation and Modulation Theory[END_REF]] or in the time-frequency domain as studied by [START_REF] Altes | Detection, estimation and classification with spectrograms[END_REF]] or Flandrin [START_REF] Flandrin | A Time-Frequency Formulation of Optimum Detection[END_REF]]. However, the binary hypotheses test is usually formulated in terms of the temporal observation. The time-frequency detection procedure we are interested in is slightly different: it is formulated at each time-frequency location (t, f ).

The aim of this detection procedure is to extract the regions of the time-frequency plane where some signal is present. This specific task reveals useful so as to determine whether a signal is present or not but also provides informations about the spectral structure of the signal and its evolution. Interpreted as a time-frequency signature of the signal, this information may be used for further processing such as classification, time-frequency sparse sources separation or estimation of the underlying physical process parameters.

Neyman-Pearson detection strategy has already been employed for this task [Huillery (2006a)]. In this paper we concentrate in the possibility to use the theoretically more optimal Bayesian approach. However due to the lack of information, a focus parameter is introduced to solve the detection problem. It turns out that the resulting procedure is equivalent to a Neyman-Pearson detection strategy, where the focus parameter plays the role of the traditional Probability of False Alarm (PFA).

The paper is organized as follows: in section 2, the Bayesian time-frequency detection procedure is introduced. In section 3, we express the probability density function (pdf) of the time-frequency coefficients. A solution for the detection task and some results on a real-life signal are presented in section 4. In section 5, we finally discuss the link between the method proposed and a Neyman-Pearson detector.

Bayesian Time-frequency detection task

The model of signal we are interested in is composed of a deterministic part d(t) embedded in a centered Gaussian perturbation p(t) with autocorrelation function Γ p (t, τ ) associated with the energy time-frequency distribution γ p (t, f ). Namely,

x(t) = d(t) + p(t).
(2.1)

The deterministic part contains the information under interest on which no a priori is known. The aim of the proposed approach is to localize the time-frequency support of this information.

The detection task consists in determining whether the energy S x (t, f ) observed at a particular time-frequency location (t, f ) originates from the perturbation only (null hypothesis H 0 ) or is also due to the deterministic part of the signal (signal hypothesis H 1 ). The two hypotheses test problem is thus formulated at each time-frequency locations as

H 0 : S x (t, f ) = S p (t, f ), H 1 : S x (t, f ) = S d+p (t, f ). (2.2)
We consider no a priori information about the probability of each hypotheses (or equivalently, we fixe them to 1/2) and equal costs for erroneous decision. In this particular case, the Bayesian detection strategy results in the maximum likelihood detector [START_REF] Van Trees | Detection, Estimation and Modulation Theory[END_REF]]. It consists in the comparaison of the likelihood ratio Λ Sx(t,f ) with 1 and writes

Λ Sx(t,f ) = p H1 (S x (t, f )) p H0 (S x (t, f )) H 1 ≷ H 0 1. (2.3)
where p H0 (.) and p H1 (.) stand for the pdfs under the null and signal hypotheses respectively. Finally, the time-frequency support of information, noted I, is defined as

I = (t, f )/S x (t, f ) > S th (t, f ) , (2.4)
where S th (t, f ) is the detection threshold obtained with the detector 2.3.

Time-frequency probability distributions

We use the spectrogram as the time-frequency representation of the signal. As a main advantage, it does not display any interference terms and remains interpretable as a physical representation of signal energy. However, these features are counterbalanced by a poor energy concentration. The pdf of spectrogram coefficients under the null hypothesis H 0 is a central χ 2 laws with δ = 2 degrees of freedom and proportionality coefficient α = γ p (t, f )/2 [Huillery (2006b)]. It writes

p H0 (s) = 1 γ p (t, f ) exp - s γ p (t, f ) . (3.1)
Under signal hypothesis H 1 , the pdf of a spectrogram coefficient is a non central χ 2 law. The noncentrality parameter is equal to the spectrogram coefficient S d (t, f ) of the deterministic part d(t) of the signal. At time-frequency location (t, f ), the pdf p H1 writes

p H1 (s) = 1 γ p (t, f ) exp - s + S d (t, f ) γ p (t, f ) I 0 2 s.S d (t, f ) γ p (t, f ) , (3.2)
where I 0 (.) stands for the zero order modified Bessel function of the first kind.

Solution to the detection problem

From the formulation given in section 2, the Maximum Likelihood (ML) detection threshold S th ML is defined as the quantile that equates the two probability distributions p H0 and p H1 . Using the expressions of probability distributions given in section 3, S th ML is solution of the equation

I 0   2 S d (t, f ).S th ML γ p (t, f )   = exp S d (t, f ) γ p (t, f ) . (4.1)
We solved this equation numerically and after linear regression, the ML detection threshold S th ML can be obtained as

S th ML (t, f ) = 0.26 × S d (t, f ) + γ p (t, f ). (4.2)
Without further developments, the detection threshold S th ML depends on two parameters, namely S d (t, f ) and γ p (t, f ). The deterministic contribution S d (t, f ) being precisely what we are looking for, an ill problem is faced.

Let us define the Local Time-Frequency Signal to Noise Ratio (local-SNR), noted ρ(t, f ), as the ratio between the signal and perturbation energies observed at a single time-frequency location (t,f). In linear scale, it writes

ρ(t, f ) = S d (t, f ) γ p (t, f ) . (4.3)
To determine the detection threshold S th (t, f ) of eq. 2.4, we propose to introduce a focus parameter, noted ρ 0 , defined as a local-SNR for which the detector will be tuned optimally in the ML sense. This parameter has to be fixed before the detection procedure and leads to a detection threshold S th (t, f ) calculated as

S th (t, f ) = (0.26 × ρ 0 + 1) γ p (t, f ). (4.4)
When ρ 0 is different from the local-SNR ρ(t, f ) (presumably most of the time), the detection threshold S th (t, f ) is not optimal. In practice, ρ 0 approximately corresponds to the minimal local-SNR on the "information support" I that is detected. We also note the dependence of the detection threshold S th (t, f ) on the perturbation spectral power γ p (t, f ) that needs to be estimated when unknown. An estimation procedure dedicated to central χ 2 laws can be found in [START_REF] Hory | Maximum likelihood noise estimation for spectrogram segmentation[END_REF]].

Figure 1 displays some detection results obtained with the French speech signal "Joyeux noël" embedded in a white gaussian perturbation of known variance. In each cases b), c), d) and e), the local-SNR of the time-frequency locations detected as "information support" are around or greater than the focus parameter ρ 0 . We note the increasing amount of false alarms as the focus local-SNR ρ 0 decreases.

Discussion

Starting from a Bayesian formulation of the detection procedure, a lack of information imposes the introduction of a detection condition so as to determine a detection threshold. We choose to formulate this condition as a local signal to noise ratio for which the detection procedure is set to be optimal. The focus parameter introduced now imposes a single value to the detection threshold (see eq. 4.4). Considering a binary hypothesis test where the probability distribution under H 1 is unknown, the Neyman-Pearson detection strategy [START_REF] Van Trees | Detection, Estimation and Modulation Theory[END_REF]] consists in fixing a priori a Probability of False Alarm (PFA). The choice of this PFA also imposes a single value to the detection threshold. The focus with ρ 0 expressed in linear scale. In figure 1, the PFA corresponding to each focus parameter ρ 0 is also noted. Figure 1-f) depicts the relation between ρ 0 and the PFA. However, as the PFA is a noise dedicated detection parameter, the focus parameter introduces a signal condition in the detection procedure. It is an answer to the problem: "I want to detect the signal time-frequency components with energy at least ρ 0 -time higher than the background perturbation.". We expect this formulation of detection condition to be of practical interest.

Figure 1 .

 1 Figure 1. Detection results for different focus parameter ρ0. Spectrogram is constructed with 50% overlapping 512 points Hanning windows, with a zero-padding of 2.


Julien Huillery
email: julien.huillery@lis.inpg.fr

Nadine Martin
email: nadine.martin@lis.inpg.fr

Time-Frequency Modeling and Detection of random non-stationary signals for Monitoring Purposes

This paper deals with the modelization and detection of non-stationary random signals in the time-frequency space. A time-frequency random model of signal is derived from a given temporal model. The time model we are interested in consists in a deterministic signal embedded in an additive centered Gaussian perturbation. This Gaussian model is characterized by two parameters, which are the mean and covariance matrix of the process. The corresponding time-frequency model depends on the time-frequency transform applied to the signal. For the spectrogram, the determinant parameters are the nature and length of the analysis window and the zero-padding. We show that for a Gaussian signal, spectrogram coefficients distribution can be approximated by a chi 2 law defined by three parameters. A time-frequency signal detection task inspired from a Neyman-Pearson strategy is performed on the basis of this probabilistic time-frequency model. The detector determines the time-frequency regions where signal energy is present. It thus provides a time-frequency signature of the signal. This information is used for structural health monitoring techniques. Extraction of the fundamental meshing frequency and harmonics of a gearbox under varying load conditions is presented.

I. Introduction

T he study presented in this paper relates to structural health monitoring through the analysis of signals.

As monitoring is linked with the observation of the behaviour of a system, it reveals an intrinsically nonstationary context. Time-Frequency (TF) research aims at the development of new analysis tools dedicated to non-stationary signals. These tools have proved great efficiency for signal analysis as they allow the observation of the evolution along time of the signal frequency structure.

A large amount of work has been done in the field of analysis of random signals, based on time series and/or spectral analysis where effective tools constructed on a theoretical framework have been developed for detection, estimation and classification. An essential aspect of probabilistic approaches for signal analysis is the choice of the structure and nature of the model used to represent the randomness of the signal. This modelization always concerns the time representation of the signal. The first aim of this paper is to present how a time random model can be transplanted into the time-frequency domain.

Similarly to the time domain, this probabilistic time-frequency model can be used as a framework for a time-frequency signal detection task. Time-frequency detection extract the time-frequency locations where a signal is present. It reveals useful for the determination of the signal temporal locations but also provides deeper information about the spectral structure of the signal and how this structure evolves. In this paper we present a procedure, based upon a Neyman-Pearson framework, that localizes the regions of the timefrequency plan where signal energy is present.

Structural health monitoring task based upon this analysis of the signal is performed as described in figure 1. We investigate vibration monitoring of a rotative machinery under fluctuating load conditions. Different loads imposed to an induction motor cause the rotative speed to fluctuate. Our aim is to recover the law of the load by extracting the fundamental meshing frequency of the system. The time-frequency detection procedure is used for this purpose.

II. Time-frequency modeling

The time random model of signal we are interested in is composed of a deterministic part d(t) and an additive random part r(t), namely

x(t) = d(t) + r(t). (1)
The deterministic part represents the support of the information under interest while the random part stands for the receiver environment. The non-stationarities originate from the deterministic process. The structure of this model allows the development of signal processing methods without prior hypotheses upon the nature or structure of the information contained in the signal. The only a priori information is the random nature of the additive perturbation process r(t). It namely consists in its law of probability that can be represented by its probability density function or equivalent. The results are here presented for a centered Gaussian process with covariance matrix R associated to the autocorrelation function Γ(τ ). As a consequence, each sample x(t) of the signal is distributed as a Gaussian variable with mean d(t) and autocorrelation function Γ(τ )

x(t) ∼ N (d(t), Γ(τ )) . (2)
Once the time modelization task of the phenomenon under investigation has been performed, the next step consists in determining the corresponding random model in the time-frequency domain.

Many time-frequency representations of signals have been proposed yielding to the need of a classification. [START_REF] Cohen | Time-Frequency Distribution -A Review[END_REF] One first distinction separates representations associated with linear or quadratic transformation of the signal. Short-Time Fourier Transform and Wavelet Transform are members of the linear group. The advantage of the quadratic group is to offer the possibility of an interpretation of time-frequency representations as physical quantities like energy. The class of Cohen gathers some of the quadratic transforms. One key aspect regarding the choice of a quadratic transform is the balance between energy concentration and presence of interference terms. The Wigner-Ville distribution (presence of interferences, high energy concentration) and the spectrogram (no interferences, poor energy concentration) correspond to the farthest transforms. To each time-frequency representation of the signal will correspond a specific time-frequency random model. The law of probability of the time-frequency coefficients will depend on the transformation applied to the signal. In this study, results are presented for the case of spectrogram.

Given a discrete analysis window w[n] of length M , the Short-Time Fourier Transform X w [n, k] of a discrete signal x[n] is formed by the successive Discrete Fourier Transforms of the windowed signal. The spectrogram S w x [n, k] corresponds to the squared modulus of the STFT or equivalently to the sum of the squares of the STFT real and imaginary parts, X r w [n, k] and X i w [n, k] respectively. We thus start from the following definitions:

S w x [n, k] = X r w [n, k] 2 + X i w [n, k] 2 , (3)
X r w [n, k] = M -1 m=0 x[n -m]w[m] cos(-2πk m N ), (4)
X i w [n, k] = M -1 m=0 x[n -m]w[m] sin(-2πk m N ). ( 5
)
where N is the length of the computed FFT and N -M corresponds to the zero-padding. The law of the random variable S w x [n, k] is thus determined by the probabilistic behaviour of the squared modulus of the vector X[n, k] defined by

X[n, k] T = [X r w [n, k] X i w [n, k]]. (6)
Equations ( 4) and ( 5) express the real and imaginary parts of the STFT as linear combinations of signal samples. As x[n] are Gaussian variables (not necessarily independent), X r w [n, k] and X i w [n, k] are also distributed as Gaussian laws. X[n, k] is thus a Gaussian vector defined by five parameters m 1 , m 2 , Σ 11 , Σ 22 and Σ 12 (for clarity of notation we drop the time-frequency location [n, k] in the probability parameter notations):

X[n, k] = X r w [n, k] X i w [n, k] ∼ N m 1 m 2 , Σ 11 Σ 12 Σ 12 Σ 22 . (7)
From equations ( 1),( 4) and ( 5), we obtain

m 1 = D r w = Re(ST F T {d(t)}) (8) m 2 = D i w = Im(ST F T {d(t)}) (9)
and

Σ 11 = W T C k RC k W (10) Σ 22 = W T S k RS k W (11) Σ 12 = W T C k RS k W (12)
where

W T = [w[0] • • • w[M -1]
] is the analysis window vector and C k (resp. S k ) is the cosine (resp. sine) diagonal matrix,

C k = diag cos(-2πk m N ) m=0,M -1 S k = diag sin(-2πk m N ) m=0,M -1
Spectrogram coefficients are thus distributed as the squared modulus of the Gaussian vector X. This law of probability is determined through the five parameters m 1 , m 2 , Σ 11 , Σ 22 and Σ 12 . This distribution can be approximated 2 with a chi 2 distribution with three parameters χ 2 (α, δ, θ). The coefficients of proportionality α, the degree of freedom δ and the uncentrality parameter θ are then defined as

δ = (Σ 11 + Σ 22 ) 2 Σ 2 11 + Σ 2 22 + 2Σ 2 12 , (13)
α = Σ 11 + Σ 22 δ , (14)
θ = m 2 1 + m 2 2 α = S w d α , (15)
where S w d is the spectrogram of the deterministic part d(t) alone.

One advantage of this time-frequency random model is the direct link between the transform that is used and the resulting distribution. Parameters of the time-frequency transformation being known, the parameters of the time-frequency model are defined by the time-frequency distribution D w [n, k] of the deterministic part and the variance-covariance matrix R of the additive random process.

III. Time-frequency signal detection

We use the results presented in the previous section to develop an algorithm that automatically detect and isolate time-frequency components of the determinist part of our model. In this work, a time-frequency component of a signal is defined as a region of the time-frequency plane where signal energy is present. Furthermore, the additive random process r(t) is assumed to be white with an unkown spectral power σ 2 . We can thus write R = σ 2 I N . For this simple case, the three parameters (α,δ,θ) of the χ 2 law (equations 13, 14 and 15) reduce to

δ = 1 2β 2 -2β + 1 , (16)
α = σ 2 δ , (17)
θ = m 2 1 + m 2 2 α = S w d α , ( 18
)
where β is a function of the analysis window w[n] and the frequency k defined by

β = 1 2 + 1 2 * M -1 n=0 w[n] 2 cos(4πk n N ). ( 19
)
Note that for most frequencies k, β equals 1 2 and the χ 2 law has two degrees of freedom (δ = 2). [START_REF] Millioz | Short-Time Fourrier Transform Probability Distribution for Time-Frequency Segmentation[END_REF] At each time-frequency location [n, k], the detection scheme can be formulated as a two hypothesis test problem. The null hypothesis H 0 corresponds to the presence of the random additive process r(t) only, and the signal hypothese H 1 is valid if the deterministic part d(t) provides energy at the considered location. That is:

H 0 : S w x [n, k] = S w r [n, k], H 1 : S w x [n, k] = S w r+d [n, k].
From section II and above, the distribution of S w x [n, k] under H 0 is a central (θ = 0) χ 2 law for every time-frequency location [n, k]. The proportionality coefficient α is equal to σ 2 /δ where the degree of freedom δ and the spectral power σ 2 remains to be estimated. Under H 1 , spectrogram coefficients are distributed as a non-central χ 2 variable. As for this case the non-centrality parameter θ is fluctuating over the timefrequency plane, a Neyman-Pearson strategy seems preferable for the detection task: a given fixed False Alarm Probability (PFA) determines the detection threshold on the probability distribution under H 0 .

The general principle of the algorithm 4 is to iteratively select higher energy connected time-frequency coefficients until the distribution of the remaining region matches a possible H 0 distribution. Doing so the algorithm iteratively estimates the unknown parameter σ 2 of the random process.

The time-frequency detection procedure is now described: In the time frequency plane, candidates to the signal class H 1 are selected through the use of a PFA on the probability distribution of the noise class H 0 . For this purpose, parameters α and δ (proportionality coefficient and degree of freedom respectively) of the central χ 2 law are estimated 5 by maximising the likelihood with the data contained in the noise class H 0 . A threshold is determined on this probability distribution according to the initialy given PFA. TF coefficients S w x [n, k] superior to this threshold are selected as candidates for the signal class H 1 .

The signal class regions then grow by clustering candidates connected in the time-frequency plane. A candidate for the signal class is assigned to the class H 1 if it is contiguous to a member of the class H 1 . The time-frequency region growing stops when the number of candidates falls under a threshold determined with the initial number of candidates and the PFA.

After each grow, the likelihood of the remaining noise class distribution with a central χ 2 distribution is evaluated. The procedure repeats iteratively until the convergence of the likelihood stops its evolution.

In fine, the limit between the noise and signal regions is determined according to the time-frequency model. The spectral power σ 2 of the additive random process is estimated. The time-frequency space is segmented into regions corresponding to the deterministic signal components and may be interpreted as a time-frequency signature.

IV. Load variations monitoring

We used the time-frequency detection procedure described in section III for the analysis of vibration signals from rotative machinery. The aim of the experiment is the monitoring of a fluctuating load applied to the motor. Excess load may then be detected so as to prevent mechanical failure or motor break down.

The experiment has been carried out using data from the benchmark "GOTIX" (see figure 2) of the Signal and Image Laboratory, Grenoble, France. This benchmark is composed of a 55kW asynchronous motor, a gearbox with a multiplicative ratio of 57/15, a speed variable controller and a DC motor that allows the simulation of variable loads. Signals are acquired from various voltage/current sensors and accelerometers placed at different locations. We used the vibration signal recorded on the motor shaft. During acquisition, the load represented by the DC motor was continuously modified. Vibration signals from rotative machinery display a harmonic structure (see figure 3-a)). The fundamental frequency corresponds to the rotation speed of the motor. Variation of the load imposed to the rotative machinery results in inversely proportional variation of the rotation speed. As a consequence, extraction of the fundamental meshing frequency of the vibration signal allows to recover the load conditions the system has been subject to.

Unfortunately, the fundamental frequency of vibration signals may have less energy than some of its harmonics and suffer from amplitude variations. Furthermore, a load variation ∆L generates a frequency variation of K∆L on the fundamental frequency and n * K∆L on the n th harmonic. As the frequency resolution of a time-frequency analysis is constant over the whole frequency range, resolution on the load variation is gained when looking at high harmonics. In the case under interest, the 7 th harmonic reveals the more energetic and of relatively stable amplitude (figure 3-b)). We thus restrict the analysis to the frequency band [5200Hz; 6100Hz]. Some knowledge about the behaviour of the investigated rotative machinery and sensors may help to determine this spectral region of interest as it remained the same for all the experiments we conducted.

Figure 3-c) depicts the time-frequency signal detection result from which the load variations are extracted (figure 3-d)) by use of the motor torque/rotation speed characteristic. We recover the load variations manually obtained during the experiment.

V. Conclusion

More than extract the temporal location of a signal, time-frequency signal detection provides information about the spectral structure of a signal and the way it evolves. In this study, a time-frequency detection procedure is presented. This procedure is based on a time-frequency probabilistic model constructed on a given Gaussian temporal model and taking into account the characteristics of the time-frequency transform used for the analysis. Monitoring of the load imposed to a rotative machinery is presented and show the applicability of the method.

Figure 1 .

 1 Figure 1. Overview of the method

Figure 2 .

 2 Figure 2. The benchmark "GOTIX" from the Signal and Image Laboratory (LIS)

Figure 3 .

 3 Figure 3. Time-frequency signal detection and load variations monitoring


A Pyrkin

R Ortega

A Bobtsov

S Aranovskiy

D Gerasimov

Ćuk Converter Full State Adaptive Observer Design

A novel approach to the problem of partial state estimation of a class of nonlinear systems is proposed. In this paper we apply the new adaptive observer design technique to power converters. For the sake of ease of exposition we preferred to concentrate on the specific example of the Ćuk power converter. The main idea is to translate the state estimation problem into one of estimation of constant, unknown parameters related to the systems initial conditions. The proposed observer is shown to be applicable for the reconstruction of the state of power converters for further using in the full-state feedback controller. Comparison with Immersion and Invariance technique for the observer design is made via numerical examples.

I. INTRODUCTION

The problem of regulating the output voltage of switched power converters has attracted the attention of many control researchers for several years now. Besides its practical relevance in modern power systems for satellites, non-civilian, industrial and consumer electronic applications, these systems are an interesting theoretical case study because they are switched devices whose averaged dynamics are described by a bilinear second order non-minimum phase system with saturated input and a highly uncertain parameters -the load resistance and the input voltage.

In this paper we apply the new adaptive observer design technique to power converters. For the sake of ease of exposition, instead of developing a-notationally cumbersomegeneral theory for a broader class of power converters, we preferred to concentrate on the specific example of the Ćuk power converter, depicted in Fig. 1.

We use a nonlinear controller proposed in [START_REF] Astolfi | Nonlinear and Adaptive Control with Applications[END_REF]. In contrast to other schemes reported in the literature [START_REF] Fliess | Regulation of non-minimum phase outputs: a flatness based approach[END_REF]- [START_REF] Sira-Ramirez | Passivity-based controllers for the stabilization of DC-to-DC power converters[END_REF] this type of controllers do not rely on the systems inversion, hence they can directly regulate the output signal, which as is wellknown is a non-minimum phase output. Furthermore, the control laws are simple static state feedback. See [START_REF] Ortega | Passivity-based Control of Euler-Lagrange Systems[END_REF] for a list of references and [START_REF] Escobar | An experimental comparison of several nonlinear controllers for power converters[END_REF] for a comparative experimental study for the Boost converter case. a.pyrkin@gmail.com, bobtsov@mail.ru, aranovskiysv@niuitmo.ru, gerasimovdn@mail.ru 2 R. Ortega is with the LSS-Supelec, 3, Rue Joliot-Curie, 91192 Gif-sur-Yvette, France : ortega@lss.supelec.fr

E L

1 C 2 L 3 C 4 G i 3 i 1 v 2 v 4 + - + - + - Fig. 1: DC-DC Ćuk converter circuit
The main problem is the adaptive observer design for Ćuk converter states that are currents and voltages. It is assumed that a part of states are measurable. The rest of state vector is to be observed and estimates should be substituted to the full-state controller. A new framework for constructing globally convergent (reduced-order) observers for a well-defined class of nonlinear systems is presented (see also [START_REF] Back | Dynamic observer error linearization[END_REF], [START_REF] Boutat | On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form[END_REF]). Instrumental to this development is to formulate the observer design problem as a problem of parameter estimation, which represents the initial conditions of the unknown part of the state. The class of averaged Ćuk converter model is suitable for the proposed observer design technique.

The remaining of the paper is organized as follows. Section II presents the problem formulation. Section III is devoted to a discussion of the main theoretical results. Section IV illustrates the application of the technique to Ćuk converter with two scenarios of work. The paper is wrappedup with simulation results and concluding remarks in Section V.

II. PROBLEM STATEMENT

The Observer Design procedure starts by writing the averaged dynamic of the power converters in the following PCH form [START_REF] Rodriguez | A new family of energy-based non-linear controllers for switched power converters[END_REF]

Ẋ = [J (u) -R] ∂H ∂X (X) + gE, (1)
where X is the state vector, consisting of charges and fluxes, E is the input voltage, u represents the duty ratio, H(x) is the total energy and J (u), R, g represent the internal interconnection, damping and external interconnection matrices, respectively. PCH models result from the network modeling of energy conserving lumped-parameter physical systems with independent storage elements [START_REF] Escobar | A Hamilt,onian viewpoint in the modeling of switching power convcrters[END_REF].

The Cuk DC-to-DC power converter shown in Fig. 1 can

1
be written in PCH form with

J =     0 -(1 -u) 0 0 (1 -u) 0 u 0 0 -u 0 -1 0 0 1 0     , (2)
R = diag{0, 0, 0, 1 R }, g = 1 0 0 0 ⊤ , (3)
and then the average model of this device is given by

L 1 di 1 dt = -(1 -u)v 2 + E (4)
C 2 v2 = (1 -u)i 1 + ui 3 (5)
L 3 di 3 dt = -uv 2 -v 4 (6)
C 4 v4 = i 3 -Gv 4 , (7)
where L 1 , C 2 , L 3 , C 4 are inductances and capacities, G = 1 R are positive constants. We refer the reader to [START_REF] Astolfi | Nonlinear and Adaptive Control with Applications[END_REF] for further details on the model. The destination of the Ćuk converter is to regulate the voltage across the load (i.e., the capacitor voltage v 4 ) to a constant value V d . One can apply the control law proposed in [START_REF] Astolfi | Nonlinear and Adaptive Control with Applications[END_REF], [START_REF] Rodriguez | A new family of energy-based non-linear controllers for switched power converters[END_REF].

III. OBSERVER DESIGN

In this paper we consider the problem when a part of state variables are unmeasurable and are to be observed. Further a theoretical discussion about this issue follows.

Consider the dynamical system

ẋ = f x (x, y, u) ẏ = f y (x, y, u), (8)
where x is an unmeasurable part of X, y are measurable states, f x : R nx × R ny × R m → R nx and f y : R nx × R ny × R m → R ny are smooth mappings. Assume that the input signal vector u : R + → R m is such that all trajectories of the system are bounded. Find, if possible, mappings

F : R n ξ × R ny × R m → R n ξ and G : R n ξ × R ny × R m → R nx , for some positive integer n ξ , such that the (partial state) observer ξ = F (ξ, y, u) x = G(ξ, y, u), (9)
ensures that ξ is bounded and

lim t→∞ |x(t) -x(t)| = 0, (10)
for all initial conditions (x(0), y(0), ξ(0)) ∈ R nx+ny+n ξ and a well defined class of input signals u ∈ U. Assumption 1: There exists three mappings

φ : R nx × R ny → R nz φ L : R nz × R ny → R nx h : R ny × R m → R nz ,
with n z ≥ n x , verifying the following conditions.

(i) (Left invertibility of φ(•, •) with respect to its first argument)

φ L (φ(x, y), y) = x, ∀x ∈ R nz , ∀y ∈ R ny .
(ii) (Transformability into cascade form)

∂φ ∂x f x (x, y, u) + ∂φ ∂y f y (x, y, u) = h(y, u). (11)
An immediate corollary of (ii) in Assumption 1 is that the partial change of coordinates

z = φ(x, y), (12)
ensures

ż = h(y, u). (13)
Moreover, the left invertibility condition (i) ensures that the partial state x can be recovered from z and y, that is,

x = φ L (z, y). ( 14
)
Define the dynamic extension

χ = h(y, u), (15)
with χ(0) ∈ R nz . From ( 13) and (15) we get ż = χ. Hence, integrating this equation yields

z(t) = χ(t) + θ, (16)
where

θ := z(0) -χ(0). (17)
Based on ( 14) and ( 16) we get the state observer

x = φ L (χ + θ, y), (18)
where θ : R + → R nz is an on-line estimate of the vector θ.

Therefore, the problem of observation of the unmeasurable state x is translated into a parameter estimation problem for the regression model that is to be found later. To complete the observer design it is necessary to ensure the existence of a consistent estimator for the unknown parameter θ.

Proposition 1: Consider the system (8) verifying Assumption 1 and the dynamic extension (15). We can compute a mapping Φ :

R nz × R ny × R m × R nz → R ny such that ẏ = Φ(χ, y, u, θ) (19) x = φ L (χ + θ, y), ( 20
)
where θ ∈ R nz is a vector of constant, unknown parameters. The proof is given by replacing (16) in ( 14) which yields (20), and the regression model ( 19) is obtained replacing (20) in [START_REF] Back | Dynamic observer error linearization[END_REF] to get f y (φ L (χ + θ, y), y, u) =: Φ(χ, y, u, θ).

IV. ĆUK CONVERTER OBSERVER DESIGN

To illustrate the generality of the approach we consider two different measurement scenarios. In the first one we assume that (v 2 , i 3 ) are measurable, while in the second one (v 2 , v 4 ) are measurable. Although from the practical viewpoint it is "easier" to measure voltages, we also consider the first one since, as shown in [START_REF] Astolfi | Nonlinear and Adaptive Control with Applications[END_REF], is the one that can be solved with immersion and invariance (I&I) observers, with which we compare our observer in simulations below.

Case I Denoting y := col(v 2 , i 3 ), x := col(i 1 , v 4 ) we get from ( 4) and ( 7)

ẋ1 = - 1 L 1 (1 -u)y 1 + E L 1 ẋ2 = 1 C 4 y 2 - G C 4 x 2 .
The right hand side of the second equation depends on x 2 , therefore we propose the mapping

φ(x, y) = x - 0 GL3 C4 y 2
, that, introducing the partial change of coordinates z = φ(x, y), yields the required form

ż = -1 L1 (1 -u)y 1 + E L1 1 C4 y 2 + G C4 uy 1 =: h(y, u).
The dynamic extension is then given by χ = h(y, u), and the regression model is of the form ẏ = Φ 0 (χ, y, u) + Φ 1 (u)θ (21)

θ := x(0) -χ(0) - 0 GL3 C4 y 2 (0) , (22)
where

Φ 0 (χ, y, u) := 1 C2 (1 -u)χ 1 + 1 C2 uy 2 -1 L3 uy 1 -1 L3 χ 2 -GL3 C4 y 2 , (23)
Φ 1 (u) := 1 C2 (1 -u) 0 0 -1 L3 . (24)
The model (21) contains the time derivative of the output y. To get a classical (static) regression model we use the standard filtering technique [START_REF] Middleton | Adaptive computed torque control for rigid link manipulations[END_REF] and define the filtered signals

(•) := α p + α (•),
where p := d dt and α > 0 is a design parameter. Applying the filter to (21) we obtain the standard linear, static regression model

ϑ = Φ 1 θ + ǫ (25)
where

ϑ := αp p + α y - α p + α Φ 0
is clearly measurable (without differentiation) and ǫ is an exponentially decaying signal that depends on the filter initial conditions and the filter time constant 1 α .

The regression model ( 25) is used for the parameter estimator, which is the classical gradient estimator

θ = ΓΦ ⊤ 1 (ϑ -Φ 1 θ),
where the adaptation gain Γ = Γ ⊤ > 0 is a design parameter.

The state observer takes the form

x = θ + χ + 0 GL3 C4 y 2 .
The regressor matrix Φ 1 (u) given in (23) has a very simple form. Indeed, the matrix is diagonal with the second term in the regression simply the constant -1 L3 . Clearly, for this case we also have U = {u : R + → (0, 1)} and consistent estimation is always guaranteed.

In Proposition 8.3 of [START_REF] Astolfi | Nonlinear and Adaptive Control with Applications[END_REF] the following I&I observer is proposed

xI&I = ζ 1 ζ 2 + C 2 γ 1 y 1 L 3 γ 2 y 2 ζ1 = 1 L 1 [-(1 -u)y 1 + E] -γ 1 [(1 -u)( ζ1 + C 2 γ 1 y 1 ) + uy 2 ] (26) ζ2 = 1 C 4 [y 2 -G( ζ2 -L 3 γ 2 y 2 )] -γ 2 [uy 1 + ζ2 -L 3 γ 2 y 2 ], (27)
where γ 1 , γ 2 > 0 are design parameters. It should be noted that in the latter reference the parameters E and G are treated as unknown and are also estimated. If they are assumed known the I&I observer takes the form given above. Simulations were carried out to evaluate the performance of the proposed observer. The simulations were done for the model ( 4)- [START_REF] Escobar | An experimental comparison of several nonlinear controllers for power converters[END_REF] in closed-loop with the certainty equivalent version of the full-state feedback I&I controller given in Proposition 8.2 of [START_REF] Astolfi | Nonlinear and Adaptive Control with Applications[END_REF]. That is, the control law was defined by

u = |V d | |V d | + E + λ G|V d |v 2 + E(i 3 -x1 ) 1 + (G|V d |v 2 + E(i 3 -x1 )) 2 (28)
where V d < 0 is the reference imposed to the output voltage v 4 and λ is chosen as

λ = λ 0 min |V d | |V d | + E , E |V d | + E , with 0 < λ 0 < 2.
The full-state version of this controller, i.e., replacing x1 by i 1 , respectively, ensures global asymptotic stability of the desired equilibrium as well as verification of the saturation constraints in the input signal. The performance of our observer was compared with the I&I observer (27) via numerical simulations which were performed with the following values of the converter parameters L 1 = 10 mH, C 2 = 22.0 µF, L 3 = 10 mH and C 4 = 22.9 µF, G = 0.0447 S and E = 12 V. The initial conditions for all simulations are set to x(0) = (1, -2), y(0) = (4, -2).

The initial set point for the output voltage is V d = -5 V, and then this is changed at t = 0.2 s to V d = -40 V, at t = 0.4 s to V d = -10 V, at t = 0.6 s to V d = -25 V, at t = 0.8 s to V d = -15 V. The simulation results are presented in Figs. 2, 3, 4 with different observer gains and the control law (28) with λ = 0.5. Case II Denoting now y := col(v 2 , v 4 ), x := col(i 1 , i 3 ) we get from (4), ( 6)

ẋ1 = - 1 L 1 (1 -u)y 1 + E L 1 ẋ2 = - 1 L 3 uy 1 - 1 L 3 y 2 .
Since the right hand side of these equations is independent of x we can directly select

φ(x, y) = x.
The dynamic extension is given by

χ = -1 L1 (1 -u)y 1 + E L1 -1 L3 uy 1 -1 L3 y 2 =: h(y, u),
and the regression form is

ẏ = Φ 0 (χ, y, u) + Φ 1 (u)θ θ := x(0) -χ(0) (29)
where

Φ 0 (χ, y, u) := 1 C2 (1 -u)χ 1 + 1 C2 uχ 2 1 C4 χ 2 -G C4 y 2 , (30)
Φ 1 (u) := 1 C2 (1 -u) 1 C2 u 0 1 C4 . ( 31
)
The state observer is defined as x = θ + χ, where the parameter estimator has the same form as in Case I.

It is important to underscore that the regressor matrix Φ 1 (u) given in (30) has an extremely simple form. Indeed, due to its upper triangular form, the estimation of the second parameter is decoupled from the first one and, moreover, the corresponding term in the regression is simply the constant 1 C4 . Also, since the matrix depends only on the input signal u the set U is defined as

U := {u : R + → (0, 1) | t+T t 1 -u(s) (1 -u(s))u(s) (1 -u(s))u(s) u 2 (s) + C 2 2 C 2 4 ds ≥ δ > 0}.
Some simple calculations show that the matrix inside the integral is positive definite for any u ∈ (0, 1). Hence, U = {u : R + → (0, 1)} and consistent estimation is always guaranteed.

The performance of our observer was illustrated via numerical simulations. They were done under the same scenario as the ones done for Case I, but now with the certainty equivalent observer that results replacing i 1 and i 3 by x1 and x2 , respectively. The controller is given by

u = |V d | |V d | + E + λ G|V d |v 2 + E(x 2 -x1 ) 1 + (G|V d |v 2 + E(x 2 -x1 )) 2 (32)
The simulation results are presented in Figs. 5.

V. CONCLUSIONS

A radically new approach to design state observers for nonlinear systems has been proposed. The key idea is to translate the state observation problem into one of parameter estimation.

The proposed technique has been shown to be applicable to position estimation of a class of power converters and application for Ćuk converter is presented. We expect to identify other classes of physical systems to which the proposed method is applicable in future.

1

  A. Pyrkin, A. Bobtsov, S. Aranovskiy and D. Gerasimov are with the Department of Control Systems and Informatics, ITMO University, Kronverkskiy av. 49, Saint Petersburg, 197101, Russia :

Fig. 2 :Fig. 3 :

 23 Fig. 2: Transients of the observation errors (a) x1 := x1i 1 , (b) x2 := x2v 4 , (c) the voltage reference V d and voltage output v 4 and (d) the control input u for the proposed observer with tuning gains α = 0.1, Γ = diag{0.01, 1}.

Fig. 4 :Fig. 5 :

 45 Fig. 4: Transients of the observation errors (a) x1 , (b) x2 , (c) the voltage reference V d and voltage output v 4 and (d) the control input u for the I&I observer (27) with tuning gains γ 1 = 15, γ 2 = 2.