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## Advanced feature extraction techniques for induction machines in non-stationary environments

It has been demonstrated that the Fourier transform-based techniques cannot not provide simultaneous time and frequency localization and are not very useful for ana-lyzing time-variant, non-stationary signals. Then, the problem of stator current spec-tral estimation in non-stationary environments has received a great deal of attention. Therefore, several techniques have been proposed to analyze faults. These techniques can be classified into three categories: parametric techniques, nonparametric tech-niques, and demodulation techniques. Nonparametric techniques include time-scale and time-frequency presentations [20, 102, 217–223]. Parametric techniques include non-stationary MLE and nonstationary subspace techniques [224]. A demodulation Several advanced combined techniques have been proposed to analyze faults in start-up and steady-state regimes [226]. A fusion between two techniques: the Com-plete Ensemble Empirical Mode Decomposition (CEEMD) and the MUSIC is proposed in [226]. In this case, the proposed methodology allows identifying time evolution of the faulty frequencies in start-up and steady-state regimes from the short data record signal buried in noise, as it is the case for inverter-fed induction motors. Another technique has been proposed in [227] to detect incipient broken rotor bar in induction motors using high-resolution spectral analysis based on the start-up current analysis. This technique is based on the short-time MUSIC algorithm that provides high-resolution and the time-frequency pseudo-representation. The proposed methods can graphically show the physical eﬀect of a broken or partially-broken rotor bar. An application of the ESPRIT and the Simulated Annealing Algorithm (SAA) has been proposed to detect broken rotor bar fault in induction motors with short-time measurement data in [143]. These proposed techniques can correctly identify the parameters of the broken rotor bars characteristic components with short-time measurement data. Another fusion be-tween two techniques: the Hilbert transform and the ESPRIT for detecting rotor fault in induction motors at low slip has been proposed in [142]. This fusion combines two main characteristics: ability to avoid spectral leakage and to achieve high-frequency resolution even with a short measurement time. A comparative study and the evalua-tion of various condition monitoring methods used for induction machines, with the aim of early detection of one partially-broken rotor bar by steady-state current spectrum analysis and diﬀerent supply conditions is proposed in [228]. The techniques consid-ered in this study are the Fast Fourier transform (FFT), Wavelet and FFT, MUSIC, Empirical Mode Decomposition (EMD) and FFT, and EMD associated with MUSIC. Broken rotor bar detection in variable speed drive-fed induction motors at start-up by high-resolution spectral analysis has been proposed in [229]. In this case, the time-frequency spectrum is able to graphically show a diﬀerent pattern for the healthy and faulty conditions.

### Hybrid Approaches

To diagnose faults, several hybrid approaches have been proposed in the literature [286–293]. Among hybrid approaches, Neuro-Fuzzy Systems (NFS) have received a great of attention to overcome the knowledge acquisition bottleneck faced by humans while designing the knowledge base of a traditional fuzzy expert system. The neural network training techniques of NFS can handle the information retrieval from data using optimization techniques. The fuzzy system representation on the other hand provides the intuitive understanding of the resulting system and establishes the possibility of integrating expert knowledge. Since the two approaches have a diﬀerent knowledge representation, their combination can be a persuasive way to fuse information from diﬀerent sources, namely human experts and experimental data. NFS can generate new rules from data or they can refine existing rules by adapting parameters within them.

#### Statistical decision theory

The decision based on the detection theory allows making an optimal decision in order to identify which hypothesis is true without need for a training database. Popular criteria defining the detection procedure with unknown signal and noise parameters are the Bayesian and the Neyman-Pearson approaches. The Bayesian approach is a detector to composite hypothesis testing. Unknown parameters are considered as realizations of random variables and are assigned a prior Probability Density Function (PDF) [294]. Unfortunately, this approach requires multidimensional integration with a dimension equal to the unknown parameter dimension. The Neyman-Pearson approach involves a maximization of the probability of detection PD for a given probability of false alarm PF a [295]. It is based on the likelihood ratio test of the PDFs under a binary hypothesis. The threshold of this test is chosen from the false alarm constraint PF a. When the likelihood ratio depends on unknown parameters, these parameters are replaced by their estimates using the MLE. This solution is known as the Generalized Likelihood Ratio Test (GLRT). Applications of Fault-detection procedures based on hypothesis testing can be found in [138, 296].

**Stator Current Parameters Estimation**

When a parametric model is considered, the objective is often the estimation and/or the detection. The main purpose of any parametric modeling is often to adjust pa-rameters of a selected model function such that the model optimizes some criterion. Generally, it is based on the measured signal fitting with a minimum possible error. In signal processing, this task is called model parameters estimation [297]. Once the stator current model has specified, the problem becomes on of determining an optimal estimator. Regarding the stator current model, the problem is a estimation of multiple sinusoids with unknown parameters in the Gaussian noise. This problem has attracted a great intention in signal processing community. It is called spectral line analysis or line spectrum analysis that the main goal is extracting information on sinusoidal signals in noise. This estimation problem has been studied in numerous applications: sonar, radar, underwater surveillance, communications, geophysical exploration, speech anal-ysis, nuclear physics and other fields [303]. The multiple sinusoids is a non linear model with unknown parameters that generates a non linear least squares problem. Then, the natural estimator to estimate parameters of this model is the Nonlinear Least Squares Estimator (NLSE) [158, 298] since the optimal minimum variance unbiased (MVU) estimator is analytically diﬃcult to obtain it or may not exist [298]. In fact, The NLSE is usually applied in situations where a precise statistical characterization of the data is unknown or where an optimal estimator cannot be found or may be too complicated to apply in practice. Estimations of NLSE are obtained by the squared deviations minimization between stator current measurements and the assumed stationary model [158, 298]. These estimation problems using NLSE can be expressed by the following cost function ÓΩ Ô = arg{Ω,◊} Î ≠ Î x H(Ω)◊ 2 , (2.4).

**Subspace Spectral Estimation Techniques**

This section proposes subspace spectral estimation techniques to avoid limitations of the MLE. These techniques are an attractive alternative to the ML-based estimation methods since they can attain nearly the same estimation performance for time-series as the NLS estimator without being based on the intractable cost function [305]. The term subspace-based refers to separating two distinct subspaces: the signal subspace and the noise subspace. To separate these subspaces, an eigendecomposition of the covariance matrix Rx is required.

**Covariance Matrix**

According to chosen stator current model, it can be seen that in the absence of noise, the N-dimensional vector x belongs to L-dimensional subspace but in case when the noise is present, it is not the case [306]. Therefore, the N-dimensional vector x belongs to P -dimensional subspace that can diﬀerentiate between two main subspaces: signal and noise subspaces. To separate these subspaces, an eigendecomposition of the covariance matrix Rx = E Ëx[n]xH [n]È is used. This matrix Rx has two main properties: it is orthogonally diagonalizable and their eigenvalues are positive and real. The covariance matrix eigendecomposition can be written as follows P Rx = UΛUH = ÿ (2.9) ⁄kukukH k=1 where P º L is the eigenvalues number, Λ = diag [⁄1, …, ⁄L], and U = [uL+1, …, uP ]. The eigenvalues ⁄m are real and positive, arranged in descending order and the corre-sponding eigenvectors uk are orthonormal. Thus, the covariance matrix can be written as a sum of the signal and the noise covariance matrices Rx = Rs + Rn = Ë S G È C 0s Λn DËS GÈ , (2.10) Λ 0 H.

**Table of contents :**

List of Figures

List of Tables

Introduction

**1 Condition Monitoring and Fault Detection of Induction Machines: State of the Art**

1.1 Introduction

1.2 Induction machine faults

1.2.1 Construction

1.2.2 Faults: types, causes, and effects

1.2.2.1 Stator faults

1.2.2.2 Broken rotor bars

1.2.2.3 Bearing faults

1.2.2.4 Air gap eccentricity

1.3 Maintenance Methods of Induction Machines

1.3.1 Corrective maintenance

1.3.2 Preventive maintenance

1.3.3 Predictive Maintenance

1.4 Existing Condition Monitoring Techniques

1.4.1 Oil Analysis

1.4.2 Vibration Monitoring

1.4.3 Acoustic Monitoring

1.4.4 Temperature Monitoring

1.4.5 Torque monitoring

1.4.6 Motor Current Signature Analysis

1.5 Induction machine faults modeling

1.5.1 Finite element modeling

1.5.2 Magnetic equivalent circuit modeling

1.5.3 MMF and permeance wave modeling

1.5.4 Multiple Coupled Circuit modeling

1.5.5 dq modeling

1.6 Advanced feature extraction techniques for induction machine fault analysis

1.6.1 Power spectrum estimation

1.6.1.1 Nonparametric techniques

1.6.1.2 Higher order spectra analysis

1.6.1.3 Parametric techniques

1.6.2 Demodulation techniques

1.6.2.1 Monodimensional techniques

1.6.2.2 Multidimensional techniques

1.6.3 Advanced feature extraction techniques for induction machines in non-stationary environments

1.7 Induction machine fault detection techniques

1.7.1 Artificial intelligence techniques

1.7.1.1 Expert Systems

1.7.1.2 Fuzzy logic

1.7.1.3 Artificial Neural Networks

1.7.1.4 Support Vector Machine

1.7.1.5 Genetic Algorithms

1.7.1.6 Hybrid Approaches

1.7.2 Statistical decision theory

1.8 Conclusion

**2 Stator Current Parameters Estimation and Fault Severity Analysis **

2.1 Introduction

2.2 Stator Current Model

2.2.1 Stator Current Model in Stationary Conditions

2.2.2 Stator Current Frequency Model

2.2.3 Stator Current Parameters Estimation

2.3 Subspace Spectral Estimation Techniques

2.3.1 Covariance Matrix

2.3.2 MUSIC Estimators

2.3.3 ESPRIT Estimators

2.3.4 Modified ESPRIT

2.4 Model Order Estimation

2.4.1 Model Order Estimation (Nonparametric Approach)

2.4.2 Model Order Selection (Parametric Approach)

2.5 Proposed Fault Detection Methodology

2.5.1 Proposed Fault Severity Criterion

2.5.2 Condition Monitoring Architecture

2.6 Simulations Results

2.6.1 Subspace Techniques Performances

2.6.2 Model Order Selection Performances

2.6.3 Fault Severity Criterion Performances

2.7 Conclusion

**3 Induction Machine Fault Detection based on the Generalised Likelihood Ratio Test **

3.1 Introduction

3.2 Statistical Decision Theory

3.2.1 Simple Binary Hypothesis Testing

3.2.2 Bayes Criterion

3.2.3 Minmax Criterion

3.2.4 Neyman-Pearson Criterion

3.2.5 Receiver Operating Characteristics

3.3 Composite Hypothesis Testing

3.3.1 BLRT Approach for Composite Hypothesis Testing

3.3.2 GLRT Approach for Composite Hypothesis Testing

3.4 Proposed Induction Machine Faults Detector

3.4.1 Problem Formulation

3.4.2 GLRT of the Stator Current Model

3.4.3 Performance of the Proposed Faults Detector

3.4.4 Proposed Induction Machine Faults Detection Architecture

3.5 Simulations Results

3.5.1 Simulations Parameters

3.5.2 ROC curves and Histogram

3.5.3 Influences of SNR, N, and PFa

3.6 Conclusion

**4 Experimental Validation **

4.1 Introduction

4.2 Experimental Setup Description

4.2.1 Setup Description for Bearing Faults

4.2.2 Setup Description for Broken Rotor Bars

4.3 Experimental Results for Bearing Faults

4.3.1 Spectral Estimation Techniques

4.3.2 Demodulation Techniques

4.3.3 Fault Severity Analysis

4.3.4 GLRT-Based Faults Detection

4.4 Experimental Results for Broken Rotor Bars

4.4.1 Spectral Estimation Techniques

4.4.2 Demodulation Techniques

4.4.3 Fault Severity Detection

4.4.4 GLRT-Based Faults Detection

4.5 Conclusion

**5 Conclusions and Recommendations for Future Research **

**References **