Classification of vectors forms dedicated to bearings fault detection of electrical machines based on PSO algorithm

The classification problem has been addressed in many contexts and by researchers in many disciplines. This reflects its broad appeal and usefulness as one of the steps in exploratory data analysis [1]. For explicit classification, it is not necessarily desirable to accurately represent the energy distribution of a signal in time and frequency. In fact, such a representation may conflict with the goal of classification, generating a time–frequency representations (TFR) that maximizes the separability of TFRs from different classes. It may be advantageous to design TFRs that specifically highlight differences between classes [2]. Since all TFRs can be derived from the ambiguity plane, no a priori assumption is made about the smoothing required for accurate classification. Thus, the smoothing quadratic TFRs retain only the information that is essential for classification [3, 4]. Technically, a feature represents a distinguishing property, a recognizable measurement, and a functional component obtained from a section of a pattern. Extracted features are meant to minimize the loss of important information embedded in the signal. In addition, they also simplify the amount of resources needed to describe a huge set of data accurately. This is necessary to minimize the complexity of implementation, to reduce the cost of information processing, and to cancel the potential need to compress the information. At the interface between the rotor and the stator, the ball bearing is also having a relatively rapid aging. Typically this type of fault is diagnosed by the spectrum of measurement acoustic or vibration [5, 6]. For improved and authentic fault diagnosis using vibration analysis techniques it is necessary that the acquired vibration signals is ‘clean’ enough that small changes in signal attributes due to an impending fault in any component can be detected [7, 8]. To tackle this problem, we have developed a method based on cloud point dispersion parameter. Since, we used the analytical signals normalized by Hilbert transform of healthy and faulty bearing of induction motor, then extracting vectors forms from time–frequency representation dependant class signal (TFRDCS). Fisher contrast is used to design the kernel nonparametric TFRDCS, It is deliberately designed to maximize separability between classes and minimize the intra-class variance. Recently, the optimization of the size of these vectors realized by particle swarm optimisation (PSO). 2. Analytic signals and Hilbert transform


Introduction
The classification problem has been addressed in many contexts and by researchers in many disciplines.This reflects its broad appeal and usefulness as one of the steps in exploratory data analysis [1].For explicit classification, it is not necessarily desirable to accurately represent the energy distribution of a signal in time and frequency.In fact, such a representation may conflict with the goal of classification, generating a time-frequency representations (TFR) that maximizes the separability of TFRs from different classes.It may be advantageous to design TFRs that specifically highlight differences between classes [2].Since all TFRs can be derived from the ambiguity plane, no a priori assumption is made about the smoothing required for accurate classification.Thus, the smoothing quadratic TFRs retain only the information that is essential for classification [3,4].
Technically, a feature represents a distinguishing property, a recognizable measurement, and a functional component obtained from a section of a pattern.Extracted features are meant to minimize the loss of important information embedded in the signal.In addition, they also simplify the amount of resources needed to describe a huge set of data accurately.This is necessary to minimize the complexity of implementation, to reduce the cost of information processing, and to cancel the potential need to compress the information.
At the interface between the rotor and the stator, the ball bearing is also having a relatively rapid aging.Typically this type of fault is diagnosed by the spectrum of measurement acoustic or vibration [5,6].For improved and authentic fault diagnosis using vibration analysis techniques it is necessary that the acquired vibration signals is 'clean' enough that small changes in signal attributes due to an impending fault in any component can be detected [7,8].To tackle this problem, we have developed a method based on cloud point dispersion parameter.Since, we used the analytical signals normalized by Hilbert transform of healthy and faulty bearing of induction motor, then extracting vectors forms from time-frequency representation dependant class signal (TFRDCS).Fisher contrast is used to design the kernel nonparametric TFRDCS, It is deliberately designed to maximize separability between classes and minimize the intra-class variance.Recently, the optimization of the size of these vectors realized by particle swarm optimisation (PSO).

Analytic signals and Hilbert transform
This representation is commonly used in image processing, where the phase of signal contains more relevant information as the module.Therefore on this principle and for the diagnostic necessary of the asynchronous machine [9] used a phase analysis of the spectrum, and concluded that the information by the phase may be relevant indicative a presence of a fault in the time domain, The Hilbert transform is the convolution of the signal with ( ) t / 1 and can underline local properties, as follows: where t is time, is obtained the amplitude of signal: The amplitude of the analytical representing the instantaneous amplitude of signal (or envelope) of signal when the signal represents the instantaneous phase, which formulas are given their by: [ ] [ ] The use of the Hilbert transform for the phase analysis is applied to the modulus of the spectrum of the Fourier transform of the signal ( ) t x .Indeed, its analytical signal is given by: The phase of the analytic signal can be expressed by:

Time-frequency analysis
The problem of diagnosis systems is that they use signals either in the time or frequency domain.In our approach, instead of using a time or a frequency approach, it is potentially more informative to use both time and frequency.Time-frequency analysis of the motor current makes signal properties, related to fault detection, more evident in the transform domain [10].
It is now well accepted that the representations of a signal jointly in time and frequency offer a real interest: they provide a description of the signals nonstationary, that is to say the analysis of laws frequencies signal behaviour over time.The relation between ambiguity plane and TFR has been recognized for a long time.Any bilinear (Cohen class) TFR can be expressed as the two-dimensional Fourier transform of the product of the ambiguity plane of the signal and a kernel function: ) where t represent time, f represent frequency, η represents continuous frequency shift, and τ represent continuous time lag.For a given signal the ambiguity plane is defined as: where ( ) present the signal at a future time ( ) present the complex conjugate of ( ) . For diagnosis, the optimization procedure of TFR Eq. ( 7) via parameter kernel is computationally very prohibitive.It would be better to use the optimal TFR that can be classified directly in ambiguity plane.
We propose to design and use the classifier directly in the ambiguity plane of Doppler delay.It is possible to view the class dependent TFR and observe the timefrequency structure being exploited by the classifier:

Feature extraction
We transform the Fisher's discriminate ratio (FDR) to opt φ kernel in a binary matrix by replacing the maximum N points with 1 and the other points with 0, is shown in Fig. 1.
Features can be extracted directly from where o is an element-by-element matrix product.The kernel has the same dimensions as the ambiguity plane.By multiplying the opt φ kernel with a certain signal's ambiguity plane, we will find k feature points for this signal.We put them into a vector in order to create the training feature vector where the mean class of ambiguity plane and:

Particle swarm optimization (PSO)
Particle Swarm Optimization (PSO) was invented by Kennedy and Eberhart1 in the mid 1990s while attempting to simulate the choreographed, graceful motion of swarms of birds as part of a socio-cognitive study investigating the notion of "collective intelligence" in biological populations.In PSO, a set of randomly generated solutions (initial swarm) propagates in the design space towards the optimal solution over a number of iterations (moves) based on large amount of information about the design space that is assimilated and shared by all members of the swarm.PSO is inspired by the ability of flocks of birds, schools of fish, and herds of animals to adapt to their environment, find rich sources of food, and avoid predators by implementing an "information sharing" approaches, hence, developing an evolutionary advantage.References 1 and 2 describe a complete chronicle of the development of the PSO algorithm form merely a motion simulator to a heuristic optimization approach [11,12].
The basic PSO algorithm consists of three steps, namely, generating particles' positions and velocities, velocity update, and finally, position update.Here, a particle refers to a point in the design space that changes its position from one move (iteration) to another based on velocity updates.First, the positions i  2).The positions and velocities are given in a vector format with the superscript and subscript denoting the th i particle at time k .In Eqs. ( 1) and ( 2), rand is a uniformly distributed random variable that can take any value between 0 and 1.This initialization process allows the swarm particles to be randomly distributed across the design space.
( ) ( ) The second step is to update the velocities of all particles at time 1 + k using the particles objective or fitness values which are functions of the particles current positions in the design space at time k .The fitness function value of a particle determines which particle has the best global value in the current swarm g k Ρ , and also determines the best position of each particle over time i Ρ , i.e. in current and all previous moves.The velocity update formula uses these two pieces of information for each particle in the swarm along with the effect of current motion i k v , to provide a search direction i k v 1 + , for the next iteration.The velocity update formula includes some random parameters, represented by the uniformly distributed variables, rand, to ensure good coverage of the design space and avoid entrapment in local optima.The three values that effect the new search direction, namely, current motion, particle own memory, and swarm influence, are incorporated via a summation approach as shown in equation 3 with three weight factors, namely, inertia factor w , self confidence factor 1 c , and swarm confidence factor 2 c , respectively [13,14]: The original PSO algorithm1 uses the values of 1, 2 and 2 for w , 1 c , and 2 c respectively, and suggests upper and lower bounds on these values as shown in Eq. ( 3) above.However, the research presented in this paper found out that setting the three weight factors w 1 c , and 2 c at 0.5, 1.5, and 1.5 respectively provides the best convergence rate for all test problems considered.Other combinations of values usually lead to much slower convergence or sometimes non-convergence at all.
The tuning of the PSO algorithm weight factors is a topic that warrants proper investigation but is outside the scope of this work.For all the problems investigated in this work, the weight factors use the values of 0.5, 1.5 and 1.5 for w , 1 c and 2 c respectively.The position update is the last step.The Position of each particle is updated using its velocity vector as shown in Eq. ( 4) and depicted in Fig. 2: Fig. 2 Depiction of the velocity and position updates in particle swarm optimization The three steps of velocity update, position update, and fitness calculations are repeated until a desired convergence criterion is met.In the PSO algorithm implemented in this study, the stopping criteria is that the maximum change in best fitness should be smaller than specified tolerance for a specified number of moves, S , as shown in Eq. ( 16) [12].In this work, S is specified as ten moves and ε is specified as ( ) ( ) The main advantages of the PSO algorithm are [13]: simple concept, easy implementation, robustness and computational efficiency compared with the mathematical algorithm and other heuristic optimization techniques.These superior features make PSO a highly viable candidate to be used to solve the multi-objective optimization problems.In this chapter we will use the PSO to optimize the size of the vectors extracted from the forms TFRDCS.

The experimental test bed PRONOSTIA
The proposed method is verified on experimental vibration signals taken from the test bed PRONOSTIA, designed and realized within FEMTO-ST institute [15].This test bed has been developed to test and validate the fault detection, diagnostic and prognostic algorithms of ball bearings.The aim of the test bed PRONOSTIA is to provide realistic data which characterize the natural degradation of bearings throughout their useful life.This test bed helps to accelerate the aging of bearings by applying severe operating conditions by varying the rotation speed and by applying a radial force greater than that one recommended by the manufacturer.These constraints allow simulating the degradation of ball bearings in few hours.
The test bed is composed of a ball bearing of type NSK 6804RS installed on a shaft, as shown in Fig. 3.The characteristics of this bearing are given in Table.

Particle memory influence
Current motion influence  method for calculating a parameter very interesting, the dispersion parameter of the cloud of points ξ .This pa- rameter is used to calculate the TFR and extraction feature vectors.

Data treatment by Hilbert transforms
We can easily conclude that the signatures induced in vibration signal analyzed by the Hilbert transform are more pronounced than those induced in the spectrum of signal.
We have proposed a method based a pretreatment Data by Hilbert transform for calculating a very interesting parameter to know the cloud points dispersion parameter ξ : where This parameter ξ is used to calculate the TFR and extraction vectors forms.
As a side from a representation of the cloud points dispersion ξ of Analytics Vibration Signal for a healthy machine and bearing faults (Fig. 5) to remark that the cloud points dispersion ξ for bearing fault are much separated compared with a healthy machine while the points of the same classes are very approximations.The optimization of analytical vibration signal will be do by PSO method, With the optimization criterion, the optimization of vectors form will be fitting and executing to extract pertinent points.We were able to reduce the size of point in the vector from twenty to ten points.Thus the analytical vibration signal with healthy (Figs. 6 and 7) or bearing failure (Figs. 8 and 9) is characterized by ten points each relevant also called scores or high contrast in the sense of Fisher.These vectors can be easily used by beings classification techniques or artificial intelligence.

Conclusion
In this paper, we have proposed a new method calculating a very interesting parameter, that dispersion of cloud parameter of points.Since we cannot use the vibration signals directly due to their very low values..This parameter is used to calculate the TFR and extraction feature vectors.
The simulation results have shown a representation of the dispersion of cloud points of Vibration Analytics Signal (AVS) for a healthy and faulty bearing that the dispersion parameter confirms the separability of classes (healthy bearing class and faulty bearing class).then we made the extraction of the vector formed by the RTF and the separation of classes by filtering (kernel Fischer) and the following we have optimized by PSO; these processing gives us a separation between healthy bearing and faulty.The main advantages of the PSO algorithm are simple concept, easy implementation, robustness and computationnal efficiency compared with the mathematical algorithm and other heuristic optimization techniques.These superior features make PSO a highly viable candidate to be used to solve the multi-objective optimization problems.

Fig. 1 .
Fig. 1 Kernel design Feature points are ambiguity plane points of locations ( ) τ η, where ( ) [ ] 1 , = τ η φ C opt k x , and velocities i k v , of the initial swarm of particles are randomly generated Ambiguity function A i Ambiguity function A i+1 Ambiguity function A n Signal i Signal i+1 Signal n Fisher's discrimination Kernel i using upper and lower bounds on the design variables values min x and max x , as expressed in Eqs.(1) and (

5 10 −
for all test problems.

Fig. 4
Fig. 4 Normal and degraded bearings 6.2.Training set The vibration signals are composed of 2560 samples recorded every 10 seconds with a sampling frequency equal to 25.6 kHz.We cannot use the vibration signals directly due to their very low values.We have proposed a

Fig. 6
Fig. 6 Vector forms of healthy machine before optimization

Fig. 10
Fig. 10 Classes position for vector forms before optimization

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. Bouguerne, A. Lebaroud CLASSIFICATION VECTORS FORMS DEDICATED TO BEARINGS FAULT DETECTION OF ELECTRICAL MACHINES BASED ON PSO ALGORITHM S u m m a r y