Bearing Faults Diagnosis by Current Envelope Analysis under Direct Torque Control Based on Neural Networks and Fuzzy Logic—A Comparative Study

Abstract

Diagnosing bearing defects (BFs) in squirrel cage induction machines (SCIMs) is essential to ensure their proper functioning and avoid costly breakdowns. This paper presents an innovative approach that combines intelligent direct torque control (DTC) with the use of Hilbert transform (HT) to detect and classify these BFs. The intelligent DTC allows precise control of the electromagnetic torque of the asynchronous machine, thus providing a quick response to BFs. Using HT, stator current is analyzed to extract important features related to BFs. The HT provides the analytical signal of the current, thus facilitating the detection of anomalies associated with BFs. The approach presented incorporates an intelligent DTC that adapts to stator current variations and characteristics extracted via the HT. This intelligent control uses advanced algorithms such as neural networks (ANN-DTCs) and fuzzy logic (FL-DTCs). In this paper, a comparison between these two algorithms was performed in the MATLAB/Simulink environment for a three-phase asynchronous machine to evaluate their effectiveness under the proposed approach. The results obtained demonstrated a high ability to detect and classify BFs, confirming the effectiveness of each algorithm. In addition, this comparison highlighted the specific advantages and disadvantages of each approach. This information is valuable in choosing the most suitable algorithm according to the constraints and specific needs of the application.

1. Introduction

The squirrel cage induction machine (SCIM) is the most demanded rotating machine in industrial companies thanks to its characteristics and advantages of robustness, standardization, and good purchase and maintenance cost [1,2,3,4].

Indeed, the availability of the SCIM in the production lines is mandatory to ensure continuity of service; however, malfunctions in parts of this machine can stop the production process completely, resulting in additional loads and heavy financial losses [2,5].

Defects in an SCIM can be mechanical, electrical, or magnetic, but mechanical defects are the most encountered, especially bearing defects (BFs), which represent more than 41% [6,7,8].

Damages to the outer bearing ring (ORBF), inner bearing ring (IRBF), balls, and bearing cage are the common manifestations of BFs [8,9].

A BF in a rotating machine can influence the different signatures of the quantities from this machine (current, speed, electromagnetic torque, temperature, sound, and vibration), and when these quantities are abnormal, this proves that there is a defect in the SCIM [1,2,10].

The main explanation for BFs in SCIMs is electromagnetic torque variations, which result in vibration and, as a result, damage to various bearing components in the turning machine [2,4,7,8,9,11,12].

Currently, controls based on power electronics for starting and controlling rotating machinery represent a source of vibration and harmonics in electrical quantities, including stator current, and these controls are still being examined by researchers to improve their performance, starting with scalar control (SC), passing through field-oriented control (FOC), and arriving at direct torque control (DTC) [13]. The DTC is considered the most efficient and does not require the installation of sensors in the air gap of the machine. Some characteristics stand out. First, it is distinguished by the absence of pulse width modulation (MLI) and instead uses methods such as torque adjustment by hysteresis. Also, DTC is based on the flux model proposed by Takahashi and Depenbrock in 1985 [14], and in order, the selection of vector voltages is carried out directly from a table of location constants established from qualitative rules describing the behavior of the machine [13,15,16,17]. However, despite improvements in control flux diagrams, such as reducing torque ripples or flux distortion, electromagnetic torque ripples remain a major drawback of direct torque control (DTC) [2,14,16,18,19,20]. In this context, this study seeks to evaluate the evolution of DTC control in terms of limiting the severity of BFs by minimizing the amplitudes of torque ripples caused by these defects. In addition, this article also aims to evaluate the impact of improved DTC control on the signal of the drawn current, which is used for the diagnosis of BFs.

Speed controls based on artificial intelligence (artificial neural networks (ANNs) and fuzzy logic (FL)) that do not require knowledge of a mathematical model have recently been proposed, so these approaches have good robustness to parametric variations and measurement noise despite their computational conditions, their development time, and the need for expert knowledge of the system to be adjusted [14,21]. The enhancement solution based on artificial neural networks (ANNs) is presented in [2,13,15,16], where hysteresis comparators and a switching panel are replaced by ANN-based intelligent controllers. In [22,23,24], fuzzy logic is introduced to DTC; it consists of replacing the hysteresis regulators and the switch table with a fuzzy logic controller (FLC). In [25,26], fuzzy logic and artificial neural networks are combined to replace truth tables, hysteresis comparators, and PID controllers, and this control uses an adaptive neuro-fuzzy inference system (ANFIS) to produce a voltage vector that allows the flux and torque to be directed to their references on a given period. These techniques have been very successful in the field of control and identification of nonlinear systems and allow the control of the switching frequency in order to obtain a fast flux and torque with fewer fluctuations [14].

When adopting these intelligent controls for an SCIM in the presence of BFs, a significant reduction in electromagnetic torque ripples is found, and also, the THD of the stator current is reduced by about 68%, which effectively contributes to the diagnosis of BFs.

In the literature, the most chosen quantities for the diagnosis of BFs from birth are current [2,10,18,27,28,29,30,31,32] and vibration [33,34,35,36,37], and recently, much research has focused on sound analysis [38,39,40] and thermography [41,42,43,44].

The most used methods in the case of BF detection are those based on vibration [37]; however, the implementation of vibration sensors is complex and cumbersome [45,46].

Motor current signature analysis (MCSA) is an effective and efficient technique for diagnosing various faults in different devices [47,48,49], and especially in SCIM, this technique has several advantages, including being non-invasive and not requiring special sensors [33,50]. However, the major challenge for this method lies in the non-stationarity of the signals to be analyzed as the line current of the machine, as this case is present when the SCIM is controlled by a closed loop control including DTC [19,51,52].

The fast Fourier transform (FFT), which is typically used in MCSA diagnosis, is impeded by DTC controls in many research works [10,18] because the variable rotating speed causes a continual variation of the supply frequency (non-stationarity) [2,12,50,53]. The adoption of advanced signal processing methods is the solution that meets this requirement of non-stationarity of electrical signals or vibration, and generally, these methods are based on three transforms widely used in the literature, namely short-term Fourier transform (STFT) [10], the continuous wavelet transform (CWT) [35,54], and discrete (DWT) [34,53,55,56] and Hilbert transform (HT) [2,18,27,51], the latter of which contributes to the extraction of the signal envelope to be processed. Envelope analysis is the technique that is less sensitive to the variation of the load torque, so its mathematical model is the simplest [27].

This article has two main contributions:

• Contribution to the diagnosis of BFs of an SCIM with a DTC control by MCSA by analyzing the spectrum of the envelope current (ECS) extracted by the HT.
• Measurement and comparison of the impact of improving DTC control by making simulations first in the case of conventional DTC (CDTC), then in the case of an ANN DTC, and finally with an FLDTC. The comparison of these three DTC control modes concerns electromagnetic torque, rotational speed, stator current THD, and ECC.

To organize and direct the reader in a process of progressive reading, this article is organized in the following form:

Section 2 presents the mathematical model of the SCIM and the BFs. Section 3 presents the HT and the process of obtaining the current envelope. Section 4 contains a summary of the CDTC control, and Section 5 and Section 6 are devoted to the description of the ANNDTC and FLDTC smart controls, respectively. The simulation results and their interpretations are discussed in Section 7. Finally, this article concludes with a conclusion and a proposal for future work in Section 8.

2. Modeling of the SCIM and BFs

2.1. SCIM Model

In this study, the modeling of BFs is carried out in the mechanical part, which concerns the load torque, so a basic model of the SCIM is useful. This model is characterized by electrical and mechanical parameters. The SCIM system is nonlinear, and it is described by the three following Equations (1)–(3) [2,24,57]:

• The electrical equations are as follows:

$$\{\begin{array}{l}{\mathrm{v}}_{\mathrm{s}\mathsf{\alpha }}={\mathrm{R}}_{\mathrm{s}}.{\mathrm{i}}_{\mathrm{s}\mathsf{\alpha }}+{\displaystyle \frac{\mathrm{d}{\mathsf{\psi }}_{\mathrm{s}\mathsf{\alpha }}}{\mathrm{d}\mathrm{t}}}\\ {\mathrm{v}}_{\mathrm{s}\mathsf{\beta }}={\mathrm{R}}_{\mathrm{s}}.{\mathrm{i}}_{\mathrm{s}\mathsf{\beta }}+{\displaystyle \frac{\mathrm{d}{\mathsf{\psi }}_{\mathrm{s}\mathsf{\beta }}}{\mathrm{d}\mathrm{t}}}\\ 0={\mathrm{R}}_{\mathrm{r}}.{\mathrm{i}}_{\mathrm{r}\mathsf{\alpha }}+{\displaystyle \frac{\mathrm{d}{\mathsf{\psi }}_{\mathrm{r}\mathsf{\alpha }}}{\mathrm{d}\mathrm{t}}}+{\mathsf{\omega }}_{\mathrm{m}}.{\mathsf{\psi }}_{\mathrm{r}\mathsf{\beta }}\\ 0={\mathrm{R}}_{\mathrm{r}}.{\mathrm{i}}_{\mathrm{r}\mathsf{\beta }}+{\displaystyle \frac{\mathrm{d}{\mathsf{\psi }}_{\mathrm{r}\mathsf{\beta }}}{\mathrm{d}\mathrm{t}}}-{\mathsf{\omega }}_{\mathrm{m}}.{\mathsf{\psi }}_{\mathrm{r}\mathsf{\alpha }}\end{array}$$

(1)

• The magnetic equations are as follows:

$$\{\begin{array}{c}{\mathsf{\psi }}_{\mathrm{s}\mathsf{\alpha }}={\mathrm{L}}_{\mathrm{s}}{\mathrm{i}}_{\mathrm{s}\mathsf{\alpha }}+{\mathrm{L}}_{\mathrm{m}}.{\mathrm{i}}_{\mathrm{r}\mathsf{\alpha }}\\ \begin{array}{l}{\mathsf{\psi }}_{\mathrm{s}\mathsf{\beta }}={\mathrm{L}}_{\mathrm{s}}{\mathrm{i}}_{\mathrm{s}\mathsf{\beta }}+{\mathrm{L}}_{\mathrm{m}}.{\mathrm{i}}_{\mathrm{r}\mathsf{\beta }}\\ {\mathsf{\psi }}_{\mathrm{r}\mathsf{\alpha }}={\mathrm{L}}_{\mathrm{r}}{\mathrm{i}}_{\mathrm{r}\mathsf{\alpha }}+{\mathrm{L}}_{\mathrm{m}}.{\mathrm{i}}_{\mathrm{s}\mathsf{\alpha }}\\ {\mathsf{\psi }}_{\mathrm{r}\mathsf{\beta }}={\mathrm{L}}_{\mathrm{r}}{\mathrm{i}}_{\mathrm{r}\mathsf{\beta }}+{\mathrm{L}}_{\mathrm{m}}.{\mathrm{i}}_{\mathrm{s}\mathsf{\beta }}\end{array}\end{array}$$

(2)

• The mechanical equations are as follows:

$$\{\begin{array}{c}{\mathrm{T}}_{\mathrm{e}\mathrm{m}}=\mathrm{p}.({\mathsf{\psi }}_{\mathrm{s}\mathsf{\alpha }}{\mathrm{i}}_{\mathrm{s}\mathsf{\beta }}-{\mathsf{\psi }}_{\mathrm{s}\mathsf{\beta }}{\mathrm{i}}_{\mathrm{s}\mathsf{\alpha }})\\ \mathrm{J}.{\displaystyle \frac{\mathrm{d}\mathsf{\Omega }}{\mathrm{d}\mathrm{t}}}+\mathrm{f}.\mathsf{\Omega }={\mathrm{T}}_{\mathrm{e}\mathrm{m}}-{\mathrm{T}}_{\mathrm{r}}\end{array}$$

(3)

Equations (1) and (2) represent the mathematical model of the SCIM, where the voltages across the stator coils depend on the stator resistance R_s, the stator current (i_sα, i_sβ)_, and the flux variation. These fluxes depend on the current and inductances.

Equation (3) shows the expressions for electromagnetic torque T_em and speed Ω, where electromagnetic torque depends on the number of pole pairs, stator currents, and stator fluxes. Speed depends on the torque itself, the inertia J, and the friction coefficient f.

2.2. Bearing Defects

Bearing failure is one of the leading causes of rotating machinery, resulting in costly downtime. Point defects are localized and can be classified according to the following affected elements as shown in Figure 1: defect of the outer ring of the bearing, defect of the inner ring of the bearing, defect of the ball, and defect of the cage. They can be similar to an incipient defect, with a chipping created by a material defect or a crater caused by bearing fatigue [46,51,58].

BFs usually cause vibrations, increases in the sound level emitted by the machine and eccentricity of the rotor, which produce an unbalanced magnetic attraction. In addition, they induce oscillations in the load torque of the machine and harmonics in the current signal [1,51,59]. The impact of these rolling defects on the torque makes it possible to model them by using Equation (4), where the load torque is represented by a continuous component of amplitude T₀ and related torque variations of BF of amplitude T_c and frequency f_c [8,51].

$${\mathrm{T}}_{\mathrm{r}}(\mathrm{t})={\mathrm{T}}_0+{\mathrm{T}}_{\mathrm{c}}.\cos (2\pi {\mathrm{f}}_{\mathrm{c}}.\mathrm{t})$$

(4)

Indeed, whenever a ball in a ball bearing encounters an isolated defect, for example, a crack, chipping, or an imprint left by a foreign body, it causes an impact. The repetition of these impacts results in the vibration of the bearing at frequencies that are determined by the location of the defect, the geometry of the bearing, and the speed of rotation of the shaft. The characteristic rolling frequencies are noted as follows:

• BPFO: ball passage frequency on the outer ring for an ORBF.
• BPFI: ball passage frequency on the inner ring for an IRBF.

These frequencies are given by the following equations [34,37,57,58,60]:

$$\mathrm{BPFO}={\displaystyle \frac{{\mathrm{N}}_{\mathrm{b}}}{2}}{\mathrm{f}}_{\mathrm{r}}\left(1-{\displaystyle \frac{{\mathrm{D}}_{\mathrm{b}}}{{\mathrm{D}}_{\mathrm{c}}}}\cos \mathsf{\alpha }\right)$$

(5)

$$\mathrm{BPFI}={\displaystyle \frac{{\mathrm{N}}_{\mathrm{b}}}{2}}{\mathrm{f}}_{\mathrm{r}}\left(1+{\displaystyle \frac{{\mathrm{D}}_{\mathrm{b}}}{{\mathrm{D}}_{\mathrm{c}}}}\cos \mathsf{\alpha }\right)$$

(6)

with the following variables:

f_r: rotation frequency.

N_b: number of marbles.

D_b: diameter of a ball.

D_c: diameter of the cage.

α: angle of contact.

Thus, the frequency of harmonics in the spectrum of current extracted by FFT is given by Equation (7) [2,58].

$${\mathrm{f}}_{\mathrm{d}}{=\mathrm{f}}_{\mathrm{s}}\pm \mathrm{k}{·\mathrm{f}}_{\mathrm{c}}$$

(7)

In Equation (7), k represents the order of the harmonic; f_c can be BPFO, BPFI, or BRF; and f_s is the frequency of the supply current.

This article focuses specifically on defects at a rolling point. This defect produces a predictable characteristic of predictable fault frequencies in current waveforms as shown in Equation (8) [34]:

$$\mathrm{i}(\mathrm{t})={\displaystyle \sum _{\mathrm{n}=1}^{\infty }{\mathrm{i}}_{\mathrm{n}}·\cos (\frac{2\mathsf{\pi }{\mathrm{f}}_{\mathrm{d}}}{\mathrm{p}}}+\mathsf{\phi })$$

(8)

where φ is the phase angle and p represents the number of pole pairs of the SCIM.

The defect is not at all apparent in the FFT spectrum, especially during continuous variation of the feed frequency and rotational speed, so accurate diagnosis of a bearing defect often requires the use of more elaborate analysis techniques than the simple acquisition of an FFT spectrum. Envelope analysis makes it possible to extract periodic impacts such as those produced during BFs. When the envelope spectrum indicates a severe defect of BFs, the severity of the defect is explained by the presence of many harmonics in the spectrum. The envelope of the signal to be processed is obtained by the famous Hilbert transform presented in the next section.

3. Hilbert’s Transform

The Hilbert transform (HT) is a signal processing tool that offers a high level of resolution. HT is also one of the most demanded advanced signal analysis techniques today in the field of machine fault diagnosis [1,2,27,36,61,62].

To obtain the instantaneous envelope and phase of the real signal x(t), we must first perform the Hilbert transform of the signal x(t). The Hilbert transform x(t) of a real signal x(t) is given by (9) [1,2,27,61]:

$$\mathrm{X}(\mathrm{t})={\displaystyle \frac{1}{\mathsf{\pi }\mathrm{t}}}\ast \mathrm{x}(\mathrm{t})={\displaystyle \frac{1}{\mathsf{\pi }}}{\displaystyle {\int }_{-\infty }^{+\infty }\frac{\mathrm{x}(\mathrm{t})}{\mathrm{t}-\mathsf{\tau }}\mathrm{d}\mathsf{\tau }}$$

(9)

By combining X(t) and x(t), the analytical signal of the real signal x(t) is obtained as shown in Equations (10) and (11) [55]:

$$\mathrm{a}\left(\mathrm{t}\right)=\sqrt{{\mathrm{x}}^2\left(\mathrm{t}\right)+{\mathrm{X}}^2\left(\mathrm{t}\right)}$$

(10)

$$\mathsf{\phi }\left(\mathrm{t}\right)={\tan }^{-1}({\displaystyle \frac{\mathrm{X}\left(\mathrm{t}\right)}{\mathrm{x}\left(\mathrm{t}\right)}})$$

(11)

where $\mathrm{a}\left(\mathrm{t}\right) \mathrm{a}\mathrm{n}\mathrm{d} \mathsf{\phi }\left(\mathrm{t}\right)$ are the instantaneous amplitude and instantaneous phase, respectively.

The presence of a defect in the DTC-controlled SCIM results in distortion of the envelope signal. Figure 2a illustrates the envelope of the current in the healthy state, where the amplitude of the ripples of this envelope is slightly less than that shown in Figure 2b. The amplitudes of the oscillations increase due to the ORBF effect.

Figure 2d shows the impact of an ORBF on the frequency content of the stator current and its envelope. The instantaneous frequencies of the harmonic spectra related to the BFs are directly provided by the envelope spectrum. On the other hand, for the current spectrum, these frequencies depend on the supply frequency. If this supply frequency varies, as is the case when using DTC, the current spectrum becomes useless, which underlines the benefit of the time harmonic analysis by the HT. In Section 6, simulation results in the frequency domain will be limited to the analysis of SCE, from which instantaneous frequencies will be extracted and interpreted.

4. Direct Torque Control

In the 1980s, several researchers, including I. Takahashi, T. Noguchi, and Depenbrok, developed a direct torque control technique for induction motors. This technique is known as DTC (direct torque control). The DTC works by directly measuring the control pulses sent to the voltage inverter switches, thus maintaining stator flux and electromagnetic torque in predetermined hysteresis zones. The application of this method makes it possible to separate the control of flux and torque. The voltage inverter produces seven phase plane positions at its output, corresponding to eight voltage vector sequences. The flux can be estimated using current and voltage measurements of the stator of the machine, which is described by the two Equations (12) and (13). Figure 3 illustrates the block diagram of the DTC applied to an SCIM [2,13,17,24].

$$\{\begin{array}{l}{\mathsf{\psi }}_{\mathrm{s}\mathsf{\alpha }-\mathrm{est}}={\displaystyle \int \left({\mathrm{v}}_{\mathrm{s}\mathsf{\alpha }}-{\mathrm{R}}_{\mathrm{s}\mathsf{\alpha }}{\mathrm{i}}_{\mathrm{s}\mathsf{\alpha }}\right)}\mathrm{dt}\\ {\mathsf{\psi }}_{\mathrm{s}\mathsf{\beta }-\mathrm{est}}={\displaystyle \int \left({\mathrm{v}}_{\mathrm{s}\mathsf{\beta }}-{\mathrm{R}}_{\mathrm{s}\mathsf{\beta }}{\mathrm{i}}_{\mathrm{s}\mathsf{\beta }}\right)}\mathrm{dt}\end{array}$$

(12)

$${\mathsf{\psi }}_{\mathrm{s}}=\sqrt{{{\mathsf{\psi }}^2}_{\mathrm{s}\mathsf{\alpha }}+{{\mathsf{\psi }}^2}_{\mathrm{s}\mathsf{\beta }}}$$

(13)

The calculation of the electromagnetic torque is obtained from the two components of the flux using Formula (14) [2,13,17,24]:

$${\mathrm{T}}_{\mathrm{e}\mathrm{m}-\mathrm{e}\mathrm{s}\mathrm{t}}=\mathrm{p}·\left({\mathsf{\psi }}_{\mathrm{s}\mathsf{\alpha }-\mathrm{est}}{\mathrm{i}}_{\mathrm{s}\mathsf{\beta }}-{\mathsf{\psi }}_{\mathrm{s}\mathsf{\beta }-\mathrm{est}}{\mathrm{i}}_{\mathrm{s}\mathsf{\alpha }}\right)$$

(14)

The three essential parameters in every DTC control, whether conventional or intelligent, are the three errors of flux, torque, and velocity (as well as the angle θ_s, ξ_ψ, ξ_T, ξ_Ω) between the reference frame (α, β) and the vector ψ_s, which determine the location of the vector ψ_s on the basis of the components ψ_sα and ψ_sβ. These four parameters are given by Equations (15)–(18) [2,13].

$${\mathsf{\theta }}_{\mathrm{s}}=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{g}\left({\displaystyle \frac{{\mathsf{\psi }}_{\mathrm{s}\mathsf{\beta }}}{{\mathsf{\psi }}_{\mathrm{s}\mathsf{\alpha }}}}\right)$$

(15)

$${\mathsf{\xi }}_{\mathsf{\psi }}={\mathsf{\psi }}_{\mathrm{r}\mathrm{e}\mathrm{f}}-{\mathsf{\psi }}_{\mathrm{e}\mathrm{s}\mathrm{t}}$$

(16)

$${\mathsf{\xi }}_{\mathrm{T}}={\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{f}}-{\mathrm{T}}_{\mathrm{e}\mathrm{s}\mathrm{t}}$$

(17)

$${\mathsf{\xi }}_{\mathsf{\Omega }}={\mathsf{\Omega }}_{\mathrm{r}\mathrm{e}\mathrm{f}}-{\mathsf{\Omega }}_{\mathrm{e}\mathrm{s}\mathrm{t}}$$

(18)

The components that make up the conventional DTC and the fault detection elements are shown in Figure 3.

Figure 3 above shows the block diagram of the CDTC control on one side and the diagnostic elements of the SCIM BFs on the other.

The CDTC consists mainly of the flux and torque estimator based on the electrical quantities current and voltage, the two hysteresis comparators, and the switching table.

The diagnostic part contains the two transforms for processing the current signal (FFT and HT).

The sector and the variations in flux and torque make it possible to choose the voltage vectors to follow the reference values of the flux and torque.

Table 1. Switching table for a CDTC.

Flux Variation H(ψs)	Torque Variations H(Tem)	Sectors Si
		S1	S2	S3	S4	S5	S6
Δψs = 1	ΔTem = 1	V2	V3	V4	V5	V6	V1
	ΔTem = 0	V7	V0	V7	V0	V7	V0
	ΔTem = −1	V6	V1	V2	V3	V4	V5
Δψs = 0	ΔTem = 1	V3	V4	V5	V6	V1	V2
	ΔTem = 0	V0	V7	V0	V7	V0	V7
	ΔTem = −1	V5	V6	V1	V2	V3	V4

shows how to choose the appropriate voltage vector [13,19].

Voltage vectors V_i depend on sector S_i, flux ψ_s, and torque T_em to follow their values. The flux variation Δψ_s is either 0 or 1, while the torque variation ΔT_em is either 0, 1, or −1 depending on the direction of rotation.

5. Direct Torque Control Based on Artificial Neural Networks

Nonlinear optimization problems, where these tools cannot guarantee to find a global minimum, are addressed. The numerical cost and accuracy of these algorithms depend on the initialization of their internal parameters, which may themselves require adjustment according to the application. The authors in [63] gave an overview of optimization techniques commonly used in electrical machines.

In the proposed DTC strategy, an artificial neural network was used to regulate rotational speed. This method was used to obtain good quality of the electric current of the SCIM stator, to minimize torque fluctuations, and to obtain a minimum value of the THD of the current. In order to ensure these improvements, it was planned to replace the hysteresis comparators, the switching table, and the derived integral proportional control (PID) with neural controllers [2,13,16]. The parameters of the artificial neural network controllers used in this work are shown in

Table 2. The learning outcomes achieved for ANN controller blocks.

Learning Outcomes	Values
	ANN_Ω	ANN_T	ANN_ψ	ANN_SW_T
Number of iterations	1000	85	209	168
Time	0:00:34	0:00:21	0:00:34	0:00:34
Performance	0.000538	0.0148	0.000506	0.000459
Gradient	4.04	2.09 × 10−5	7.31 × 10−6	1.18 × 10−5
Him	0.01	10−9	10−9	10−5
Validation checks	0	6	6	6
Best validation performance	7.5373 × 10−4 at epoch 1000	1.2627 × 10−3 at epoch 79	5.4767 × 10−3 at epoch 203	3.9404 × 10−4 at epoch 162

. Figure 4 illustrates the block diagram of the ANN-DTC control, which integrates these neural controllers [2,13].

5.1. ANN-Based Comparators, Cruise Control, and Switchboard Controls

ANNs are highly connected networks of elementary processors running in parallel. Neural network training data are one of the most important features of ANNs for learning and improving their functioning [13]. The neuron is the basis of the ANN and consists of the vertices and activation functions shown in Figure 5.

The mathematical equations of a neuron are described as follows [13]:

$${\mathrm{y}}_{\mathrm{i}}={\mathrm{F}}_1\left(\mathrm{s}\right)\ast \left\{{\sum }_{\mathrm{i}=1}^{\mathrm{N}}\left({\mathrm{x}}_{\mathrm{i}}{\ast \mathrm{w}}_{\mathrm{i}}+\mathrm{b}\right)\right\}\mathrm{with}{\mathrm{F}}_1\left(\mathrm{s}\right)={\displaystyle \frac{{\mathrm{e}}^{\mathsf{\alpha }\mathrm{s}}-{\mathrm{e}}^{-\mathsf{\alpha }\mathrm{s}}}{{\mathrm{e}}^{\mathsf{\alpha }\mathrm{s}}+{\mathrm{e}}^{-\mathsf{\alpha }\mathrm{s}}}}$$

(19)

$${\mathrm{O}}_{\mathrm{i}}={\mathrm{F}}_2\left(\mathrm{s}\right)\ast \left\{{\sum }_{\mathrm{i}=1}^{\mathrm{N}}\left({\mathrm{y}}_{\mathrm{i}}\ast {\mathrm{w}}_{\mathrm{i}}+\mathrm{b}\right)\right\}\mathrm{with}{\mathrm{F}}_2\left(\mathrm{s}\right)=\mathsf{\beta }\mathrm{s}$$

(20)

where input signals (x_i), synaptic input signal weights (w_i), and bias and neuron output signals (y_i) are the main parameters. Also, a linear activation function F₂(s) and a hyperbolic nonlinear activation function F₁(s) are brought into play, where α and β are gains.

The neural network in this paper is trained using the feed-forward backpropagation technique until the RMS between the desired model and the output model is very small [2,13].

The total number of neurons for each controller is considered in a hidden layer for network training, and the input x_i has a synaptic weight w_i. The output of the hidden layer is where b is the value of the neuron’s bias.

$${\mathrm{v}}_{\mathrm{i}}={\sum }_{\mathrm{i}=1}^{\mathrm{N}}{{\mathrm{e}}_{\mathrm{a}}\mathrm{w}}_{\mathrm{i}}+\mathrm{b}$$

In this paper, the x_i input of ANN neural controllers of speed, stator flux, and torque is the error between the reference setpoint and the estimated setpoint, so the ANN switching table contains three neurons in its input layer, as shown in Figure 4 ξ_Ω, ξ_ψ, ξ_T.

5.2. Values and the Algorithm Chosen for ANN

Among the algorithms available in the Matlab toolbox, three of them can be identified: gradient descent, the gradient descent algorithm with momentum, and the Levenberg–Marquardt algorithm, which are represented by the traingd, traingdm, and trainlm functions, respectively. Two performance indices are also available: the MSE (RMS error) to be minimized and the regression value R, which is used to estimate the correlation between outputs and desired targets [13].

The results of the tests carried out on these algorithms demonstrated that the Levenberg–Marquardt method is distinguished by its speed of convergence towards the minimum of the quadratic error, thus surpassing the other two methods and offering good performance.

The learning outcomes obtained for ANN controller blocks are shown in Table 2. They correspond to those of the Levenberg–Marquardt method that were finally chosen to be applied in our ANN-DTC order. Also, for all controllers, the chosen adaptation learning function is Trainlm, and the activation function adopted is Transig.

The summary of the optimal values adopted in our order is illustrated in

Table 3. Specification of the optimum proposed ANN models.

ANN Parameters	Values
	ANN_Ω	ANN_T	ANN_ψ	ANN_SW_T
Neural network	Three-layer feed-forward network with sigmoid hidden neurons
Number of hidden layer nodes	10	10	10	10
Number of neurons in the input layer	1	1	1	3
Number of neurons in the second hidden layer	1	1	1	3
Number of neurons in the output layer	1	1	1	3
Learning rate	0.5	0.5	0.5	0.5
Number of epochs	20	100	100	200

.

6. Direct Torque Control Based on Fuzzy Logic

Currently, fuzzy logic occupies an important place in the field of control, especially in electrical engineering. It offers the possibility of designing controllers capable of generating effective control laws, even in the absence of precise knowledge of the process to be controlled [23,24,64]. Unlike approaches using precise mathematical expressions, this type of controller relies on inferences made from rules based on linguistic variables [24,64].

The integration of fuzzy logic into direct torque control has significant advantages in terms of the performance and robustness of the control system. Thus, the use of a fuzzy logic controller replaces traditional switching boards and hysteresis controllers, thus improving system performance and reducing fluctuations in electromagnetic flux and torque [23,24], Figure 6 illustrates the DTC structure based on fuzzy logic for an SCIM powered via a voltage inverter.

In this system, the fuzzy logic controller considers three inputs, including flux error, torque error, and angle θ_s. These parameters are all taken into consideration to perform the calculations and inferences necessary to generate the appropriate control signals [24].

Generally, the control by fuzzy logic is carried out in three steps: fuzzification, the use of a rules table to determine the output based on inputs, and defuzzification [23].

The purpose of the fuzzification process is to transform input variables into linguistic variables by defining membership functions for each input variable. Each input and output of the fuzzy controller is divided into a specified number of fuzzy sets [23,24]. The two input variables (torque error and flux error) are divided into two linguistic variables, where the language variable N is assigned to the negative error, the language variable P is assigned to the positive error, the language variable Z is assigned to the zero error in the case of torque. The position of the stator flux vector is considered the third input variable of the fuzzy controller. This variable is defined in a discursive universe divided into six fuzzy sets (θ₁ to θ₆), whose membership functions adopt triangular shapes, as illustrated below. The output variable that represents the switching state of the inverter is divided into three output groups (S₁, S₂, and S₃). Each output group is defined in a discursive universe divided into two fuzzy sets (zero and one), where the membership functions adopt a singleton form [23].

Control rules should be established based on input and output variables in accordance with the switching table of the conventional DTC. The structure of the fuzzy logic controller is shown in Figure 7. The fuzzy relationships that determine the output variables of the controller as a function of the input variables are grouped in

Table 4. Set of fuzzy relationships.

ζψ	ζTem	i1	i2	i3	i4	i5	i6
P	N	V6 (101)	V1 (100)	V2 (110)	V3 (010)	V4 (011)	V5 (001)
P	Z	V7 (111)	V0 (000)	V7 (111)	V0 (000)	V7 (111)	V0 (000)
P	P	V2 (110)	V3 (010)	V4 (011)	V5 (001)	V6 (101)	V1 (100)
N	N	V5 (001)	V6 (101)	V1 (100)	V2 (110)	V3 (010)	V4 (011)
N	Z	V0 (000)	V7 (111)	V0 (000)	V7 (111)	V0 (000)	V7 (111)
N	P	V3 (010)	V4 (011)	V5 (001)	V6 (101)	V1 (100)	V2 (110)

[24].

The output variable, which cannot be directly executed by physical actuators, must be transformed into a numerical quantity, which is the objective of the defuzzification step. In this work, the maximum value method is used as a defuzzification criterion, the with output value being chosen as the abscissa corresponding to the maximum value of the resulting membership function. The control algorithm then has 36 relations. The inference method used is the Mamdani method based on the max–min decision because it has the advantage of being easy to implement on the one hand and giving better results on the other hand [23,24]. Control rules are expressed in terms of input and output variables as shown in Table 4.

7. Simulation Results and Discussions

In order to obtain the necessary results for the diagnosis of BFs and to compare them, a MATLAB/Simulink simulation model was developed for SCIM in different situations (healthy state and with BFs) and for the three control modes: (i) with CDTC, (ii) with ANNDTC control, (iii) with FLDTC control. The computer used is equipped with a 7th-generation i5 processor with 16 GB RAM. The simulation time chosen is 10 s to observe the behavior of the machine, with a sampling rate of 10 kHz. The SCIM engine used is 1.5 kW, the parameters of which are given in

Table A1. Parameters of SCIM.

Variable	Value
Nominal power	Pu = 1.5 kW
Stator and rotor resistances	Rs = 4 Ω; Rr = 0. 8 Ω
Self-inductances	Ls = 0.39045 H
Rotor inductance	Lr = 0.02139 H
Maximum of mutual inductance	M = 0.08736 H
Viscous frictions	f = 0.004565 Kg·m2/s
Total inertia	J = 0.0147 Kg·m2
Pair pole number	P = 2

in the Appendix A. The reference ball bearing chosen is 62052RSC3, with its technical characteristics provided in

Table A2. Technical characteristics of the bearing.

Type of Bearing	Inside Diameter	Outer Diameter	Width	Angle Contact α
62052RSC3	25 mm	52 mm	15 mm	0

, and the dimensions necessary to determine the characteristic frequencies of the BFs are given in

Table A3. The values used to determine BFs frequencies.

Number of Balls	Diameter of a Ball	Diameter of the Cage	Cosa
9	7.94 mm	38.5 mm	1

. The load torque oscillations caused by nascent BFs have a frequency, fc, and an amplitude of 1 Nm. In all situations, the SCIM starts empty, and then at time t = 5 s, the nominal load (T_n = 10 Nm) is applied. To illustrate the comparison between the three DTC control modes, the signals extracted for measurement and evaluation are as follows: (i) electromagnetic torque and rotational speed, (ii) harmonic distortion rate (THD), and (iii) current envelope spectrum (SCE). For the latter, the graphs are represented on a scalar scale, where the amplitudes are weighted according to the length of the modulus of the signal.

7.1. Rotational Speed

Based on Figure 8a–c, the improved DTC control has a clear impact on speed performance. Indeed, in the healthy state of the SCIM, the CDTC allows a reference speed monitoring n_ref with an overshoot that reaches 1 Rpm; however, when we adopt the ANNDTC, this exceedance is almost 0.5 Rpm, and it becomes almost zero (0.1 Rpm) with the FLDTC. These data show the advantage of improving the CDTC based on artificial intelligence.

According to Figure 9a–c, a defect in the bearing induces oscillations of the mechanical quantities speed and torque. In the case of ORBF, the amplitude of these oscillations for the speed signal reaches 3.2 Rpm with the CTDC, 2.5 Rpm with the ANNDTC, and 1.5 Rpm with the FLDTC. Also, as illustrated in the Figure 10a–c in the case of IRBF, the values of these amplitudes become 2.5 Rpm with the CTDC, 1.75 Rpm with the ANNDTC, and 0.9 Rpm with the FLDTC.

These measurements of the level of oscillations in the case of each BF and for each type of control show the importance of the FLDTC, which dampens and controls the amplitude of these oscillations by about three times compared to the CDTC and two times compared to the ANNDTC; hence, the improvement of the performance of the DTC has a good impact on the SCIM because it minimizes the influence of mechanical defects by reducing their severity and keeping the SCIM running as soon as possible. This improvement then makes the DTC tolerant to BFs.

7.2. Electromagnetic Torque

Regarding torque performance, the DTC control is well known for its major disadvantage, which lies in the ripples of the electromagnetic torque. When a mechanical defect appears at the SCIM rotor, it could increase the amplitude of these ripples as well as the vibrations of the machine.

According to Figure 11a–c, the CDTC causes ripples of T_em of an order of 2.5 Nm amplitude, and the impact of ORBF or IRBF is no longer apparent because the proportion of amplitude due to BFs is negligible before CDTC control. Meanwhile, the minimization of these ripples by the ANNDTC (Figure 11b, Figure 12b, and Figure 13b) or FLDTC (Figure 11c, Figure 12c, and Figure 13c) makes it possible to clearly illustrate the impact of BFs on the signal of T_em, especially when the specific frequency of BF is low as in the case of ORBF.

In the measurement of the influence of each DTC control on the signal of the electromagnetic couple, we noticed in the healthy state that the ripples are better controlled by the ANNDTC (amplitude is about 0.45 Nm) followed by the FLDTC (amplitude is about 0.9 Nm).

The results obtained for T_em show the benefit of ANNDTC compared to CDTC and FLDTC, unlike the speed at which FLDTC is the most advantageous compared to CTDC and ANNDTC in controlling this magnitude.

From the six figures above, it can be seen that the two ANNDTC and FLDTC controls offer better control of mechanical quantities even in the presence of BFs by contributing to the preservation of SCIM.

Despite this major benefit of intelligent DTC controls on torque and speed, they can hide the presence of nascent BFs because they dampen their effects. Therefore, the confirmation must be based on the electrical quantities and especially the supply current of the SCIM.

7.3. The Harmonic Distortion Rate (THD) of Stator Current

From Figure 14, Figure 15 and Figure 16, it is clear that ANNDTC and FLDTC have a tendency of lowering total harmonic distortion (THD) compared to CDTC. Indeed, in the healthy state of the SCIM ordered by CDTC, this rate is worth 12.71% against 3.78% with the ANNDTC and 3.81% with the FLDTC.

The appearance of ORBF or IRBF generates an increase in this THD with a value that does not reach 0.9%. It can also be seen that this rate in the case of BFs increases when the characteristic frequency of mechanical defect is seminude (for example, 4.66% for ORBF against 4.38% for IRBF). Despite the remarkable differences of this THD, it remains insignificant to conclude on the presence of BFs.

In work that is concerned with diagnosis, and during the analysis of the electric current, often, the adoption of FFT is the preferred solution to extract spectra that allow the localization of faults in electrical machines, but with the presence of closed-loop controls, FFT must be accompanied by an advanced method of signal processing such as HT because electrical signals are not stationary.

7.4. Envelope Current Spectrum (ECS)

The analysis of the envelope of the current extracted by the HT is the suitable method to illustrate the SCEs in relation to the BFs; however, a signal rich in harmonic can hide the spectra indicating the defects. Indeed, according to Figure 9, Figure 10 and Figure 11, which show the spectral content of the envelope of the current in the healthy state, we notice that the CDTC and the ANNDTC are the most harmonious and the spectra caused by these controls are large amplitudes, while those created by the FLDTC are almost zero.

The improved intelligent control acts as an additional filter to properly target BF-dependent CESs.

From Figure 17, Figure 18 and Figure 19, it is noted that the FLDTC guarantees an almost clear spectral content, allowing one to have spectra indicating ORBF or IRBF as intended. On the other hand, CDTC and ANNDTC have other spectra in the range of 0 to 150 Hz, which can bias the location of BFs.

Despite the minimization of the amplitude of SCE by the two controls ANNDTC and FLDTC with respect to the CDTC, the ratio between the amplitude of the spectra indicating BFs and that of the fundamental is better for the two intelligent controls compared to the CDTC.

Table 5. Impact of DTC controls on the number and amplitude of spectra in the presence of ORBF.

Nature of DTC	Amplitudes from the Fundamental to the Healthy State	Amplitudes of the Fundamental in the Presence of ORBF	Amplitudes SCE de ORBF	ORBF Report	Number of Spectra in the Range 0 to 150 Hz in the Healthy State
CDTC	1461	1502	1744	1.16	4
ANNDTC	621	837	1652	1.97	5
FLDTC	443	1031	1108	1.07	3

and

Table 6. Impact of DTC controls on the number and amplitude of spectra in the presence of IRBF.

Nature of DTC	Amplitudes from the Fundamental to the Healthy State	Amplitudes of the Fundamental in the Presence of IRBF	Amplitudes SCE de IRBF	IRBF Report	Number of Spectra in the Range 0 to 150 Hz in the Healthy State
CDTC	1461	1493	955	0.64	4
ANNDTC	621	796	915	1.15	5
FLDTC	443	732	653	0.9	3

show these amplitudes, the ratio evoked, and the number of spectra in the healthy state in the range of 0 to 150 Hz having an amplitude of the order or one that goes beyond the fundamentals.

According to the two tables (Table 5 and Table 6) below, we notice that the presence of BFs automatically causes an increase in the amplitudes of the fundamental. This increase is very important in the case of the FLDTC (130% with ORBF and 65.2% with IRBF). For the ANNDTC, this increase is 34.8% for the ORBF and 28.2% for the IRBF, while the CDTC gives only a slight increase in this amplitude of the fundamental (2.8% with ORBF and 2.2% with IRBF). Hence, there is an advantage of these two intelligent controls compared to the CDTC, and these two graphs below illustrate this advantage.

Also, in the last two columns of these two tables, the ratio between the amplitude of the spectra indicating BFs and that of the fundamental and between the ratio evoked and the number of spectra in the healthy state in the range 0 to 150 Hz having an amplitude of the order or exceeding the fundamental represents another argument for the advantage of the FLDTC and the ANNDTC in the problem of diagnosis in relation to the CDTC. Indeed, even if the ANNDTC has the largest number of harmonics in the low-frequency interval of 0 to 150 Hz, it has the best ratio.

The two figures (Figure 20 and Figure 21) below summarize the impact of each control on the destination of the harmonic spectra in relation to the BFs.

A brief comparison with related publications that primarily concentrate on broken rotor bar (BRB) faults is presented in

Table 7. Comparison between this proposed technique and some works that have adopted the HT.

Publication Reference	Control Type	Signal Processing Tool	Faults	Is the Control Helping to Diagnose (Yes/No)?
[2]	ANN-DTC	HT	BFs	Yes
[18]	CDTC	HT	BRB	No
[19]	ANN-FL-DTC	HT-DWT-ANNs	BRB	Yes
[61]	Variable frequency drives	HT	BRB	No
Proposed technique	ANN-DTC/FL-DTC	HT	BFs	Yes

below. But by addressing the BFs and the ANN-DTC control, this study is distinct. Furthermore, while it addresses BRB issues, a study authored by Aissa et al. [19] has yielded improvements in both diagnostic and control aspects.

8. Conclusions

This article compares three DTC methods—CDTC, ANNDTC, and FLDTC—for diagnosing BFs. FLDTC offers the best-quality current envelope spectrum but requires more simulation time, while ANNDTC offers good mechanical performance, with a spectrum still affected by noise. The Hilbert transform is effective for processing non-stationary signals. The contribution of this work lies in the value of advanced DTC controls not only for improving the mechanical performance of SCIM but also for accurately detecting the faulty bearing part.

Future work will validate these results experimentally for other rotor defects and explore sound analysis for non-invasive fault diagnosis.