domain: spi info

text:
Younes Azzoug
email: younes.azzoug@univ-biskra.dz

Abdelkarim Ammar
email: a.ammar@univ-boumerdes.dz

Remus Pusca
email: remus.pusca@univ-artois.fr

Raphael Romary
email: raphael.romary@univ-artois.fr

Mohamed Sahraoui
email: m.sahraoui@univ-biskra.dz

Antonio J Marques Cardoso

A Single Observer for Currents Estimation in Sensor's Fault-Tolerant Control of Induction Motor Drives

Keywords: fault tolerant control, fault detection, current estimation, induction motor, direct torque control, adaptive observer. I

This paper proposes a fault-tolerant control technique against current sensors failure in direct torque controlled induction motors drives. Several important aspects are covered in the proposed algorithm, such as the detection of sensors failure, the isolation of faulty sensors, and the reconfiguration of the control system by a correct estimation. A logic circuit ensures fault detection by analyzing the residual signal between the measured and estimated quantities, while a single modified Luenberger observer performs the task of estimating the line currents, even when all the sensors are in the faulty state. In addition, a decision logic circuit isolates the erroneous signal and simultaneously selects the appropriate estimated current signal. The proposed scheme is tested on MATLAB/Simulink environment and experimentally validated in a laboratory prototype mainly containing a dS1104 card and 4 kW induction motor.

INTRODUCTION

Fault Tolerant Control (FTC) methods are crucial in variable speed drives used in critical and high-sensitivity applications, such as, space exploration, nuclear power plants, electric vehicles, air and rail transportation; in order to ensure system security, and uninterrupted operation, even with degraded performance, when an unexpected failure occurs in the variable speed drive. Consequently, a fault tolerant control strategy must be implemented to guarantee that faults are handled in such a way that there will be no damage.

Generally, variable speed Induction Motor (IM) drives need feedback information from a speed/position transducer, two or three current sensors and a dc-link voltage sensor at least [START_REF] Berriri | Easy and Fast Sensor Fault Detection and Isolation Algorithm for Electrical Drives[END_REF], [START_REF] Yu | Current sensorless direct predictive control for induction motor drives[END_REF]. Unfortunately, sensors are very prone to failures, and for this reason several research results have been published regarding sensors fault-tolerant control, as well as Fault Detection, Isolation, and Reconfiguration (FDIR) methods [START_REF] Gajanayake | Sensor fault detection, isolation and system reconfiguration based on extended Kalman filter for induction motor drives[END_REF]- [START_REF] Azzoug | Fault Tolerant Control for Speed Sensor Failure in Induction Motor Drive based on Direct Torque Control and Adaptive Stator Flux Observer[END_REF].

The study in [START_REF] Gajanayake | Sensor fault detection, isolation and system reconfiguration based on extended Kalman filter for induction motor drives[END_REF] presents a sensor FTC of an induction motor drive, based on the extended Kalman filter, and a reduced number of adaptive observers, to keep the operation of the system under the occurrence of sensors failures. In [START_REF] Dybkowski | Stator current sensor fault detection and isolation for vector controlled induction motor drive[END_REF], a fault-tolerant vector control is proposed. A logic circuit is used to identify the faulty current sensor. The missing current information is replaced by an appropriate stator current, which value is calculated from algebraic equations and from the other measured healthy currents. However, this approach is not practical in the event of a successive failure in the different sensors. A fault-tolerant control scheme is proposed in [START_REF] Lee | Instrument fault detection and compensation scheme for direct torque controlled induction motor drives[END_REF], in which an adaptive fuzzy logic-based observer is used to generate residuals, which analysis allows the detection, and isolation of the faulty sensor. However, the dynamic response of this scheme for isolating defective sensors is too slow. In [START_REF] Najafabadi | Detection and Isolation of Speed-, DC-Link Voltage-, and Current-Sensor Faults Based on an Adaptive Observer in Induction-Motor Drives[END_REF], it is proposed a sensor fault detection and isolation scheme for an IM drive, based on currents estimation and rotor resistance variations, which are then used in a decision unit for the identification and isolation of the faulty sensor. In [START_REF] Yu | Speed and Current Sensor Fault Detection and Isolation Based on Adaptive Observers for IM Drives[END_REF], a FTC technique against speed and current sensors failures in IM drives is proposed, and based on three adaptive observers with different inputs. However, this method is not suitable to achieve an efficient FTC in case of failures in two or three sensors. The results obtained in [START_REF] Yu | Current Sensor Fault Diagnosis and Tolerant Control for VSI-Based Induction Motor Drives[END_REF] offer a solution to the succession of faults in the different sensors but the number of the used adaptive observers is still three. Another type of FTC methods has been also presented in the literature, in which the sensors FTC algorithm changes the control according to the type of the faulty sensor, as proposed in [START_REF] Stettenbenz | Smooth switching controllers for reliable induction motor drive operation after sensor failures[END_REF]- [START_REF] Aguilera | A fault tolerant system for current sensors in induction motor drives[END_REF]. A smooth FTC of IM drives with sensor failures is presented in [START_REF] Stettenbenz | Smooth switching controllers for reliable induction motor drive operation after sensor failures[END_REF], [START_REF] Liu | Smooth Fault-Tolerant Control of Induction Motor Drives with Sensor Failures[END_REF]. In the healthy state, the IM drive's operation is accomplished using direct torque control; indirect field oriented control is then used when the dc-link voltage sensor fails, and v/f control is utilized during current sensors failure. The major drawback of this method, as reported in [START_REF] Tabbache | A Control Reconfiguration Strategy for Post-Sensor FTC in Induction Motor-Based EVs[END_REF], is that these techniques are experimentally difficult to implement.

This paper proposes a direct torque fault-tolerant control strategy for the current sensors failure in induction motor drives. Some features are distinct from the other proposed methods. First, the estimation of stator currents is performed using a single observer. Then, the method proposed here allows the detection of faulty sensors even when several sensors fail successively, case in which a decision unit, based on a logic circuit, performs the task of reconfiguring and isolating the effects corresponding to the faulty sensor. Finally, the proposed technique can be reorganized in an The 3rd International Conference on Applied Automation and Industrial Diagnostics (ICAAID, 25-27 September 2019, Elazig, Turkey) adaptive way in the event of the loss of one or two sensors. The proposed scheme is simulated in MATLAB/Simulink environment, and experimentally implemented in a laboratory test bench under different operating conditions. The remainder of the paper is organized into V sections. Section II analyses the direct torque control. Afterwards, section III describes the concept of the proposed FTC. Simulation and experimental results are presented and discussed in section IV. Finally, section V presents the main conclusion of the paper.

II. DIRECT TORQUE CONTROL OF THE INDUCTION MOTOR

The system equation below gives the induction motor model used in the present work.

( , given that is the mutual inductance.

The electromagnetic torque is expressed by the following equation:

(2)

The stator flux components are given in the system equation below:

(3)

The Direct Torque Control (DTC) basic idea is to estimate flux and torque instaneous values only from the stator variables. The estimated torque is compared with the reference one given by the speed controller (PI controller in the present work), the resulting torque error signal is sent to an hysterisis controller (three-levels hysterisis controller is adopted in the present work) to get a control signal , which fed a switching look-up table. Likewise to the estimated flux, which is compared with fixed value of a stator flux magnitude, the resulting flux error signal is fed an hysterisis controller (two-levels hysterisis controller is adopted in the present work) to get a control signal for the switchin look-up table. This latter uses flux and torque control signals in order to generate three control signals , and for the three-phase inverter (see Fig. 4).

III. CONCEPT OF THE PROPOSED FAULT-TOLERANT CONTROL

In the proposed fault-tolerant direct torque control, two current sensors in phase-a, and phase-b are used in addition to a dc-link voltage sensor, and an incremental encoder. Furthermore, using the control signals , and , a voltage synthesizer construct the three-phase stator voltages applied to the machine from the voltage measured in the dclink.

A. Stator Currents Estimation

In order to estimate the line currents, a modified adaptive observer is used, which gives high-performance estimation stator currents, based on the IM model, which is described by the following state equation: [START_REF] Dybkowski | Stator current sensor fault detection and isolation for vector controlled induction motor drive[END_REF] where:

and:

as well as:

The idea of the proposed stator currents estimator is based on the general theory of Luenberger observer [START_REF] Luenberger | An introduction to observers[END_REF], which allows the estimation of variable or unknown parameters of a system. The equation of the Luenberger observer is expressed in [START_REF] Lee | Instrument fault detection and compensation scheme for direct torque controlled induction motor drives[END_REF], where the symbol "^" denotes the estimated values: [START_REF] Lee | Instrument fault detection and compensation scheme for direct torque controlled induction motor drives[END_REF] where:

Since there is no information about the measured currents, the vector becomes:

The gain matrix is defined as follows:

where: with a positive gain.

The estimation is performed throw the conservation of the state model ( 4) of the IM with the gain matrix . This matrix ensures the convergence of the observer, and has been determined by the classical pole placement procedure. Then, replacing the adaptive mechanism by the measured rotational speed, as well as, feeding the observer with the stator voltages provided by the voltage synthesizer. This gives the stator currents estimation, as illustrated in Fig. 1.

B. Fault Detection, Isolation and System Reconfiguration

The logic-circuit presented in Fig. 2 is intended to ensure the fault detection, isolation, and system reconfiguration. The detection of failures is performed by analyzing the residual signal between the measured and estimated quantities, passing through a Low Pass Filter (LFP) to extract the useful signal that will be compared to a welldefined threshold (Th=0.8), then a proper estimation is used for the isolation and reconfiguration of the system. As mentioned previously, two current sensors (in phasea and phase-b) were used in the proposed method for measuring a and b line currents. The axis transformation presented in Fig. 3 developed by [START_REF] Chakraborty | Speed and Current Sensor Fault Detection and Isolation Technique for Induction Motor Drive Using Axes Transformation[END_REF] and used in both [START_REF] Manohar | A robust current sensor fault detection scheme for sensorless induction motor drive[END_REF], and [START_REF] Manohar | Current Sensor Fault-Tolerant Control for Direct Torque Control of Induction Motor Drive Using Flux-Linkage Observer[END_REF], which is also used in the proposed FTC, in order to have the component currents using only the measurements of components. As shown in Fig. 3, the β-axis is aligned with the b-axis. Considering the equation of transformation from to , which is given in [START_REF] Najafabadi | Detection and Isolation of Speed-, DC-Link Voltage-, and Current-Sensor Faults Based on an Adaptive Observer in Induction-Motor Drives[END_REF], it is clear that depends on the two-phase currents and , whereas the current depends only on the current . Therefore, if the phase-a current sensor becomes faulty, only that will be affected. But, if the phase-b current sensor becomes faulty, both currents and will be affected.

So, the faulty measured current components must be replaced by the corresponding estimated currents. As a consequence, Table 1 shows the identification of the faulty sensors and the correct components of the estimated currents that will replace the erroneous components. As presented in Fig. 2 and defined in Table I, it appears that in the case of sensor-a failure, fault detection part of the logic circuit generates an impulse Za=1, whereas, in the case of sensor-b failure, fault detection logic circuit generates an impulse Zb=1. Once the faulty sensor is identified, it is isolated, and the system is reconfigured, using the fault isolation and system reconfiguration part of the logic circuit. The logic gate OR output is 1 when any one of its inputs Za or Zb or both is 1 due to the effect of the failure of sensor-b on current. As a consequence, in the case of sensor-b failure, is isolated and replaced by .

IV. SIMULATION AND EXPERIMENTAL RESULTS

The proposed current sensor direct torque fault-tolerant control for induction motor drive is simulated in Matlab\Simulink environment using 4 kW induction motor parameters (see the Appendix). Fig. 4 presents the block diagram of the proposed method.

B. Test of the proposed FTC when sensor-b becomes faulty and sensor-a remains healthy

Figs. 8 and9 give another test of the suggested FTC method with simulation and experimentation respectively, where a failure in sensor-b is performed, whereas sensor-a remains healthy. Simulation and experimental tests were started in healthy conditions until t=1s for simulation test and t=9.8s for experimental test, where sensor-b becomes faulty. Considering Figs. 8(a) and 9(a), it can be seen that is slightly affected by the failure of sensor-b, while is significantly affected, which is explained previously in equation [START_REF] Najafabadi | Detection and Isolation of Speed-, DC-Link Voltage-, and Current-Sensor Faults Based on an Adaptive Observer in Induction-Motor Drives[END_REF]. The fault detection algorithm reacts quickly by generating an impulse Zb=1, indicating that sensor-b is faulty, unlike Za which remains at 0, meaning that sensor-a gives proper measurement. At this moment, switch-b switches to input 1 to replace by , and according to the output of the gate OR (see Table 1) switch-a switches to input 1 to replace by . Figs.

C. Test of the proposed FTC when both sensor-a and sensor-b becom faulty

Figs. 10 and 11 illustrate respectively the simulation and experimentaion results obtained by applying the proposed FTC when the both sensors fail. The sensor-a failure is simulated at t=1s, while the sensor-b fault is simulated at t=1.1s as shown in the Figs. 10(a) and 10(b). In the experimental test, sensor-a failure is appeared at t=8.75s, while sensor-b failure is appeared at t=14.35s as can be seen in Figs. 11(a When the defect is suddenly introduced in the sensor-a, the fault detection mechanism has reacted quickly by generating a Za index equal to 1, indicating that sensor-a is faulty. Based on equation ( 6), the current will not be affected, which is visible in Figs. 10(b 6), we can see that the currents and both depend on the state of the measurment of sensor-b, which is confirmed in Figs. 10(a) and 11(a) where the degree of severity of the fault in the current is increased. Therefore, the isolation and reconfiguration algorithm chooses and (see Table 1). Figs. 10(d), 11(d) and 10(e), 11(e) clearly show that speed and torque maintain their performance and remain uninterrupted, even under the worst possible conditions, when the both current sensors fail. In addition, Figs. 10(f) and 11(f) show the constant estimate of stator flux. The 3rd International Conference on Applied Automation and Industrial Diagnostics (ICAAID, 25-27 September 2019, Elazig, Turkey)

V. CONCLUSION This paper presents a direct torque fault-tolerant control diagram for current sensors failures in the induction motor drives. For the identification of a sensor failure, a logic circuit is used in the phases where the current sensors are located, which is able to detect the failures of the used current sensors. A current estimator is used to estimate the stator currents. Immediately after the failure of the current sensor, a decision logic circuit automatically selects the correct current signal to maintain the operation of the system. Simulation and experimental results show the effectiveness of the proposed method, where the fault detection is accomplished with fast dynamics, and the drive is performing successfully, without compromising the control system.

APPENDIX

Fig. 1 .

 1 Fig. 1. Proposed stator currents estimator.

Fig. 2 .

 2 Fig. 2. Fault detection, isolation and system reconfiguration scheme

Fig. 3 .

 3 Fig. 3. Axis transformation model.

Fig. 4 .

 4 Fig. 4. Block diagram of the proposed FTC.

Fig. 5

 5 Fig. 5 displays the laboratory test bench in which the proposed control technique has been implemented and tested experimentally, which is composed mainly of a 4 kW threephase IM whose parameters are the same used in simulation model, dS1104 card, two current sensors, and dc-voltage sensor.

Fig. 5 .

 5 Fig. 5. Laboratory experimental test bench. All simulation and experimental tests are performed at 1000 rpm with 20 N.m load torque. A. Test of the proposed FTC when sensor-a becomes faulty and sensor-b remains healthy Figs. 6 and 7 show simulation and experimental results obtained respectively, applying the proposed FTC. The drive was started with healthy current sensors in both simulation and experimentation. At t=1s in simulation test and t=10.6s in experimental test, current sensor-a fails. As a result and corresponding to the used axis transformation equation (6), becomes faulty such is depicted in Figs. 6(a) and 7(a), and remains healthy as can be seen in Figs. 6(b) and 7(b). At the moment of failure occurrence, Za sensor-a state indicator becomes 1, judging sensor-a faulty, but Zb sensorb state indicator remains 0, affirming that sensor-b is healthy, as illustrated in Figs. 6(c) and 7(c). As a consequence of Za=1, switch-a switches to input 1 to replace the erroneous measured current by the estimated one as explained previously in Table 1. Figs. 6(d), 7(d) and 6(e), 7(e) illustrate respectively the high-performance speed tracking of the IM drive, and the uninterrupted fulfillment of electromagnetic torque requirement under both healthy and faulty conditions, and Figs. 6(f), 7(f) depict stator flux circular trajectory.

Fig. 6 .

 6 Fig. 6. Simulation results when sensor-a is faulty and sensor-b remains healthy: (a) currents, (b) currents, (c) fault indicator, (d) reference and actual speeds, (e) load and electromagnetic torque, and (f) stotor flux trajectory.

Fig. 7 .

 7 Fig. 7. Experimental results when sensor-a is faulty and sensor-b remains healthy: (a) currents, (b) currents, (c) fault indicator, (d) reference and actual speeds, (e) load and electromagnetic torque, and (f) stator flux trajectory.

  8(c) and 9(c) demonstrate sensor state indicators Za and Zb. Figs. 8(d) for simulation and 9(d) for experimentation show the speed tracking performance of the IM drive before and after the failure occurrence.

Fig. 8 .

 8 Fig. 8. Simulation results when sensor-b is faulty and sensor-a remains healthy: (a) currents, (b) currents, (c) fault indicator, and (d) reference and actual speeds.

Fig. 9 .

 9 Fig. 9. Exprimental results when sensor-b is faulty and sensor-a remains healthy: (a) currents, (b) currents, (c) fault indicator, and (d) reference and actual speeds.

  ) and 11(b). While, Figs. 10(c) and 11(c) show the fault indicator signals (Za for sensor-a and Zb for sensor-b).

Fig. 10 .

 10 Fig. 10. Simulation results when both sensor-a and sensor-b becom faulty: (a) currents, (b) currents, (c) fault indicator, (d) reference and actual speeds, (e) load and electromagnetic torque, and (f) stotor flux trajectory.

Fig. 11 .

 11 Fig. 11. Exprimental results when both sensor-a and sensor-b becom faulty: (a) currents, (b) currents, (c) fault indicator, (d) reference and actual speeds, (e) load and electromagnetic torque, and (f) stotor flux trajectory.

TABLE I .

 I FAULTY SENSOR IDENTIFICATION AND CORRECT COMPONENT SELECTION

	Sensor-a	Sensor-b	Za	Zb	OR	Proper selected components
	Healthy	Healthy	0	0	0	and
	Faulty	Healthy	1	0	1	and
	Healthy	Faulty	0	1	1	and
	Faulty	Faulty	1	1	1	and

ACKNOWLEDGMENT This work has been achieved within the framework of CE2I project (Convertisseur d'Energie Intégré Intelligent). CE2I is co-financed by European Union with the financial support of the European Regional Development Fund (ERDF), French State and the French Region of Hauts-de-France. As well, this work was supported by the European Regional Development Fund (ERDF) through the Operational Program for Competitiveness and Internationalization -COMPETE 2020, and also by National Funds through the Portuguese Foundation for Science and Technology (FCT), under Project POCI-01-0145-FEDER-029494 and Project UID/EEA/04131/2013.


text:
Younes Azzoug
email: younes.azzoug@univ-biskra.dz

Arezki Menacer
email: menacer_arezki@hotmail.com

Rémus Pusca
email: remus.pusca@univ-artois.fr

Raphaël Romary
email: raphael.romary@univ-artois.fr

Tarek Ameid

Abdelkarim Ammar
email: abdelkarim.ammar@univ-boumerdes.dz

Fault Tolerant Control for Speed Sensor Failure in Induction Motor Drive based on Direct Torque Control and Adaptive Stator Flux Observer

Keywords: fault tolerant control, fault detection and isolation, direct torque control, induction motor, sensorless control, variable speed drives. I

This paper deals with a speed sensor fault tolerant control (FTC) scheme for induction motor (IM) drive based on direct torque control (DTC) and adaptive stator flux observer (ASFO). Many important points are treated in the proposed algorithm, which are the detection of the sensor failure, isolation of the faulty sensor and the reconfiguration of the control system by proper estimation. An adaptive stator flux observer is used instead of classical flux and torque estimator to perform the stator flux components, electromagnetic torque, flux magnitude, rotor position and the estimated rotor speed. Furthermore, a fault detection and isolation (FDI) logic circuitry is used to detect speed sensor fault and isolate it and simultaneously generate a logic impulse to switch to the sensorless mode. The proposed algorithm is simulated in MATLAB/Simulink.

INTRODUCTION

Induction Motors (IMs) are broadly used in different applications for variable speed, simplicity of construction, less maintenance requirement, ruggedness, high performances and remarkable reliability are advantages that convince manufacturers to use this type of engines [START_REF] Ameid | Broken rotor bar fault diagnosis using fast Fourier transform applied to fieldoriented control induction machine: simulation and experimental study[END_REF] [START_REF] Ameid | Discrete wavelet transform and energy eigen value for rotor bars fault detection in variable speed field-oriented control of induction motor drive[END_REF]. To get high performances in variable speed control applications, many strategies of control are used, among them Sliding Mode Control, Backstepping Control, Field Oriented Control (FOC) and Direct Torque Control (DTC). The FOC method allows the decoupling between the flux and the torque to get an independent control like direct current machine [START_REF] Blaschke | The Principle of Field Orientation as Applied to the NEW Transvector Closed-Loop System for Rotating-Field Machines[END_REF]. On the other hand, FOC strategy needs transformations of current and voltage vectors [START_REF] Panda | A Three-Level Fuzzy-2 DTC of Induction Motor Drive Using SVPWM[END_REF] which is not the case in the DTC of IM drive, so the estimated flux and electromagnetic torque are directly used to generate the control pulses for voltage source inverter through a welldefined switching table [START_REF] Manohar | Current Sensor Fault-Tolerant Control for Direct Torque Control of Induction Motor Drive Using Flux-Linkage Observer[END_REF].

The closed loop control like DTC which is presented in Fig. 1 needs feedback from speed and voltage sensors and two current sensors at least. Here, reliable operation of all control constituents gives satisfactory performances of all the drive system, but failure of any one of these components can cause the failure of the drive system. The failures in the drive systems are broadly classified as converter faults (semiconductors failure, power supply faults...), sensor faults (connection problems, noise, offset, the voltage drop of the battery...) and machine faults which are divided on two types, mechanical faults (eccentric rotor, bearing failures, shaft misalignment...) and electrical faults (broken bars, shorted circuit windings, insulation deterioration...).

In the last few years, there has been a growing interest on the Fault Tolerant Control (FTC) which is indispensable especially in the high-tech sectors such as air and railway transportation, astronomy, nuclear, medical industry... FTC techniques are designed for the purpose of detecting failures in the control chain and keeping a minimum dynamics of the control system after the occurring of the faults. However, sensors are the most components susceptible to defects [START_REF] Yu | Speed and Current Sensor Fault Detection and Isolation Based on Adaptive Observers for IM Drives[END_REF]. This work offers a Fault Detection and Isolation and fault tolerant control to speed/position sensor problems which is a basic component in adjustable speed drives, he gives informations that can be used to perform speed, position or torque control.

Several publications have been published in recent years documenting the speed sensor FTC, one of the examples of speed sensor FDI is presented in [START_REF] Chakraborty | Speed and Current Sensor Fault Detection and Isolation Technique for Induction Motor Drive Using Axes Transformation[END_REF], the authors in this work propose a speed and current sensor FDI, whereas, the currents are estimated using d-and q-axis currents and the speed is estimated by using model reference adaptive system (MRAS), through a mechanism based on logic decisions to select the appropriate value and reconstruct the drive system. A FTC which depend on voting algorithms (VAs) is discussed in [START_REF] Boukhnifer | Fault tolerant control to mechanical sensor failures for Induction Motor drive: A comparative study of voting algorithms[END_REF] [START_REF] Tabbache | Virtual-Sensor-Based Maximum-Likelihood Voting Approach for Fault-Tolerant Control of Electric Vehicle Powertrains[END_REF] [10] while, the VAs choose the most correct output signal given by virtual sensors like sliding mode observer (SMO), adaptive Luenberger observer (ALO) and extended Kalman filter (EKF) to ensure the best behavior in the faulty mode. After all, a paper which resume presented approaches have been presented by [START_REF] Bourogaoui | Speed/position sensor fault tolerant control in adjustable speed drives -A review[END_REF], giving a bibliographical review about speed/position FDI and FTC in variable speed drives.

Therefore, this paper focuses on speed sensor direct torque fault tolerant control technique in induction motor drive systems. In the proposed algorithm, the detection of the speed sensor fault is accomplished by a logic circuitry which can detect the fault in the speed loop. Once, the sensor fault is identified, process of speed estimation becomes active. In this regard, an adaptive flux observer based on machine model is used to estimate the rotor speed moreover stator fluxes and electromagnetic torque. The logic circuitry performs the task of fault reconfiguration and isolation of the speed sensor at the faulty state. The proposed speed sensor FTC system can be globally used in any speed control strategies of IM. The proposed scheme is simulated and tested in MATLAB/Simulink under different operating conditions. This paper is organized into VI sections; section II describes briefly the induction motor model and the direct torque control. Section III analysis the design of the adaptive stator flux observer. The adopted fault tolerant control method is presented in section IV. Afterwards, section V discusses the simulation results. Finally, section VI concludes the paper.

R R R i i V dt L L L L L L di R R R i i V dt L L L L L L d V R i dt d V R i dt                                                                                  (1)
Where:

 

, where M is mutual stator-rotor inductance.

The electromagnetic torque is expressed by the following equation:

  e s s s s T p i i       (2)
The DTC basic idea is to calculate flux and torque instaneous values only from the stator variables.

III. ADAPTIVE STATOR FLUX OBSERVER

The IM model given in ( 1) is described by the following state equation [START_REF] Ammar | Nonlinear SVM-DTC for induction motor drive using input-output feedback linearization and high order sliding mode control[END_REF]:

() X A X BU Y CX         (3)
Where

T s s s s X I I           , T s s Y I I        , T ss U V V    
             , 1 0 1 0 00 00 s s L B L              , 1 0 0 0 0 1 0 0 C     Which   1 1 1 s r a T T                , 2 s r r M a L L T   , 3 s r M a L L   , 4 r M a T  , 5 1 r a T  , s s s L T R  , r r r L T R



The adaptive stator flux observer scheme is presented in Fig. 2 with parameters considered in (4). ASFO equation is given by [START_REF] Saberi | Improved sensorless direct torque control method using adaptive flux observer[END_REF]:

ˆ() ˆry X A X BU K Y CX           (4)
The symbol ^ denote estimated value and K is the observer gain matrix, using the Lyapunov theory to deduce the mechanism of adaptation speed [START_REF] Ammar | Adaptive MRAC-based direct torque control with SVM for sensorless induction motor using adaptive observer[END_REF]. The stator current and stator flux estimation error is given by [START_REF] Djamila | Simultaneous Estimation of Rotor Speed and Stator Resistance in Sensorless Indirect Vector Control of Induction Motor Drives Using a Luenberger Observer[END_REF]:

    ê A KC e A X     (5)
with     Considering the following Lyaponov function:

  2 / T r V e e     (9)
Where,  is a positive constant.

The derivative of ( 9) is as follows:

    2 1 T T r de dV de d ee dt dt dt dt                 (10)        
                  (11)
With ˆr  is the rotor velocity estimated by the adaptive observer. The rotor speed estimation is given by the Eq. 12, obtained through equality between the second and the third term of Eq. 11:

The expression ( 12) is established for a constant velocity, so as to ameliorate the feedback of the rotor position estimation in various condition, the law of the control is modified according to the following relation:

ˆˆˆˆ( ) ( ) s s s s r p I s I s i I s I s K e e K e e dt                   (13)
With p K and i K are the positive gains of the PI controller. The return loop of the gain matrix K is selected to ensure the high dynamics and performances of the global ASFO.

1 2 3 4 2 1 4 3 T K K K K K K K K K      (14) with 1 2 3
,, K K K and 4 K are given by:

1 2 2 3 4 1 1 ( 1) ( 1) 1 1 1 1 ( 1) ( 1) ( 1) s r r s s s r r r r s r s r r Kl T T K l L M L M M K l l T T L T L T T LM K l L                                                                    
where l is a positive constant. IV. SPEED SENSOR FAULT TOLERANT CONTROL By using the adaptive stator flux observer instead of the torque and flux estimator in the direct torque control scheme (Fig. 5), we can get the electromagnetic torque and stator flux components, in addition, estimated rotor speed. To detect speed sensor failures and isolate them, a thresholdbased scheme (Fig. 3) is proposed to allow to the motor drive system to switch to speed sensorless control mode in the case of speed sensor failure.

As it is seen from Fig. 3 and described in Table I, before the speed sensor failure, the IM drive system rest on DTC with speed sensor feedback, FDI generates a signal Z=0. In the case of failure in the speed sensor, FDI generates an impulse at the faulty instant Z=1 to switch from measured speed to estimated speed. The FDI is based on calculation of the error (Er) between estimated and measured speed which will be pass through a low-pass filter (LPF) in order to extract the useful signal, this error is compared to a threshold (Th) well defined according to the reference speed.

V. SIMULATION RESULTS

The behavior of the proposed fault tolerant control for speed sensor failures in induction motor drive based on direct torque control and adaptive stator flux observer is verified in simulation with MATLAB/Simulink at rated load (5 N.m) using nominal motor parameters presented in Appendix A. Fig. 6 represents simulation results characterizing the performances of the FTC under the cited condition. At the healthy state, a step of 1000 rpm in speed command is applied at 0s. Total failure of the speed sensor is simulated using a switch circuit in Simulink at 1s. Fig. 6(a



) and the speed chosen by FDI algorithm ( r  ) according to system state, which depict an uninterrupted speed tracking by the estimated speed in the event of total speed sensor failure. The fault indicator Z explained in Table I is given in Fig. 6(b). From this figure it can be seen that, at the healthy state fault indicator Z is 0 but at t=1s, fault indicator Z is 1 which means that the speed sensor is in the faulty state. As can be seen from Fig. 6(c), the FDI algorithm makes the decision from calculating error (Er) between reference and measured speeds compared to the defined threshold (Th). Considering Fig. 7, this plots the behavior of the fault tolerant control system at low speed (200 rpm) versus total failure of sensor speed. Initially the system was operating with a healthy speed sensor and at t=1s; the output of sensor speed becomes 0 as given by Fig. 7(a). Here we can see that the estimated speed ensure the continuity of drive system. Fault indicator Z can be found in Fig. 7(b), Z becomes 1 when sensor speed made to 0. Fig. 7(c) gives another example about the process of the FDI algorithm by calculating the error and the threshold which is in this case 10% of reference speed (20 rpm) and compares them to give the decision to switch from sensor mode to the sensorless mode. Electromagnetic torque, stator currents, flux magnitude and flux components are shown in Fig. 7(d), 7(e), 7(f) and Fig. 7(g). We can see that the adaptive stator flux observer gives a satisfactory estimation of these quantities which are necessary to preserve the principle of the DTC. To investigate the high performances of the system, the fault tolerant control algorithm was tested in variable speed, Fig. 8 represent simulation results in case of Benchmark industrial test. The scenario of speed sensor fault is tested by switching off the speed sensor at t=1.1s when the IM is running at 600 rpm, after failure, a step change in reference speed is applied. Results shown in Fig. 8(a) and 8(b) indicate a successful and high performance fault tolerant control system response.

V

   are the stator voltage components.

R

  stator and rotor resistance.



  rotor angular speed.

Fig. 1 .

 1 Fig. 1. DTC scheme of IM drive used in the presente paper.

And

Fig. 2 .

 2 Fig. 2. Overall diagram of the adaptive observer.

Fig. 3 .

 3 Fig. 3. Scheme of the fault detection and isolation system.

Fig. 4

 4 Fig. 4 shows the details flowchart for the proper selection of measured or estimated speed in the different states of the speed sensor, while, Fig. 5 illustrates the block diagram of DTC scheme for speed sensor FTC.

Fig. 4 .

 4 Fig. 4. Flowchart for the selection of measured and estimated speed.

Fig. 5 .

 5 Fig. 5. Block diagram of DTC scheme for speed sensor fault tolerant control.

  Fig.6represents simulation results characterizing the performances of the FTC under the cited condition. At the healthy state, a step of 1000 rpm in speed command is applied at 0s. Total failure of the speed sensor is simulated using a switch circuit in Simulink at 1s. Fig.6(a) shows

Fig. 6 (Fig. 6 .

 66 Fig. 6. Simulation results with total failure of the speed sensor at 1000 rpm and rated load (5 N.m): (a) reference, measured and estimated speed, (b) fault indicator, (c) threshold and error, (d) electromagnetic torque, (e) three-phase stator currents, (f) stator flux magnitude, (g) stator flux components.

Fig. 7 .

 7 Fig. 7. Simulation results with total failure of the speed sensor at low speed (200 rpm) and rated load (5 N.m): (a) reference, measured and estimated speed, (b) fault indicator, (c) threshold and error, (d) electromagnetic torque, (e) three-phase stator currents, (f) stator flux magnitude, (g) stator flux components.

Fig. 8 .

 8 Fig. 8. Simulation results with total failure of the speed sensor following Benchmark trajectory: (a) reference, measured and estimated speed, (b) fault indicator.

TABLE I .

 I IDENTIFICATION OF THE STATE OF THE SPEED SENSOR

			Speed sensor state
			Healthy state	Faulty state
	Fault indicator Z		0	1
	However, after several tests on the studied system, the
	threshold (Th) is defined as follows:
	Th	     	** ** 0.045 0.1 rr rr if if   	    200 200 rpm rpm 

* r  : reference speed

VI. CONCLUSION

The main concern of the paper is the fault tolerant control for speed sensor failure in induction motor drive based on direct torque control and stator flux adaptive observer. For detecting the failures in speed sensor, a fault detection and isolation developed algorithm was employed. FDI system in the speed loop can detect any failure in the speed measurement by calculating error between reference and measured speed and compare this error to a well defined threshold. An adaptive stator flux observer is adopted to estimate rotor speed and stator flux components. In the moment of failure in speed sensor, a decision-making logic circuit automatically selects the sensorless mode to continue the drive uninterrupted system. The analytic model is presented in the paper and the simulation results ensure the satisfactory operation of the fault tolerant control proposed scheme.

APPENDIX A

Specifications