Fault Tolerant Control based Backstepping for ...

ICIEM’14, International Conference on Industrial Engineering and Manufacturing Batna University, Algeria May, 11-13, 2014

Fault Tolerant Control based Backstepping for Permanent Magnet Synchronous Motors H. Mekki1, 3, O. Benzineb2, 4, D. Boukhetala3, M. Tadjine3, M.E.H. Benbouzid4 1

Université de M’sila. B.P 166 Ichbilia M’sila, Algérie. [email protected] 2 Université de Blida Route de Soumaa, Algérie. [email protected] 3 Laboratoire de Commande des Processus, Ecole Nationale Polytechnique. 10 Av, Hassan Badi, BP 182, Alger, Algérie. [email protected], [email protected] 4 Université de Brest, EA 4325 LBMS Rue de Kergoat, Brest, France. [email protected] Abstract—The aim of this paper is to design and to apply a fault tolerant control approach based backstepping control strategy to permanent magnet synchronous motors (PMSM). After giving the PMSM model, we give also the stator mechanical faults model; in this case the nominal control (backstepping) present a robustness because it permits to compensate both parametric and load torque disturbance but this control approach can’t reject the mechanical faults effect. In order to design FTC block an additive control is thus added to the nominal control this additive control illustrated from the internal model which is activated automatically as of appearance of the faults to compensate its effect. Simulations tests have been carried out to evaluate the consistency and the performance of the proposed fault tolerant control approach. Keywords— Fault tolerant control, Backstepping control, Permanent magnet synchronous motors, Internal model, Mechanical faults.

I. INTRODUCTION As automated systems become more complex, a key challenge is how to achieve (at worst) graceful degradation in performance in the event of a fault associated with an actuator, sensor or component subsystem [1]. Under these circumstances, it is important for the system to be kept stable with an acceptable closed loop control performance when faults occur. Ideally, in applications where continuity of operation is a key feature, the closed loop system should be capable of maintaining its pre-specified performance in terms of quality of service, safety, and stability despite the presence of faults [2]. This procedure is rendered possible thanks to the fault tolerant control (FTC) design [3]. Fault tolerance has become an increasingly interesting topic in the last decade where the automation has become more complex. The objective is to give solutions that provide fault accommodation to the most frequent faults and thereby reduce the costs of handling the faults [4]. In general, the FTC approaches can be classified into two types: the passive approach and the active approach. The survey papers [5] and [6] review respectively the concepts and the state of the art in the field of FTCs, in [7] review and present the FTC for PMSM; then comparative study between this two FTCs approaches and the recent advances have been reported in [8] and [9]. Permanent magnet synchronous motor (PMSM) drive is nowadays widely used in the industry applications due to

172

their high efficiency and high power/torque density [10]. These motors are used in many applications such as traction with variable speeds in transportation [11]. But a breakdown in these motors can cause the stop of the production facility or require the use of redundant equipment to circumvent the problem. Several failures can affect electrical motor drives [12] and can appear on the level of rotor or stator of this later. They can be electric or mechanic and their causes very varied. Indeed, many studies [11-14] showed that each faults revealed harmonics at specific frequencies in the currents of the machine. This frequencies signature dependent on the PMSM structural’s parameters. Starting from the work presented in [15] where authors take the FOC as nominal control in the FTC strategy. In [16] and [17] author’s present the FTC based Backstepping Control for induction motor. In [10] a FTC based Direct Torque Control for PMSM was presented. In this paper we take nonlinear Backstepping Control as the nominal control which presents a remarkable robustness control scheme based on Lyaunov theory in order to obtain a desired response. Our objective in this paper is to design a robust Backstepping technique which is able to steer the current and the speed variables to their desired references and to compensate the load torque disturbances. After giving the stator mechanical faults model in order to design FTC block, we associated the nominal control with an internal model which generates an additive term to compensate the faults effect. The level of the compensation is an indicator of the faults severity and the nature of the compensation is a help with the diagnosis. The paper is organized as follows. Section 2 describes the permanent magnet synchronous motors systems (PMSM) healthy oriented model. Section 3, is devoted to the design of the robust backstepping control technique, which is able to steer the direct current and speed variables to their desired references and to compensate the load torque at the end we studies the stability of the closed-loop system by Lyapunov function theory. In Section 4, the PMSM faulty model also the additive control low based internal model are presented, then the global FTC are analyzed in detail, followed by the design of the compensation controller for the faults. To illustrate the proposed FTC methodology, a numerical simulation is given in Section 5. Finally, some concluding remarks are made in Section 6.

ICIEM’14, International Conference on Industrial Engineering and Manufacturing Batna University, Algeria May, 11-13, 2014

class of feedback linearisable nonlinear systems exhibiting constant uncertainty, and it guarantees global regulation and tracking for the class of nonlinear systems transformable into the parametric-strict feedback form. The backstepping design alleviates some limitations of other approaches [18, 20, 21 ]. It offers a choice of design tools to accommodate uncertainties and nonlinearities and can avoid wasteful cancellations. The idea of backstepping design is to select recursively some appropriate functions of state variables as pseudocontrol inputs for lower dimension subsystems of the overall system. Each backstepping stage results into a new pseudo-control design, expressed in terms of the pseudocontrol designs from the preceding design stages. When the procedure terminates, a feedback design for the true control input results and achieves the original design objective by virtue of a Lyapunov function, which is formed by summing up the Lyapunov functions associated with each individual design stage [21, 17, 22]. The control objective in this case is to force the PMSM speed ( ω r = x3 ) to follow its reference x3* and maintain in the same time the direct current ( id = x1 ) to zero under load torque disturbance. The application of the backstepping control strategy to the PMSM in this case is divided into two steps (see [18, 20]).

II. PMSM HEALTHY MODEL The setting in the state form of the PMSM model allows the simulation of this latter. In the rotor rotating ( d − q ) reference frame, the PMSM stator current model is described as follows [18], [19]:

 x = f ( x) + Bu + DTL  T T  x = [x1 x2 x3 ] = id iq ωr  T u d = Vd   b1 0 0 u =  u = V  ; B =   q  0 b2 0  q   T  D = [0 0 d ]

[

]

(1)

With the following expression of field vector f (x) :

 f1 ( x) = a1 x1 + a2 x2 x3   f 2 ( x) = a3 x2 + a4 x3 + a5 x1 x3  f ( x) = a x + a x + a x x 6 2 7 3 8 1 2  3

(2)

The components of this vector are expressed according to the PMSM parameters as follows:

ϕf Lq  Rs R L ; a2 = ; a3 = − s a4 = − ; a5 = d a1 = − L L L L Lq d d q d   n 2p ϕ f n 2p ϕ f f  ; a7 = − ; a8 = ( Ld − Lq ) a6 = − J J J  np  1 1 ; b1 = ; b2 = − d = − J Ld Lq   Where : id , iq

1. Speed regulator: This first step consists to identify the error eω which represents the error between real speed ωr = x3 and reference ω r* = x3* . In this case we control x3 by x2 . Let the Lyapunov function:

: d, q axis stator current;

Vd , Vq : d, q axis stator voltage;

V1 =

Ld , Lq : d, q axis stator inductance; Rs

: Stator resistance;

ϕf

: Rotor permanent magnet flux.

ωr

: Mechanical rotor speed ( ωr = n p Ω )

f LT J

1 2 1 eω = ( x3 − x3* ) 2 2 2

(3)

Whose derivative is:

V 1 = eω eω = ( x3 − x3* )( x3 − x3* )

(4)

The error derivative is given by:

: Viscous friction coefficient

eω = a6 x2 + a7 x3 + dTL − x3*

: Load torque : Moment of Inertia

(5)

If we selected stabilizing functions as follows:

x2* =

As presented in the appendix we take in this paper in PMSM with smooth poles Ld = Lq = L in this case (a8 = 0)

1 (−a7 x3 − K1eω − K 2 sign (eω ) + x3* ) a6

(6)

Where k1 and k 2 are positive constants.

The use of the classical controllers such as the proportional and integral controller (PI) is insufficient to provide good speed tracking performance. To overcome these problems, a robust controller based on backstepping control approach is proposed.

Then the derivative of Lyapunov function V 1 is written as: V1 = − K1eω2 − K 2 eω + eω dTL

(7)

If we take − K1 >> dTL then we get:

III. BACKSTEPPING CONTROL TECHNIQUE The Backstepping is a systematic and recursive design methodology for nonlinear feedback control. This approach is based upon a systematic procedure for the design of feedback control strategies suitable for the design of a large

173

V1 < − K1eω2 < 0

(8)

ICIEM’14, International Conference on Industrial Engineering and Manufacturing Batna University, Algeria May, 11-13, 2014

This guarantees convergence of the speed ω r to its

x3*

reference disturbance.

eω eω = a6 ( x2 − x2* )( x3 − x3* ) + V1

with robustness respect to load torque

The second step consists to control the currents id = x1 and

iq = x2 by the voltages u d = Vd

Otherwise if one chooses:

uq =

2. Direct and Quadrature currents regulator:

(19)

1 (− f 2 ( x) + x2* − K 4 ( x2 − x2* ) − a6 ( x3 − x3* )) (20) b2

From the second term of (14)

and u q = Vq ; where

eq eq = ( x2 − x2* )( f1 ( x) + b2u q − x 2* )

x1 → x1* = 0 and x2 → x2*

(21)

By replacing (20) in (21) we get:

Consider the following Lyapunov function:

1 1 1 V = eω2 + eq2 + x12 2 2 2

eq eq = − K 4 ( x2 − x2* ) 2 − a6 ( x3 − x3* )( x2 − x2* )

(9)

(22)

Where K 4 > 0 . Finally, by grouping terms (19) and (22) we obtain:

Where eω = ( x3 − x3* ) and eq = ( x2 − x2* ) The derivative of V with respect to time is: V = eω eω + eq eq + x1 x1

e = − K 4 ( x2 − x2* ) 2 − a6 ( x3 − x3* )( x2 − x2* ) + a ( x − x * )( x − x * ) + V

(10)

6

From (10) and in order to control x2 by u d . The term of the derivative V can be written as: x1 x1 = x1 ( f1 ( x) + b1u d )

(11)

2

2

3

3

By simplification:

e = − K 4 ( x2 − x2* ) 2 + V1

e = − K 4 ( x2 − x2* ) 2 − K1 ( x3 − x3* ) 2 (12)

(25)

Finely, from the foregoing, it is clear that it suffices to properly select the different gains K i ( i = 1, 2, 3, 4 ) for the set-negativity of the derivative of the complete Lyapunov function ( V ≤ 0 ) overall V defined by (25). This implies that all the error variables are globally uniformly bounded and maintain the system closed loop performance in presence of load torque disturbances.

Then this term is written:

x1 x1 = − K 3 x12 < 0

(24)

From (8) and (24) we get:

If we take the first control low as:

1 u d = (− f1 ( x) − K 3 x1 ) b1

(23)

1

(13)

Where K 3 > 0 then the convergence of x1 to 0 is ensuring. The remaining terms of (10) Let e = eω eω + eq eq as:

IV. DESIGN OF FAULT TOLERANT CONTROL

e = ( x3 − x3* )( x3 − x3* ) + ( x2 − x2* )( x 2 − x 2* )

(14)

1. PMSM faulty model

The first term can be written as:

eω eω = ( x3 −

x3* )(a6 x2

+ a7 x3 + dTL −

x3* )

In this section we briefly review how the PMSM model will be modifies in presence of faults which can be both of mechanical and electrical nature. The faults dealt with in this paper can be summarized in the class of Stator asymmetries, mainly due to static eccentricity as presented in [13] and [14]. Eccentricity related fault is the condition of unequal air gap that exist between the stator and rotor. Air gap eccentricity fault can occur due to inaccurate positioning of the rotor with respect to the stator, bearing wear, stator core movement, shaft deflection etc [12]. Following the theory in [12], it turns out that the presence of stator faults generates asymmetries in the PMSM, yielding some slot harmonics (sinusoidal components) in the stator currents (see [11], [13] and [14]). In [14], the amplitude of side-band components at frequencies (1±(2k-1)/P)fs, where k is an integer number, has been employed for static eccentricity diagnosis in PMSMs. Despite, the PMSMs under eccentricity fault have been investigated in a few papers and performance of the

(15)

or by adding and subtracting the term a6 x 2* we get:

eω eω = ( x3 − x3* )(a6 ( x2 − x2* ) + a6 x2 + a7 x3 + dTL − x3* ) (16) By simplification :

eω eω = a6 ( x2 − x2* )( x3 − x3* ) + ( x3 − x3* )(a6 x2* + a7 x3 + dTL − x3* )

(17)

By replacing the term a6 x 2* presented in (6) in this last equation we get:

eω eω = a6 ( x2 − x2* )( x3 − x3* ) + ( x3 − x3* )(− K1eω − K 2 sign (eω ) + dTL )

(18)

From (6) and (18) we get:

174

ICIEM’14, International Conference on Industrial Engineering and Manufacturing Batna University, Algeria May, 11-13, 2014

 x1 = a1 x1 + a2 x2 x3 + b1u d + Γd w   x 2 = a3 x2 + a4 x3 + a5 x1 x3 + b2u q + Γq w   x3 = a6 x2 + a7 x3 + dTL

faulty motor has been analyzed, no criterion has been so far recommended for eccentricity fault diagnosis. id → id + A sin (ω1t + ϕ )  iq → iq + A cos(ω1t + ϕ )

(26)

Where id and iq denote the stator currents in the (d − q )

 Γd  With: Γ( w) = −  w  Γq 

(30)

Γd = −(a1Qd + a2Qq x3 + Qd S )  Γq = −(a3Qd + a5Qq x3 + Qd S )

reference frame. The pulsations ω1 of the harmonic components depend on the kind of fault (due to the stator asymmetries). The amplitude A and the phases ϕ are unknown; they depend on the stator faults entity.

In this work the pulsations ω1 are assumed to be unknown. In the presence of stator faults the PMSM model becomes:

The sinusoidal components generated by the presence of the stator faults can be modeled by the following exosystem [21]:

2. Control reconfiguration

w = S (ω1 ) ⋅ w w ∈ ℜ

2n f

(27)

With: S (ω1 ) is the vector of the pulsations.

ω1   0  S (ω1 ) =   − ω1 0  Where ω1 is the pulsation of the harmonic generated by the stator faults The amplitudes and the phases of the harmonics are unknown; they depend on the initial state w(0) of the exosystem. Then, the additive sinusoidal terms in (22) can be as a suitable combination of the exosystem state, i.e:

id → id + Qd w  iq → iq + Qq w

(31)

The principal of this FTC system is presented in the fig.1. In this figure the compensation term u c resulting from the equation (30) is known which is useful to compensate the undesirable terms, which makes it possible to give an adequate form to the error dynamics, on the basis of which we calculate the unknown term u ad this additive control is added to the nominal control and setting to compensate the faults effect (FTC aspect). This additive control results from the internal model whose role is to reproduce the signal representing the faults effect (FDI aspect). The faults effects resulting from a stable autonomous system called exosystem. The load torque is compensated by the nominal control. For this (31) becomes: x = f ( x) + Bu + Γ( w)

(28)

(32)

The new control law is expressed by:

Qd = (1 0 1 0  1 0)  Qq = (0 1 0 1  0 1) Recalling the current dynamics in the un-faulty operative condition reported in the previous section, a simple computation shows that, once the perturbing terms Qd w and Qq w are added, by deriving (28) the ( id − iq ) modify as:

 did  dt = x1 = a1 x1 + a2 x2 x3 + b1u d  − (a1Qd + a2Qq x3 + Qd S ) w    diq = x = a x + a x + a x x + b u 2 3 2 4 3 5 1 3 2 q  dt  − (a3Qd + a4Qq x3 + Qd S ) w 

x = f ( x) + Bu + dTL + Γ( w)

(29)

u = u nom + u ad + uc  u d  u d nom  u d ad  u d c   + +  u = u  = u  q   q nom  u q ad  u q c  

(33)

1 *  * u1c = b ( x1 + k3 x1 )  1  1 u = a6 ( x3 − x3* ) 2c  b 2 

(34)

Where:

On the basis of which we calculate the unknown term u ad with the expression which we retained from (31) and from the nominal control presented in (12) and (20). The instantaneous difference between the state derivative of the system and the reference becomes:

Bearing in mind the dynamics of the stator currents in the normal (i.e., in the absence of faults) operative conditions, it is also simple to get the PMSM dynamics after the occurrence of a fault. As a matter of fact, taking (29) it is readily seen that the PMSM healthy model given by (1) and (2). In faulty condition (presence of stator faults) will be given by:

* *  ed   x1   x1   f1 ( x) + b1u d − x1 − Γd w    x =  eq  =  x 2  −  x 2*  =  f 2 ( x) + b2u q − x 2* − Γq w (35) eω   x3   x3*   f ( x) − x *    2 3

From (12), (20), (33) and (34) after replacing in (35) we get

175

ICIEM’14, International Conference on Industrial Engineering and Manufacturing Batna University, Algeria May, 11-13, 2014

 ed  − k3ed + b1u d ad − Γd w  x =  e  = − k e + b u 4 q 2 q ad − Γq w  q eω  a e − K e − K sign (e ) 1 ω 2 ω  6 q

Its derivative compared to time takes this form:

e = ξ − w = S (ϖ )ξ + N ( ~ x ) + S (ϖ ) w

(36)

The equations describing the dynamics of the errors in closed loop are thus:

 In the third equation if eq → 0 ⇒ eω → 0

~  ~ x = A ⋅ ~ x + Γ⋅e  e = S (ϖ ) e + N ( ~ x)

Let us notice that the first two equations do not depend on the variable eω In the continuation, for the determination of u ad let us consider the subsystem: x  e  ~ ~ x =  ~1  =  d  x  2  ed 

(37)

From system (38) we can write it in a matrix form: ~ ~ x = H ( ~ x ) + B ⋅ u ad − Γ ⋅ w Compensation term

References Nominal controller

unom +

ξ

Faults FDI

+ uc

u

Faults Output PMSM

+ uad Internal model

1 ~T ~ 1 T x ⋅ x + e ⋅e 2 2 After develop of calculates V becomes: ~ V = ~x T ⋅ A ⋅ ~x + eT ⋅ ΓT ⋅ ~x + eT ⋅ N ( ~x )

(38)

(39)

(49)

Finally V is written:

~  ~ ~ ~ ~ −Vk=3 ~ x T ⋅0A ⋅ ~ x ≤0  H ( x ) = A ⋅ x and A =   − 0 k  The system (46) becomes:4   Γd 0  ~ bd 0  B  =  0 bq  and Γ =  0 Γq       Γ ⋅ e 0 =   (40) e = S (ϖ ) e

(41)

(42)

Consider the systems (39) and the additive term given by (42) in this case we have: (43)

The new error variable is considered: e = (ξ − w)

(48)

N (~ x ) = −Γ T ~ x

Then uad is chosen like [11]:

~x = H ( ~x ) + Γ ⋅ (ξ − w)

(47)

In this case the N (x~ ) choice is given by:

In this case for the determination of the internal model we introduce a resent implicit fault tolerant control approach which does not rest on the resolution of the Sylvester equation problem [23] as proposed in [11]. To solve this problem we propose in this paper an internal based Lyapunov theory which takes the following form:

~ u ad = B −1Γξ

It is necessary to find the expression of N (x~ ) which cancels the error of observation of the faults e and makes it possible at the same time to reject their effect for it cancels also ~ x.

V=

Fig.1 Fault tolerant control structure

ξ = S (ϖ )ξ + N ( ~ x)  dim(ξ ) = dim( w) = 2n f

(46)

That is to say the Lyapunov function of the system (46):

Whose dynamics results from the system (36)

w = S (ϖ ) ⋅ w  ~ x1  − k3 x1 + b1u d ad − Γd w  ~  ~ = x =    ~  ~  x1   − k 4 x2 + b2u q ad − Γq w 

(45)

(44)

176

(50)

(51)

ICIEM’14, International Conference on Industrial Engineering and Manufacturing 120

3

100

2

80

1 Speed Error

Speed (rad/sec)

Batna University, Algeria May, 11-13, 2014

60 40

-1 Wr ref

20 0

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-2 -3

0.8

0

0.2

0.1

0.6

0.8

0.7

100 Vd

Id

8

80

Iq Iq ref

Vq

60 Voltages (V)

6 Currents (A)

0.5

Time (Sec)

10

4 2 0

40 20 0 -20

-2 -4 0

0.4

0.3

Time (Sec)

-40 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-60

0

0.1

0.2

0.4

0.3

Time (Sec)

0.5

0.6

0.7

0.8

Time (Sec)

The objective of the control is achieved by adopting the procedure suggested and we able to compensate the faults effect on the system ( x → 0 ) and to reproduce ( e → 0 ) thanks to the internal model.

APPENDIX RATED DATA OF THE SIMULATED PMSM

Rated Values

V. SIMULATION RESULTS In this section we taste the proposed controller by simulation where the parameters of the PMSM used are given in Table 1. In Fig.2 after association between PMSM and PWM inverter controlled by the backstepping technique we start the simulation without any load torque, then at t=0.4 sec the application of a load torque equal to the nominal torque ( TL = 0.05 Nm ) is presented, then we introduce after that the effect of stator fault at t =0.6 sec. From these simulations we can noticed that backstepping controller (nominal controller) which we synthesized present a robustness compared to the load torque disturbance, but proves to be insufficient in the event of fault. This is checked by simulations represented above when the internal model is not active.

177

Power

22

w

Frequency

50 2

Hz

Rs Ld

3.4

Ω

0.0121

H

Lq

0.0121

H

ϕf

0.4212 

H

J

0.0001

Kg.m 2

f

0.0005

IS

np Rated parameters

ICIEM’14, International Conference on Industrial Engineering and Manufacturing Batna University, Algeria May, 11-13, 2014

For Fig.3 we simulate the global closed loop system with the FTC based backstepping approach. The FTC approach (when the internal model is active) which we synthesized rejects the effect of the load torque disturbances and also the stator faults effect. 7

120

6 5 4

80 Speed Error

Speed (rad/sec)

100

60 40

2 1 0

Wr ref

20 0

3

-1 -2 -3

0

0.1

0.2

0.3

0.5

0.4

0.6

0.7

0.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.6

0.7

0.8

Time (Sec)

Time (Sec)

100

10 8

Id

6

Iq

Vd Vq

50

Id ref

Voltages (V)

Currents (A)

4 2 0

0

-2

-50

-4 -6 -8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-100

0.8

0

0.1

0.2

0.3

0.4

0.5

Time (Sec)

Time (Sec)

VII. REFERENCES VI. CONCLUSION

[1]

In this paper a backstepping control based FTC approach for PMS Motors has been presented. In un-faulty condition the backstepping controller permits to steer the direct current and the speed variables to their desired references and to reject the load torque disturbances, however the presence of stator mechanical faults degraded the performances of the PMSM. In order to compensate the faults effect a FTC approach can be designed starting with generating from the internal model state, an additive term wish we add to the nominal control (backstepping) to compensate the faults effect. The simulation results show the robustness and the effectiveness of the proposed FTC control scheme.

[2]

[3]

[4]

[5]

[6]

[7]

178

C. Edwards, C. Pin Tan, “Sensor fault tolerant control using sliding mode observers”, Control Engineering Practice 14 pp 897–908. 2006. R.J. Paton, "Fault Tolerant Control Systems: The 1997 Situation", Proc. IFAC Safe process, Hull, United Kingdom, pp.1033-1055, 1997. A. Fekih, et al, “A robust fault tolerant control strategy for a class of nonlinear uncertain systems”, Proceeding of the American Control Conference Minneapolis, Minnesota, USA, pp. 5474-5480, June 2006. Ahmad Akrad, et al “Design of a Fault-Tolerant Controller Based on Observers for a PMSM Drive”, IEEE Transactions on Industrial Electronics, vol. 58, no. 4, pp. 1416-1427, April 2011. Isermann, R., Schwarz, R., Stolzl, S.: ‘Fault-tolerant drive-by-wire systems-concepts and realization’. Proc. IFAC Sym. Fault Detection, Supervision and Safety for Technical Processes: SAFEPROCESS, pp. 1–15, 2000. Prashant, M., Jinfeng L., Panagiotis D.C. ‘Fault-Tolerant Process Control Methods and Applications’ (Springer-Verlag London, 2013) A. El-Refaie, “Fault-tolerant permanent magnet machines: a review,” Electric Power Applications, IET, vol. 5, no. 1, pp. 59 –74, January 2011.

ICIEM’14, International Conference on Industrial Engineering and Manufacturing Batna University, Algeria May, 11-13, 2014 [8]

[9]

[10]

[11]

[12]

[13]

[14]

[15] [16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

Zhang, Y.M., Jiang, J. “Bibliographic review on reconfigurable faulttolerant control systems”, Annual Reviews in Control, Vol 32, Issue 2, pp 229–252, Dec 2008. Jin Jiang., Xiang Yu, “Fault-tolerant control systems: A comparative study between active and passive approaches”, Annual Reviews in Control, Vol 36, Issue 1, pp 60–72, April 2012. Qingfang Teng, Jianguo Zhu, Tianshi Wang, Gang “Lei Fault Tolerant Direct Torque Control of Three-Phase Permanent Magnet Synchronous Motors”, WSEAS Transactions on Systems, Issue 8, Volume 11, pp 465- 476 August 2012 A. Rezig and M. R Mekideche “effect of rotor eccentricity faults on noise generation in permanent magnet synchronous motors”, Progress In Electromagnetics Research, Vol. 15, pp 117-132, 2010. Vas, P. “Parameter estimation,condition monitoring and diagnosis of electrical machines”. Oxford, UK:Oxford Science Publications. 1994. M. Akar and İ. Çankaya“ diagnosis of static eccentricity fault in permanent magnet synchronous motor by on-line monitoring of motor current and voltage”, Journal of Electrical & Electronics Engineering vol. 9, no. 2 2009. B. M. Ebrahimi, J. Faiz, and M. J. Roshtkhari, "Static, dynamic and mixed eccentricity fault diagnosis in permanent magnet synchronous motors," IEEE Transactions on Industrial Electronics, vol. 56, no. 11, pp. 4727-4739, Nov 2009. C. Bonivento, et al, “Implicit Fault Tolerant Control: Application to Induction Motors,” Automatica 40, pp. 355-371, 2004, H. Mekki et al “Implicit Fault Tolerant Control Technique Based Backstepping Application to Induction Motor” Journal of Electrical Systems: special issue N° 2,(2010), PP 109-124. 2010 Benzineb, O., Tadjine, M., Benbouzid, M.E.H., Diallo, D. “Sur la commande tolérante aux défauts des machines asynchrones. Une approche implicite,” European Journal of Electrical Engineering, Revue Internationale de Génie électrique, Vol. 15/6 – 2012, pp. 633-657, 2012. A. Lagrioui, H. Mahmoudi, “Nonlinear Adaptive Backstepping Control of Permanent Magnet Synchronous Motor (PMSM),” Journal of Theoretical and Applied Information Technology Vol. 29 No.1 July 2011. Wei Gao and Zhirong Guo “Speed Sensorless Control of PMSM using Model Reference Adaptive System and RBFN,” Journal of Networks, Vol. 8, No. 1, January 2013 J. Zhou, Y. Wang, “Real-time nonlinear adaptive backstepping speed control for a PM synchronous motor,” Control Engineering Practice, Vol. 13, pp. 1259-1269, 2005. Lin, F. J., and Lee, C. C., “Adaptive backstepping control for linear induction motor drive to track period references, IEE Proc. Electr. Power Appl., Vol. 147, (6), 2000, pp 449-458. M. Azizur Rahman, D. M. Vilathgamuwa, M. N. Uddin and K. J. Tseng, “Nonlinear control interior permanent-magnet synchronous motor,” IEEE Trans. On Ind. Appl., Vol. 39, No2, pp. 408-416, 2003. Lin Y. et al. “Condition numbers of the generalized Sylvester equation,” IEEE Trans. Automatic Control, Vol. 52, N°12, pp. 2380-2385, December 2007.

179