Fuzzy Logic Speed Control for Sensorless Indirect ...

International Review of Automatic Control (I.RE.A.CO.), Vol. 6, N. 3 ISSN 1974-6059 May 2013

Fuzzy Logic Speed Control for Sensorless Indirect Field Oriented of Induction Motor Using an Extended Kalman Filter A. Bennassar, A. Abbou, M. Akherraz, M. Barara Abstract – In this paper, we present the speed sensorless of indirect field oriented control (IFOC) of induction motor (IM). The speed estimator has been designed using extended Kalman filter technique (EKF). According to this method, the rotor speed is obtained from the directly measurable stator voltages and currents. A fuzzy logic controller (FLC) used for the speed and sliding mode controllers (SMC) used for stator currents control to allow a good performance and robustness against the load disturbances. Simulation results are illustrated and demonstrate the effectiveness of the proposed strategy. Copyright © 2013 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Speed Sensorless, Indirect Field Oriented Control, Induction Motor, Extended Kalman Filter, Fuzzy Logic Control, Sliding Mode Control

very low speeds [2], [3]. The main techniques of sensorless control of induction motors are: Model reference adaptive systems (MRAS), Kalman and Luenberger observers. In this paper, the extended Kalman filter (EKF) is used for the estimation of the rotor speed of an induction motor. This estimator can be used for the joint state and parameter estimation of a non-linear dynamic system in which the process noise and measurement noise are considered. In the EKF, which is a stochastic observer, the speed to estimate is considered as a state. The EKF uses two stages, in the first step, the state are predicted by using the mathematical model of induction motor and in the second step, the predicted state are corrected by using feedback correction scheme. The induction motor is traditionally controlled by the proportional-integral controller, but when there are load disturbances and parameter variations, the PI controller does not satisfy response. In the recent literature, several tools are proposed from which we quote the fuzzy logic control and sliding mode control to overcome the above problems. Fuzzy theory was first proposed by Prof. Zadeh [4]. The main advantages of the FLC that it does not require any exact mathematical model of the system. Fuzzy logic is based on the linguistic rules by means of IF-THEN rules with the human language. The sliding mode control is a type of variable structure system characterized by high simplicity and robustness against insensitivity to parameters variation and disturbances. This approach utilizes discontinuous control laws to drive the system state trajectory onto a sliding or switching surface in the state space. The dynamic of the system while in sliding mode is insensitive to model uncertainties and disturbances [5]. However, the discontinuous control presents a major drawback and

Nomenclature usα,β isα,β Φsα,β usd,q isd,q Φsd,q  ωr ωsl ωs p Rs, Rr Ls, Lr, Lm Tr σ Ts

Stator voltage in the stationary α, β axis Stator current in the stationary α, β axis Rotor flux in the stationary α, β axis Stator voltage in the stationary d, q axis Stator currents in the stationary d, q axis Rotor flux in the stationary d, q axis Rotor speed Rotor electrical speed Slip frequency Stator frequency Number of pole pairs Stator and rotor resistances Stator, rotor and mutual inductances Rotor time constant Leakage coefficient Sampling time

I.

Introduction

The induction motor is one of the most widely used machines in various industrial applications due to its high reliability, relatively low cost, and modest maintenance requirements [1]. Several methods of control are used to control the induction motor among which the field orientation control that allows a decoupling between the flux and the torque, in order to obtain an independent control of the flux and the torque like dc motors. The sensorless speed control of induction motor drives has received over the last few years a great interest. Thus it is necessary to eliminate the speed sensor to reduce hardware and increase mechanical robustness [1]-[17]. In traditional approaches, the flux and the speed are estimated using the stator voltage and currents, but this can provide a large error in speed estimation, including Manuscript received and revised April 2013, accepted May 2013

Copyright © 2013 Praise Worthy Prize S.r.l. - All rights reserved

332

A. Bennassar, A. Abbou, M. Akherraz, M. Barara

constitutes the main criticism to the sliding mode control techniques. This paper is organized as follow: an introduction in first section, the induction motor model is described in section two, section three reviews the indirect field oriented control strategy, in section four the extended Kalman filter algorithm is discussed, section five and six develop the fuzzy logic controller for the rotor speed and sliding mode controller for the stator currents, the section seven is devoted to simulation results of this control strategy, a conclusion, reference list and author’s information end the paper.

s  sl  r

(3)

where sl is calculated from the reference values:

sl 

* 1 isq Tr i*sd

(4)

Furthermore, r is expressed as follow:

r  p

(5)

Thus the space angle of the rotor flux is given by:

II.

Induction Motor Model

A dynamic model of the induction motor in stationary reference frame can be expressed in the form of the state equation as shown:

x  Ax  Bu

(1)

y  Cx

(2)

s  r  

IV.

t

t

t

where x, u and y are the state vector, the input vector and the output vector respectively:

  -    0 A  Lm   Tr   0 

0

 Tr

-

- r

0

-

Lm Tr

1 Tr

r

 r   1   L    s Tr   ; B   0 -r   0   1   0 -  Tr 

 0   1   Ls  0   0 

Extended Kalman Filter for Rotor Speed Estimation

x  f  x,u   w

(7)

y  h  x  v

(8)

where:

   is     is  f  x,u          

1 0 0 0  C  0 1 0 0  with:



(6)

The extended Kalman filter can be used to estimate state and parameters for nonlinear systems. In this case, the rotor speed of induction motor is considered as a state and parameter to be estimated and is increased to the state vector from which the state model becomes nonlinear. Considering the process noise w and the measurement noise v, the dynamic behavior of the induction motor can be given by the following system [6]-[7]:

where:

x  is is r  r   ; u  us us  ; y  is is 

* 1 isq Tr i*sd

Rs 1   L 1 L2  ;  ;   1  m ; Tr  r  Ls  Tr  Lm Rr Ls Lr

 1  r  rr   us  Tr  Ls    1  rr  r   us   Tr  Ls   Lm 1 is  r  rr   Tr Tr   Lm 1 is  rr  r   Tr Tr   r  

h  x   is is 

III. Indirect Field Oriented Strategy

t

The covariance matrices Q and R of these noises are defined as:

The behavior of the induction motor subjected to rotor flux oriented control is illustrated in Fig. 1. The angular frequency s of the rotor flux is obtained as the sum of the slip frequency sl and rotor electrical speed r:

 

 

Q  cov  w   E wwt ; R  cov  v   E vv t

Copyright © 2013 Praise Worthy Prize S.r.l. - All rights reserved

International Review of Automatic Control, Vol. 6, N. 3

333

A. Bennassar, A. Abbou, M. Akherraz, M. Barara

4) Estimation of state variables:



ˆxk 1|k 1  ˆxk 1|k  K k 1 yk 1  H k ˆxk 1|k



(14)

5) Update of error covariance matrix:

Pk 1|k 1  Pk 1|k  K k 1 H k Pk 1|k

V.

From the above dynamic model, the rotor speed can be estimated by the following extended Kalman filter algorithm: 1) Prediction of state variables:





Fuzzy Logic Control

Fig. 2 shows the block diagram of fuzzy logic controller system. The FLC consists of four major blocks, Fuzzification, knowledge base, inference engine and defuzzification.

Fig. 1. Block diagram of proposed scheme

ˆxk 1|k  f xk|k ,uk

(15)

(9)

2) Estimation of error covariance matrix: Fig. 2. Block diagram of a fuzzy logic controller

Pk 1|k  Fk Pk|k Fkt  Q

(10)

There are two inputs, the speed error e(k) and the change of speed error ce(k). The two input variables are calculated at every sampling time as:

where:

Fk 

 1 - Ts     0  Fk   Lm Ts  Tr   0   0 

0 1  Ts  0 Ts



f xk|k ,uk

Lm Tr 0

xk

Ts

(11)

Ts r 1 Tr

Tsr

Ts r Ts

 Tr

Tsr 1  Ts

0

1 Tr

0

 Ts r     Ts r    Tsr     Tsr    1 

K k 1  Pk 1|k H kt  H k Pk 1|k H kt  R 

ce  k   e  k   e  k  1

(17)

V.1.

-1

(12)

V.2.

h  xk 

Fuzzification

The crisp input variables are e(k) and ce(k) are transformed into fuzzy variables referred to as linguistic labels. The membership functions associated to each label have been chosen with triangular shapes. The following fuzzy sets are used, NL (Negative Large), NM (Negative Medium), NS (Negative Small), ZE (Zero), PS (Positive Small), PM (positive Medium), and PL (Positive Large). The universe of discourse is set between – 1 and 1. The membership functions of these variables are shown in Figs. 3, 4 and 5.

where:

xk

(16)

where *(k) denotes the reference speed, (k) is the actual speed and e(k-1) is the value of error at previous sampling time.

3) Kalman filter gain:

Hk 

e  k   *  k     k 

xk  ˆxk|k

 Tr

1  Ts



(13)

Knowledge Base and Inference Engine

The knowledge base consists of the data base and the rule base. The data base provides the information which is used to define the linguistic control rules and the fuzzy data in the fuzzy logic controller. The rule base specifies the control goal actions by means of a set of linguistic control rules [8].

xk  ˆxk|k

1 0 0 0 0  Hk    0 1 0 0 0

Copyright © 2013 Praise Worthy Prize S.r.l. - All rights reserved

International Review of Automatic Control, Vol. 6, N. 3

334

A. Bennassar, A. Abbou, M. Akherraz, M. Barara

NL

NM NS ZE PS PM

PL

1

ce/e NL NM NS ZE PS PM PL

Degree of membership

0.8

0.6

0.4

0.2

0 -1

-0.8

-0.6

-0.4

-0.2

0 e

0.2

0.4

0.6

0.8

NL NL NL NL NL NM NS ZE

NM NL NL NL NM NS ZE PS

TABLE I FUZZY RULES BASE NS ZE PS NL NL NM NL NM NS NM NS ZE NS ZE PS ZE PS PM PS PM PL PM PL PL

PM NS ZE PS PM PL PL PL

PL ZE PS PM PL PL PL PL

There are many defuzzification techniques to produce the fuzzy set value for the output fuzzy variable. In this paper, the centre of gravity defuzzification method is adopted here and the inference strategy used in this system is the Mamdani algorithm.

1

Fig. 3. Membership function for input e NL

NM NS ZE PS PM

VI.

PL

1

Sliding mode technique is a type of variable structure system (VSS) applied to the non-linear systems. The sliding mode control design is to force the system state trajectories to the sliding surface S(x) and to stay on it by means a control defined by the following equation [9]: u  ueq  un (18)

0.8 Degree of membership

Sliding Mode Control

0.6

0.4

0.2

0 -1

-0.8

-0.6

-0.4

-0.2

0 ce

0.2

0.4

0.6

0.8

where ueq and un represent the equivalent control and the discontinue control respectively:

1

s un  k  sat    

Fig. 4. Membership function for input ce NL

NM NS ZE PS PM

PL

here  defines the thickness of the boundary layer and

1

s sat   is a saturation function. To attract the trajectory   of the system towards the sliding surface in a finite time, un(x) should be chosen such that Lyapunov function, satisfies the Lyapunov stability:

0.8 Degree of membership

(19)

0.6

0.4

S  x  S  x   0

0.2

0 -1

-0.8

-0.6

-0.4

-0.2

0 u

0.2

0.4

0.6

0.8

The general equation to determine the sliding surface proposed is as follow [10]:

1

d  S  x       dt 

Fig. 5. Membership function for output u

The inference engine evaluates the set of IF-THEN and executes 7×7 rules as shown in Table I. The linguistic rules take the form as in the following example:

n 1

e

(21)

here, e is the tracking error vector, λ is a positive coefficient and n is the system order. To control the stator currents of the IM, the sliding surfaces are defined as follows:

IF e is NL AND ce is NL THEN u is NL V.3.

(20)

S  isd   i*sd  isd

Defuzzification

In this stage, the fuzzy variables are converted into crisp variables.

 

S isq  i*sq  isq

Copyright © 2013 Praise Worthy Prize S.r.l. - All rights reserved

(22) (23)

International Review of Automatic Control, Vol. 6, N. 3

335

A. Bennassar, A. Abbou, M. Akherraz, M. Barara

120

Take account the equations of the stator currents defined in the IM model above, we obtain:

real estimated 100

1 n S  isd    vsd  Ls

 

1 n  vsq  Ls

Rotor speed (rad/s)

S isq

(24)

(25)

80

60

40

20

We take: eq n vsd  vsd  vsd

(26)

eq n vsq  vsq  vsq

(27)

0

0

0.1

0.2

0.3

0.4

0.5 0.6 Time (s)

0.7

0.8

0.9

1

0.7

0.8

0.9

1

Fig. 6. Rotor speed 4

During the sliding mode and in permanent regime n S  isd   S  isd   0 , S isq  S isq  0 , vsd  0 and

 

3.5

 

3 Error speed (rad/s)

n vsq  0 , where the control equivalent can be expressed

as: eq vsd

  *    Ls  i*sd   i*sd  s isq  rd  T  r 



eq * vsq   Ls i*sq   i*sq  s isd  r rq



(28)

2 1.5 1 0.5

(29)

0 -0.5

The discontinuous control actions can be given as:

 s i   n vsd  kvsd  sat  sd   vsd 

2.5

0

0.1

0.2

0.3

0.4

0.5 0.6 Time (s)

Fig. 7. Error speed

(30) 1.5

 rd

  

 s isq  kvsq  sat   vsq 

 

 rq 1

(31) Rotor flux (Wb)

n vsq

To verify the system stability condition, kvsd and kvsq are strictly positive coefficients.

0.5

0

VII.

Simulation Results

A series of simulation tests were carried out on indirect field oriented control of induction motor drive based on the extended Kalman filter and using both fuzzy logic control and sliding mode control for the speed and stator currents respectively. Simulations have been realized under the Matlab/Simulink. The parameters of induction motor used are indicated in Table A1. VII.1.

-0.5

0

0.1

0.2

0.3

0.4

0.5 0.6 Time (s)

0.7

0.8

0.9

1

Fig. 8. Rotor flux

VII.2.

Operating with Reverse Speed Rotation

In this case, a test of robustness of the control is realized by the reverse of the speed rotation between 100 rad/s and 100 rad/s.

Operating with No and Full Load

The Figures 6, 7, 8 and 9 represent the simulation results obtained from a no load operating. The application of a torque with 10 Nm is realized at t = 0.5s. The reference of the rotor flux is set to 1 Wb.

VII.3.

High Speed Functioning

The Figs. 12 and 13 illustrate simulation results when a rotor speed of 157 rad/s is imposed.

Copyright © 2013 Praise Worthy Prize S.r.l. - All rights reserved

International Review of Automatic Control, Vol. 6, N. 3

336

A. Bennassar, A. Abbou, M. Akherraz, M. Barara

40

0.7

30

0.6

20

Error speed (rad/s)

Stator phase current (A)

0.5 10 0 -10 -20 -30

0.4 0.3 0.2 0.1

-40

0

-50 -60

0

0.1

0.2

0.3

0.4

0.5 Time (s)

0.6

0.7

0.8

0.9

-0.1

1

-0.2

0

0.5

1

Fig. 9. Stator phase current

1.5

Time (s)

150

Fig. 13. Error speed

real estimated 100

Rotor speed (rad/s)

VII.4. 50

Low Speed Functioning

In this case, the estimated speed is carried out for low speeds ± 10 rad/s (Figs. 14, 15).

0

15 real estimated

-50 10

-150

0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

Rotor speed (rad/s)

-100

4

Fig. 10. Rotor speed

5

0

-5

1 -10

0.8 0.6

-15

0

0.2

0.4

0.6

0.8

Error speed (rad/s)

0.4

1 Time (s)

1.2

1.4

1.6

1.8

2

1.4

1.6

1.8

2

0.2

Fig. 14. Rotor speed

0 0.8

-0.2 -0.4

0.6

-0.6 0.4

-1

0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

Error speed (rad/s)

-0.8 4

Fig. 11. Error speed

0.2

0

-0.2

160 -0.4

140

-0.6

Rotor speed (rad/s)

120 100 80

0

0.2

0.4

0.6

0.8

1 Time (s)

1.2

Fig. 15. Error speed

60

With the results above, we can notice the estimated speed tracking performance test in different working in high and low speed. We can observe some estimated speed oscillations at low speed. The fuzzy logic controller does have any effect by the change in load.

40 20 0 -20

real estimated 0

0.5

1

1.5

Time (s)

Fig. 12. Rotor speed

Copyright © 2013 Praise Worthy Prize S.r.l. - All rights reserved

International Review of Automatic Control, Vol. 6, N. 3

337

A. Bennassar, A. Abbou, M. Akherraz, M. Barara

[9]

The stator phase current remains sinusoidal and takes appropriate value. It is evident from the results that the fuzzy logic controller gives better responses in terms of overshoot, static error and fast response.

[10] [11]

VIII. Conclusion The extended Kalman filter for sensorless indirect field oriented of induction motor has been presented in this paper. The drive system was simulated using fuzzy logic controller for the speed and sliding mode controllers for the stator currents. Simulation results show the good performance of the speed estimation at high and low speed. The fuzzy logic controller gives better tracking speed and robustness against the load torque disturbance. The parameters of control and of the discontinuous control of the variable structure system have been chosen to reduce the chattering phenomenon.

[12]

Appendix

[16]

[13]

[14]

[15]

TABLE A1 INDUCTION MOTOR PARAMETERS Rated power 3 kW Voltage 380V Y Frequency 50 Hz Pair pole 2 Rated speed 1440 rpm Stator resistance 2.2 Ω Rotor resistance 2.68 Ω Inductance stator 229 mH Inductance rotor 229 mH Mutual inductance 217 mH Moment of Inertia 0.047 kg m2

[17]

Authors’ information LEEP, Electric Engineering Department, Mohammed V University, Mohammadia School’s of engineers, Rabat, Morocco. E-mail: [email protected]

References [1]

[2]

[3]

[4] [5]

[6]

[7]

[8]

V. Utkin, Variable structure systems with sliding modes, IEEE Transactions on Automatic Control, Vol .22(Issue 2):212–222, 1977. JJE. Slotine, W. Li, Applied nonlinear control, New Jersey, Prentice-Hall, Inc, 1991. B. Sahu, KB. Mohanty, S. Pati, A comparative study on fuzzy and PI speed controllers for field-oriented induction motor drive, International Conference on Industrial Electronics, Control and Robotics 2010, pp. 191-196, 27-29 Dec, 2010. K. Yahia, S.E. Zouzou, F. Benchabane and D. Taibi, Comparative study of an adaptive Luenberger observer and extended Kalman filter for a sensorless direct vector control of induction motor, Vol. 50:99-107, 2009. M. M. Krishan, Fuzzy Sliding Mode Control with MRAC Technique Applied to an Induction Motor Drives, (2008) International Review of Automatic Control (IREACO), 1 (1), pp. 42-48. Lalalou, R., Bahi, T., Bouzekri, H., Sensorless indirect vector controlled induction motor using fuzzy logic control of speed estimation and stator resistance adaptation, (2010) International Review on Modelling and Simulations (IREMOS), 3 (3), pp. 325330. Salehi, S.J., Manoochehri, M., A nonlinear sliding mode control of induction motor based on second order speed and flux sliding mode observer, (2011) International Review on Modelling and Simulations (IREMOS), 4 (3), pp. 1057-1065. Boulghasoul, Z., Elbacha, A., Elwarraki, E., Real time implementation of fuzzy adaption mechanism for MRAS sensorless indirect vector control of induction motor, (2011) International Review of Electrical Engineering (IREE), 6 (4), pp. 1636-1653. Aydin, M., Gokasan, M., Bogosyan, S., Comparison of two EKF based observers optimized online by both simulated annealing and Big Bang-Big Crunch methods for sensorless estimations in Induction Motor, (2011) International Review on Modelling and Simulations (IREMOS), 4 (5), pp. 2111-2121.

Abderrahim Bennassar, was born in Casablanca, Morocco in 1987. He received Master degree in treatment of information from Hassan 2 University, Casablanca in 2011. Currently, he is pursuing PhD degree at Mohammadia School of Engineering, Rabat. His researcher interests include the control strategies for AC Drives, especially Induction Motor Drives and Sensorless Control. E-mail: [email protected]

A. Abbou, T. Nasser, H. Mahmoudi, M. Akherraz, and A. Essadki, Induction motor controls and implementation using dspace, WSEAS Transactions on Systems and Control, Vol. 7:2635, 2012. H. Nakano, and I. Takahashi, Sensorless field oriented control of an induction motor using an instantaneous slip frequency estimation method, IEEE Power Electronics Specialists Conference, Vol. 2, pp. 847-854, 11-14 April 1988. T. Ohtani, N. Takada, and K. Tanaka, Vector control of induction motor without shaft encoder, IEEE industry Applications Society Annual Meeting Conference Record, Vol. 1, pp. 500-507, 1-5 Oct, 1989. LA. Zadeh, Fuzzy sets, Information and Control, Vol. 8:338-353, 1965. C. Vecchio, Sliding mode control: theoretical developments and applications to uncertain mechanical system, Degli Studi Pavia University. YR. Kim, SK. Sul, and MH. Park, Speed sensorless vector control of induction motor, using extended Kalman filter, IEEE Transactions on Industry Applications, Vol. 30(Issue 5):12251233-1994. KL. Shi, TF. Chan, Y.K. Wong, and S. L. Ho, Speed estimation of an induction motor drive using an optimized extended Kalman filter, IEEE Transactions on Industrial Electronics, Vol. 49:124133, 2002. V. Peter, Sensorless vector and direct torque control, Oxford New York Tokyo, Oxford University Press, 1998.

Ahmed Abbou received the ‘’ Agrégation Génie Electrique’’ from Ecole Normale Supérieur de l’Enseignement Technique ENSET Rabat in 2000. He received the ‘’Diplôme des Etudes Supérieurs Approfondies’’ in industrial electronics from Mohammadia School’s of engineers, Rabat in 2005. He received with Honors the Ph.D. degree in industrial electronics and electrical machines, from Mohammadia School’s of engineers Rabat in 2009. Since 2010, he has been a Professor of power Electronic and Electric drives at the Mohammadia School’s of engineers, Rabat. He has presented papers at national and international conference on the Electrical machine, Power Electronic and electric drives. His current area of interest is related to the innovative control strategies for Ac machine Drives, renewable energy. E-mail: [email protected]

Copyright © 2013 Praise Worthy Prize S.r.l. - All rights reserved

International Review of Automatic Control, Vol. 6, N. 3

338

A. Bennassar, A. Abbou, M. Akherraz, M. Barara

Mohammed Akherraz Graduated from the Mohammadia School’s of engineers Rabat morocco, in 1980. In 1983 he was graduated a Fulbrighr scholarship to pursue his postgraduate studies. He earned the Ph.D degree in 1987 rom UW, Seattle. He joined the EE department Of the Mohammadia School’s of engineers, Rabat Morocco, where he‘s presently a professor of power electronics and Electric drives. He published numerous papers in international journal and conferences. His areas of interests are: power electronics, Electric drives, Computer Modeling of power Electronics circuit, and systems drives. E-mail: [email protected]

Copyright © 2013 Praise Worthy Prize S.r.l. - All rights reserved

International Review of Automatic Control, Vol. 6, N. 3

339