A Takagi-Sugeno fuzzy control of induction motor …...A Takagi-Sugeno fuzzy control of induction motor drive: experimental results 49 3 Observer-based fuzzy control design 3.1 T-S

44 Int. J. Automation and Control, Vol. 12, No. 1, 2018

Copyright © 2018 Inderscience Enterprises Ltd.

A Takagi-Sugeno fuzzy control of induction motor drive: experimental results

Habib Ben Zina*, Moez Allouche, Mansour Souissi and Mohamed Chaabane Laboratory of Science and Technique of Automatic Control and Computer Engineering (Lab-STA), National School of Engineering of Sfax, Tunisia Email: [email protected] Email: [email protected] Email: [email protected] Email: [email protected] *Corresponding author

Larbi Chrifi-Alaoui Laboratory of Innovative Technology, University of Picardie Jule Verne, Cuffies, France Email: [email protected]

Maha Bouattour Laboratory of Science and technique of Automatic Control and Computer Engineering (Lab-STA), National School of Engineering of Sfax, Tunisia Email: [email protected]

Abstract: This paper studies an observer-based H∞ tracking control problem for induction motor in order to guarantee the field-oriented control (FOC) performances. Firstly, the physical model of the induction motor is approximated by the T-S fuzzy technique in the synchronous d-q frame rotating. Then, a fuzzy observer-based feedback control is synthesised to guarantee the control performances. The proposed controller is based on a T-S reference model in which a desired trajectory has been specified. The performances of the trajectory tracking are analysed using the Lyapunov theory and the L2 optimisation. The gains of the observer and the controller are obtained by solving a set of LMIs constraint in a single step. To highlight the effectiveness of the proposed strategy experimental results are presented for a 1.5 KW induction motor.

A Takagi-Sugeno fuzzy control of induction motor drive: experimental results 45

Keywords: induction motor; Takagi-Sugeno; Lyapunov theory; LMIs constraints.

Reference to this paper should be made as follows: Ben Zina, H., Allouche, M., Souissi, M., Chaabane, M., Chrifi-Alaoui, L. and Bouattour, M. (2018) ‘A Takagi-Sugeno fuzzy control of induction motor drive: experimental results’, Int. J. Automation and Control, Vol. 12, No. 1, pp.44–61.

Biographical notes: Habib Ben Zina received his Master in Automatic Control from the University of Tunis in 2012. Currently, he is a PhD candidate in Electrical Engineering at the National Engineering School of Sfax (ENIS). His research interests include robust observer and fault tolerant control for Takagi-Sugeno models.

Moez Allouche received his MSc degree from National Engineering School of Sfax, Tunisia in 2006. Also, he obtained his PhD in Electrical Engineering from the National Engineering School of Sfax, Tunisia in 2010. Currently, he is an Assistant Professor in High Institute of Industrial Systems of Gabes. His research interests include robust control, renewable energies (solar energies), nonlinear control, fault tolerant control, fault detection and isolation.

Mansour Souissi received his PhD in Physical Sciences from the University of Tunis, Tunisia in 2002. He is a Professor in Automatic Control at Preparatory Institute of Engineers of Sfax, Tunisia. Since 2003, he is holding a research position at Automatic Control Unit, National School of Engineers of Sfax, Tunisia. His research interests include robust control, optimal control, fuzzy logic, linear matrix inequalities, and applications of these techniques to agriculture systems.

Mohamed Chaabane received his PhD in Electrical Engineering from the University of Nancy, France in 1991. Currently, he is a Professor in National School of Engineers of Sfax and Editor-in-Chief of the International Journal on Sciences and Techniques of Automatic Control and Computer Engineering (IJSTA). His research interests include robust control, delay systems and descriptor systems.

Larbi Chrifi-Alaoui received his PhD in Automatic Control from the Ecole Centrale de Lyon. Since 1999, he has a teaching position in automatic control in Aisne University Institute of Technology, UPJV, Cuffies-Soissons, France. His research interests are mainly related to linear and nonlinear control theory including sliding mode control, adaptive control, robust control, with applications to electric drive and mechatronics systems.

Maha Bouattour received his Master in Electrical Engineering from the University of Sfax, Tunisia, in 2005 and PhD in Electrical Engineering from the University of Picardie Jules Verne, France, and the from the University of Sfax, in 2010. She is an Associate Professor with the Laboratory of Sciences and Techniques of Automatic and Computer Engineering Lab-STA, National Engineering School of Sfax ENIS, University of Sfax. Her current research interests include fuzzy control and fault tolerant control.

46 H. Ben Zina et al.

1 Introduction

Thanks to their reliability, robustness and low cost; the induction motor becomes the most popular electromechanical actuator in the industrial applications such as in the electric drive traction (Ferranti et al., 1993). However, the control of the induction motor is complex due to its nonlinear dynamic. Several control strategies have been proposed to control the induction motor. Field-oriented control (FOC) is considered to be the most popular of these strategies (Palma and Dente, 1992; Blaschke, 1972). The advantage of this control technique is that it guarantees that the torque and flux controls are decoupled and it is easily implemented in induction motor drive.

In the last years, tracking control for dynamical system has attracted many efforts (Ren and Beard, 2004). This control approach has been successfully applied to induction motor. Marino et al. (1993, 1998) presented a direct adaptive controller for speed regulation in which the motor model is input-output decoupled by a feedback-linearising technique. In Cauet et al. (2001), a H2/H∞ approach with a reference model is used to ensure the tracking performance and the load torque rejection. Wai and Chang (2002) presented an adaptive system with an inverse rotor time-constant observer, based on a model reference adaptive system (MRAS) theory. Abdelhalek and Bachir (2015) propose a high gain observer-based nonlinear control for induction motor. Sliding mode control technique have been integrated in tracking control (Fu and Xie, 2005; Jamousi et al., 2013; Salih et al., 2015), but the chattering phenomenon can degrade the control performances.

The success of the previous methods depends on the model complexity of induction motor. Fortunately, Takagi-Sugeno (T-S) technique is considered as an efficient way to represent complex system. This technique decompose the model of the nonlinear system in a series of linear models involving nonlinear weighting functions. The equivalent fuzzy model describes the dynamic behaviour of the system (Tanaka and Sugeno, 1985). This approach has been successfully integrated in nonlinear system modelling and control (Jeung and Lee, 2014; Lee et al., 2015; Tseng et al., 2001).

In the last years, fuzzy techniques have been successfully extended in induction motor modelling and control (Hammoudi et al., 2015; Gunabalan and Subbiah, 2015). Allouche et al. (2013) present a fuzzy tracking control schemes for induction motor. The main aim of the presented control is to reject the externals disturbances and to achieve tracking performances. The developed design method uses two steps for resolving the stability problem.

In order to benefit from some recent advances in observer-based tracking control developed by the control community, in our works we are interested to the synthesis of tracking control law for induction motor drive which achieve the H∞ performance. This control technique can guarantee the FOC performances. The main result of this work is to simplify the design method of the tracking control of induction motor drive. With more accurately the controller and the observer gains are given on a single step and the stability conditions became less conservative. A fuzzy augmented system containing the tracking error the estimation error and the reference state is constructed. Then a fuzzy observer-based tracking controller is designed. Using the Lyapunov theory the stability condition of the observer-based H∞ tracking control are given in terms of LMIs and can be resolved on a single one step.


This paper is organised as follow: Section 2 introduce an open-loop control strategy. A fuzzy observer-based tracking control is considered in Section 3. Finally, experimental results are provided to demonstrate the design effectiveness.

Notation: the symbol (*) denotes the transpose elements in the symmetric positions.

2 Open loop control

2.1 Physical model of induction motor

By using the assumptions of the linearity of the magnetic circuit, the dynamic model of the induction motor in the synchronous d-q reference frame can be described as

( ) ( ( )) ( ( )) ( ) ( )x t f x t g x t u t w t= + + (1)

where

[ ]( ) Ψ Ψ Tsd sq rd rq mx t i i ω=

( )

( )

( )

1( ( ))

1

ssd s sq rd s p m rq

r

ss sd sq s p m rd rq

r

sd rd s p m rqr r

sq s p m rd rqr r

prd sq rq sd m

r

Ki ω i ψ K n ω ψT

Kω i i K n ω ψ ψT

M i ψ ω n ω ψf x tT TM i ω n ω ψ ψT T

n M fψ i ψ i ωJL J

⎡ ⎤− + + +⎢ ⎥⎢ ⎥⎢ ⎥− − + +⎢ ⎥⎢ ⎥⎢ ⎥− + −= ⎢ ⎥⎢ ⎥⎢ ⎥− − −⎢ ⎥⎢ ⎥⎢ ⎥− −⎢ ⎥⎣ ⎦

α

α

1 0 0 0 0( ( ))

10 0 0 0

T

s

s

σLg x t

σL

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

[ ]( ) Tsq sdu t u u=

( ) 0 0 0 0T

rCw tJ

⎡ ⎤= −⎢ ⎥⎣ ⎦

21

s r

MσL L

= −

,s rs r

s r

L Lτ τR R

= =


ss r

MKσL L

=

1 1s r

σστ στ

−⎛ ⎞= +⎜ ⎟⎝ ⎠

α

where ωm is the rotor speed, ωs is the electrical speed of stator, (ψrd, ψrq) are the rotor fluxes, (isd, isq) are the stator currents, and (usd, usq) are the stator voltages. The load torque Cr is considered as an unknown disturbance. The motor parameters are moment of inertia J, stator and rotor resistances (Rs, Rr) stator and rotor inductances (Ls, Lr), mutual inductance M, friction coefficient f and number of pole pairs np.

2.2 Open loop control

In this section, we present the open loop control strategy. If we replace the state variables of the IM x(t) = [isd isq Ψrd Ψrq ωm]T by the corresponding reference [isdc isqc ψrdc 0 ωmc] in equation (1), we obtain

( )

( )

1

0

1

1

ssdc sdc s sqc rdc sdc

r s

ssqc s sdc sqc s p mc rdc rqc

r

s p mc rdc sqcr

rdc sqc rdcr r

pmc rdc sqc mc r

r

d Ki i ω i ψ udt τ σLd Ki ω i i K n ω ψ ψdt τ

Mω n w ψ iτ

d Mψ i ψdt τ τ

n Md fω ψ i ω Cdt JL J J

⎧ = − + + +⎪⎪⎪ = − − − +⎪⎪⎪ = − − −⎨⎪⎪

= −⎪⎪⎪

= − −⎪⎩

α

α

(2)

The open-loop reference of the stator current and electrical speed can be written as follows:

rdc rsdc rdc

r rsqc mc mc

p rdc

ψ τ di ψM M dt

JL C f di ω ωn Mψ J J dt

⎧ = +⎪⎪⎨ ⎛ ⎞⎪ = + +⎜ ⎟⎪ ⎝ ⎠⎩

(3)

sc p mc sqcr rdc

Mω n ω iτ ψ

= + (4)

ssdc s sdc sdc sc sqc rdc

r

sqc s sqc sqc sc sdc s p mc rdc

d Ku σL i i ω i ψdt τdu σL i i ω i K n ω ψdt

⎧ ⎛ ⎞= + − −⎜ ⎟⎪⎪ ⎝ ⎠⎨

⎛ ⎞⎪ = + + +⎜ ⎟⎪ ⎝ ⎠⎩

α

α (5)


3 Observer-based fuzzy control design

3.1 T-S fuzzy induction motor model

When the field-oriented strategy is implemented, we can ensure independent control of the torque and of the rotor flux. Then the dynamics of the induction motor is similar to the separately excited DC motor. Indeed, the rotor flux vector (ψrd, ψrq) is aligned to the d-axis and we can obtain

0rd rdc

rq

ψ ψψ

=⎧⎨ =⎩

(6)

To ensure the performances of the FOC, the electrical speed of the stator in the rotating synchronous d-q frame must be chosen as

s p m sqr rd

Mω n ω iτ ψ

= + (7)

If we replace the electrical speed of the stator (7) in the physical model (1), the nonlinear model of the induction motor can be expressed as

( ) ( ) ( ) ( )( ) ( )

x t Ax t Bu t w ty t Cx t

= + +⎧⎨ =⎩

(8)

where

0

0

10 0

10 0

0 0

ss s p m

r

ss s p m

r

sqr r r rdc

sqr r rdc r

p psq sd

r r

Kω K n ωτ

Kω K n ωτ

M M iAτ τ τ ψ

M M iτ τ ψ τ

n M n M fi iJL JL J

⎡ ⎤−⎢ ⎥⎢ ⎥⎢ ⎥− − −⎢ ⎥⎢ ⎥⎢ ⎥−= ⎢ ⎥⎢ ⎥⎢ ⎥− −⎢ ⎥⎢ ⎥⎢ ⎥

− − −⎢ ⎥⎣ ⎦

α

α

[ ]( ) Ψ Ψ Tsd sq rd rq mx t i i ω=

1 0 0 0 0

10 0 0 0

T

s

s

σLB

σL

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦


1 0 0 0 00 1 0 0 00 0 0 0 1

C⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

The fuzzy model can be constructing using the well-known sector nonlinearity technique. The system (8) contains the following three nonlinearities:

1

2

3

( ) ( )( ) ( )( ) ( )

sd

sq

m

z t i tz t i tz t ω t

=⎧⎪ =⎨⎪ =⎩

(9)

Thus we can transform the nonlinear terms under the following shape:

1 max 2 min( ) ( ) ( ) ; {1, 2, 3}j j j j jz t F t z F t z j= + = (10)

where

min1

max min

max2

max min

( )( )

( )( )

j jj

j j

j jj

j j

z t zF t

z zz z t

F tz z

−⎧ =⎪ −⎪⎨ −⎪ =⎪ −⎩

(11)

The fuzzy model described by fuzzy rules if-then will be used to deal the control design problem for the induction motor. The ith rule of the fuzzy model for the induction motor is of the following form:

3.1.1 Rule Ri

If (z1(t) is Fi1) and (z2(t) is Fi2) and (z3(t) is Fi3) then ( ) ( ) ( ) ( ),ix t A x t Bu t w t= + + i = 1, 2, …, 8.

The global fuzzy model can be written in the following form:

( )8

1

( ) ( ( )) ( ) ( ) ( )i ii

x t h z t A x t Bu t w t=

= + +∑ (12)

where

8

1

( ( ))( ( ))( ( ))

ii

ii

μ z th z tμ z t

=

=

∑ (13)

( )3

1

( ( )) ( )i ik kk

μ z t F z t=

=∏ (14)

( ( )) 0ih z t >

and


8

1

( ( )) 1ii

h z t=

=∑

3.2 Reference model

In order to specify the desired trajectory, we consider the following reference model

( ) ( ) ( )r r rx t A x t r t= + (15)

where xr(t) = [isdr isqr Ψrdr Ψrqr ωmr]T is the reference state of the closed loop system.

1 2

1

0

0

10 0

10 0

0 0

sr s p mr

ssr s p m

r

sqrr r r r rdc

sqrr r rdc r

p psqr sdr

r r

ζ ω ζ K n ωKω ζ K n ωτ

M M iA τ τ τ ψ

M M iτ τ ψ τ

n M n M fi iJL JL J

−⎡ ⎤⎢ ⎥⎢ ⎥− − −⎢ ⎥⎢ ⎥⎢ ⎥−

= ⎢ ⎥⎢ ⎥⎢ ⎥− −⎢ ⎥⎢ ⎥⎢ ⎥− − −⎢ ⎥⎣ ⎦

ζ1 = (α + KΨ), 2fs

r s r

MKKζτ L τ

⎛ ⎞= −⎜ ⎟⎝ ⎠

and KΨ and Kf are positive constants introduced to

improve the dynamic of the IM. r(t) is a bounded reference input given as:

[ ]( )( )rU

r t B Iw t⎡ ⎤

= ⎢ ⎥⎣ ⎦

(16)

where Ur = [usdr usqr]T

( )

( )

Ψand

Ψ

ψ s ssdr s sdc i sdc sc sqc rdc

R s

sqr s sqc f sqc sc sdc s p mc rdc

MK L KdU σL i K i ω idt τ L

dU σL i K i ω i K n ωdt

⎧ ⎡ − ⎤⎛ ⎞= + + − +⎪ ⎢ ⎥⎜ ⎟⎪ ⎝ ⎠⎣ ⎦⎨⎡ ⎤⎪ = + + + +⎢ ⎥⎪ ⎣ ⎦⎩

α

α

To attenuate the external disturbances, we consider the H∞ performances related to the tracking error xr(t) – x(t) as follow:

[ ] [ ]( ) ( )20 0

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )f ft tT T T

r rx t x t x t x t γ r t r t w t w t dt− − ≤ +∫ ∫ (17)

For this, the following fuzzy observer is constructed to estimate the immeasurable state of the induction motor.


( )( )8

1

ˆ ˆ( ( )) ( ) ( ) ( ) ( )

ˆ ˆ( ) ( )

i i ii

x h z t A x t Bu t L y t y t

y t Cx t=

⎧= + + −⎪

⎨⎪ =⎩

∑ (18)

The observer error is define as

ˆ( ) ( ) ( )e t x t x t= − (19)

From equations (12) and (18), we obtain

( )[ ]8

1

( ) ( ( )) ( ) ( )i i ii

e t h z t A L C e t w t=

= − +∑ (20)

The structure of the fuzzy tracking control is defined as follow.

( )8

1

ˆ( ) ( ( )) ( ) ( )i i ri

u t h z t K x t x t=

= −∑ (21)

Figure 1 Control strategy (see online version for colours)

The controller design methodology is illustrated by the scheme illustrated in Figure 1. After manipulation, an augmented system can be expressed in the following form:

( )8 8

1 1

( ) ( ( )) ( ) ( ) ( )i j r iji j

x t h z t h z t A x t F t= =

⎡ ⎤= +⎣ ⎦∑∑ φ (22)

where

( )( )

( ) ( ) , 0 , ( )( )

( ) 0

t

r

e t I Iw t

x t e t F I tr t

x t I

−⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥= = = ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

φ


0 00 0

i j j i rj

ij i i

rj

A BK BK A AA A L C

A

+ − −⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦

Then, the H∞ performances (17) related to the tracking error can be modified as follow:

20 0

( )f ft t

Tx xdt γ t dt≤∫ ∫ φ (23)

Now, the objective is to determine the controller and observer gains Ki and Li for the augmented model (22) with the guaranteed tracking performance (23) for all ( ).tφ

Theorem 1: For a given positive scalar μ, the closed loop fuzzy system in equation (22) is asymptotically stable and the H∞ performance is guaranteed with an attenuation level γ, if there exists some matrices 1 1 0,TX X= > 2 2 0,TP P= > 3 3 0,TP P= > Yj, Ji and the scalar γ satisfy the following LMI min γ such that

11 1

1

66 2

77 3

Ω 0 0 0 02 0 0 0 0 0 0 0

2 0 0 0 0 0 02 0 0 0 0 0

2 0 0 0 00

Ω 0 0 0Ω 0 0

0 00

j i rjBY A A I I XμX μI

μI μIμI μI

μI μIP

PγI

γII

− − −⎡ ⎤⎢ ⎥∗ −⎢ ⎥⎢ ⎥∗ ∗ −⎢ ⎥∗ ∗ ∗ −⎢ ⎥

⎢ ⎥∗ ∗ ∗ ∗ −⎢ ⎥ <∗ ∗ ∗ ∗ ∗⎢ ⎥

⎢ ⎥∗ ∗ ∗ ∗ ∗ ∗⎢ ⎥∗ ∗ ∗ ∗ ∗ ∗ ∗ −⎢ ⎥

⎢ ⎥∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −⎢ ⎥⎢ ⎥∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −⎣ ⎦

(24)

where 1 1 1

11 2 1, ,j j i iK Y X L P J P X− − −= = =

( )11 1 1 1Ω TTj jiA X X A BY BY= + + +

( )66 2 2Ω TTi i iiP A A P J C J C= + − −

77 3 3Ω Trj rjP A A P= +

Proof: Consider the following candidate Lyapunov function for the augmented system (22)

( )( ) ( ) ( )Tv x t x t Px t= (25)

To achieve the performance required by equation (23) and the closed loop stability of equation (22) the following inequality must hold

( ) 2( ) ( ) ( ) ( ) ( ) 0TV x t x t x t γ t t+ − <φ φ (26)


The time derivative of the candidate Lyapunov function is:

( ) ( ) ()

8 8

1 1

( ) ( ( )) ( )

Φ Φ 0

T Ti j r ijij

i j

T T Ti i

V x t h z t h z t x A P PA I x

x PF F Px= =

⎡ ⎤= + +⎣ ⎦

+ + <

∑∑ (27)

After we substitute ( )V x from equation (27) in equation (26), we obtain:

8 8

21 1

0Φ Φ

T Tij iij

Tii j

x xA P PA PFF P γ I= =

⎛ ⎞⎡ ⎤+⎡ ⎤ ⎡ ⎤⎜ ⎟ <⎢ ⎥⎢ ⎥ ⎢ ⎥⎜ ⎟−⎣ ⎦ ⎣ ⎦⎣ ⎦⎝ ⎠∑∑ (28)

Equation (28) is satisfied if the following condition holds

20

Tij iij

Ti

A P PA I PFF P γ I

⎡ ⎤+ +<⎢ ⎥−⎣ ⎦

(29)

P is structured as follows:

1

2

3

0 00 00 0

PP P

P

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

(30)

Then after manipulation and using Ji = P2Li the inequality (29) can be written as follows:

( )11 1 1 1 1

22 2

33 32

2

0 0Θ 00 0

0

j i rj

ij

P BK P A A P PP

Pγ I

γ I

⎡ϒ − − − ⎤⎢ ⎥∗ ϒ⎢ ⎥⎢ ⎥= <∗ ϒ⎢ ⎥∗ ∗ ∗ −⎢ ⎥

⎢ ⎥∗ ∗ ∗ ∗ −⎣ ⎦

(31)

where

( )11 1 1 1 1TT

i j jiP A A P P BK BK P Iϒ = + + + +

( )22 2 2TT

i i iiP A A P J C J Cϒ = + − −

33 3 3T

rj rjP A A Pϒ = +

The condition (31) contains nonlinear terms. Now, the goal is to formulate it as an LMI problem.

Hence, Θij can be partitioned as

11 12

22

Θ ΘΘ

Θij⎡ ⎤

= ⎢ ⎥∗⎣ ⎦ (32)

Θ11 = ϒ11, Θ12 = [–P1BKj P1(Ai – Arj) P1 – P1] and Θ22 is the lower right block of equation (31).


To effect the necessary change of variable the congruence lemma is required. Pre and post multiplying equation (32) by

11

11

0 0 00 0 0 0

where0 0 0 0

0 0 0

PP I

Q XX I

I

−

−

⎡ ⎤⎢ ⎥⎡ ⎤ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎢ ⎥⎣ ⎦

Then, it follows that

1 1 111 121 1 1

22

Θ Θ0

ΘP P P X

X X

− − −⎡ ⎤<⎢ ⎥∗⎣ ⎦

(33)

Equation (33) implied that XΘ22X is negative then we obtain

2 122 22Θ 2 ΘX X μX μ −≤ − − (34)

By substituting equation (34) into equation (33) and by using the Schur complement, we obtain

1 1 111 121 1 1

22

Θ Θ 02 0

Θ

P P P XμX μIμI

− − −⎡ ⎤⎢ ⎥∗ − <⎢ ⎥⎢ ⎥∗⎣ ⎦

(35)

After substituting Θ11, Θ12, Θ22 in equation (35) the LMI presented in Theorem 1 is obtained.

4 Experimental results

In order to verify the performances of the proposed method, the experimental testes were carried out using the test benchmark, of Laboratory of Innovative Technology (LTI), University of Picardie Jules Verne at Cuffies France, presented in Figure 2. The IM stator is fed by a SEMIKRON converter (4 KW, IGBT modules) controlled by the dsp 1104 board. The dspace board receives the measured current and the actual position through the current transducer board LA-55NP and a 5,000 points incremental encoder. The interface is used to provide galvanic isolation to all signals connected to the dspace controller. The induction motor is characterised by the parameters described in Table 1.

The performances of the developed approach are tested for different form of speed:

1 step with 90 rad/s value and a load of 3 N·m at t = 10 s.

2 step change from 100 rad/s to 120 rad/s.


Table 1 Induction motor parameters

Description Parameter Value Units

Stator resistance Rs 5.72 Ω Rotor resistance Rr 4.2 Ω Stator inductance Ls 0.462 H Rotor inductance Lr 0.462 H Mutual inductance M 0.4402 H Moment of inertia J 0.0049 kg·m2 No. of pairs pole np 2 -

Figure 2 The test benchmark (see online version for colours)

Figure 3 Rotor speed (see online version for colours)


Figure 4 d-axis stator current (see online version for colours)

Figure 5 q-axis stator current (see online version for colours)

Figure 6 d-axis rotor flux (see online version for colours)


Figure 7 q-axis rotor flux (see online version for colours)

Figure 8 Rotor speed (see online version for colours)

Figure 9 d-axis stator current (see online version for colours)


Figure 10 d-axis rotor flux (see online version for colours)

Figure 11 q-axis rotor flux (see online version for colours)

The simulation results illustrated in Figures 3–11 show the convergence of the estimated rotor speed toward the real speed and then to the desired trajectory with a small tracking error in the two case of reference speed in spite of the load torque being applied and in spite the change of speed. Figures 4 and 9 show the d-axis stator current, the q-axis stator current is illustrated in Figure 5, whereas Figures 6 and 10 represent the d-axis rotor flux. Furthermore the q-axis rotor flux is illustrated in Figures 7 and 11. From these results, we can see that when the rotor speed reference change, the response of the IM currents and flux undergo a weak fluctuation and remain near its reference value and the decoupling control characteristic is achieved. Then, we can conclude that the experimental results confirm the effectiveness of the proposed fuzzy state feedback controller and the performances of the control scheme.


5 Conclusions

In this work, a fuzzy tracking control has been designed for the field oriented induction motor drive with external disturbance. The T-S fuzzy model is used to represent the induction motor in the synchronous rotating d-q frame. A fuzzy observer that based on a fuzzy controller is developed to guarantee H∞ tracking performance. The stability condition has been proposed in terms of LMIs, which offers more degree of freedom in guaranteeing the tracking performance. Finally, Experimental results demonstrated the effectiveness of the proposed fuzzy controller.

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A Takagi-Sugeno fuzzy control of induction motor …...A Takagi-Sugeno fuzzy control of induction motor drive: experimental results 49 3 Observer-based fuzzy control design 3.1 T-S