adaptive fuzzy sliding mode controller for indirect vector

Volume II, Issue VII, July 2013 IJLTEMAS ISSN 2278 - 2540

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Adaptive Fuzzy Sliding Mode Controller for Indirect

Vector Control of Induction Motor Drive

Barkha Rajpurohit, Arti Gosain Anil Kumar Chaudhary

Department of Electrical Engineering Assistant Professor

Mandsaur Institute of Technology Dept. of Electrical Engineering

Madhya Pradesh, India Mandsaur Institute of Technology

[email protected] [email protected]

Abstract— In this paper a fuzzy sliding mode control is proposed

for speed control of indirect field-oriented induction motor drive.

First a indirect field-oriented control introduced briefly. Then a

sliding mode control is investigated. The proposed control design

uses a fuzzy logic technique for implementing a fuzzy hitting

control law to remove completely the chattering phenomenon on

a conventional sliding mode control. Here to adjust the

fuzzy parameter for further assuring robust and optimal control

performance, an adaptive algorithm which is derived in the

sense of Lyapunov stability theorem is utilized. The

proposed fuzzy sliding-mode controller is compared with

sliding mode controller with external load perturbation using

periodic speed command. The simulation results shows that fuzzy

sliding mode controller is robust for tracking the periodic

command free from chattering.

Keywords-Indirect vector control;sliding mode controlg;fuzzy

sliding mode control; speed control;induction motror

I. INTRODUCTION

Sliding mode controller (SMC) is one of the effective ways

for controlling electric drive system. It is a robust control

because the high-gain feedback control input cancels non-

linearities, parameter uncertainties and external disturbance. It

also offers a fast dynamic response and a stable control system

[4].The first step of SMC design to select a sliding surface that

models the desired closed-loop performance in state variable

space. In the second step, design a hitting control law such that

the system state trajectories are forced toward the sliding

surface and stay on it. The system state trajectory in the period

of time before reaching the sliding surface is called the

reaching phase. Once the system trajectory reaches the sliding

surface, it stays on it and slides along it to the origin. The

system trajectory sliding along the sliding surface to the origin

is the sliding mode. However this control strategy produces

some drawbacks associated with large control chattering that

may wear coupled mechanisms and excite unstable system

dynamics. Though introducing a boundary layer may reduce

the chatter amplitude, the stability inside the boundary layer cannot be guaranteed and poor selection of boundary layer will

result in unstable tracking responses[2].In order to remedy this

phenomenon an fuzzy sliding mode control is introduced in

which a fuzzy hitting control law is embedded into SMC

system to the sliding surface and an adaptive algorithm derived

in the sense of the Lyapunov stability theorem is utilized to

adjust the fuzzy parameter. This method can leads to stable

close loop system with avoiding chattering problem.

This paper presents a adaptive fuzzy sliding-mode control

scheme (AFSMC).A indirect vector control is reported in

section-II.A fuzzy sliding-mode control is discussed in section-

III.Test results are discussed in Section-IV and finally some

concluding remarks are stated in section-V.

II. INDIRECT FIELD-ORIENTED INDUCTION MOTOR DRIVE

The block diagram of an indirect field-oriented induction

motor drive is shown in fig.1.Here the induction motor is fed

by a hysteresis current controlled pulse width modulated

(PWM) inverter.

1

Lm

Lm Rr

ψ r Rr

Fig.1.Indirect vector controlled Induction Motor Dive

The torque component of current iqs* is generated by speed

error with the help of PI or any intelligent controller. The flux

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= K

component of current ids* is obtained from the desired rotor ∧

fluxψ r is determined from the following equation,

∧

investigated in this paper to enhance the robustness of the IM

drive for high performance application.

ψ r = L m i d s (1) Now assume, the parameters without external load disturbance,

The slip frequency ωsl* is generated by the current iqs* is

determined from the equation,

rewriting (8) represents the nominal model of the IM drive

system

&

ω = Lm Rr i

X ( t ) = A p n ω r ( t ) + B p n U ( t ) (9) sl ∧ qs

ψ Lr (2)

Where

− −

A p n = − B / J and

− −

B p n = K t /

J

are the nominal

The slip speed signal ωsl* added with feedback rotor speed

signal ωr to generate frequency signal ωe.The slip speed together with the rotor speed is integrated to obtain the stator

reference space vector position θe.

values of Ap and Bp . By considering parameter variations and

external load disturbance, the equation (9) can be modified as

X& (t) = (Apn + ∆A)ωr (t) + (Bpn + ∆B)U(t) + CpTL

θe = ∫ ωe dt = ∫ (ωr + ωsl ) (3)

= ( Apn )ωr (t ) + (Bpn )U (t ) + L(t ) (10)

The vector rotator converts the two phase d-q axis reference .

currents iqs*and ids*to three phase currents ia*,ib*,ic*.The

reference currents are compared with the actual currents ia,ib,ic

from induction motor. The currents errors are fed to the

hysteresis current controller. The hysteresis current controller

allows the induction motor currents to vary with in a hysteresis

band such that the required performance of the machine is

obtained.

The mechanical equation of an IM drive system can be

represented as

Where A and B denote the uncertainties due to system

parameters J and B,U(t) is the speed command, ωr is the

feedback rotor speed. L(t) is the lumped uncertainty and

defined as

L (t ) = ∆Aωr (t ) + ∆BU (t ) + C pTL (11)

III. DESIGN OF ADAPTIVE FUZZY SLIDING MODE

CONTROLLER FOR INDUCTION MOTOR DRIVE

The overall scheme of sliding mode controller (SMC) is

shown in fig.2 below, in which a simplified indirect field- J ω& r (t)+ Bωr (t) + TL = Te (4)

Where ωr is the rotor speed, J is the moment of inertia, B is the

damping coefficient and TL is the external load disturbance.Te

denotes electromagnetic torque is given by

∗

oriented IM drive is used to represent the real controlled plant

[2].

Te = Kt iqs

Where, Kt is the torque constant is defined as

(5)

3n p L2

m *

t

2

L

i ds

r (6)

Substituting equation (5) into equation (4) The mechanical

equation of an IM drive system can be represented as

1

JS + B

ω& r ( t ) = − B

ω J

r

( t ) + K t i*

J q s

( t ) − 1

T J

L

(7)

X& ( t ) = A p ω r ( t ) + B p U ( t ) + C p

TL

(8)

Where x(t) = ωr (t), Ap = -B/J, Bp = Kt / J, Cp =-1/J, U(t) =

i*qs is the control effort. The system uncertainties

including parameter variations, external load disturbance

influence the IM seriously, though the dynamic behaviour of

IM is like that of separately excited motor. Therefore a

SMC system is

Fig.2. Block Diagram of AFSMC



The control aim to design a suitable control law so that the

motor speed ωr can track desired speed commands ωr*.In

sliding mode control, the system is controlled in such a way

that the tracking error, „e‟ and rate of change of error „ e& ‟ always move towards a sliding surface. The sliding surface is

defined in the state space by scalar equation

Where γ is the width of the boundary layer. Stability inside the

layer cannot be ensured and the inadequate selection of the

boundary layer may result in unstable tracking response.

Therefore a fuzzy sliding mode control system, in which a

fuzzy logic mechanism is used to follow the hitting control

s(e, e&, t ) = 0

Where, the sliding variable,S is

s(t) = e&(t) + λe(t)

(12) (13)

law is used. Here the sliding surface S be the input linguistic variable and

fuzzy hitting control law Uf be the output linguistic variable.

Where λ is a positive constant that depends on the bandwidth of

the system, e(t) = ωr*- ωr is the speed error, in which ωr*is the

reference speed and ωr is the actual speed. Take the

derivative of the sliding surface with respect time and use equation (10), then

s&(t) = &e&(t) + λe& (t) ∗

The proposed controller uses the following variables P (positive),N( negative),Z (zero) for the input variable S

PE (positive Effort),NE (negative Effort),ZE (zero effort) for

the ouput variable Uf.

The rule base involved in the fuzzy sliding mode system is

S& (t) = ω& r (t) − Apn ωr (t) − Bpn U(t) − L(t) + λe& (t)

(14) given as follows

Referring to (14), the control effort being derived as the

solution of s&(t ) = 0 without considering the lumped uncertainty (L(t)=0) is to achieve the desired performance under nominal

model and it is referred to as equivalent control effort as

follows

Rule1:If S is P,then Uf is PE.

Rule2: If S is Z thenUf is ZE.

Rule3:If S is N then Uf is NE.

−1 ∗

Ueq (t) = Bpn ω& r (t) − Apn ωr (t) + λe&

(t)

(15)

However ,the indirect vector control is highly parameter

sensitive. Unpredictable parameter variatios,external load

disturbance,unmodelled and nonlinear dynamics adversely

affect the control performance of the drive system. Therefore

the control effort cannot ensure the favourable control

performance. Thus auxiliary control effort should be designed to

eliminate the effect of the unappreciable disturbances. The

auxiliary control effort is referred to as hitting control effort as

follows

(a) (b)

Fig.3.Membership function (a) Input fuzzy sets for S. (b)

Output fuzzy sets for Uf

Then a fuzzy hitting control law can be estimated by fuzzy

logic inference mechanism as follows:

U h (t ) = gh sgn(S (t )) (16)

Where 0 ≤ ω1 ≤1,0 ≤ω2 ≤1,and 0 ≤ω3 ≤1 are the firings

strengths of rules 1,2, and 3; respectively(r1=r),(r1=0)

and(r3=0) are the centre of total membership functions Where gh is a hitting control gain concerned with upper bound of uncertainties, and sgn(.) is a sign function.

Now, totlly sliding mode control law as follows

PE,ZE,and NE respectively, r is a fuzzy parameter. The

relation ω1+ω2+ω3=1 is valid according to the special case of

triangular membership functions. Moreover, the fuzzy hitting U SMC (t ) = U eq (t ) + U h (t ) (17) control effort Uf can be further analysed as the following four

conditions and only four conditions will occur for any value

But this controller gives unacceptable performance due to high

control activity, resulting in chattering of control variable and

system states. To reduce chattering a boundary layer in

generally introduced into SMC law, then the control law of

of S according to fig.3(a)

Condition 1: When rule 1 is triggered (S > Sa ; ω1=1; ω2=

ω3= 0)

equation (17) can be rewritten as U f (t) = r (20)

U h (t ) =

gh S (t )

S (t ) + γ

(18)

Condition 2: When rules 1 and 2 are triggered

simultaneously.(0 <S ≤ Sa;0 < ω1 , ω2 ≤1; ω3= 0)

ω2; ω3<1)

U f (t) = r3ω3 = −rω3 (22)

Condition 4: when rule 3 is triggered (S≤Sb; ω1= ω2=0; ω3=1)



L ( t ) ( ω1 − ω2 B p n )

1 3

U (t) = U (t) + U (t) = U (t) + r(t )(ω − ω )

Choose a Lyapunov candidate function as

2 2

U (t) = −r

(23) s(t) + αBpn r%

(t)

f V (S(t), r%(t)) =

U f (t) = r3ω3 = −rω3 (21)

rewritten as

Condition 3: when rules 2 and 3 are triggered simultaneously ˆ

AFSMC eq ˆ

f eq ˆ

1 3

(29)

(Sb < S ≤ 0; ω1= 0;0 ≤

(30)

From all four possible conditions, it can be seen that b 2

S(t)( ω1- ω3) = S(t) (ω1- ω3) ≥0

Now, total fuzzy sliding mode control (FSMC) law can be

represented as

Where α is a positive constant. Take the derivative of

Vb (S(t), r%(t)) with respect to time, and using (14) and (29)

we obtain

U (t) = U (t) + U

(t) = U (t) + r(ω − ω )

(24) AFSMC eq f eq 1 3

& & &

Define a Lyapunov candidate function as

S(t)2

V(S(t), %r(t)) = S(t)S(t) + αBpn %r(t)r(t)

= −S(t)Bpn r(t)(ω1 − ω3 ) − S(t)L(t) + αBpn %r(t)r&(t)

Va (t) = 2

(25)

Take the derivative of lyapunov function with respect to time =−S(t)B r(t)(ω −ω )+

L(t) +r*(ω −ω )−r*(ω −ω ) +αB r%(t)&r(t)

and using (14 ) and (24), it is obtained

pn 1 3 1 3 1 3 pn

Bpn

V& a (t) = S(t)S& (t) = −S(t)Bpn r(ω1 − ω3 ) − S(t)L(t) = −S(t)Bpn r(t)(ω1 − ω3 ) − S(t)

= −Bpn r S(t)

ω1 − ω3 − S(t)L(t)

×Bpn (ω1 − ω3 )

L(t)

+ r*

+ αBpn r%(t)&r(t)

(31)

≤ −Bpn r S(t)

ω1 − ω3 + S(t) L(t) Bpn (ω1 − ω3 )

L(t)

If the adaption law is designed as

= −Bpn S(t) r ω1 − ω3 − B

r(t) =

S(t)(ω1 − ω3 )

(32)

& pn

α

If the following inequality

r >

(26)

Then (31) can be represented as

L ( t ) *

V& b (S ( t ), r% ( t )) = − S ( t ) B p n ( ω1 − ω 3 ) ×

B ( ω − ω ) + r

holds, then sliding condition V& a (t) = S(t)S& (t) ≤ 0

can be p n 1 3

satisfied. According to() there exists an optimal value r*

as

follows to achieve minimum control efforts and match the

≤ S( t )(ω − ω ) L ( t )

− S( t ) B

(ω − ω

) r *

sliding condition: 1 3

(ω − ω ) p n 1 3

= −S(t)Bpn (ω1 − ω3 ) ε (32)

r* =

L(t) + ε

(27) According to the inequality S(t)(ω1 − ω3 ) ≥ 0 ,it is obtained

( ω1 − ω3 Bpn )



Refernce Speed

Actual Speed

Reference Speed

Actual Speed

Speed

(Rad

) Speed(R

ad)

Torq

ue(

N-m

)

40

Because Vb (S ( 0 ), r% ( 0 )) is bounded, and V b (S ( t ), r% ( t )) is 50

non increasing and bounded, the following results is

obtained t 30

li m ∫ P ( T ) d T < ∞ t → ∞ 0

(34)

20

Also, P ( t ) ≥ 0 is a positive function and P& (t) is bounded for 10

all time, so by Barbalat‟s Lemma ,it can be shown that

li m P ( t ) = 0 t → ∞

according to (34).That is S ( t ) → 0 as 0

t → ∞ ,so moreover the tracking error e(t) will converge to

zero.

IV. SIMULATIONS RESULTS AND DISCUSSION The

induction motor drive system in indirect vector control

mode is simulated in MATLAB environment using power

system block set each with sliding mode controller (SMC) and fuzzy sliding mode control (FSMC).The controller‟s

performance are tested and compared for speed tracking. The

tracking response of speed depicted in fig.4.Here it is observed

that in case of SMC ,the actual speed is not exactly track the reference command speed, it has some error where as in case of

FSMC it has been exactly track the reference command speed.

The tracking response of torque depicted in fig. 5. From

fig.5(a),it is observed that in case of SMC ,the actual

torque tracks the reference torque, but chattering phenomenon is present. Chattering is occurring due to large

control gain in the hitting control law. However in case of

FSMC, fig.5 (b) chattering is absent and actual torque

smoothly tracks the reference torque. The alpha-beta stator

current for SMC and FSMC is depicted in fig.6.From fig.6,it has been observed that alpha-beta axis current is completely

decoupled in both the case., but in case of SMC fig.6(a)

chattering is present, where as in FSMC fig.6(b) it is

purely decupled without any chattering.

-10

-20

-30

-40

-50

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time(Sec)

(a)

50

40

30

20

10

0

-10

-20

-30

-40

-50

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time(Sec)

(b)

Fig.4.Speed response for periodic command with (a) sliding

mode controller (b) fuzzy sliding mode controller

30

20

10

0

-10

-20

-30

Actual Torque

Referenc e Torque

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time(Sec)



Ac tual Torque

Refernce Torque

S t

a t

o r

A

l p

h a

- B

e t

a

A x

i s

C

u r

r e

n t (

A )

T

orq

ue(N

-m)

S t

a t

o r

A l

p h

a -

B e

t a

A

x i

s

C u

r

r e n

t

30

20

10

0

-10

-20

(a) 30

20

10

0

-10

-20

-30 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time(Sec)

(b)

-30 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Time(sec)

(b)

Fig.5.Torque response for periodic command with

(a)sliding mode controller (b)fuzzy sliding mode controller

30

20

10

0

-10

-20

Fig.6.d-q axis stator current response for periodic command

with (a) sliding mode control(b) fuzzy sliding mode controller

V.CONCLUSIONS

This paper has successfully demonstrated the application of

the proposed adaptive fuzzy sliding mode control system to an indirect field-oriented induction motor drive for tracking

periodic commands. First, the description of the classical

sliding mode controller (SMC) is presented in detail. Then, the

fuzzy logic control is used to mimic the hitting control law to

remove the chattering. Compared with the conventional

sliding mode control system, the fuzzy sliding mode

control system results in robust control performance

without chattering. The chattering free improved

performance of the AFSMC makes it superior to

conventional SMC, and establishes its suitability for the induction motor drive.

-30 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Time(Sec)

(a)

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adaptive fuzzy sliding mode controller for indirect vector

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