Volume II, Issue VII, July 2013 IJLTEMAS ISSN 2278 - 2540
www.ijltemas.in Page 119
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
Volume II, Issue VII, July 2013 IJLTEMAS ISSN 2278 - 2540
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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
Volume II, Issue VII, July 2013 IJLTEMAS ISSN 2278 - 2540
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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)
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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 )
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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)
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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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