1. Publication of Little Lion Scientific R&D, Islamabad PAKISTAN
Journal of Theoretical and Applied Information Technology
15th July 2011. Vol. 29 No.1
© 2005 - 2011 JATIT & LLS. All rights reserved.
ISSN: 1992-8645
www.jatit.org
E-ISSN: 1817-3195
NONLINEAR ADAPTIVE BACKSTEPPING CONTROL OF
PERMANENT MAGNET SYNCHRONOUS MOTOR (PMSM)
1
A. LAGRIOUI, 2H. MAHMOUDI
1,2
Department of Electrical Engineering, Powers Electronics Laboratory
Mohammadia School of Engineering- BP 767 Agdal- Rabat -Morocco
E-mail:
1
lagrioui71@gmail.com , 2 mahmoudi@emi.ac.ma
ABSTRACT
In this paper, a nonlinear adaptive speed controller for a permanent magnet synchronous motor (PMSM)
based on a newly developed adaptive backstepping approach is presented. The exact, input-output feedback
linearization control law is first introduced without any uncertainties in the system. However, in real
applications, the parameter uncertainties such as the stator resistance and the rotor flux linquage and load
torque disturbance have to be considered. In this case, the exact input-output feedback linearization
approach is not very effective, because it is based on the exact cancellation of the nonlinearity. To
compensate the uncertainties and the load torque disturbance, the input-output feedback linearization
approach is first used to compensate the nonlinearities in the nominal system. Then, nonlinear adaptive
backstepping control law and parameter uncertainties, and load torque disturbance adaptation laws, are
derived systematically by using adaptive backstepping technique. The simulation results clearly show that
the proposed adaptive scheme can track the speed reference s the signal generated by a reference model,
successfully, and that scheme is robust to the parameter uncertainties and load torque disturbance.
Keywords: Feedback input-output linearization – Adaptive Backstepping control, Permanent Magnet
Synchronous Motor (PMSM)
cancellations.
In
addition,
the
adaptive
backstepping approach [11][16] is capable of
keeping almost all the robustness properties of the
mismatched
uncertainties.
The
adaptive
backstepping is a systematic and recursive design
methodology for nonlinear feedback control. The
basic idea of backstepping design is to select
recursively some appropriate functions of state
variables as pseudo control inputs for lower
dimension subsystems of overall system. Each
backstepping stage results in a new pseudo control
design, expressed in terms of the pseudo control
designs from preceding design stages. When the
procedure terminates, a feedback design for the true
control input results in achieving of a final
Lyapunov function by an efficient original design
objective. The latter is formed by summing up the
Lyapunov functions associates with each individual
design stage [3][6][9].
In this paper, the backstepping approach is used
to design state feedback non linear control, first
under an assumption of known electrical
parameters, and then with the possibility of
1. INTRODUCTION
Permanent magnet synchronous motors (PMSM)
are widely used in high performance servo
applications due to their high efficiency, high
power density, and large torque to inertia ratio [1],
[2]. However, PMSMs are nonlinear multivariable
dynamic systems and, without speed sensors and
under load and parameter perturbations, it is
difficult to control their speed with high precision,
using conventional control strategies.
Linearization and/or high-frequency switching
based nonlinear speed control techniques, such as
feedback linearization control and sliding mode
control, have been implemented for the PMSM
drives [13]. However, it is more efficient to use a
nonlinear control method that is based on
minimizing a cost function and allows one to
tradeoff between the control accuracy and control
effort.
The adaptive backstepping design offers a choice
of design tools for accommodation of uncertainties
an nonlinearities. And can avoid wasteful
1

2. Publication of Little Lion Scientific R&D, Islamabad PAKISTAN
Journal of Theoretical and Applied Information Technology
15th July 2011. Vol. 29 No.1
© 2005 - 2011 JATIT & LLS. All rights reserved.
ISSN: 1992-8645
www.jatit.org
parametric uncertainly and bounded disturbances
included. The steady state performances of the
backstepping based controller and the problem of
the disturbance rejection are enhanced via the
introduction of an integral action in the controller.
c3 =
did
= a1id + a2 iq Ω + a3 vd
dt
diq
= b1iq + b2 id Ω + b3 Ω + b4 vq
dt
dΩ
= c1Ω + c2 id iq + c3iq + c4TL
dt
The dynamic model of a typical surface-mounted
PMSM can be described in the well known (d-q)
frame through the park transformation as follows
[5][7][11]:
Lq
did
R
v
= − s id + pΩiq + d
dt
Ld
Ld
Ld
dt
=−
REPRESENTATION [11],[12]
Φf
Rs
L
v
id − d pΩid −
pΩ+ d
Lq
Lq
Lq
Ld
⎡id ⎤
⎢ ⎥
The vector [ X ] = ⎢iq ⎥ choice as state vector is
⎢Ω ⎥
⎣ ⎦
(1)
justified by the fact that currents and speed are
measurable and that the control of the instantaneous
torque can be done comfortable via the currents i sd
Where
v d , v q Direct-and quadrature-axis stator voltages
and/or i sq . Such representation of the nonlinear
Direct-and quadrature-axis stator currents
state corresponds to:
Ld , Lq Direct -and quadrature-axis inductance
P
Rs
Φf
Ω
ω
f
TL
J
d[X ]
= F [ X ] + [ G ].[ U ]
dt
[ y] = H [X ]
Number of poles
Stator resistance
(3)
With
Rrotor magnet flux linkage
⎡ x1 ⎤ ⎡id ⎤
⎢ ⎥
[ X ] = ⎢ x 2 ⎥ = ⎢iq ⎥ Three order state vector
⎢ ⎥
⎢ x 3 ⎥ ⎢Ω ⎥
⎣ ⎦ ⎣ ⎦
Viscous friction coefficient
Load torque
Moment of Inertia
a1 = −
Lq
Rs
; a2 = p
;
Ld
Ld
b1 = −
Φf
L
Rs
; b2 = − p d ; b3 = − p
Lq
Lq
Lq
a3 =
1
Ld
Control vector
(5)
⎡ y1 ⎤ ⎡i ⎤
[ y] = ⎢ ⎥ = ⎢ d ⎥
⎣ y 2 ⎦ ⎣Ω ⎦
p.Ω )
(4)
⎡v d ⎤
[U ] = ⎢ ⎥
⎣v q ⎦
Eelectrical rotor speed
Mechanical rotor speed ( ω =
We assume :
b4 =
(2)
3. PMSM NONLINEAR STATE
T
f
dΩ 3p
= (Φ f iq + (Ld −Lq )id iq ) − Ω+ L
dt 2J
J
J
id , iq
3p
1
.Φ f et c 4 = −
2J
J
The (1) equation become:
2. MODEL OF THE PMSM
diq
E-ISSN: 1817-3195
Output vector
⎤
⎡ f1(x) ⎤ ⎡a1.x1 + a2.x2.x3
⎥
⎢ f (x)⎥ = ⎢b .x + b .x .x + b .x
F[ X ] = ⎢ 2 ⎥ ⎢ 1 2 2 1 3 3 3
⎥
⎢ f3(3) ⎥ ⎢c1.x3 + c2.x1.x2+c3.x2 + c4.Cr ⎥
⎣
⎦ ⎣
⎦
1
f
3p
; c1 = −
; c2 =
( Ld − Lq )
Lq
J
2J
2
(6)

3. Publication of Little Lion Scientific R&D, Islamabad PAKISTAN
Journal of Theoretical and Applied Information Technology
15th July 2011. Vol. 29 No.1
© 2005 - 2011 JATIT & LLS. All rights reserved.
ISSN: 1992-8645
www.jatit.org
&
⎡vd ⎤
⎡v1 ⎤ ⎡ y1 ⎤
⎢v ⎥ = ⎢ && ⎥ = [ξ ( x)] + [D( x)].⎢v ⎥
⎣ 2 ⎦ ⎣ y2 ⎦
⎣ q⎦
⎤
⎡ f1 ( x) ⎤ ⎡a1.id + a2 .iq .Ω
⎢
⎥
⎢ f ( x)⎥ = b .i + b .i .Ω + b .Ω
⎢1 q 2 d
⎥ (7)
2
3
⎢
⎥
⎢ f 3 ( x)⎥ ⎢c1.Ω + c2 .id .iq + c3 .iq + c4 .Cr ⎥
⎣
⎦ ⎣
⎦
(12)
Where
⎡ f1 ( x)
⎤
[ξ ( x)] = .⎢
⎥
⎣c 2 x 2 f1 ( x) + (c 2 x1 + c3 ) f 2 ( x) + c1 f 3 ( x)⎦
0
⎡ a
⎤
[ D( x)] = .⎢ 3
a 3 c 2 x 2 b4 (c 2 x1 + c 3 )⎥
⎣
⎦
Now, the standard pole-assignment technique can
be adopted to design the state feedback control
output as follows:
Nonlinear functions of our model:
⎡ g11 g12 ⎤ ⎡a3 0⎤
[G] = ⎢ g 21 g 22 ⎥ = ⎢0 b4 ⎥ Control matrix (8)
⎥
⎥ ⎢
⎢
⎢ g31 g 32 ⎥ ⎢0 0 ⎥
⎦ ⎣
⎣
⎦
⎡h (x) ⎤ ⎡x1 ⎤ ⎡i ⎤
H[X] = ⎢ 1 ⎥ = ⎢ ⎥ = ⎢ d ⎥ Output vector
⎣h2(x)⎦ ⎣x3⎦ ⎣Ω⎦
E-ISSN: 1817-3195
d
v1 = ki (idref −id ) + idref
dt
(13)
2
d
d
v2 = 2 Ωref + kΩ1. (Ωref −Ω) + kΩ2.(Ωref −Ω)
dt
dt
Where the idref and Ωref are the commands for the
(9)
The model set up in (2) can be represented by the
following functional diagram (Figure1)
speed and the d-axis current, respectively.
Figure 1: Nonlinear diagram block of PMSM in
d-q frame.
4.
INPUT-OUTPUT FEEDBACK
LINEARIZATION
Figure2 : simulation scheme of input-output feedback
linearization control
The input-output feedback linearization control for
a PMSM is introduced first. The control objective is
to make the mechanical speed follow the desired
speed asymptotically.
5. NONLINEAR BACKSTEPPING
CONTROLLER
In order to make the actual speed follow The
desired one and avoid the zero dynamic, w and isd
are chosen as the control outputs. Then the new
variables are defined as follows:
5.1 Non adaptive case
It is assumed that the engine parameters are known
and invariant.
y1 = id
equation (1) the mathematical model of the
machine. The objective is to regulate the speed to
its reference value . We begin by defining the
tracking errors:
By choosing
(10)
y2 = Ω
Then the new state equations in the new state
coordinates can be written as follows:
.
[i
d
]
T
iq Ω as variable states and
ew = Ω ref − Ω
ed = idref − id with idref = 0
.
y1 = id = f1 ( x) + a3 .vd
&
&
(11)
y 2 = Ω = f 3 ( x)
&&
&&2 = Ω = c1 f 3 ( x) + c2 iq ( f1 ( x) + a3 vd ) +
y
(14)
eq = eqref − iq
And its dynamics derived from (1):
(c 2 id + c3 )( f 2 ( x) + a 4 .v q )
dew dΩ ref dΩ
dΩ
=
−
=−
dt
dt
dt
dt
To linearize the equations, define the new control
variable as follows:
We replace (1) in (15), we obtain:
3
(15)

4. Publication of Little Lion Scientific R&D, Islamabad PAKISTAN
Journal of Theoretical and Applied Information Technology
15th July 2011. Vol. 29 No.1
© 2005 - 2011 JATIT & LLS. All rights reserved.
ISSN: 1992-8645
www.jatit.org
dew
f
T
3p
= − [(Ld − Lq )idiq + Φf iq ] + Ω+ L
dt
J
J
2J
Its derivative along the trajectories (21) (22) and
(23) is:
(16)
We define the following quadratic function:
1 2
V1 = ew
2
&
&
&
&
V2 = e w e w + e d e d + e q e q
(17)
Lq
⎤
⎡
v
R
3p
ed ⎢kd ed − d + s − Ω iq + (Ld − Lq )ewiq ⎥
Ld Ld
Ld
2J
⎦
⎣
⎡
2(k J − f ) ⎛ 3 pΦ f
⎜
eq +
+ eq ⎢kq eq + w
3 pΦ f ⎜ 2J
⎢
⎝
⎣
&
&
V1 = ewew
T ⎤ (18)
f
⎡ 3p
= ew.⎢− [(Ld − Lq )id iq + Φ f iq ] + Ω+ L ⎥
J
J⎦
⎣ 2J
Utilizing the backstepping design method, we
consider the d-q axes currents components id and iq
as our virtual control elements and specify its
desired behavior, which are called stabilising
function in the backstepping design terminology as
follows:
vq
3p
⎞ 3 pΦ f
(Ld − Lq )ed iq − kwew ⎟ +
ew − +
2J
2J
Lq
⎠
Φf ⎤
Rs
L
iq + Ω d id + Ω ⎥
Lq
Lq
Lq ⎥
⎦
idref = 0
iqref
Where
The expression (25) found above
following control laws:
(19)
vd = kd Ld ed + Rs i d −ΩLq iq +
k w is a positive constant
(20)
dew 3 pΦ f
3p
=
eq + (Ld − Lq )ed iq − kwew (21)
dt
2J
2J
1
Furthermore the dynamics of the stabilizing errors
Park-
SVMPWM
(ed , eq ) can be computed as:
vd
vq
L
de di
di
di
v R
d
= dref − d = − d = − d + s id −Ω q iq (22)
Ld
dt dt dt
dt Ld Ld
=
diqref
−
diq
wref
id
iq
ia
i b PMSM
ic
Park
w
In the controller development in the previous
section, it was assumed that all the system
parameters are known. However, this assumption is
not always true. The flux linkage varies nonlinearly
with the temperature rise and, also, with the
external fields produced by the stator current due to
the nonlinear demagnetization characteristics of the
magnets. The winding resistance may vary due to
heating. In addition, working condition changes
Φf
Rs
L
(23)
iq + Ω d i d + Ω
Lq Lq
Lq
Lq
To analyze the stability of this system we propose
the following Lyapunov function:
1 2
2
2
(e w + e d + e q )
2
Inverter
5.2 Adaptive case
+
V2 =
Equation
(26)
sa
sb
sc
Figure3: Simulation scheme of non-adaptive case
=
dt
dt
dt
3 pΦ f
⎤
2(k w J − f ) ⎡
3p
eq +
( Ld − Lq )ed iq − k w ew ⎥ −
⎢
3 pΦ f ⎣ 2 J
2J
⎦
vq
3 pLd
(Ld − Lq )ewi q
2J
vq =
Substituting (14) and (19) into (16) the dynamics of
the tracking error of the velocity becomes:
deq
requires the
2Lq (kwJ − f ) ⎛ 3 pΦ f
⎞
3p
⎜
⎜ 2J eq + 2J (Ld − Lq )ed iq − kwew ⎟ +
⎟
3 pΦ f
⎝
⎠
(26)
3 pΦ f Lq
ew + Rsiq + ΩLd id + ΩΦf + kq Lqeq
2J
With this choice the derivatives of (24) become:
&
V2 = − k w e w − k d e d − k q e q ≤ 0
(27)
Substituting (19) in (18) the derivative of V1
becomes:
2
&
V1 = − k w ew
(25)
2
2
2
= −k wew − kd ed − kq eq +
Its derivative along the solution of (16), is given by:
2
( fΩ + TL + k w Jew )
=
3 pΦ f
E-ISSN: 1817-3195
(24)
4

5. Publication of Little Lion Scientific R&D, Islamabad PAKISTAN
Journal of Theoretical and Applied Information Technology
15th July 2011. Vol. 29 No.1
© 2005 - 2011 JATIT & LLS. All rights reserved.
ISSN: 1992-8645
www.jatit.org
such as load torque and inertia mismatch inevitably
impose parametric uncertainties in control system
design. Hence, it becomes necessary to account for
all these uncertainties in the design of high
performance controller. We will see how we
efficiently handle these uncertainties through stepby-step adaptive backstepping design and
parameter adaptations.
In ( ) , we do not know exact value of the load
torque TL ; then , it is necessary to estimate them
adaptively, and replace them with their estimates
)
TL .
(
)
)
2
f ω + T L + k ω Je ω
3 pΦ f
So from ( ) and ( ) the following speed error
dynamics can be deduced :
)
Or i sqref =
deω
1
=
dt
J
&
&
&
&
V2 = e w e w + ed ed + eq eq +
1 ~~
1 ~ ~
1 ~ ~
&
&
&
TL TL +
Rs Rs + Φ f Φ f
γ1
(28)
⎤ ~ ⎡1 ~
3p
1
~⎡1 ~ 1
&
&
+ Rs ⎢ Rs + id ed − iqeq ⎥ + Φ f ⎢ Φ f − eωeq −
2J
Ld
Lq
⎣γ 3
⎢γ 2
⎥
⎣
⎦
⎤
(kω J − f ) 2 1
eq − Ωeq ⎥
ˆ
Lq
JΦ f
⎥
⎦
According to the above equation (33) , the
control laws are now designed as:
3 pLd
( Ld − Lq )ew i q
2J
(34)
ˆ
⎞
2Lq (kwJ − f ) ⎛ 3pΦf
3p
⎜
vq =
eq + (Ld − Lq )ediq − kwew ⎟ +
⎟
ˆ
2J
3pΦf ⎜ 2J
⎝
⎠
ˆ
3pΦf Lq
ˆ
ˆ
ew + Rsiq + ΩLdid + ΩΦf + kqLqeq
2J
ˆ
v d = k d Ld ed + Rs i d −ΩLq iq +
Φf
Rs
L
2 ⎛f
⎞~
iq + Ω d id + Ω +
⎜ − kω ⎟TL
(31)
Lq Lq
Lq
Lq 3 pΦ f ⎝ J
⎠
Let us define a new Lyapunov function the closed
loop system as follows:
~
~2 Φ2 ⎞
~2
1⎛ 2
2
2
(32)
⎜ e w + e d + e q + TL + R s + f ⎟
V2 =
2⎜
γ1
γ2
γ3 ⎟
⎠
⎝
+
Consequently (33) is simplified to :
2
2
2
&
V 2 = −k w e w − k d e d − k q e q +
2 feq
e ⎤
~ ⎡ 1 ~ 2k ω eq
&
+ TL ⎢ TL −
+
− ω ⎥+
ˆ
ˆ
J ⎥
3 pΦ f 3 pJΦ f
⎢γ 1
⎦
⎣
eω , ed , eq , the
~
error TL , the resistance
Where the stabilizing errors
~
~
~
rotor magnetic flux linkage estimation error Φ
f
ˆ
= Φ
f
− Φ
γ 2 and γ 3 are
f
(35)
⎤
1
1
~ ⎡1 ~
&
R s ⎢ Rs +
i d ed − i q eq ⎥ +
Ld
Lq
⎢γ 2
⎥
⎣
⎦
ˆ
estimation error R s ( R s = R s − R s and the
~
(Φ
γ3
2
q q
ˆ
Φ f ⎤ ~ ⎡ 1 ~ 2kωeq
2 feq
e ⎤
Rs
L
&
+
− ω⎥
iq + Ω d id + Ω ⎥ + TL ⎢ TL −
ˆ
ˆ
Lq
Lq
Lq ⎥
3 pΦ f 3 pJΦ f J ⎥
⎢γ 1
⎣
⎦
⎦
diq
load torque estimation
2
d d
Lq
⎤
⎡
v
R
3p
ed ⎢kd ed − d + s − Ω iq + (Ld − Lq )ewiq ⎥
2J
Ld Ld
Ld
⎣
⎦
ˆf
⎡
2(k J − f ) ⎛ 3 pΦ
⎜
+ eq ⎢kqeq + w
eq +
ˆ
3 pΦ f ⎜ 2J
⎢
⎝
⎣
ˆ
vq
3p
⎞ 3 pΦ f
(Ld − Lq )ed iq − kwew ⎟ +
ew − +
2J
2J
Lq
⎠
=
−
=
dt
dt
dt
⎤
2(kwJ − f ) ⎡3 pΦ f
3p
eq + (Ld − Lq )ediq − kwew ⎥ −
⎢
3 pΦ f ⎣ 2J
2J
⎦
vq
(33)
= −k e − k e − k e +
⎛ ~ 3 pΦ f
⎞
3p
⎜ − TL +
eq +
( Ld − Lq )ed iq − kω Jeω ⎟
⎜
⎟
2
2
⎝
⎠
diqref
γ2
2
w w
)
~
(29)
where TL = TL − TL
As a result, the d-q currents errors dynamics can be
rewritten as :
Lq
de d
di
v
R
(30)
= − d = − d + s id − Ω
iq
dt
dt
Ld Ld
Ld
deq
E-ISSN: 1817-3195
f
⎤
(k J − f ) 2 1
3p
~ ⎡1 ~
&
Φf ⎢ Φf −
eω eq − ω
e q − Ωe q ⎥
ˆ
2J
Lq
JΦ f
⎢γ 3
⎥
⎣
⎦
) are all included. And γ 1 ,
an adaptation gain. Tracking the
The parameter adaptation laws are then
chosen as :
derivative of (32) and substing (29) , (30) and (31)
into this derivative we can obtain:
5

6. Publication of Little Lion Scientific R&D, Islamabad PAKISTAN
Journal of Theoretical and Applied Information Technology
15th July 2011. Vol. 29 No.1
© 2005 - 2011 JATIT & LLS. All rights reserved.
ISSN: 1992-8645
www.jatit.org
⎡ 2k ω e q
2 feq
e ⎤
~
&
TL = γ 1 ⎢
−
+ ω⎥
ˆ
ˆ
J ⎥
⎢ 3 pΦ f 3 pJΦ f
⎣
⎦
⎡ 1
⎤
1
~
&
Rs = γ 2 ⎢−
id ed +
iq eq ⎥
Lq
⎢ Ld
⎥
⎣
⎦
E-ISSN: 1817-3195
6
(36)
4
Iabc (A )
2
(37)
0
-2
-4
-6
⎡3p
⎤
(k J − f ) 2 1
~
&
eq + Ωeq ⎥ (38)
Φ f = γ 3 ⎢ eω eq + ω
ˆ
Lq
JΦ f
⎢ 2J
⎥
⎣
⎦
Figure3: iabc currents without uncertainties
Then (35) can be rewritten as follows
200
&
V2 = −k e − k e − k e ≤ 0
2
d d
2
q q
(39)
Define the following equation
2
2
2
Q (t ) = k w e w + k d e d + k q e q ≥ 0
(40)
vq v d
SVMPWM
Equation
(34)
Equats (36),
(37) and (38)
id
iq
0.3
0.35
0.4
0
0
0.05
0.1
0.15 time(s) 0.2
0.25
0.3
0.35
0.4
Figure4: speed response without uncertainties
14
ia
Inverter
0.25
50
Tload
T
12
i b PMSM
10
ic
T, TL(N.m )
Park
-1
0.15 time(s) 0.2
0.1
100
Furthermore, by using LaSalle Yoshizawa’s
theorem [11], its can be shown that Q(t) tends to
zero as t →∞, ew, ed and eq will converge to
zero as. Consequently, the proposed controller
is stable and robust, despite the parameter
uncertainties.
sa
sb
sc
0.05
150
w re f , w (ra d / s )
2
w w
0
Park
w
8
6
4
Figure4: simulation scheme of an adaptive case
2
6. SIMULATION RESULTS
0
0
6.1 PMSM parameter’s
0.15 time(s)0.2
0.25
0.3
0.35
0.4
0.18
phid
phiq
0.16
value
0.14
300 v
3000 tr/s to 150 Hz
14.2 N.ms
0.4578 Ω
4
3.34 mH
3.58 mH
0.001469 kg.m2s
0.0003035 Nm/Rad/s
0.171
0.12
phidq(w b)
Maximal voltage of food
Maximal speed
Nominal Torque ;Tn
Rs
Number of pair poles :p
Ld
Lq
The moment of inertia J
Coefficient of friction viscous f
Flux of linquage Фf
0.1
Figure5: Torque response without uncertainties
TABLE I : PARAMETERS OF PMSM
parameter
0.05
0.1
0.08
0.06
0.04
0.02
0
time(s)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
figure6: d-q axis flux without uncertainties
6.2 Results
For a trajectory wref=200 rad/s at 0s, =100rad/s at
t=0.15s, =200rad/s at t=0.25s, the following figures
( 1 to 7) show the performance of the input output
linearization control.
6
0.4

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E-ISSN: 1817-3195
14
12
iq
Id
Iq
12
id
iq
TL
10
10
8
8
Idq(A )
6
6
4
4
2
2
id
0
0
-2
0
0.05
Time(s)
0.15
0.2
0.1
0.25
0.3
0.35
-2
0
0.4
Figure7: d-q axis currents response without
uncertainties
0.1
0.2
0.3
0.4
0.5
0.6
figure10: d-q axis currents with step load and
without uncertainties.
The figures below (figures 8, 9 and 10) shows
the evolution of speed, electromagnetic torque
and d-q axis currents in the presence of a
disturbance load torque.
The figures 10 to 13 show the robustness of
adaptive backstepping controller compared with
that of non-adaptive case and feedback input-output
linearization.
200
200
w
150
100
w (ra d / s )
w (ra d /s ) a n d T L (N . m )
150
50
TL
50
0
-50
time(s)
0
0.05
0.1
0.15
0.2
100
0.25
0.3
0.35
0
0.4
time(s)
0
0.1
0.2
0.3
0.4
0.5
0.6
Figure11: speed response of the proposed adaptive
controller with step load (+12Nm at 0.2s ,
+8Nm at 0.4s)
Figure8: response speed with step load at t= 0.2s
12
iq
10
8
200
6
adaptive control case
non adaptive control case
150
2
S peed
4
id
100
0
50
-2
0
0.1
0.2
0.3
0.4
0.5
0.6
0
0
Figure9: electromagnetic torque with step load
time(s)
0.1
0.2
0.3
0.4
0.5
0.6
Figure12: speed response of the non-adaptive and
adaptive controller with step load
7

8. Publication of Little Lion Scientific R&D, Islamabad PAKISTAN
Journal of Theoretical and Applied Information Technology
15th July 2011. Vol. 29 No.1
© 2005 - 2011 JATIT & LLS. All rights reserved.
ISSN: 1992-8645
www.jatit.org
REFRENCES:
250
[1] Guy GRELLET & Guy CLERC « Actionneurs
électriques » Eyrolles–November 1996
[2] Melicio, R; Mendes, VMF; Catalao, JPS
“Modeling, Control and Simulation of FullPower Converter Wind Turbines Equipped
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Permanent
Magnet
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[3] Gholamian, S. Asghar; Ardebili, M.;
Abbaszadeh, K “Selecting and Construction of
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[4] Muyeen, S. M.; Takahashi, R.; Murata, T.;
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[5] K.Paponpen and M.Konghirum “ Speed
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[6] S.Hunemorder, M.Bierhoff, F.W.Fuchs ‘’
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[7] J. Thomas K. René A.A. Melkebeek “Direct
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Motors- An Overview” 3RD IEEE – April
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[8] Shahnazari, M.; Vahedi, A. “Improved
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‘issue6) pg 1248-1258 Nov-Dec 2009
[9] Youngju Lee, Y.B. Shtessel, “Comparison of a
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[10] A.Lagrioui, H.Mahmoudi « Contrôle Non
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200
S peed
150
Feedback linearization
Adaptive Control case
non-adaptive control case
100
50
0
-50
E-ISSN: 1817-3195
time(s)
0
0.1
0.2
0.3
0.4
0.5
0.6
Figure13: speed response of the input-output
linearization- non-adaptive and adaptive
controller with variation of parameters
( Rs = RS 0 ± ∆RS )
250
Non adaptive case
Adaptive case
200
input output linearization
Speed
150
100
50
0
Time(s)
0
0.1
0.2
0.3
0.4
0.5
0.6
Figure14: speed response of the input-output
linearization- non-adaptive and adaptive
controller with variation of parameters
( Φ f = Φ f ± ∆Φ f ) at t=0.3 s
0
6. CONCLUSION:
A method of nonlinear backstepping control has
been proposed and used for the control of a PMSM.
The simulation results show, with a good choice of
control parameters, good performance obtained
with the proposed control as compared with the
non-linear control by state feedback. From the
speed tracking simulations results, we can find that
the nonlinear adaptive backstepping controllers
have excellent performance. The direct axis current
id is always forced to zero in order to orient all the
linkage flux in the d-axis and to achieve maximum
torque per ampere. The q-axis current will approach
a constant when the actual motor speed achieves
reference
speed.
The show simulation results, fast response without
overshoot and robust performance to parameter
variations and disturbances throughout the system
8

9. Publication of Little Lion Scientific R&D, Islamabad PAKISTAN
Journal of Theoretical and Applied Information Technology
15th July 2011. Vol. 29 No.1
© 2005 - 2011 JATIT & LLS. All rights reserved.
ISSN: 1992-8645
www.jatit.org
[12] A. Lagrioui, Hassan Mahmoudi “Modeling and
simulation of Direct Torque Control applied to
a Permanent Magnet Synchronous Motor”
IREMOS- Journal – August 2010.
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Luenberger” The 2nd International Conference
on Multimedia Computing and Systems
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[15] D M. Rajashekar, I.V.Venu Gopala swamy,
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Permanent Magnet Synchronous Motor Drive”
JATIT-Vol 24. No. 2 – 2011
[16] B.Belabbes,
M.K.Fellah,
A.Lousdad,
A.Maroufel, A.Massoum ‘’ Speed Control by
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PMSM’’ Acta Electrotechnica et Informatica
No:4 Vol:6 -2006
9
E-ISSN: 1817-3195