Download Modélisation de la MSAP

Current and Speed Control for the PMSM
Using a Sliding Mode Control
Ahmed Lagrioui
Département Génie Electrique
Laboratoire d’Electronique de Puissance
Ecole Mohammadia d’Ingénieurs
Avenue Ibn Sina B.P 767 Agdal Rabat, Maroc
Hassan Mahmoudi
Département Génie Electrique
Laboratoire d’Electronique de Puissance
Ecole Mohammadia d’Ingénieurs
Avenue Ibn Sina B.P 767 Agdal Rabat, Maroc
[email protected]
[email protected]
Abstract-- In this article, we present the mathematical model of
the synchronous motor to permanent magnet (MSAP) permitting
the simulation of his dynamic behavior under the
MATLAB/SIMULINK environment. This model is
based on
the transformation of Park. This paper proposes a realization of
robust speed and current control for the Permanent Magnet
Synchronous Motor (PMSM) using a PI-Sliding mode control.
The PI control has a good performance in the dynamic system
while the sliding mode control has robustness against the system
uncertainties.
parametric variations and well adapted to the modelled
systems [2][8]. To this end, we are interested in the
application of sliding mode for the control of the PMSM
decoupled by the singular perturbation technique.
II. MODELING OF THE PMSM
A. Electrical and Mechanical equations of the PMSM
The electric equations of the MSAP in the plane d-q
I.
INTRODUCTION
The Permanent Magnet Synchronous Motor (PMSM)
have attracted increasing interest in recent years for industrial
drive application. The high efficiency, high steady state
torque density and simple controller of the PM motor drives
compared with the induction motor drives make them a good
alternative in certain applications[1][6].
The Technique of the vectorial control allows comparing
the PMSM to the DC machine with separate excitation from
the point of the view torque. The flux vector must be
concentrated on the d axis with the Isd current null [5].
However the exact knowledge of the rotoric flux position
gives up a precision problem. Thus, it is possible to control
independently the speed and the forward current Isd. The
traditional algorithm of control (PI or PID) proves to be
insufficient where the requirements performances are very
savere. Various nonlinear analysis tools have been used by
many authors to investigate the speed control of PMSM
such as sliding-mode control technique [4][5][7], adaptive
backstepping method [6][7], Input-Output linearization
Control by Poles placement[7].
The Sliding Mode Control is a nonlinear algorithm
which give the robustness properties with respect to the
d sd
  sq
dt
d sq

  sd
dt
u sd  R s i sd 
Keywords: Permanent Magnet Synchronous Motor (PMSM),
Sliding Mode Control (SMC), Proportional Integral Control (PIC), Variable Structure System (VSS).
u sq  R s i sq
(1)
Equations of fields
 sd  Ld isd   f
(2)
 sq  Lqisq
Electromagnetic Torque
The electromagnetic Torque is given by
Ce 

3
p. ( Ld  Lq ).isd isq   f isq
2

(3)
and the Mechanical Equation
J
d
 f .  Ce  Cr
dt
where
Rs
: Stator resistance
Ld , Lq : Stator d and q axis inductance
f
: Viscous friction coefficient
J
: Rotor moment of inertia
(4)
speed (Ω) are measurable, and that the control of the
instantaneous torque can be done comfortably through the
intermediary of currents isd and/or isq.
: Number of pairs pole
p
f
: Permanent magnet flux

: Motor speed
  p.
: Inverter frequency
i sd , i sq
: d-q axis currents
usd , usq
: d-q axis voltages
Ce
: Electromagnetic Torque
Cr
: Load Torque
The representation of non linear state of the PMSM can be
written as follows
d[ X ]
 F [ X ]  [G].[U ]
dt
[ y]  H [ X ]
With

 f1 ( x)  a1.x1  a2 .x2 .x3


 
F[ X ]   f 2 ( x)  b1.x2  b2 .x1.x3  b3 .x3

 f 3 (3)  c1.x3  c2 .x1.x 2 c3 .x2  c4 .Cr 
B. Non Linear Model of the PMSM
If we consider isd, isq and Ω as a variable of states, and
use the previous equations, we get the following nonlinear
system
dt
 (
f
Rs
L
1
)i sq  ( p. d )i sd   ( p. )  u sq
Lq
Lq
Lq
Lq
 g11 g12  a3 0
[G]   g 21 g 22   0 b4 
 g 31 g 32  0 0 
We set
Lq
Rs
1
, a2  p
, a3 
Ld
Ld
Ld
b1  
f
Rs
L
, b2   p d , b3   p ,
Lq
Lq
Lq
c1  
b4 
(9)
Non linear functions of our model
(5)
f
3p
d
1
 ( )  [( Ld  Lq )i sd i sq   f i sq ]  C r
dt
J
2J
J
a1  
(8)

 f1 ( x)  a1.isd  a2 .isq .


 

 f 2 ( x)  b1.isq  b2 .isd .  b3.

 f 3 ( x)  c .  c .i .i  c .i  c .C 
2 sd sq
3 sq
4 r

 1
Lq
disd
R
1
 ( s )i sd  ( p. )i sq   u sd
dt
Ld
Ld
Ld
disq
(7)
(10)
Matrix of order
h1 ( x)   x1  isd 

    
H [ X ]  h2 ( x)   x2   isq 
h3 ( x)   x3   
 
1
Lq
(11)
Vector of exit
1
Rs
3p
3p
, c2  ( Ld  Lq ) , c3  . f , c4  
2J
J
2J
Lq
The model establishes in equation (7) can be represented
below by the functional diagram (Figure 1)
The equation (5) becomes
disd
 a1isd  a2isq  a3u sd
dt
disq
 b1isq  b2isd   b3  b4u sq
dt
d
 c1  c2isd isq  c3isq  c4Cr
dt
d[ X ]
dt
[U]
[G]
++
∫
[X]
[Y]
H[X]
(6)
III. STATE REPRESENTATION OF THE PMSM [6],[7]
 x1  i sd 
   
The choice of the vector  X   x 2  i sq  as vector of
 
 x 3   
state is justified by the fact that the currents (isd, isq) and the
F[X]
Cr
Figure 1. Non linear Diagram of the PMSM in reference mark d-q
IV.
CLASSICAL CONTROL USING A PI REGULATOR
A. decoupling and compensation
For purposes of rotor magnetizing-flux oriented vector
control, the direct-axis stator current isd (rotor field
component) and the quadrature-axis stator current isq (torqueproducing component) must be controlled independently.
However, the equations of the stator voltage components are
coupled. The direct axis component usd depends on isq, and
the quadrature axis component usq depends on isd. The stator
voltage components usd and usq cannot be considered as
decoupled control variables for the rotor flux and
electromagnetic torque. The stator currents isd and isq can
only be independently controlled (decoupled control) if the
stator voltage equations are decoupled, therefore the stator
current components are indirectly controlled by controlling
the terminal voltages of the synchronous motor.
NB: in steady state, it’s assumed that the iq current loop is
fast enough compared to the speed loop to be considered
equivalent to a gain B.
Id
Iqréf
Ωréf
RΩ
c2
L
cq)
3
Lq
)
Figure 4. The cascade control
relating to q axis
The equations of the stator voltage components in the d, q
reference frame can be reformulated and separated into two
components: linear components (usd, usq) and decoupling
components( Ed, Eq)
V.
u * sd  u sd  E d
(12)
u * sq  u sq  E q
With
u sq  R s i sq  Lq
di sd
dt
di sq
(13)
dt
E d   pLq .i sq
yréf +
Functional diagram
PId
+
-
u sd
+
1
Rs  Ld .S
isd
eq
+
-
a r 1
PIq
u *sq
+
-
er 1
+
+
a2
e2
+
e1
a1
Figure 5. Basic Sliding Mode Controller
Figure 2. id current loop using a PI controller
iqréf
+
ed
u *sd
Plant
Order: n
Rank: r
r≤n
u
-Umax
-
-
S
Derivative Estimator
in figures 2, 3 and 4.
idréf
u
S +Umax
e0 +
ki
and the functional diagrams are represented
s
kp 
u sq
Rs
1
 Lq .S
Ω
Lq
)
SLIDING MODE CONTROL
A. Principle
For the components of the stator current, we choose
regulators PI. While for speed, we choose a regulator PI with
anti-windup in order to control this variable during the
transition phase. The PI regulator choice contributes to find
the decoupling quality between the two axes d and q. The
quadrature current reference iq* is provided by a speed PI
regulator. The reference limitation prevents the torque to
exceed the fixed maximal value. The PI regulators are of the
form:
1
f  J .s
The variable structure system (VSS) differs from other
control systems in that it changes its control structures
discontinuously. In the usual controls systems, the controls
structures are fixed in the process of controller, even through
the coefficients are changed continuously according to the
adaptation systems mechanism. The same structures are
preserved through the control process. The control actions
provide the switching between subsystems, which give a
desired behaviour of the closed loop system.
E q  pLd i sd  p f
B.
B
RΩ is a PI regulator with anti-windup.
The equations are decoupled as follows:
u sd  R s i sd  Ld
Cr
Iq
isq
Figure 3. iq current loop using a PI controller
With:
dy
,
e1 
dt
e2 
d2y
dt 2
and e r 1 
d r 1 y
dt r 1
Switching boundary in r-1 dimensional error space:
S( e0, e1, …… er-1)
u=+umax
n region
S(e)<0
u=-umax
u=-umax
P region
S(e)>0
u=+umax
S(e)=0
Figure 6. State trajectory in SMC
y

In the sliding mode, ideally, S(e)=0
y
y réf
( s) 
External loop (relative to speed)
1
1  a1 s  ...  a r 1 s r 1
(14)
B. Application to current and speed control of the PMSM
The SMC is applied to PMSM model, in such a way to
obtain simple surfaces. Figure 7 shows the proposed control
scheme in a cascade from in which the surfaces are required.
The internal loop allows controlling the direct current id,
whereas the external loop provides the speed regulation.
ed  i sdref  i sd
(15)
We choose the sliding surface
S (i sd )  i sdref  i sd
S (isd )  isdref  isd  0
 u sd  u sdc 
(16)
(17)

1 
i sdref  a1 x1  a 2 x 2 x3
a3

(18)
(28)
 c x c C

ref
1 3
4 r
(29)
c2 x1  c3
S ().S ()  0
(30)
 i sqn  K  .Sgn( S ())
(31)
So that it results the output command of the quadratic
current is
 isqref  isqc  isqn
 c x c C

ref
1 3
4 r

c2 x1  c3
 K  .Sgn( S ()) (32)
3) Stability factor determination
The functions coefficients ‘Sgn(S(Xi))’ must be quite
selected to ensure the stability of the system and to satisfy
the sliding mode condition
(33)
i sd , i sq , 
So tha it result the output command of the direct current is
 u sdref  u sdc  u sdn
k   max
Cr ,

1 

i sdref  a1 x1  a 2 x 2 x3  K d .Sgn( S (i sd ))
a3
(19)
C r  f
p f
Udc
θ
vabc
2) Speed regulation
MLI
Inverter
PARK-1
It have two loops of cascades on the q-axis
 internal loop (relative to iq-current)
(20)
S (i sq )  i sqref  i sq
(21)
S (isq )  isqref  isq  0
(22)


PARK
idreéf=0
iq
Ωréf
(24)
So that it results the output command of the quadratic
current is
Figure 7. Scheme of simulation
 u sqref  u sqc  u sqn

1 
isqref  b1 x 2  b2 x1 x3  b3 x3  K q .Sgn( S (isq ))
b4
(25)
Ω
id
SMCq
SMCΩ
S (isq ).S (isq )  0 by choosing
u sqn  K q .Sgn( S (i sq ))
SMCd
iqref
1 
 usq  usqc 
isqref  b1x2  b2 x1x3  b3 x3 (23)
b4
During the convergence mode we have to satisfies the
PMSM
Usd
Usq
eq  iqref  i sq

 
 0
S ()  
ref
k q   max Rs isq  Ld isd p  p f
u sdn  K d .Sgn(S (isd ))

(27)
i sd , i sq , 
condition S (i sd ).S (i sd )  0 by choosing
condition
S ()  ref  
k d   max Rs isd  Lq isq p
During the convergence mode we have to satisfies the

(26)
 isq  isqc 
The sliding surface for each loop is chosen as follows:
1) Direct current regulation
When the direct current error ed is
e   ref  
θ
∫
Figure 8. Tracking performance in the presence of external disturbance
VI.
(Cr=10Nm at t=0s, Cr=6Nm at t=0.2s, Cr=10Nm at t=0.4s)
and ( wréf=300 rad/s at t=0s , wréf =300 rad/s at t=0.6s )
RESULTS OF SIMULATION
A. Parameters of the PMSM
Table I. parameter of the PMSM
parameter
value
Maximal voltage of food
Maximal speed
Nominal Torque ;Cenom
Rs
300 vs
3000 tr/s to 150 Hz
14.2 N.ms
0.4578 Ω
Number of pair poles :p
Ld
Lq
The moment of inertia J
Coefficient of friction viscous f
Flux of linquage Фf
(8.a) Electromagnetic and Load torque (Ce, Cr)
(8.b) d-and-q axis current (isd,isq)
(8.c) speed (Ω)
(8.d) Stator current ( ia, ib, ic)
2)
Using a SMC Controller
4
3.34 mHs
3.58 mHs
0.001469 kg.m2s
0.0003035 Nm/Rad/s
0.171 wbs
(9.a)
B. Results
1) Using a PI Regulator
(9.b)
(8.a)
(9.c)
(8.b)
(9.d)
(8.c)
Figure 9. Tracking performance in the presence of external disturbance
Cr=10Nm at t=0s, Cr=6Nm at t=0.2s, Cr=10Nm at t=0.4s)
and ( wréf=300 rad/s at t=0s , wréf =300 rad/s at t=0.6s )
(9.a) Electromagnetic and Load torque (Ce, Cr)
(9.b) d-and-q axis current (isd,isq)
(9.c) speed (Ω)
(8.d)
(9.d) Stator current ( ia, ib, ic)
VII.
CONCLUSION
We presented in this paper the performance the sliding
Mode Control compared with a classical control (regulator
PI).
The SMC is unfeeling to parameters variation, such as the
stator resistor.
(10.a)
The chattering phenomenon is been successfully eliminated
from speed control. The effectiveness of the proposed
control system was proved by simulation results and their
comparison with conventional regulator (PI).
REFERENCES
(10.b)
[1]
Guy GRELLET & Guy CLERC « Actionneurs électriques »
Eyrolles–November 1996
[2]
K.PAPONPEN and M.KONGHIRUM “ Speed Sensorless
Control Of PMSM using An Improved Sliding Mode Observer
With Sigmoid function” ECTI- Vol5.NO1 February 2007.
[3]
B.Le PIOUFLE, G.GEORGIOU, I.P.LOUIS « Application of
the NLS orders for the regulation quickly and in position da
plots it synchronous autopilotée » Magazine Phy.Appliquée
1990.
[4]
B.Le PIOUFLE « Comparison of speed non linear control for
tea survomotor » electric Machines Power and systems -1993.
[5]
M.BODSON, J.CHAISSON “Diffrential Geometric methods
for Control of Motors electric”, Vol.8 - 1998
[6]
A.LAGRIOUI, H.MAHMOUDI « Contrôle Non linéaire en
vitesse et en courant de la machine synchrone à aimant
permanent » ICEED’07-Tunisia
[7]
A.LAGRIOUI, H.MAHMOUDI « Nonlinear Tracking Speed
Control for the PMSM using an Adaptive Backstepping Method
» ICEED’08-Tunisia
[8]
Youngju Lee, Y.B. SHTESSEL, “Comparison of a feedback
linearization controller and sliding mode controllers for a
permanent magnet stepper motor,"
ssst, pp.258, 28th
Southeastern Symposium on System Theory (SSST '96), 1996.
[9]
Freescale Semiconductor Literature Distribution Center :
“Sensorless PMSM Vector Control with a Sliding Mode
Observer for Compressors Using MC56F8013” DRM099
Figure 10. Test of robustness of the SMC controller and PI regulator for
different values of the Stator resistor Rs
(10.a) Response of the PI regulator
(10.b) Response of the SMC controller
(11.a)
(11.b)
Figure 11. comparaison result of PI and SMC
(11.a): Mechanical velocity response
(11.b): Electromagnetic Torque response