PIERS Proceedings, Cambridge, USA, July 5–8, 2010
300
Permanent Magnet Synchronous Motor Decoupling Control Study
Based on the Inverse System
Xiaoning Li1 , Xumei Mao1 , and Weigan Lin2
1
School of Mechanical Engineering, University of Electronic Science and Technology of China
No. 4, Section 2, North Jianshe Road, Chengdu 610054, China
2
School of Electronic Engineering, University of Electronic Science and Technology of China
No. 4, Section 2, North Jianshe Road, Chengdu 610054, China
Abstract— In this parer, multivariable,nonlinearity inverse system method is applied on the
PMSM control system, which is complicated nonlinearity, strong coupled, and we have realized
global linearization, at the same time, the motor rator speed and electromagnetic torque have
been dynamic decoupled. The validity of this method can be proved by the simulation result.
1. INTRODUCTION
The PMSM is a multivariable, nonlinearity, strong, coupled controlled object. It is very hard to
control its speed and electromagnetic torque by extra signal. Only can we precisely control it
after the speed and torque have been dynamic decoupled. Vector control adopted the coordinate
transform to realize decouple based on the motor electromagnetic field theory, but vector controlling
just achieve the static decoupling not dynamic doing. Inverse system theory is a new strategy. Its
essence is to realize decoupling of multivariable, nonlinearity, strong coupled system by feedback
linezation. It can realize linearization in a global range, and then we can decouple the nonlinear
system. The inverse system method don’t need profound mathematical knowledge, so it is very fit
for engineering application. In this paper, multivariable, nonlinearity inverse system method was
applied on PMSM control system, by which motor rator speed and electromagnetic torque can be
dynamically decoupled. The validity of this method can be proved by the simulation result.
2. PMSM DECOUPLING CONTROL INVERSE SYSTEM DESIGN
About PMSM, one effective control tactics is vector control. We can decompose the staror current
into the id and iq , when we keep id = 0, the electromagnetic torque of PMSM is directly related
to the iq , so we can realize the linear control to PMSM. If we reglect the damping coefficient to
simple analyses, the PMSM model can be described by the following equtations based on d-q axis:
d axis staror voltage equation
did
+ Rid − ωr Lq iq
dt
(1)
diq
+ Riq + ωr Ld id + ωr ψf
dt
(2)
ud = Ld
q axis stator voltage equation
uq = Lq
q axis staror magnetic flux linkage equation
ψq = Lq iq
(3)
ψd = Ld id
(4)
Te = Pn [Lmd if d iq + (Ld − Lq )id iq ]
(5)
d axis staror magnetic flux equation
electromagnetic torque equation
Here, ud is d axis stator voltage; uq is q axis stator voltage; id is d axis stator current, iq is q
axis stator current; ψd is d axis stator magnetic flux linkage, ψq is q axis stator magnetic flux
linkage, ψf is permanent magnet magnetic flux linkage, Ld is stator winding d axis inductance, Lq
is stator winding q axis inductance; R is stator resistance, ωr is rator angle speed, Lmd is d axis
Progress In Electromagnetics Research Symposium Proceedings, Cambridge, USA, July 5–8, 2010
301
inductance between stator and rotor, if d is permanent magnet equivalent d axis excitation current,
Te is electromagnetic torque. In order to simplified model, we set two parameter uind and uinq :
dψd
+ R · id
dt
dψq
=
+ R · iq
dt
uind =
(6)
uinq
(7)
Then the PMSM voltage equation can be simplified as:
ud = uind − ωr Lq iq
uq = uinq + ωr Ld id + ωr ψf
(8)
(9)
We can prove reversibility by utlizing the Interactor arithmetic. At first, we set up the PMSM
state equation:
Select input variable:
U = [u1 , u2 ]0 = [uind , uinq ]0
(10)
Select Output variable:
Select State variable:
Y = [y1 , y2 ]0 = [id , ωr ]0
(11)
X = [x1 , x2 , x3 ]0 = [id , iq , ωr ]0
(12)
Then the state equation can be expressed:
 0
1
 x1 = − R
L x1 + L u1
R
0
x = − L x2 + L1 u2
 20
x3 = σx2 − TJL
(13)
In this formula, σ = 3Pn · ψf · (2J)−1 , Pn is magnetic poles
Output equation is:
y = [id , ωr ]0 = [h1 (X), h2 (X)]0 = [x1 , x2 ]
(14)
First step we calculate first order deriveate of y1
R
1
(1)
y1 = x01 = − x1 + u1
L
L
We set Y1 = y10 , then
µ
t1 = rank
∂Y1
∂U T
¶
(15)
¸
1
= rank
,0 = 1
L
·
(16)
Namely Jacobian matrix full rank, so α1 = 1.
Second step we calculate first order deriveate of y2
y20 = σ · x2 −
TL
J
(17)
Because there is no include U , we need to calculate second order deriveate of y2
µ
¶
R
1
00
y2 = σ · − · x2 + · u2
L
L
Obviously, U is included in y200 , so set Y2 = [Y1 , y200 ]T , then
¶
·1
µ
∂Y2
= rank L
t2 = rank
0
∂U T
Jacobian matrix full rank, so α2 = 2.
Because
µ
¶
· 1
∂Y2
det
= rank L
0
∂U T
0
σ
L
0
σ
L
(18)
¸
=2
(19)
¸
6= 0
(20)
PIERS Proceedings, Cambridge, USA, July 5–8, 2010
302
The system vector relative degree is
α = [α1 , α2 ]0 = [1, 2]0
(21)
The highest order number of y1 and y2 is separately one and two, so the system nature order is
ne = [1, 2]T
(22)
Clearly, the system nature order is equal to vector relative degree. According to Interactor arithmetic, so the system is reversible, PMSM inverse system decoupling control structure diagram is
shown in Figure 1.
We embedded the PMSM inverse system before the original system to build a compounded
pseudo linear system, then we can convert it to one first order line integration subsystem and one
two order line integration subsystem so that it can be better controlled. So for stator current and
speed, we can discriminablily design current regulator and speed regulator to build compounded
controller together with inverse system in order to realize PMSM decoupling control graph, after
we add in a inverse system, complicated PMSM nonlinear system can be reduced to a linear
system. Stator current is corresponding to a first-order integrating object, so we adopt generally PI
controller, but rotor speed is corresponding to a second order integrating object, and PD controller
is used usually.
3. SIMULATION ANALYSE OF PMSM VECTOR CONTROL SYSTEM BASED ON THE
INVERSE SYSTEM
In this paper, we build PMSM inverse system decoupling control model and proceed simulation test
based on MATLAB. By tuning, we set kp = 30, ki = 0.1 as PI controller parameter and kp = 1500,
kd = 50 as PD controller parameter. The PMSM motor parameter is:
Stator resistance R = 2.875 Ω, turning intertia J = 8 × 10−4 kg · m2 , magnetic poles p = 4,
stator inductance Ld = Lq = L = 8.5 mH. To verify the decoupling characteristics, we set load
torque TL = 5 N · m during 0 to 0.2 s, and suddenly load increase 5 N · m in 0.4 s, simultaneously,
id * + -
PI
S-1
ud
S-1
Inverse
System
ωm*
+ -
PD
S-1
S-1
uq
PMSM
Speed
Control
System
id
ωr
Figure 1: The structure of PMSM based on the inverse system.
Figure 2: The response of torque when speed mutate.
Figure 3: The response of speed when torque mutate.
Progress In Electromagnetics Research Symposium Proceedings, Cambridge, USA, July 5–8, 2010
303
we set rotor speed ωr = 60 rad/s during 0 to 0.4 s, and suddenly speed increase 60 rad/s in 0.4 s,
from the Figure 2 and Figure 3, we can see that the rator speed keep constant ωr = 60 rad/s when
load torque varied suddenly from 5 N · m to 10 N · m, similarly, the load torque remain uncharged
when the rator speed have a sudden charge from 60 rad/s to 120 rad/s from the simulation result,
we can see that the PMSM have been depouled by inverse system, we can separately control the
speed and torque. So we respectively control speed and torque of PMSM.
4. CONCLUSION
In this paper, we adopt inverse system to realize the speed and torque dynamic decoupling control
of permanent magnet synchronous motor. We build the inverse system and verified the system by
simulation, The validity can be proved by the simulation result.
ACKNOWLEDGMENT
This work was supported by:
1. National Natural Science Foundation of China (60971037).
2. University of Electronic Science Technology of China youth fund accented term (JX0792).
REFERENCES
1. Bose, B. K., Modern Power Electronics and AC Drives, China Machine Press, Beijing, 2005.
2. Levin, A. U. and K. S. Narendra, “Control of nonlinear dynamical systems using neural networks: Controllability and stabilization,” IEEE Transactions on Neural Networks, Vol. 2,
No. 4, 192–206, 1993.
3. Wang, J., T. Li, K. M. Tsang, et al., “Differential algebraic observer based nonlinear control of
permanent magnet synchronous motor,” Proceedings of the CSEE, Vol. 2, No. 25, 87–92, 2005
(in Chinese).
4. Zhang, C., F. Lin, W. Song, et al., “Nonlinear control of induction motors based on direct
feedback linearization,” Proceedings of the CSEE, Vol. 2, No. 23, 99–107, 2003 (in Chinese).