Robust Nonlinear Speed Control of PM Synchronous
Motor Using Boundary Layer Integral Sliding Mode
Control Technique
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 8, NO. 1, 47~54, JANUARY 2000
In-Cheol Baik, Kyeong-Hwa Kim, Associate Member, IEEE, and Myung-Joong Youn, Senior Member, IEEE
Student: Jia-Je Tsai
Adviser: Ming-Shyan Wang
Date : 31th-Dec-2008
Department of Electrical Engineering
Southern Taiwan University
Outline
Abstract
I. INTRODUCTION
II. NONLINEAR SPEED CONTROL OF PMSM USING INPUT–OUTPUT
LINEARIZATION
Modeling of PMSM
Nonlinear Speed Control of PMSM Using Input–Output Linearization
Asymptotic Load Torque Observer
III. QUASI-LINEARIZED AND DECOUPLED MODEL AND PROPOSED
CONTROL STRATEGY
Quasi-linearized and Decoupled Model
Proposed Control Strategy for the Quasi-linearized and Decoupled Model
IV. EXPERIMENTAL RESULTS
V. CONCLUSION
REFERENCES
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Abstract
A digital signal processor (DSP)-based robust nonlinear speed control of a
permanent magnet synchronous motor (PMSM) is presented.
A quasi-linearized and decoupled model including the influence of parameter
variations and speed measurement error on the input–output feedback
linearization of a PMSM is derived.
Based on this model, a boundary layer integral sliding mode controller is
designed and compared to a feedback linearization-based controller that uses
proportional plus derivative (PD) controller in the outer loop.
To show the validity of the proposed control scheme, DSP-based
experimental works are carried out and compared with the conventional
control scheme.
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INTRODUCTION
Permanent magnet synchronous motor (PMSM) drives are being increasingly
used in a wide range of applications due to their high power density, large
torque to inertia ratio, and high efficiency.
This paper deals with the nonlinear speed control of a surface mounted
permanent magnet synchronous motor with sinusoidal flux distribution.
However, this approximate linearization leads to the lack of torque due to the
incomplete current control during the speed transient and reduces the control
performance in some applications such as industrial robots and machine tools.
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INTRODUCTION
The nonlinear control method, called a feedback linearization technique, is
applied to obtain a linearized and decoupled model and the linear design
technique is employed to complete the control design.
The feedback linearization deals with the technique of transforming the
original system model into an equivalent model of a simpler form, and then
employs the well-known and powerful linear design technique to complete the
control design.
In this paper, a quasi-linearized and decoupled model including the influence
of parameter variations and speed measurement error on the nonlinear speed
control of a PMSM is first derived and then the Robust control scheme
employing a boundary layer integral sliding mode is designed to improve the
control performance.
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Modeling of PMSM
The machine considered is a surface mounted PMSM and
the nonlinear state equation in the synchronous d-q
reference frame can be represented as follows:
dx
 f ( x)  Gu
dt
where
 x1   id 
   
x   x2    iq 
 x  
 3  


f ( x)  


(2)
 u   vd 
u   1    
 u 2   vq 
(1)
(3)
Lq


R



x1  P
x2 x3
L
L

d
d
f1 ( x)  
 
Lq
R
 
f 2 ( x)     P
x1 x3  x2  P x3 
Ld
Lq
Lq 
f 3 ( x)  


3 
F
TL
P
x

x

2
3


2 J
J
J


 1

 Ld

G  0

 0



0 

1 
Lq 

0 


(4)
(5)
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Nonlinear Speed Control of PMSM Using Input–Output Linearization
From (1) and the assumption that the load torque is
constant, the relationship between the outputs and inputs
of the model can be obtained as follows :
Where
 did 


v 
dt
 2   B  A d 
v 
d 
 q
 2 
 dt 




2
B3 1 
 ,
 2 J  Pf 2  3 Ff3  



f1
(6)
 1

L
A d

 0




3 P 
2 Lq J 
0
(7)
The nonlinear control input which permits a linearized
and decoupled behavior is deduced from this relationship
as follows:

 vd 
 v1  
1
   A   B    
v 


 v2  
 q

(8)
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Nonlinear Speed Control of PMSM Using Input–Output Linearization
Where V1 and V2 are the new control inputs. By
substituting (8) into (6), the linearized and decoupled
model can be given as
did
 v1
dt
d 2
 v2
2
dt
(9)
(10)
As the control laws for the new control inputs, the linear
controller employed by Le Pioufle becomes as follows:

v1  K11 id*  id

d 2 *
d *
*
v2 

K




K


21
22
2
dt
dt



(11)

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(12)
8
Nonlinear Speed Control of PMSM Using Input–Output Linearization
The following error dynamics can be obtained as
de1
 K11e1  0
dt
2
d e2
de2
 K 21
 K 22e2  0
2
dt
dt
(13)
(14)
where
e1  id*  id
e2    
*
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Nonlinear Speed Control of PMSM Using Input–Output Linearization
For the control schemes employed in this paper, an information on the
acceleration (dΩ/dt) is needed for the state feedback and can be
calculated from (1).
The 0-observer is derived under the assumption that the time variation
of the unknown and inaccessible input is zero.
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Asymptotic Load Torque Observer
the inaccessible load torque can be assumed as an unknown constant.
For a PMSM, the system equation for a disturbance torque observer
can be expressed as follows:
dz
 Dz  Ew,
dt
Where
y    Cz
F
    z1 
z      , D   J

 TL   z2 
 0
(15)
1 

J 
0 
 3P 


E   2 j  , C  1,0 , w i q
 0 


For this system,(D,C) is observable. The well-known asymptotic load
torque observer can be designed as


dz


 D z  Ew  L y  C z 
dt


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(16)
11
Quasi-linearized and Decoupled Model
The actual nonlinear control input which employs the nominal
parameter values and measured mechanical speed is expressed as
follows:

 vd 
 v1  
1
(17)
   A   B   
v 
 q
0


0
 v  
 2 
By substituting (17) into (6), a quasi-linearized and decoupled model
can be obtained as follows:
Lq
did Ro  R

id  P iq o     v1  f n1  X   v1
(18)
dt
Ld
Ld

d 2
F
  3   Ro  R
Ld
P
 P 
id   o  o    P id  o   
   f 3  f 3o
2

Lq
Lq
dt
J
 o  2 J  Lq


 Jo
v2  f n 2  X   bv2
o J
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(19)
12
Quasi-linearized and Decoupled Model
The unwanted nonlinear terms, f n1 ( X ) and f n 2 ( X ), are not exactly


known but can be estimated as f n1 ( X ) and f n 2 ( X ), and the estimation
errors are bounded by some known functions, Fn1 ( X ) and Fn 2 ( X )
The control input gain b is also unknown but its bound can be
deduced.
the feedback linearization technique is considered as a modelsimplifying device for the robust control, and the control laws for the
new control inputs v1 and v2 are derived using a boundary layer
integral sliding mode control technique to overcome the drawbacks of
the conventional nonlinear control scheme.
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Proposed Control Strategy for the Quasi-linearized and Decoupled
Model
Assume the bounds of parameter variations and speed measurement
error as follows:
R  Ro ,  min ( 1)     max ( 1.5)
J  J o ,  min ( 1)     max ( 4)
   o ,  min ( 0.8)     max ( 1.2)
(20)
   o ,  min ( 0.95)     max ( 1.05)
Obtain the minimum value for f n1 ( X ) by calculating the minimum
value of each term of f n1 ( X ) using (20) and summing up each term.
Obtain the maximum value for f n1 ( X ) by calculating the
maximum value of each term of f n1 ( X ) using (20) and summing
up each term.
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Proposed Control Strategy for the Quasi-linearized and Decoupled
Model

Obtain the estimate f n1 ( X ) by calculating the average value of
minimum and maximum values.
Obtain the estimation error bound Fn1 ( X ) by calculating the absolute
difference between minimum(or maximum) value and f n1 ( X ) .

Repeat Steps using (20) to obtain the estimate f n 2 ( X )and the estimation
error bound Fn 2 ( X ) for f n 2 ( X ) .
The bound on the control input gain is
  min

  max




bmin  
 0.2   b  bmax  
 1.2 
  min

  max

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(21)
15
Proposed Control Strategy for the Quasi-linearized and Decoupled
Model
The boundary layer integral sliding mode controller is considered to
avoid the chattering phenomenon and the reaching phase problem
The sliding surface s1 is chosen for the input–output decoupled
t
form of (18) which is second-order relative to  e1dt as follows:
0
t
d
 t
(22)
s1    1   e1dt  e1  1  e1dt  e1 (0)
0
0
 dt


The control law for v1 is designed to guarantee s1 s1  1 s1 as
 s1 
v1  v1  k1sat  
 

where

(23)

v1   f n1  1e1 , k1  Fn1  1
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Proposed Control Strategy for the Quasi-linearized and Decoupled
Model
In (23), sat(.) is the saturation function described as
  1, if s1  1
 s1  
sat    s1 , if - 1  s1  1
 1    1, if s  
1
1

(24)
From the bound on the control input gain b of (21), the geometric
mean bm can be defined as
bm  bmax bmin 
1/ 2
  min  max
 
  min  max
1/ 2



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(25)
17
Proposed Control Strategy for the Quasi-linearized and Decoupled
Model
The bound on b can then be written as
bm
 

b
1
(26)
where
 bmax
  
 bmin
1/ 2



 max  max
 
 min  min
1/ 2



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Proposed Control Strategy for the Quasi-linearized and Decoupled
Model
The sliding surface s2 is chosen for thet input-output decoupled form of
(19) which is third-order relative to 0 e2 dt as follows:
2
de2
d
 t
s2    2   e2 dt 
dt
 dt
 0
t
de2
2
 22 e2  2  e2 dt 
t  0  22 e2 (0)
0

dt
(27)
The control law for v2 is designed to guarantee s2 s2  2 s2 as

 s2  
v2  b  v2  k 2 sat   
 2  

1
m
(28)
where



v 2   f n 2  22 de2 / dt   e , k 2   Fn 2   2     1 v 2
2
2 2
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System Configuration
TABLE I
SPECIFICATIONS OF PMSM
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System Configuration
Fig. 3. Configuration of the DSP-based experimental system
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Nonlinear Speed Control of PMSM Using Input–Output Linearization
Fig. 1. Block diagram of the conventional nonlinear control scheme
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Fig. 2. Block diagram of the proposed robust nonlinear control scheme
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EXPERIMENTAL RESULTS
The design parameters used for the conventional nonlinear control
scheme are selected as K11=2700, K21=900 , and K22=810000
For the proposed robust nonlinear control scheme, the design
parameters are selected as λ1=2700, λ2=900, η1=1 , η2=10 , ψ1=0.005 ,
and ψ2=1000.
The observer gains are selected as l1=796.67 and l2=-21.024 to locate
the double observer poles at -400 when there are no parameter
variations
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EXPERIMENTAL RESULTS
(a) Conventional control scheme
J  Jo
(b) Proposed control scheme
Fig. 4. Speed response and q-axis current under no inertia variation
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EXPERIMENTAL RESULTS
(a) Conventional control scheme
J  3J o
(b) Proposed control scheme
Fig. 5. Speed response and q-axis current under +200% inertia variation
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EXPERIMENTAL RESULTS
(a) Conventional control scheme
(b) Proposed control scheme
Fig. 6. Speed response and q-axis current under +200% inertia and +20% flux
variations
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EXPERIMENTAL RESULTS
Fig. 7. Values of sliding surfaces s1 and s2 during speed transient under +200%
inertia and +20% flux variations.
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CONCLUSION
Based on a quasi-linearized and decoupled model , the design methods for the
proposed control scheme have been given using the boundary layer integral
sliding mode control technique
Compared with the conventional nonlinear control scheme, the proposed
robust nonlinear control scheme provides good transient responses under the
inertia and flux variations.
For the proposed control scheme, the chattering phenomenon and the reaching
phase problem can be avoided by introducing the boundary layer integral
sliding mode control technique.
It can be said that the proposed control scheme has the robustness
against the unknown disturbances. Therefore, it can be expected that
the proposed control scheme can be applied to the high-performance
applications such as the machine tools and industrial robots.
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REFERENCES
[1] G. Champenois, P. Mollard, and J. P. Rognon, “Synchronous servo drive: A special
application,” in IEEE-IAS Conf. Rec., 1986, pp. 182–189.
[2] M. Fadel and B. De Fornel, “Control laws of a synchronous machine fed by a
PWMvoltage source inverter,” presented at the EPE, Aachen, RFA, Oct. 1989.
[3] W. Leonhard, Control of Electrical Drives. Berlin, Germany: Springer-Verlag, 1985,
pp. 240–246.
[4] T. Rekioua, F. M. Tabar, J. P. Caron, and R. Le Doeuff, “Study and comparison of
two different methods of current control of a permanent magnet synchronous motor,” in
Conf. Rec. IMACS-TC1, vol. 1, Nancy, France, 1990, pp. 157–163.
[5] B. Le Pioufle and J. P. Louis, “Influence of the dynamics of the mechanical speed of
a synchronous servomotor on its torque regulation, proposal of a robust solution,” in
EPE, vol. 3, Florence, Italy, 1991, pp. 412–417.
[6] J. J. Carroll, Jr., and D. M. Dawson, “Integrator backstepping techniques for the
tracking control of permanent magnet brush DC motors,” IEEE Trans. Ind. Applicat.,
vol. 31, no. 2, pp. 248–255, Mar./Apr. 1995.
[7] B. Le Pioufle, “Comparison of speed nonlinear control strategies for the synchronous
servomotor,” in Electric Machines and Power Systems (EMPS). Philadelphia, PA: Taylor
& Francis, 1993, vol. 21, pp. 151–169.
[8] A. Isidori, Nonlinear Control Systems: An Introduction. Berlin, Germany: SpringerVerlag, 1985, pp. 156–163.
[9] J. J. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: PrenticeHall, 1991, pp. 207–208, 277–291.
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Thanks for your attention
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