ABSTRACT A nonlinear position controller based on backstepping control technique is proposed for a hybrid stepper motor in micro-step operation. Backstepping control approach is adapted to derive the control scheme, which is robust to parameter uncertainties and external load disturbance. Simulation results clearly show that the proposed controller can track the position reference signal successfully under parameter uncertainties and load torque disturbance rejection

Ali Selk Ghafari

Aria Alasty

CiteSeerX

DownloProceedings of IMECE2005 
2005 ASME International Mechanical Engineering and Exposition 
November 5-11, 2005, Orlando, Florida USA 
IMECE2005-80437 
HYBRID STEPPER MOTOR BACKSTEPPING CONTROL IN MICRO-STEP OPERATION 
 
 
Ali Selk Ghafari 
a_selkghafari@mehr.sharif.edu 
Aria Alasty 
aalasti@sharif.edu 
Center of Excellence in Design, Robotics and Automation (CEDRA) 
School of Mechanical Engineering, Sharif University of Technology 
P.O. Box 11365-9567, Tehran, Iran 
 
Proceedings of IMECE2005 
2005 ASME International Mechanical Engineering Congress and Exposition 
November 5-11, 2005, Orlando, Florida USA 
 
 
IMECE2005-80437 
 
 
 
ABSTRACT 
A nonlinear position controller based on backstepping 
control technique is proposed for a hybrid stepper motor in 
micro-step operation. Backstepping control approach is adapted 
to derive the control scheme, which is robust to parameter 
uncertainties and external load disturbance. Simulation results 
clearly show that the proposed controller can track the position 
reference signal successfully under parameter uncertainties and 
load torque disturbance rejection. 
 
INTRODUCTION 
Put introduction here. 
NOMENCLATURE 
The subject of the control of nonlinear systems with 
uncertainties has been attracting the attention of researchers for 
years. The two major classes of controllers which are capable 
of dealing with nonlinear uncertain systems are adaptive and 
robust controllers. Backstepping control is an approach to 
nonlinear control design which has attracted a great deal of 
research interest in recent years. It is mainly applicable to 
systems having a cascaded or triangular structure. The central 
idea of the approach is to recursively design controllers for 
subsystems in the structure and ‘‘step back’’ the feedback 
signals towards the control input. This differs from the 
conventional feedback linearization in that it can avoid 
cancellation of useful nonlinearities in pursuing the objectives 
of stabilization and tracking. In addition, by utilizing the 
control Lyapunov function, it also has the flexibility in 
introducing appropriate dynamics to make the system behave in 
a desired manner. The presentations of backstepping control in 
the literature are mostly in pure mathematical settings [1].  
This paper presents design and simulation of a nonlinear 
backstepping position controller for HSM in micro-step 
operation. Dynamic model of HSM was developed and model 
parameter uncertainties have been considered. The theoretical 
analysis and the theorems needed for the proposed controller  
aded From: https://proceedings.asmedigitalcollection.asme.org on 06/29/2019 Terms of Us 
are described in details. Simulation results show that the 
proposed controller provide high performance dynamic 
characteristics and robustness with regard to plant parameter 
uncertainties and external load disturbance.  
NOMENCLATURE 
vd, vq   motor voltages in d-q axis 
id, iq   motor phase currents in d-q axis 
θ    motor angular position 
ω    motor angular velocity 
B   motor viscous friction coefficient 
R   motor winding resistance 
L   motor phase winding inductance 
N   number of rotor teeth 
J   rotor and load inertia 
Kt   motor torque constant 
TL   load torque  
BN   nominal viscous friction coefficient 
KtN   nominal motor torque constant 
JN   nominal rotor and load inertia 
TLN   nominal load torque 
∆B   viscous friction uncertainty 
∆Kt   motor torque constant uncertainty 
εJN   rotor and load inertia uncertainty 
∆TL   load torque uncertainty 
ymi   reference states (i=1, 2, 3) 
kmi   controller gains (i=1, 2, 3) 
refθ    angular reference position  
idref   d-axis reference current command 
HSM DYNAMICAL MODEL 
Stepper Motors are widely used in positioning applications. 
Hybrid Stepper Motors (HSM) contain a permanent magnet 
mounted along the axis with toothed end caps and the stator is a 
salient pole structure as shown in Figure1. The stator poles are 
provided with windings which are energized sequentially to 1 Copyright © 2005 by ASME 
e: http://www.asme.org/about-asme/terms-of-use
Downproduce the stepping motion of the motor, and therefore 
eliminating the need for mechanical brushes [2]. Stepper 
motors have the potential to convert a conventional stepper 
motor with a fixed step-size into one with a programmable step 
angles, thereby motor positioning resolution is improved. Using 
micro-step mode the currents in the motor coils are modified 
continuous with the shape of a sine and cosine wave 
respectively. The method allows the motor to transition 
continuously from one pole (whole step position) to the 
next. The rotor advances smoothly and output torque is 
continuous at any given position, because the current is always 
correctly proportioned between the two coils. This results in 
less vibration, and makes noiseless stepping possible down to 0 
Hz. The most important characteristics of micro-stepping 
control are improved position resolution, reduction in ripple 
torque, especially at lower motor speeds [3].  
 
Figure1: HSM cross section [4] 
Basically, the model of Hybrid Stepper Motors consists of 
two parts, the electrical and mechanical parts. The mechanical 
model can be expressed in d-q state-space form as follows [3], 
[5] and [6]: 
⎪
⎪
⎪
⎪
⎪
⎩
⎪⎪
⎪
⎪
⎪
⎨
⎧
=
−−−
=
−+−
=
+−
=
ωθ
ωω
ωω
ω
dt
d
J
TBiK
dt
d
L
KiNLRiv
dt
di
L
iNLRiv
dt
di
Lqt
tdqqq
qddd
][
][
][
 
(1) 
From Eq. (1), it is obvious that the dynamic model of HSM is 
highly nonlinear due to the coupling between the speed and the 
electrical currents of d-q axis. In addition to, the model 
parameter such as viscous friction coefficient, motor torque 
constant and load inertia may vary and load torque is also 
possible to be changed, in this case there are uncertainties in 
model parameters. Thus, both the nonlinearity and uncertainties 
as well as the load torque disturbance have to be taken into 
account in design procedures of high performance position 
controller. 
NONLINEAR ADAPTIVE BACKSTEPPING CONTROL 
Adaptive backstepping is a recently developed control 
method for nonlinear systems with constant parameter 
uncertainties. The most appealing feature of adaptive 
backstepping technique is that the final controller as well as the 
adaptation laws can be derived systematically step by step [7]. 
The control objective is to design an asymptotically stable  
loaded From: https://proceedings.asmedigitalcollection.asme.org on 06/29/2019 Terms of Usespeed controller for HSM Eq. (1) to make the position and d-
axis current follow the reference signals satisfactory.  
It is assumed that the viscous friction coefficient, motor 
torque constant, load inertia and load torque have parameter 
uncertainties and considered as follows: 
⎪
⎪
⎩
⎪
⎪
⎨
⎧
∆+=
+=
∆+=
∆+=
LLNL
NN
ttNt
N
TTT
JJJ
KKK
BBB
ε
 
(2) 
Nonlinear backstepping position controller can be designed 
systematically based on suitable Lyapunov function and 
adaptation laws. The compact form of the system Eq. (1) with 
uncertainties can be considered as follows: 
qd vXgvXgXfXfX )()()()(
~
21 ++∆+=&  (3) 
where: 
⎪
⎩
⎪
⎨
⎧
=
=
=
T
T
qd
LXg
LXg
iiX
]0010[)(
]0001[)(
][
2
1
θω
 
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
+
−∆+−∆+−∆
−
∆
−
=∆
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−−−
+−+−
+−
=
0
)1(
)()()(
0
)(
)(~
ε
εωεε
ω
ω
ω
ω
ω
ω
N
LNLNqtNt
t
N
LNNqtN
d
tNq
q
d
J
TTBBiKK
L
K
Xf
J
TBiK
iN
L
K
L
Ri
iN
L
Ri
Xf
 
(4) 
Define the new variables as follows: 
⎪
⎩
⎪
⎨
⎧
==
=
==
.)(
)(
)(
23
1~2
11
d
f
ixhy
xhLy
xhy θ
 (5) 
The state space equation of system (3) can be written as: 
⎪
⎪
⎩
⎪⎪
⎨
⎧
+++=
+++=
+=
∆
∆
∆
.2222~3
1~1~1~1
2
~2
121
21
21
hLvhLvhLhLy
hLLvhLLvhLLhLy
hLyy
fqgdgf
ffqfgdfgf
f
&
&
&
 
(6) 
Three scalar new uncertain parameters are defined as follows: 
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
+
∆+
−=
+
∆+
−=
+
∆−
−=
)1(
)(
)1(
)(
)1(
)(
3
2
1
ε
α
ε
α
ε
εα
N
LLN
N
ttN
N
N
J
TT
J
KK
J
BB
 
(7) 2 Copyright © 2005 by ASME 
: http://www.asme.org/about-asme/terms-of-use
 
T
o
b
c
b
m
e
a
t
d
L
V
c
L
V
t
s
T
t
V
w
I
Downloadehe system Eq. (6) can be written as follows: 
⎪
⎪
⎩
⎪⎪
⎨
⎧
+=
+++=
=
dgf
qf
uhLhLy
iyhLy
yy
22~3
32211
2
~2
21
1
&
&
&
ααα  (8) 
Now, a reference model below is used to assign the desired 
utput dynamic behavior for the nonlinear system: 
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−−=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
dref
ref
m
m
m
m
m
m
mm
m
m
m
ik
k
y
y
y
k
kk
y
y
y θ
3
1
3
2
1
3
21
3
2
1
0
0
0
0
00
0
010
&
&
&
 (9) 
Using the reference model, performance of the system can 
e evaluated easily, because the tracking problem could be 
hanged to regulation problem. Define the error variables 
etween the dynamic model (8) of HSM and the reference 
odel (9) as: 
T
mmm
T yyyyyyeee ][][ 332211321 −−−==  (10) 
nd use the following transformation: 
⎪⎩
⎪
⎨
⎧
−+=
+=
drefmmmdd
mmmmq
ikyku
L
u
ykyku
333
2211
1~
~
 (11) 
hen the differential equation of the errors are given as follows: 
⎪
⎪
⎩
⎪⎪
⎨
⎧
+=
++++=
−=
.~
~
2~3
32211
2
~2
121
df
qqf
refm
uhLe
iyuhLe
kee
&
&
&
ααα
θ
 
(12) 
The nonlinear backstepping controller can be easily 
esigned through suitable Lyapunov candidate. Define the 
yapunov candidate V as: 
2
3
2
2
2
1 2
1
2
1
2
1 eee ++=  (13) 
onsidering the error dynamics (12), the derivative of the 
yapunov candidate in (13) becomes as follows: 
]~[
]~[
][
2~3
32211
2
~2
121332211
df
qqf
refm
uhLe
iyuhLe
keeeeeeee
+
+++++
+−=++=
ααα
θ&&&&
 
(14) 
o ensure 0≤V& , the simplest way is to make the items in first, 
econd and third brackets equal to 2211 , ekek −− and 33ek− . 
hen the following results can be obtained: 
⎪⎩
⎪
⎨
⎧
−−=
−−−−−=
2~33
32211
2
~22
~
~
hLeku
iyhLeku
fd
qfq ααα  (15) 
hus the derivative of the Lyapunov candidate becomes as:  
0K)( 233
2
22
2
11 <−=++−= eeekekek
T&  (16) 
here 3,2,1,0],[K 321 =>= ikkkkdiag i , and 0=V&  if .0=e  
t follows that V in (13) is Lyapunov function of (12).  d From: https://proceedings.asmedigitalcollection.asme.org on 06/29/2019 Terms of UsSTABILITY ANALYSIS 
Considering the fact from 0<V& , it follows that all the error 
e1, e2 and e3 are bounded. Define the following new function: 
0)( ≥= KeetM T . 
Integrating (16) gives that: 
)())0(),0(()(
0
tVeVdM
t
−=∫ θττ . 
Since 0)( ≥tV , it gives that 
∞<≤∫∞→ ))0(),0(()(lim
0
θττ eVdM
t
t
, 
for all t. Through Barbalat’s Lemma [8], it can be obtained that 
∞→→ tastM 0)( .  
Therefore it can be concluded that all the error variables 
are bounded and converge to zero asymptotically. Further, 
since the motor angular position and d-axis current tracking 
objectives are equivalent to that the error variables converge to 
zero, it can be concluded that the design objectives have been 
satisfied.  
SIMULATION STUDY AND RESULTS 
The block diagram of the nonlinear backstepping controller 
and Hybrid Stepper Motor (HSM) are given in Figure 2. The 
parameters of the HSM under our simulation study are shown 
in Table 1, and system parameters uncertainties are considered 
as in Table 2.  
In this paper, the reference model (9) is adapted to make 
the HSM follows the command signals in the specified manner. 
Pole placement technique is used to determine the gains. 
Actually from Eq. (9), there are two decoupled subsystems, one 
is a second-order mechanical subsystem for motor angular 
position, and other one is a first-order electrical subsystem for 
d-axis current. For the mechanical subsystem, the gains km1 and 
km2 can be chosen according to the expected transient response 
as km1=155 and km2=20. km3 is chosen as 50 the d-axis current 
reach the step command within 0.1 s without any overshoot. 
For the nonlinear controller, k1, k2 and k3 may be chosen for the 
nominal system, which means they are obtained by forcing the 
uncertainties 21,αα and 3α in error dynamics (12) to be zero. k1, 
k2 and k3 are chosen 1500, 250 and 100 respectively. Simulation 
results for tracking problem with constant 0.1 N.m disturbance 
load torque, in various micro-steps and angular velocities are 
shown in Figures 3 to 6. In these Figures tracking problem 
consist of three stages, in the first two stages, motor track the 
desired position profile in different micro-steps and speeds in 
CW direction, in the last stage motor rotates CCW and tracks 
the desired profile with different speeds to reach the initial 
position. As shown in these Figures tracking error for tracking a 
desired position profile with constant disturbance load torque 
has maximum value less than a micro-step (0.45 degree) and  is 
attenuated rapidly and the steady state error is zero for all these 
cases. Simulation results form Figures 3 to 6 show good 
performance of the proposed nonlinear backstepping controller 
for micro-step control of the HSM in tracking and load 
disturbance rejection. 3 Copyright © 2005 by ASME 
e: http://www.asme.org/about-asme/terms-of-use
Downloa 
Figure 2 Block diagram of the HSM nonlinear 
backstepping controller system 
Table1 HSM parameters under simulation study [3] 
24109849.1 KgmJ N
−×= -the rotor and load moment of inertia 
sKgmBN /0223.0
2= -the nominal viscous friction coefficient 
AmNKtN .187.0= -the nominal motor torque constant 
mNTLN .1.0=  -the nominal load torque 
Ω= 2.2R  -the motor windings resistance 
mHL 2.3=  -the motor windings inductance 
Table 2 HSM parameters uncertainties [3] 
25.00
283.0153.0
047.0009.0
100857.2100666.1 44
<<
<<
<<
×<<× −−
L
t
T
K
B
J
 
 
Figure 3 Tracking problem for 15 and 15 micro-steps (in 0.05, 
0.1 rps) in CW and 30 micro-step (0.2 rps) in CCW  
ded From: https://proceedings.asmedigitalcollection.asme.org on 06/29/2019 Terms of Us 
Figure 4 Tracking problem for 25and 30 micro-steps (in 0.2 , 
0.05 rps) in CW and 55 micro-step (0.2 rps) in CCW 
 
Figure 5 Tracking problem for 18 and 64 micro-steps (in 0.05 , 
0.15 rps) in CW and 82 micro-step (0.5 rps) in CCW 
 
Figure 6 Tracking problem for 158 and 250 micro-steps (in 
0.25 , 0.5 rps) in CW and 408 micro-step (0.3 rps) in CCW 4 Copyright © 2005 by ASME 
e: http://www.asme.org/about-asme/terms-of-use
DownloCONCLUSIONS 
In this study, a nonlinear backstepping position controller 
for HSM in micro-step operation has been proposed. It has been 
shown that the proposed controller can provide the properties of 
insensitivity to uncertainties and external load disturbance. 
Simulation results of closed loop system with nonlinear 
backstepping position controller, demonstrates the high 
tracking and disturbance rejection accuracy in HSM nominal 
model upon the effect of load torque disturbance and model 
uncertainties.  
APPENDIX 
)1(
)()()(
0
0
0
1
0
0
1~
1
2
~
1~
1~
2
2
2
1
2~
1~
2
1
2
1
ε
εωεε
ω
ω
ω
+
∆−+∆−+∆−
=
−−−
=
=
=
=
=
=
=
+−=
=
∆
∆
∆
N
NNqttN
ff
N
LNNqtN
f
fg
fg
g
g
f
f
q
d
f
f
J
TTBBiKK
hLL
J
TBiK
hL
hLL
hLL
hL
L
hL
hL
hL
iN
L
RihL
hL
 
REFERENCES 
[1] H. K. Khalil, “Nonlinear Syatems”, Second Edition, 
New Jersey: Prentice Hall, 1996. 
[2] L. Chrifi-Alaoui and A. Lebrun, “H-Infinity Feedback 
Control of a Permanent Magnet Stepper Motor”, 
IEEE, 1994, pp. 108-113.  
[3] A. Selk Ghafari, G.R. Vossoughi, “Simulation and 
Experimental Study of Real-Time Robust Control of 
Hybrid Stepper Motor with QFT Method in Micro-
Step Operation”, CD-Rom Proceedings of IEEE 
International Conference on Mechatronics, June 3-5, 
2004, Istanbul, Turkey 
[4] http://www.altareu.com.tr. 
[5] P.P. Acarnley , “Stepping motors: a guide to modern 
theory and practice”, 3rd Edition, Peter Peregrinus 
Ltd. Publication, 1992. 
[6] Takashi Kenjo and Akira Sugawara, “Stepping 
motors and their microprocessor controls”, Second 
Edition, Oxford Press, 1994. 
[7] H.J. Shieh, K.K. Shyu, “Nonlinear Sliding Mode 
Torque Control with Adaptive Backstepping 
Approach for Induction Motor Drive”, IEEE 
Transactions on Industrial Electronics, 46(2), 1999, 
pp. 380-389. 
[8] J.J. Slotine , W. Li, “Applied Nonlinear Control”, 
New Jersey: Prentice-Hall, 1999. 
 
 
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aded From: https://proceedings.asmedigitalcollection.asme.org on 06/29/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use


IMECE2005-80437 HYBRID STEPPER MOTOR BACKSTEPPING CONTROL IN MICRO-STEP OPERATION

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IMECE2005-80437 HYBRID STEPPER MOTOR BACKSTEPPING CONTROL IN MICRO-STEP OPERATION

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