IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 1
Disturbance/Uncertainty Estimation and Attenuation
Techniques in PMSM Drives–A Survey
Jun Yang, Member, IEEE, Wen-Hua Chen, Senior Member, IEEE, Shihua Li, Senior Member, IEEE,
Lei Guo and Yunda Yan, Student Member, IEEE
Abstract—This paper gives a comprehensive overview on
disturbance/uncertainty estimation and attenuation (DUEA) tech-
niques in permanent magnet synchronous motor (PMSM) drives.
Various disturbances and uncertainties in PMSM and also other
alternating current (AC) motor drives are first reviewed which
shows they have different behaviors and appear in different
control loops of the system. The existing DUEA and other relevant
control methods in handling disturbances and uncertainties
widely used in PMSM drives, and their latest developments are
then discussed and summarized. It also provides in-depth analysis
of the relationship between these advanced control methods in
the context of PMSM systems. When dealing with uncertainties,
it is shown that DUEA has a different but complementary
mechanism to widely used robust control and adaptive control.
The similarities and differences in disturbance attenuation of
DUEA and other promising methods such as internal model
control and output regulation theory have been analyzed in
detail. The wide applications of these methods in different AC
motor drives (in particular in PMSM drives) are categorized and
summarized. Finally the paper ends with the discussion on future
directions in this area.
Index Terms—Disturbances, uncertainties, PMSM drives, esti-
mation, robustness, robust control, adaptive control.
I. INTRODUCTION
W
ITH the increasing demands of higher precision ma-
chine drives, AC machine drives, which are widely
considered as a substitute of direct current (DC) machine
drives, are deemed as the most prevailing components of mod-
ern motion control systems due to many distinctive features
they offer. Among various AC machine drives, PMSM has
been receiving abundant attention because of its advantageous
features including high efficiency, high power density, large
torque-to-inertia ratio, low noise, and free maintenance [1]–
[5]. As such, PMSM drives have been extensively applied to
Manuscript received October 9, 2015; revised January 11, 2016, February
21, 2016, April 11, 2016 and May 03, 2016; accepted May 17, 2016.
Copyright (c) 2015 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be
obtained from the IEEE by sending a request to pubs-permissions@ieee.org.
This work was supported in part by National Natural Science Foundation
of China under Grants, 61203011, 61473080 and 61573099, PhD Program
Foundation of Ministry of Education of China under Grant 20120092120031,
Natural Science Foundation of Jiangsu Province under Grant BK2012327,
China Postdoctoral Science Foundation under Grants 2013M540406 and
2014T70455, and a research grant from the Australian Research Council.
J. Yang, S. Li and Y. Yan are with School of Automation, Southeast Univer-
sity, Nanjing 210096, China (e-mails: j.yang84@seu.edu.cn, lsh@seu.edu.cn
and yyd@seu.edu.cn).
W.-H. Chen is with the Department of Aeronautical and Automotive
Engineering, Loughborough University, Leicestershire, LE11 3TU, UK (e-
mail: w.chen@lboro.ac.uk).
L. Guo is with the School of Electrical Engineering and Automation,
Beihang University, Beijing, China (e-mail: lguo@buaa.edu.cn).
a variety of industrial sectors, such as robotics, machine tools,
electrical vehicles, power generations and aerospace [4].
Despite many advantages described above, high precision
control of PMSM drives is rather challenging because the
motion dynamics of PMSM are complicated and intrinsically
nonlinear, and, in addition, subject to various sources of distur-
bances and uncertainties [6]–[10]. Aiming to achieve desired
servo control performance, apart from classical proportional-
integral-derivative (PID) controllers, plenty of advanced con-
trol algorithms have been put forward for AC machine drives,
for example, model predictive control [5], [11], [12], robust
and adaptive control [13]–[19], internal model control [20],
output regulation [2], [21], disturbance observer-based control
(DOBC) [4], [5], [22]–[32] and active disturbance rejection
control (ADRC) [12], [33]–[36], to name but a few.
It has been widely recognized that a crucial task of con-
troller design for PMSM systems is to reject various external
disturbances and improve robustness in the presence of a
wide range of uncertainties. The disturbances/uncertainties in
AC machine drives, which usually exhibit different features,
are generated from a wide range of sources including the
changes of load, operational environments and the mechanical
or electrical parts in the motor systems. In order to design
a successful control algorithm and achieve desirable control
performance, it is important to have a comprehensive un-
derstanding of the features of the disturbances/uncertainties
in AC servo systems first. Consequently, the first focus of
this paper is to provide a comprehensive overview of var-
ious kinds of disturbances/uncertainties in typical types of
AC machine drives including PMSM, induction motor (IM)
and brushless direct current motor (BLDCM). Among many
advanced control strategies in dealing with disturbances and
uncertainties, disturbance/uncertainty estimation and attenua-
tion (DUEA) techniques have received considerable attention
in AC machine drives during the past several decades. Here
DUEA represents a category of algorithms/methods sharing
a similar fundamental idea; that is, an observation mecha-
nism is designed to estimate disturbances/uncertainties and
corresponding compensation is then implemented by making
use of the estimate [37]. Both DOBC and ADRC mentioned
earlier belong to the category of DUEA. Since DUEA exhibits
promising performance in handling disturbances and uncer-
tainties [38], and it is not as well known as other methods
for handling disturbances and uncertainties such as internal
model control (IMC) and robust control, the second focus of
this paper is to give a brief survey on the applications of DUEA
and related techniques in PMSM drives.
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 2
It is also noticed that understanding the relationships be-
tween different DUEA methods and other well established
control methods in handling disturbances and uncertainties
particularly in the context of motion control is of significant
importance. It not only provides insight into the differences
and similarities of those methods, but also, more importantly,
guides researchers and engineers to identify and select most
appropriate methods for applications in their hands. Accord-
ingly, the next focus of this paper is to develop and present
the relationships among these techniques. This part aims to
not only give insight of the differences and similarities among
different DUEA methods, but also discuss their links with
many well known control methods. We will start with a general
discussion on the basic features of DUEA in the comparison
with robust control and adaptive control that are both proven
to be effective in dealing with uncertainties, and then focus
on two well established and closely related advanced control
methods, namely internal model control (IMC) [39] and non-
linear output regulation (NOR) [40]. The last focus of the
paper is on the application of the DUEA in several types of
popular AC motor drives and a number of typical applications
have been summarized.
The organization of the paper is as follows. In Section II,
various disturbances and uncertainties existing in AC machine
drives will be revisited. An overview of DUEA and related
techniques for AC machine drives will be provided in Section
III. Section IV focuses on exploring the relationships between
DUEA methods and their relationships with other well known
disturbance/uncertainty attenuation methods. In Section V,
various applications of DUEA approaches in AC machine
drives are presented, followed by concluding remarks and
future directions in Section VI.
II. DISTURBANCES/UNCERTAINTIES IN AC MACHINE
DRIVES
In most parts of the paper, we mainly focus on discussing
PMSM system as a benchmark AC drive system. However, it
is noticed that PMSM shares many similarities of the problem
formulation, disturbance/uncertainty properties and control al-
gorithms with a number of other advanced AC motor drives,
such as IM, BLDCM, and switched reluctance motors [41].
For example, the differences and similarities among three most
popular AC machine drives (PMSM, IM and BLDCM) are
listed in Table I. In this section as well as several subsequent
sections, attention to other AC motors will be paid for the
interest of readers in related areas.
A generic d − q dynamic model of PMSM drives is given
with respect to its rotor reference frame as
di
d
dt
=
1
L
d
(u
d
− R
s
i
d
+ n
p
ωL
q
i
q
) ,
di
q
dt
=
1
L
q
(u
q
− R
s
i
q
− n
p
ωL
d
i
d
− n
p
ωψ
f
) ,
dω
dt
=
1
J
T
e
− d
fric
T
− d
load
T
,
T
e
=
3
2
n
p
[ψ
f
i
q
+ (L
d
− L
q
)i
d
i
q
] , d
fric
T
= B
v
ω,
(1)
TABLE II
DISTURBANCES AND UNCERTAINTIES IN PMSM
Symbol Meaning Unit
d
cog
T
Cogging Torque N · m
d
flux
T
Flux Harmonic Torque N · m
d
dead
dV
, d
dead
qV
Distortion Voltage V
d
offset
dC
, d
offset
qC
Current Offset Errors A
d
fric
T
Friction Torque N · m
d
load
T
Load Torque N · m
where ω is the rotor speed, i
d
and i
q
are stator currents in
d − q frame, u
d
and u
q
denote stator voltages in d − q frame,
d
load
T
represents the load torque disturbance, d
fric
T
is the friction
torque disturbance, T
e
is the electromagnetic torque, B
v
is
the frictional coefficient, L
d
and L
q
are stator inductances in
d − q frame, R
s
is the stator resistance, ψ
f
is the magnetic
flux linkage, J is the moment of the total inertial (rotor and
load), and n
p
is the number of poles.
A basic field oriented control framework consisting of a
speed loop and two current loops is shown in Fig. 1. The dy-
namics of PMSMs are essentially nonlinear subject to a wide
range of disturbances/uncertainties in many high-performance
applications. In this section, the disturbances/uncertainties in
PMSM systems, which are classified as unmodelled dynamics,
parametric uncertainties and external disturbances, are briefly
reviewed first. The symbols, physical meanings and units of
various disturbances/uncertainties are listed in Table II for the
sake of clarity.
Speed
controller
Current
controller
( -axis)
I
G
B
T
PMSM
Encoder
Frame
Transform
Speed and angle
estimation
Frame
Transform
&
SVPWM
Current
controller
( -axis)
*
w
q
d
*
q
i
*
d
i
q
i
d
i
q
u
d
u
e
q
Speed
loop
Current loop
( -axis) ( -axis)
d
q
w
Fig. 1. Schematic diagram of PMSM based on vector control.
A. Unmodeled Dynamics
1) Motor Body Structure Induced Torque: Due to utilization
of different rotor materials in AC motors, the body structure
may induce various pulsating torques.
• Cogging Torque: As a kind of pulsating torque, cogging
torque is basically generated by the interaction of the
rotor magnetic flux and angular variations in the stator
magnetic reluctance [42]. Note that cogging torque even
exists when the system is disconnected from the power
source. In general, this torque can be expressed as [6]
d
cog
T
(N · m) =
∞
i=1
d
cogi
T
sin (iN
c
θ
e
) ,
where N
c
is the least common multiple between the
number of slots and pole pairs, θ
e
is the electrical angle
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 3
TABLE I
DIFFERENCES AND SIMILARITIES BETWEEN PMSM, IM AND BLDCM B ASED ON FIELD ORIENTED CONROL
Motor
Type
Motor Body
IM
PM material
Rotor Type
Dynamic
Models
Disturbances/Uncertainties
Caused by Motor Body Structure Others
PMSM
4th
order
Similar
3rd
order
Control
Rotor Flux Position
Measurement
Direct/Indirect
Control
Algorithms
Similar
BLDCM
Direct
1
Cogging Torque/
2
Flux Harmonics
Skewed Slot Torque
1
Cogging Torque/
2
Commutation
Torque/
3
Nonideal Trapezoidal Back
Electromotive Force
Aluminum/ cuprum windings
(
1
wound-rotor
2
squirrel cage)
denoted by θ
e
(t) = θ
e
(t
0
) +
t
t
0
n
p
ω(τ)dτ and d
cogi
T
is
the amplitude of the ith-order harmonic cogging torque.
There is no cogging torque in induction motors. However,
the skewed slot torque in IM has a similar structure and
property as the cogging effects in PMSM [43].
• Flux Harmonics: The most widely used material in the
magnet of PMSM is Neodymium Iron Boron and its flux
density is easily affected by the temperature variation.
The resultant demagnetization phenomenon of permanent
magnets due to temperature rise has a significant impact
on the maximum torque capability and the efficiency of
PMSM [44]. The flux linkage between the rotor and stator
magnets can be expressed as [7], [45]
ψ
f
=
∞
i=0
ψ
fi
cos (6iθ
e
),
where ψ
fi
is the amplitude of the 6ith-order harmon-
ic flux. According to the definition of electromagnetic
torque T
e
, it is indicated by (1) that the effect of flux
harmonics can be represented as follows
d
flux
T
(N · m) =
3
2
n
p
i
q
∞
i=1
ψ
fi
cos (6iθ
e
).
• Others: For BLDCM systems, in addition to cogging
torque, there are mainly two other causes of pulsating
torque. One is commutation torque which is caused by the
different current slew rates between switching-in phase
and switching-out phase [46]–[48]. Another is nonideal
trapezoidal back electromotive force due to stator winding
action, the magnetization direction of rotor permanent
magnets, and imperfections in machine manufacturing
[49], [50].
2) Dead-Time Effects: The dead-time causes a loss of
a portion of the duty cycle and distortion of the voltage
applied to the drives [8], [51]–[53]. Note that such effects
become extreme severe near the zero crossing of the current.
The resultant current deterioration finally leads to the ripples
in the electromagnetic torque. The dead-time effect can be
represented on stator voltage channels in the d − q frame as
follows [8], [53]
d
dead
dV
(V ) =
∞
i=0
d
di
sin (6iθ
e
) ,
d
dead
q V
(V ) =
∞
i=0
d
q i
cos (6iθ
e
) ,
where d
di
and d
q i
are the amplitudes of 6ith-order harmonic
signals of d
dead
dV
and d
dead
qV
, respectively.
3) Measurement Error Effects: In AC servomotors, the
errors in measurements of either position or current inevitably
cause torque ripples. As an example, the offset error in current
measurements is taken to illustrate the adverse effects of
measurement errors. The offset errors superimposing directly
on the phase currents via the transform of Clarke and Park,
cause ripples on stator currents in d − q frames [7], [9], [42],
which can be modeled as [7]
d
offset
dC
(A) = d
offset
sin (θ
e
+ α) ,
d
offset
qC
(A) = d
offset
cos (θ
e
+ α) ,
where α is a constant angular displacement and d
offset
is the
amplitude of ripple.
B. Parametric Uncertainties
1) Mechanical Parameters: The inertia J of a PMSM
system, including both rotor and load, is usually a constant
during a short-term operation process. However, in some
special applications, e.g., electric winding machine, the inertia
of the whole system is time-varying, for example, increase as
time goes by in [1], [20]. If the inertia of the system increases
to several times of the original inertia, the speed response will
have a bigger overshoot and a longer settling time [1], [20].
2) Electrical Parameters: The thermal model of AC ma-
chines has a significant impact on the motor controller design
[54], [55]. The thermal model of AC machines is incorporated
in the nominal models used for control design, and improves
the performance of disturbance/uncertainty estimation. For
example, the stator resistance R
s
varies primarily with winding
temperature while a small amount of skin and stray loss effects
are neglected [2]. The resistance R
s
at temperature T can be
written as [44]
R
s
= R
0
(235 + T )/(235 + T
0
),
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 4
where R
0
is the resistance at temperature T
0
. It can be
observed from the PMSM model (1) that the stator resistance
affects the plant bandwidth of the current loop directly. As
such, the variation of stator resistance has a great impact on the
current-loop regulation performance. Moreover, the resistance
effect becomes much severe at low speeds or in high load
torque conditions [2], [3], [56].
In addition, the stator inductances L
d
and L
q
are hard to
precisely obtain [57], and they are usually functions of current
magnitude and current phase angle of the motor [2], [4]. For
instance, the effects of cross saturation [58] generally result
in variations of stator inductances which affects both the plant
gain and the open-loop electrical time constant of the motor,
and hence, the performance of the drives at different operating
conditions.
C. External Disturbances
1) Friction Torque: Friction is the tangential reaction force
between two surfaces in contact [59]. It can be represented
by static models (the Coulomb model, the Stribeck model,
the Karnppp model, etc.) and dynamic models (the Dahl
model, the Bristle model, the reset integrator model, and the
LuGre model, etc.) [59]. The nonlinear effects of friction are
unavoidable and widely exist in servo systems, which may
cause steady-state errors, tracking lags and limit cycles in
position regulation [59], [60]. Taking the Stribeck model as
an example, the friction torque d
fric
T
is expressed as [59]
d
fric
T
(N · m) =
T
c
+ (T
s
− T
c
) e
−
(
ω
ω
s
)
2
sign (ω) + B
v
ω,
where T
s
is the static friction torque, T
c
is the Coulomb
friction torque, ω
s
is the Stribeck velocity, and sign (•) is the
standard signum function.
2) Load Torque: Torque on the load side is generally
deemed as one of the most severe disturbances affecting the
dynamic performance. For example, since the transmission
mechanism is not an ideal rigid body, mechanical resonance
can be easily excited due to the load torque [10]. Speed is
inevitably changed when load torque is imposed on the motor.
3) Mechanical Factors: The effects raised by mechanical
characteristics such as torsional vibrations, backlash, and un-
certainties generated by misalignment of shaft, broken shaft
and twisted shaft are another branch of causes restricting servo
performance improvement of motor drives [61]. It is estimated
that the misalignment of shaft causes over 70% of rotating
machinery’s vibration problems in industrial motor drives [62].
Distribution of Disturbance/Uncertainty in PMSM: The
disturbance/uncertainty discussed above affect the PMSM sys-
tem via different control loops. Malposed compensation of
the disturbance/uncertainty definitely results in performance
degradation of the servo system. As such, it is of great
importance to know the distribution of disturbance/uncertainty
in PMSMs. A diagram of the distribution of unmodeled dy-
namics and external disturbances in PMSM systems is shown
in Fig. 2 (the scenario of d frame current loop is ignored
for simplicity). Clearly, the cogging torque, flux harmonics,
friction torque, and load torque affect the PMSM system in
the speed loop. The dead-time effect and offset error of Hall
current sensors affect the PMSM system in the current loops.
As mentioned above, disturbances/uncertainties are distributed
in different control loops (speed and current loops) and appears
in different forms (e.g., slowly-varying and periodic), which
imposes challenges in disturbances/uncertainties attenuation
for PMSM.
Generally speaking, disturbances/uncertainties attenuation
for PMSMs can be divided into two groups [7], [42]: im-
proving motor design and using advanced control strategies.
This paper focuses on the later one.
Speed
controller
Current
controller
( -axis)
Current loop
plant( -axis)
*
w
q
*
q
i
q
i
q
u
e
T
w
q
w
1
Js
cog
T
d
flux
T
d
offset
qC
d
fric
T
d
load
T
d
dead
qV
d
Fig. 2. The distribution of disturbance/uncertainty in PMSM systems.
III. AN OVERVIEW ON DUEA AND RELATED
TECHNIQUES IN PMSM DRIVES
This part will give a brief overview on DUEA and related
techniques in PMSM drives. Section III-A is devoted to
DUEA while other widely used control methods for handling
disturbances and uncertainties in PMSM drives are covered in
Section III-B.
A. DUEA Methods in PMSM Drives
A number of DUEA techniques have been proposed since
1960s based on different disturbance/uncertainty estimators,
including
• DOBC [63];
• extended state observer-based control (ESOBC, also
called ADRC) [64];
• unknown input observer-based control (UIOBC) [65];
• generalized proportional integral observer-based control
(GPIOBC) [66];
• equivalent input disturbance-based control (EIDBC) [67];
• uncertainty and disturbance estimator-based control
(UDEBC) [68];
• sliding mode disturbance observer-based control (SM-
DOBC) [69];
• intelligent disturbance observer-based control (IDOBC)
[70];
and so on. Different from other control methods in handling
disturbances and uncertainties, some of which are going to
be discussed in the next section, this group of algorithms is
featured with a built-in estimation mechanism for the lumped
disturbance and the influence of uncertainties [37]. In this
survey paper, we aim to elaborate the similarities/differences
among different DUEA and related methods rather than their
detailed algorithms. The readers are referred to see the survey
paper [37] for a comprehensive overview on these DUEA tech-
niques. Among those DUEA techniques, DOBC and ADRC
have received most extensive investigations and applications
in AC machine drives.
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 5
1) DOBC: As a key component of DOBC, DOB was
initiatively put forward by K. Ohnishi and his colleagues in
1980s to improve disturbance rejection and robustness in DC
motors [63]. The early DOBC was designed on the basis of
frequency-domain control theory. With the prevalence of state
space design approaches for DOBC, nonlinear DOBC has
attracted a great deal of attentions so as to further enhance the
control performances of essential nonlinear dynamic systems.
In what follows, we will try to illustrate the basic principles
of typical frequency-domain DOBC and nonlinear DOBC via
two benchmark design examples of the speed regulation of the
PMSM system.
Frequency-Domain DOBC (FDDOBC): The nominal model
utilized for FDDOBC design is generally linear one, while the
nonlinearities of the motor plant are usually treated as a part of
the lumped disturbances, which are estimated and attenuated in
the same manner as external disturbances. A design example of
FDDOBC for PMSM speed regulation is presented as follows.
Example 1 (FDDOBC): Taking the speed regulation prob-
lem of the PMSM system (1) as a benchmark example, the
dynamic model of the speed loop is given by
J ˙ω = K
t
i
q
− B
v
ω +
3
2
n
p
(L
d
− L
q
)i
d
i
q
− T
L
+ ε
ω
, (2)
where K
t
= 3n
p
ψ
f
/2, and ε
ω
denotes the unmodeled dynam-
ics in speed channel. Denoting the current control signal in q
frame as i
∗
q
, we have
J ˙ω = K
t
(i
∗
q
+ d) − B
v
ω, (3)
with the disturbance term
d(t) =
1
K
t
K
t
(i
q
− i
∗
q
) +
3
2
n
p
(L
d
− L
q
)i
d
i
q
− T
L
+ ε
ω
,
where ε
ω
denotes the unmodeled dynamics in the speed loop.
Let y = ω and u = i
∗
q
. The transfer function model of (3) is
written as
y(s) = G(s) [u(s) + d(s)] , G(s) = K
t
/(Js + B
v
).
The frequency-domain DOBC for the above speed regulation
system is designed and shown in Fig. 3.
Frequency-domain disturbance observer
Speed loop model
*
w
e
w
*
q
i
ˆ
l
d
l
d
w
n
( )
C s
( )
Q s
( )
1
n
G s
-
w
*
w
*
d
i
Controller
u
( )
g x d
x
PMSM
PMSM
( )
1
g x
]
2 1
l g p z f g u
( )
p x
ò
z
z
p
ˆ
d
PMSM
d
w
b
q
u
Controller
ˆ
( )Ax Bu L Cx+ + -
ò
ˆ
x
ˆ
x
1/ b
3
E
ˆ
d
*
w
*
w
w
*
q
i
( )
C s
( )
R s
PMSM
d
( )
G s
( )
F s
d
Fig. 3. Structure diagram of FDDOBC for a PMSM.
Besides the disturbance term d(t), the parameters J, B
v
and K
t
also have uncertainties in practice. Consequently, as
shown in Fig. 3 the nominal model
G
n
(s) = K
to
/(J
o
s + B
vo
),
with the nominal values of J
o
, B
vo
and K
to
is usually
employed in DOB design.
As shown by Fig. 3, the frequency-domain DOB is em-
ployed to estimate the total disturbances/uncertainties includ-
ing load toque disturbance, additional nonlinear dynamics,
parametric uncertainties, and even couplings from the current
loop of PMSM in the above example. It follows from Fig. 3
that
d
l
(s) =
G
−1
n
(s) − G
−1
(s)
y(s) + d(s) − G
−1
n
(s)n(s).
(4)
The first item relates to the mismatching between the physical
system G(s) and the nominal model G
n
(s), the second the
external disturbance d(s), and the last the measurement noise
n(s). Therefore, d
l
captures all the disturbance and uncertainty
influence. After letting it pass a filter Q(s), the estimate of the
lumped disturbance in DOB is given by
ˆ
d
l
(s) = G
u
ˆ
d
(s)u(s) + G
y
ˆ
d
(s)¯y(s), (5)
where
G
u
ˆ
d
(s) = −Q(s), G
y
ˆ
d
(s) = G
−1
n
(s)Q(s). (6)
The design of the filter Q(s) plays a central role in DOB. In
general, the design criterion of filter Q(s) can be summarized
as follows:
• the filter Q(s) shall be designed as a low-pass filter with
its relative degree (i.e. the order difference between the
denominator and the numerator of its transfer function)
higher than that of the nominal motor plant G
n
(s);
• the cut-off frequency of the low pass filter Q(s) is crucial
in trading off between different factors (e.g. stability and
performance requirements, frequency characteristic of the
load toque disturbance and measurement noise, the size
and characteristic of nonlinearities and uncertainties). In
general, high cut-off frequency increases the disturbance
attenuation and robustness, but demands large control
action and increases sensitivity to the sensor noise.
Various guidelines and methods for designing the filter Q(s)
can be found in [63], [71], [72] to name but only a few. The
utilization of frequency-domain DOBC to improve robustness
and disturbance rejection performance of AC servo systems
has been extensively investigated in literatures (see [22]–[26]
for recent advances).
Nonlinear DOBC (NDOBC): The nonlinear part of a AC
motor drive system can be deliberately ignored and deemed
as a part of “lumped” disturbances, and then the above
FDDOBC techniques can be applied to estimate the influence
of the ignored nonlinearities so an appropriate action can be
generated to compensate for that. However, for most of AC
motor drive systems, the nonlinear dynamics may be known or
at least partially known. The estimation and attenuation of real
external unknown disturbance and uncertainty (or unmodeled
dynamics) can be significantly improved if the (known) non-
linear dynamics could be exploited in design. This motivates
the development of state-space DOBC, in particular NDOBC
techniques for AC motor drives with nonlinear dynamics.
Example 2 (NDOBC): Here again, we consider the speed
regulation problem of the PMSM system (1) as a benchmark
example to illustrate the basic idea of NDOBC. The dynamic
