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Research Article
Fractional active disturbance rejection control
Dazi Li
a,
n
, Pan Ding
a
, Zhiqiang Gao
b
a
Institute of Automation, Beijing University of Chemical Technology, Beijing 100029, PR China
b
Center for Advanced Control Technologies, Cleveland State University, Cleveland, OH 44115, USA
article info
Article history:
Received 3 June 2015
Received in revised form
5 January 2016
Accepted 29 January 2016
Available online 28 February 2016
Keywords:
Fractional active disturbance rejection con
trol (FADRC)
Fractional extended state observer (FESO)
Fractional proportionalderivative con
troller
Linear fractional order system (FOS)
abstract
A fractional active disturbance rejection control (FADRC) scheme is proposed to improve the performance
of commensurate linear fractional order systems (FOS) and the robust analysis shows that the controller
is also applicable to incommensurate linear FOS control. In FADRC, the traditional extended states
observer (ESO) is generalized to a fractional order extended states observer (FESO) by using the fractional
calculus, and the tracking differentiator plus nonlinear state error feedback are replaced by a fractional
proportionalderivative controller. To simplify controller tuning, the linear bandwidthparameterization
method has been adopted. The impacts of the observer bandwidth
ω
o
and controller bandwidth
ω
c
on
system performance are then analyzed. Finally, the FADRC stability and frequencydomain characteristics
for linear singleinput singleoutput FOS are analyzed. Simulation results by FADRC and ADRC on typical
FOS are compared to demonstrate the superiority and effectiveness of the proposed scheme.
&2016 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Fractional calculus is the generalization of ordinary integer
order calculus. Systems described by fractional order calculus are
known as fractional order systems (FOS). Fractional calculus pro
vides a preferable method to describe complicated natural objects
and dynamical processes such as electrical noises, chaotic system,
and organic dielectric materials [1–6]. As a consequence, scientists
show more and more interests in identiﬁcation of FOS [7,8].
Commensurate linear FOS is a special kind of FOS, with a simple
model and proportional orders [9,10].
Controllers with fractional order operator are naturally suitable
for the FOS [11,12]. There are mainly four kinds of fractional order
controllers, which are CRONE (Contrôle Robuste d’Order Non
Entier) controller, TID (Tilt Integral Derivative) controller, frac
tional order PID controller, and fractional order leadlag compen
sator [13–19]. Considering the industrial universal controller
design requirements, such as compact structure, repeatability,
model independence, easy parameter turning and strong robust
ness, active disturbance rejection control (ADRC) provides an
alternative paradigm for FOS control [20–22]. The central objective
of ADRC is to treat the internal and external uncertainties as the
total disturbance and to reject them actively. Compact frame,
effortless turning and sufﬁciently good performance make ADRC
popular in the world of industrial control [23–25]. ADRC was
ﬁrstly used to control FOS in [26], where fractional order is
regarded as a part of the total disturbances, and an extended state
observer (ESO) is used to estimate and reject it. Because the
known or available model information is neglected and under
used, it would require higher observer bandwidth for accurate
state estimation. In this paper, a distinct fractional active dis
turbance rejection control (FADRC) is proposed as a generalized
and enhanced ADRC solution for the FOS. ESO is redesigned as a
fractional one according to the highest fractional order of FOS. The
modiﬁed fractional extended states observer (FESO) not only
accurately estimates the total disturbance but also the fractional
order dynamic states, leading to a reduced observer bandwidth. In
addition, a fractional order PD controller is used to replace the
tracking differentiator and the nonlinear state error feedback.
Although FADRC is designed for commensurate linear FOS ori
ginally, the robustness analyses demonstrate that FADRC is also
appropriate for incommensurate linear FOS. Simulation results
show that FADRC has more inherent superiority and potential for
FOS control.
Due to the difﬁculty brought by the nonlinearity and uncer
tainty, theoretical studies of ADRC are still lagging behind its
industrial applications. Recent research focuses on time domain
convergence, frequency response and describing function in ana
lyzing nonlinearity [27–29]. Stability analysis has been sub
stantially studied for FOS in [30]. An extended root locus method
by Patil [31] provides a simple way to construct root locus of
general FOS and is employed for FADRC analysis and design.
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
http://dx.doi.org/10.1016/j.isatra.2016.01.022
00190578/&2016 ISA. Published by Elsevier Ltd. All rights reserved.
n
Correspondence to: School of Information Science and Technology, Beijing
University of Chemical Technology, P.O. box No. 81, Beijing 100029, PR China.
Tel.: þ86 10 64434930; fax: þ86 10 64437805.
Email address: lidz@mail.buct.edu.cn (D. Li).
ISA Transactions 62 (2016) 109–119
The FOS is translated into its integer order counterpart and then
analysis method of general integer system can be directly adopted.
The rest of this paper is organized as follows. In Section 2,an
introduction of commensurate linear FOS and fractional order
state observer with fulldimensionality are presented. In Section 3,
the framework of FESO and FADRC and the corresponding algo
rithm are introduced. Section 4 presents the stability and
frequencydomain characteristics of FADRC. Simulation results of
FADRC and ADRC are then compared in Section 5. Finally, con
clusions are given in Section 6.
2. Fractional order systems and fractional order state observer
with fulldimensionality
The conventional integer order single input single output
(SISO) transfer function can be extended to the case of the FOS.
There exist various alternative deﬁnitions of the fractional deri
vative, and the Grünwald–Letnikov (GL), Riemann–Liouville (RL)
and Caputo deﬁnitions are mainly used. In all of the three deﬁni
tions, the fractional operator acts as a nonlocal operator, and that
is to say fractional derivatives have a memory of the past values.
It is difﬁcult to directly implement the fractional order operator
in timedomain for the complicated FOS by using the standard
deﬁnitions. To solve this problem, the normative integerorder
operators are applied to approximate fractional order operators. A
lot of works and researches have been done in this area. Piché
gives the discretetime approximations of fractional order opera
tors based on numerical quadrature [32]. Freeborn proposed a
method to reduce the secondorder approximation ripple error of
the fractional order differential operator [33]. The approximation
listed in [34] is adopted in this paper, which is based on network
theory approximations, and the approximation can give desired
accuracy over any frequency band.
2.1. Transfer function representation
A SISO linear FOS can be described as [35]:
ytðÞþX
n
i¼0
a
it
0
D
t
ϕ
i
ytðÞ¼X
m
j¼0
b
jt
0
D
t
φ
j
utðÞ;ð1Þ
where utðÞis the input, ytðÞis the output, ϕ
i
1rirnðÞand φ
j
0rjrmðÞare real positive numbers, and ϕ
1
oϕ
2
⋯oϕ
n
;φ
1
o
φ
2
⋯oφ
m
φ
m
oϕ
n
. Model coefﬁcients a
i
1rirnðÞand b
j
0rjð
rmÞare constants.
t
0
D
α
t
α¼φ
i
or α¼φ
j
is a fractional order
differential operator, t
0
and tdenote the upper and lower limit of
the integral interval, respectively. The Caputo's fractional deriva
tive of order αwith variable tand starting point t
0
¼0isdeﬁned as
follows:
0
D
α
t
ytðÞ¼ 1
Γ1γ
Z
t
0
y
mþ1ðÞ
τðÞ
tτðÞ
γ
dτ;ð2Þ
where ΓZðÞis Euler's gammafunction, and α¼mþγ;mAN
þ
;
0oγr1. In the fractional differential Eq. (1), if the order
differentiations are integer multiples of a single based order: i.e.
ϕ
i
¼iα;φ
j
¼jα, the system will be termed as commensurate order
and takes the following form:
ytðÞþX
n
i¼0
a
i0
D
ti
α
ytðÞ¼X
m
j¼0
b
j0
D
tj
α
utðÞ:ð3Þ
With zero initial conditions, the Laplace transform of Eq. (3)
becomes
GsðÞ¼YsðÞ
UsðÞ¼P
m
j¼0
b
j
s
j
α
1þP
n
i¼1
a
i
s
i
α
:ð4Þ
2.2. Fractional order state observer with fulldimensionality
As is well known, a commensurate FOS admits the following
statespace representation:
0
D
α
t
xðtÞ¼AxðtÞþBuðtÞ
yðtÞ¼CxðtÞ;
ð5Þ
where matrix A;B;and Care constants. A new fractional order
state observer with fulldimensionality for the commensurate
linear FOS is obtained by generalizing the classical Luenberger
state observer [36], and the structure of the state observer is
shown in Fig. 1. where Lis the undetermined coefﬁcient matrix,
and 1=s
α
represents the fractional order integer operator. The
observer error can be expressed as E¼X^
X, where Xis the actual
state and ^
Xis the estimated state.
0
D
α
t
E¼ALCðÞE:ð6Þ
When the eigenvalues of matrix (A–LC) stay in the stable
region, Eq. (6) will be asymptotically stable (refer to Section 4 for
details).
3. The structure of the fractional active disturbance rejection
controller
The traditional ADRC consists of three main parts: the tracking
differentiator (TD), the ESO, and the nonlinear state error feedback
(NLSEF). TD is used to provide the transient process and its deriva
tive of the input signal. ESO is used to estimate the states plus the
total disturbance. After the aforementioned state variables have
been obtained, NLSEF is applied to combine them and obtain the
control signal. By the efforts of Gao, the bandwidth
parameterization method was proposed to linearize the ESO and
PD controller without losing high precision and efﬁciency [37,38].In
this section, the structure of FADRC is presented. Compared with
traditional ADRC, ESO is replaced by a FESO while NLSEF is replaced
by a linear fractional PD controller. A secondorder FADRC is shown
in Fig. 2,wherev
0
tðÞand ytðÞrepresent the setpoint and output,
respectively, while u
0
tðÞdenotes the output of linear fractional PD
controller, utðÞis the control signal, and wtðÞis the external dis
turbance. Particularly, z
1
tðÞ,z
2
tðÞand z
3
tðÞare the outputs of FESO,
and _
bis a systemdependent coefﬁcient.
3.1. Design of fractional extended states observer
A second order linear FOS with commensurate order
α
is
assumed as follows
YsðÞ
UsðÞ¼b
s
2
α
þa
2
s
α
þa
1
:ð7Þ
1/s
α
+
+
+
+
+
X
ˆ
X
U
Y
B
A
C
1/s
α
L
B
ALC
Fig. 1. Block diagram of fractional order state observer.
D. Li et al. / ISA Transactions 62 (2016) 109–119110
where a
2
;a
1
;band α0oαo1ðÞare constants. Its differential
equation form is
y
2
α
ðÞ
¼a
2
y
α
ðÞ
a
1
y
þbu ¼fy
α
ðÞ
;y;t
þbu:ð8Þ
Particularly, let x
1
¼y;x
2
¼y
α
ðÞ
and x
3
¼fy
α
ðÞ
;y;t
, among
which x
1
;x
2
represent the system states and x
3
is the external
state. Then, the augmented state space form of Eq. (7) can be
represented as:
x
α
ðÞ
¼AxþBuþEh
y¼Cx ;
(ð9Þ
where
x
α
ðÞ
¼
x
1
α
ðÞ
x
2
α
ðÞ
x
3
α
ðÞ
2
6
43
7
5;x¼
x
1
x
2
x
3
2
6
43
7
5;A¼
010
001
000
2
6
43
7
5;B¼
0
b
0
2
6
43
7
5;
C¼100
;E¼
0
0
1
2
6
43
7
5;and h¼f
α
ðÞ
∙ðÞ
A linear FESO is designed to estimate the states x
1
,x
2
, and x
3
in
the following.
z
α
ðÞ
¼Azþ_
BuþLy^
y
^
y¼Cz ;
(ð10Þ
w(t)
Linear fractional
PD controller
Fractional Extended
State Observer
Plant
1
b

0(t)u(t)u(t)y
()
0
vt
()
1
zt ()
2
zt
()
3
zt
Fig. 2. Conﬁguration of a secondorder FADRC.
ωo=5
0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1.0
1.2
t
output
output
FESO z1
0 1 2 3 4 5
15
10
5
0
t
output
disturb ance
FESO z3
ωo=50
ωo=500
ωo=1000
0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1.0
1.2
t
output
output
FESO z1
0 1 2 3 4 5
15
10
5
0
t
output
disturbance
FESO z3
0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1.0
1.2
t
output
output
FESO z1
0 1 2 3 4 5
15
10
5
0
t
output
disturbance
FESO z3
0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1.0
1.2
t
output
output
FESO z1
0 1 2 3 4 5
20
10
0
10
20
t
output
disturbance
FESO z3
Fig. 3. Tracking responses with different ω
o
. (a) ω
o
¼5, (b) ω
o
¼50, (c) ω
o
¼500, and (d) ω
o
¼1000.
D. Li et al. / ISA Transactions 62 (2016) 109–119 111
where
z
α
ðÞ
¼z
1
α
ðÞ
z
2
α
ðÞ
z
3
α
ðÞ
hi
T
;z¼z
1
z
2
z
3
T
;_
B¼0_
b0
hi
, and
L¼β
1
β
2
β
3
hi
T
are observer gains. z
1
,z
2
and z
3
are the outputs
of FESO: z
1
is the estimation of the state x
1
,z
2
is the estimation of
the state x
2
, and z
3
is used to estimate the total disturbance x
3
.In
addition, _
bis the estimated value of b, and _
bb. To simplify the
tuning process, the bandwidthparameterization method [39] is
employed. According to wplane mapping (mentioned in Section
4), the FOS is translated into an integer order system in wplane.
By placing the poles of the translated characteristic equation λwðÞ
in one location, the following is obtained
λwðÞ¼w
3
þβ
1
w
2
þβ
2
wþβ
3
¼wþω
o
ðÞ
3
ð11Þ
where, the observer gains can be linearized as
β
1
¼3ω
o
β
2
¼3ω
o2
β
3
¼ω
o3
;
8
>
<
>
:ð12Þ
For the integer order system, variable
ω
o
is referred to as the
bandwidth of ESO. When it comes to FOS,
ω
o
possesses the
bandwidth characteristics. In order to facilitate distinction, the
variable
ω
o
in (11) is considered to be the wplane bandwidth of
FESO. The main objective of FESO is to estimate the total dis
turbance in real time, and wider wplane bandwidth will result in
faster response. In practice, however, upper limitation of the
bandwidth is related with the sampling ratio, and exceeding the
limit will magnify sensor noises and dynamic uncertainties. A
welltuned
ω
o
must therefore make a balance between rapidity
and stability [26,39,40]. The following Eq. (13) is used to test the
FESO single parameter
ω
o
. Both of the two poles 0:31257
1:0735istay in the stable region (refer to Section 4 for details), that
means the system of Eq. (13) is openloop stable. Parameters
ω
c
¼20,_
b¼b¼1:25 and
ω
o
is set as {5, 50, 500, 1000}. The
tracking responses are shown in Fig. 3
YsðÞ
UsðÞ¼1:25
s
1:8
þ0:625s
0:9
þ1:25:ð13Þ
Fig. 3 shows that estimating ability of FESO is strengthened
with the increase of the observer wplane bandwidth. However,
outputs of FESO become unstable when the wplane bandwidth is
beyond the toplimit. The upper limit is related to the sampling
rate, and a higher sampling rate leads to a higher upper limit.
3.2. Design of fractional PD controller
Referring to Fig. 3, a welltuned observer can track the extended
state fy;y
α
ðÞ
;ω
accurately. The control law can be designed as
u¼z
3
þu
0
_
b;ð14Þ
to obtain a desired response, where u
0
is a common linear frac
tional PD control:
u
0
¼k
p
ðv
0
z
1
Þþk
d
v
0
α
ðÞ
z
2
;ð15Þ
where k
p
and k
d
are controller gains. The parameters tuning is
further simpliﬁed using the method in [39]
k
d
¼2ω
c
k
p
¼ω
2
c
(ð16Þ
where
ω
c
is the wplane bandwidth of the controller. Time
derivative of the setpoint is omitted to avoid the pulse in [39].
However, because the fractional order differentiation of the step
signal is a gradual process rather than the pulse signal, v
0
α
ðÞ
cannot
be omitted in (15). Then Eq. (7) becomes a commensurate cascade
fractional order integrator form as
y
2
α
ðÞ
u
0
:ð17Þ
Considering Eqs. (15) and (17), the following closedloop
transfer function can be obtained, namely, the desired response
of FADRC is
YsðÞ
RsðÞk
d
s
α
þk
p
s
2
α
þk
d
s
α
þk
p
:ð18Þ
0 1 2 3
0
0.2
0.4
0.6
0.8
1.0
1.2
t
output
Fig. 4. Desired response and system outputs with different ω
o
.
0 1 2 3
0
0.2
0.4
0.6
0.8
1
1.2
t
output
Fig. 5. System outputs with different ω
c
.
Re w
Im w
sheet 1
sheet 2
sheet 3
sheet v
sheet v1
απ
Fig. 6. The correspondence between the wplane and sRiemann sheets.
D. Li et al. / ISA Transactions 62 (2016) 109–119112
Eq. (13) is taken as an example again, with ﬁxed parameter
_
b¼1:25. Fig. 4 shows the desired response and the output results
with different
ω
o
(
ω
c
¼20 is ﬁxed). Fig. 5 shows the output
responses with different controller wplane bandwidth
ω
c
(
ω
o
¼200
is ﬁxed).
As Fig. 4 shows, the output results tracks closely to the desired
response with the increase of
ω
o
. This is because the larger
observer wplane bandwidth can ensures precise disturbance
estimation and then compensator can be applied to reject the total
disturbance. Fig. 5 shows that small
ω
c
makes the system response
slower, while large
ω
c
makes it faster.
In a practical application, the design procedure consists of two
stages. In the ﬁrst stage, a linear FESO is designed and a welltuned
ω
o
is selected to ensure accurate estimations. In the second stage,
a fractional order PD controller is designed under the assumption
that the total disturbance is well estimated, and all the existing
methods could be used in this stage for designing linear or non
linear controllers.
4. Root locus and frequency domain analysis for fractional
active disturbance rejection control
A FOS can be generally expressed in the following form:
HsðÞ¼
b
m
s
m
v
þ⋯þb
1
s
1
v
þb
0
a
n
s
n
v
þ⋯þa
1
s
1
v
þa
0
:ð19Þ
where a
k
k¼0;⋯;nðÞ;b
k
k¼0;⋯;mðÞare constants, and v41. The
Riemann sheets are determined by using
s¼s
jj
e
j
ϕ
:ð20Þ
where 2kþ1ðÞ
πoθo2kþ3ðÞπ;and k¼1;0;⋯;v2. The Rie
mann sheet is named as the Principal Riemann Sheet when
k¼1:Note that only roots lying on the Principal Riemann Sheet
can determine the timedomain behavior and stability perfor
mance. These sheets are mapped to the wplane as deﬁned by
w¼wjje
j
θ
ð21Þ
where w¼s
α
and α¼1=v. The sheets can be projected to the w
plane by
α2kþ1
ðÞ
πoθoα2kþ3
ðÞ
πð22Þ
The correspondence between the wplane and sRiemann
sheets is shown in Fig. 6. With the transformation of the wplane,
the stability of FOS can be predicted by the trend of root locus. The
region of instability απ=2rarg wðÞrαπ=2
in the wplane
corresponds to the right half plane π=2rarg sðÞrπ=2
in the s
plane. The root locus branches never enter the unstable region,
which implies the system remains stable. Otherwise, if the root
locus branches never enter the stable region, the system remains
unstable. If the branches move from the stable region to the
unstable region (or move from the unstable region to the stable
region), then the range of grain can be determined [30,31,41,42].
4.1. Root locus of linear fractional order systems
The steps to plot the root loci of FOS are listed as follow:
a) Attain the open loop transfer function of FOS;
b) Transform FOS into an integer order system on the wplane;
c) Obtain the root locus of the transformed system;
d) Identify the Principal Riemann Sheets and the unstable
region on the wplane;
e) Perform stability analysis from the root locus on the wplane.
In order to analyze the stability of FADRC, the FADRC time
domain conﬁguration shown in Fig. 2 is changed into a frequency
domain block diagram form [42].AsFig. 7 shows, V
0
(s) and Y(s)
represent the setpoint and output, respectively;
Ω
denotes the
external disturbance; G
p
(s) is the transfer function of a commen
surate linear FOS.
The other transfer functions in the blocks are listed in Appendix A.
By the equivalent transformation of the block diagram, the closeloop
transfer function can be obtained as follows when the effect of the
external disturbance
Ω
is omitted.
Φ
r
sðÞ¼YsðÞ
V
O
sðÞ¼G
r1
G
c
G
p
þG
r2
G
p
_
bþG
p
G
f1
þG
c
G
p
G
f2
:ð23Þ
and
Taking Eq. (7) as an example, the parameters of FADRC are
designed using Eqs. (12) and (16). Then, the open loop transfer
function of Φ
r
sðÞcan be written as
Φ
o
sðÞ¼ Φ
r
sðÞ
1Φ
r
sðÞ
:ð25Þ
and
Φ
o
sðÞ¼ k
d
s
4
α
þk
p
þk
d
β
1
s
3
α
þk
p
β
1
þk
d
β
2
s
2
α
þk
d
β
3
þk
p
β
2
s
α
þk
p
β
3
_
b
G
p
s
3
α
þ
β
1
þk
d
s
2
α
þ
β
1
k
d
þ
β
2
þk
p
s
α
k
d
s
4
α
k
p
þk
d
β
1
s
3
α
þ
β
3
s
2
α
ð26Þ
For different kinds of objects, two categories are studied to
discuss their root loci:
4.1.1. For the commensurate fractional order system
When G
p
sðÞ¼
b
s
2
α
þa
2
s
α
þa
1
and
_
b
b
1, the open loop transfer
function can be written as
Fig. 7. Block diagram of the FADRC in frequency domain.
Φ
r
sðÞ¼ k
d
s
4
α
þk
p
þk
d
β
1
s
3
α
þk
p
β
1
þk
d
β
2
s
2
α
þk
d
β
3
þk
p
β
2
s
α
þk
p
β
3
_
b
G
p
s
3
α
þβ
1
þk
d
s
2
α
þβ
1
k
d
þβ
2
þk
p
s
α
þk
p
β
1
þk
d
β
2
þβ
3
s
2
α
þk
p
β
2
þk
d
β
3
s
α
þk
p
β
3
ð24Þ
D. Li et al. / ISA Transactions 62 (2016) 109–119 113
Fig. 8 shows the root locus when
α
¼0.9,a
1
¼1.25, a
2
¼0.625,
_
b¼b¼1.25,
ω
o
¼100 and
ω
c
¼20, the open loop poles and zeros
are given in Table 1.
4.1.2. For the incommensurate fractional order system
When G
p
sðÞ¼
b
s
2
α
þa
2
s
α
þ
δ
þa
1
and
_
b
b
1, the open loop transfer
function can be written as
Fig. 9 showstherootlocuswhen
α
¼0.9,
δ
¼0.3 a
1
¼1.25,
a
2
¼0.625,_
b¼b¼1.25,
ω
o
¼100 a nd
ω
c
¼20, the open loop poles
and zeros are given in Table 1.Fig. 10 shows the root locus
when
α
¼0.9,
δ
¼0.3 a
1
¼1.25, a
2
¼0.625, _
b¼b¼1.25,
ω
o
¼100
and
ω
c
¼20, and the open loop poles and zeros are given in
Table 1.
From the root loci in Figs. 8–10, it is clear that all the curves are
in the stable region and have no intersections with the stability
boundary. In the other words, the system remains stable for all
gain values. Although the objects are not the standard commen
surate FOS (
δ
a0), the proposed method is also applicable.
4.2. Frequencydomain characteristics analysis
Frequency domain characteristics are widely used to analyze
automatic control systems in classical control theory. Bode
250 200 150 100 50 050
40
30
20
10
0
10
20
30
40
Root Locus
Real Axis (seconds1)
Imaginary Axis (seconds1)
Stability
Boundary
Fig. 8. Root locus when α¼0.9, a
1
¼1.25, a
2
¼0.625, _
b¼b¼1.25, ω
o
¼100 and
ω
c
¼20, where the dotted line is the stability boundary.
Table 1
Open loop zeros and poles for different objects.
Objects Instability region Zeros Poles
α¼0:9
a
1
¼1:25
a
2
¼0:625
δ¼0(Fig. 8)
9
20
πrarg wðÞr
9
20
πw
1;2;3
¼100
w
4
¼10
w
1
¼0
w
2
¼121:4
w
3;4
¼89:6720:5i
w
5
¼0:05
b¼1:25
_
b¼1:25
δ¼0:3(Fig. 9)
3
20
πrarg wðÞr
3
20
πw
1;2;3
¼4:64
w
4;5;6
¼2:32þ4:02i
w
7;8;9
¼2:324:02i
w
10
¼2:15
w
11;12
¼1:0871:87i
w
1;2;3
¼0
w
4;5
¼4:7670:17i
w
6
¼4:40
w
7;8
¼2:4774:14i
w
9;10
¼2:1474:11i
w
11;12
¼2:3773:80i
w
13
¼0:38
w
14;15
¼0:1870:33i
ω
o
¼100
ω
c
¼20
δ¼0:3(Fig. 10)
3
20
πrarg wðÞr
3
20
πw
1;2;3
¼4:64
w
4;5;6
¼2:32þ4:02i
w
7;8;9
¼2:324:02i
w
10
¼2:15
w
11;12
¼1:0871:87i
w
1;2;3
¼0
w
4;5
¼2:4374:51i
w
6;7
¼2:8373:73i
w
8;9
¼1:6673:88i
w
10;11
¼4:9670:42i
w
12
¼3:92
w
13
¼0:38
w
14;15
¼0:1970:32i
12 10 8 6 4 2 0 2 4 6
10
5
0
5
10
Root Locus
Real Axis (seconds1)
Imaginary A xis (seconds1)
Stability Boundary
Fig. 9. Root locus when α¼0.9, δ¼0.3 a
1
¼1.25, a
2
¼0.625, _
b¼b¼1.25, ω
o
¼100
and ω
c
¼20, where the dotted line is the stability boundary.
Φ
o
sðÞ¼ k
d
s
4
α
þk
p
þk
d
β
1
s
3
α
þk
p
β
1
þk
d
β
2
s
2
α
þk
d
β
3
þk
p
β
2
s
α
þk
p
β
3
s
5
α
þβ
1
þa
2
s
4
α
þβ
2
þβ
1
a
2
þk
d
a
2
þa
1
s
3
α
þβ
1
k
d
a
2
þβ
2
a
2
þk
p
a
2
þβ
1
a
1
þk
d
a
1
þβ
3
s
2
α
þβ
1
k
d
a
1
þβ
2
a
1
þk
p
a
1
s
α
ð27Þ
Φ
o
sðÞ¼ k
d
s
4
α
þk
p
þk
d
β
1
s
3
α
þk
p
β
1
þk
d
β
2
s
2
α
þk
d
β
3
þk
p
β
2
s
α
þk
p
β
3
_
b
b
s
2
α
þa
2
s
α
þ
δ
þa
1
s
3
α
þβ
1
þk
d
s
2
α
þβ
1
k
d
þβ
2
þk
p
s
α
k
d
s
4
α
k
p
þk
d
β
1
s
3
α
þβ
3
s
2
α
ð28Þ
D. Li et al. / ISA Transactions 62 (2016) 109–119114
diagram is an important component for the frequencydomain
analysis. The closedloop stability, rapidity and accuracy can be
analyzed based on the bode diagram. A basic feedback frame can
be obtained from the frequencydomain block diagram of Fig. 7,
and the equivalent openloop transfer function is described in Eq.
(26). Taking Eq. (13) as an example, the Bode diagram with dif
ferent
ω
c
(
ω
o
¼200 is ﬁxed), is shown in Fig. 11(a) and the bode
diagram with different
ω
o
(
ω
c
¼50 is ﬁxed) is shown in Fig. 11(b).
Fig. 11(a) shows that the crossover frequency becomes bigger
with the increase of
ω
c
. This means that a bigger
ω
c
makes the
system response more quickly. In addition, the phase stability
margin keeps almost unchanged, which ensures the stability of the
system. It can be seen from Fig. 11(b) that the system almost has
almost the same bode diagram in the low frequency interval for
different
ω
o
, which means the observer wplane bandwidth
ω
o
has little effect on the stability.
5. Simulation and discussion
In this section, simulation results for four different FOS are
implemented to verify the superiority and effectiveness of the
proposed method. The methods in [43] and [26] are adopted to
design ADRC for FOS control (see Table 2). In Table 2, comparative
results with method in [43] are listed in No. 1 and comparative
results with method in [26] are listed in No. 2–5. For the sake of
fairness, the transition process of the setpoint generated by TD is
added in FADRC and ADRC, and the second order TD can be
designed as [21]
15 10 5 0510
15
10
5
0
5
10
15
Root Locus
Real Axis (seconds1)
Imaginary A xis (seconds1)
Stability Boundary
Fig. 10. Root locus when α¼0.9, δ¼0.3 a
1
¼1.25, a
2
¼0.625, b¼1.25, ω
o
¼100 and
ω
c
¼20. The dotted line is the stability boundary.
Fig. 11. Bode diagrams with different parameters. (a) Bode diagram with different ω
c
(ω
o
¼200 is ﬁxed). (b) Bode diagram with different ω
o
(ω
c
¼50 is ﬁxed).
D. Li et al. / ISA Transactions 62 (2016) 109–119 115
_
v
1
¼v
2
_
v
2
¼f han v
1
v
0
;v
2
;r;hðÞ
;
(ð29Þ
where v
0
is the setpoint, v
1
is the transition process of v
0
,v
2
is the
differential trajectory of v
1
,rand hare adjustable parameters, and
the function f han ∙ðÞis deﬁned as:
f han x
1
;x
2
;r;hðÞ¼
rsign aðÞ;ajj4d
r
a
d
;ajjrd;
(ð30Þ
where aand dare given as follows:
d¼rh;
d
0
¼hd;
y¼x
1
þhx
2
;
a
0
¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
d
2
þ8ry
q;
a¼x
2
þ
a
0
dðÞ
2
sign yðÞ;y
4d
0
x
2
þ
y
h
;y
rd
0
:
8
<
:
8
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
:
ð31Þ
All the parameters and assessment indices are listed in Table 2,
where Tis the sampling time; ISE is the short for integral square
error: ISE tðÞ¼R
t
0
ytðÞv
1
tðÞðÞ
2
dt.
5.1. A commensurate nonlinear fractional order system
The following commensurate nonlinear FOS is used to compare
the performance between FADRC and the Gao's method [43].
D
0:8
x
1
ðtÞ¼x
2
ðtÞ
D
0:8
x
1
ðtÞ¼ sin x
2
ðtÞðÞþx
2
ðtÞþuðtÞ
yðtÞ¼x
1
ðtÞ
:
8
>
<
>
:ð32Þ
The reference input is v(t)¼5, and the parameters of FADRC
and Gao's method are listed in Table 2.Fig. 12(a) shows the tran
sition process and closed loop output of Eq. (32), and Fig. 12
(b) shows the control signals.
As Fig. 12 shows, FADRC can track the transition process with
less oscillation, and FADRC's control signal is also reasonable. It can
be seen that the commensurate nonlinear FOS is well controlled by
FADRC. In addition, the total variation(TV) of the control effort is
used to compare the performance of different controllers. The
deﬁnition of TV index is given as follows:
TV ¼X
1
i¼1
ju
iþ1
u
i
j:ð33Þ
TV obtained for FADRC is 26.1 and that for Gao's method is 44.1,
which shows that FADRC owns smaller control effort. This is
beneﬁcial for practical application.
5.2. Gas turbine plant
The model of the fractional order gas turbine plant, which
converts the fuel energy into an useful form, is given by Nataraj
[44]. The input and output of the gas turbine are fuel rate and
turbine speed, respectively. For the operating regime at 90% rated
speed, the fractional order model is
G
90%
sðÞ¼ 103:9705
0:00734s
1:6807
þ0:1356s
0:8421
þ1:ð34Þ
For another operating regime at 93% rated speed, the fractional
order model is
G
93%
sðÞ¼ 110:9238
0:0130s
1:6062
þ0:1818s
0:7089
þ1:ð35Þ
Fig. 13 shows the output responses for the operating regime at
90% and 93% of rated speed demand. The welltuned parameters
of FADRC and ADRC are listed in Table 2.
Fig. 13(a) and (c) shows FADRC has faster adjustment capability.
Fig. 13(b) and (d) shows the corresponding control signals.
5.3. Heatsolid model
A heat solid model is described by the following fractional
order transfer function [45]
GsðÞ¼ 1
39:69s
1:26
þ0:598:ð36Þ
Table 2
Controller parameters and performance indices.
No. Objects Parameters {ω
o
,ω
c
,_
b, T}ISE (10
5
)
FADRC Comparative method FADRC Comparative method
1. Commensurate nonlinear FOS {30, 5, 1, 0.001}ω
o
¼30, k
p
¼10, k
i
¼2, k
d
¼1λ¼0.8, _
b¼1, T ¼0.001 3000 45,00 0
2. Gasturbine model (at 90% rated speed) {100, 10, 14165, 0.001}6701000
3. Gasturbine model (at 93% rated speed) {100, 10, 8533, 0.001} 340 660
4. Heatsolid model {300, 10, 0.0252, 0.0001} 35.3 37.5
{30, 10, 0.0252, 0.0001} 35.3 120
5. Solidcore active magnetic bearing {4000, 200, 5600, 0.0001} 1.2 68
0 2 4 6 8 10
0
2
4
6
t
output
transition process
FADRC
Gao's Method
The output y
Control signal
0 2 4 6 8 10
15
10
5
0
5
10
t
control signal
FADRC
Gao's Method
Fig. 12. Outputs comparison between FADRC and Gao's method. (a) The output y.
(b) Control signal.
D. Li et al. / ISA Transactions 62 (2016) 109–119116
The input and output are voltages, and least squares method is
used to identify the unknown parameters. A FADRC is designed
with the commensurate order
α
¼0.63, and the total disturbances
is f
FADRC
¼0:598=39:69y.Aﬁrstorder ADRC/secondorder ESO is
used for comparison. The total disturbance is f
ADRC
¼y
1ðÞ
y
1:26ðÞ
0:598=39:69y. Parameters of FADRC and ADRC are listed in
Table 2. Simulation results with
ω
o
¼300 and
ω
o
¼30 are shown in
Fig. 14(a) and (b), respectively.
It can be seen from Fig. 14 and Table 2 that FADRC has a better
response. It is also important to note that FADRC is able to respond
quickly even with a relatively small bandwidth (see Fig. 14(b)),
which makes the solution more practical.
5.4. Solidcore active magnetic bearing
Due to the eddy current effect, a solidcore active magnetic
bearing (AMB) shows some fractional order characteristics and the
traditional magnetic equivalent circuit model has signiﬁcant
amounts of errors when predicting the actual system. A weighted
leastsquares method was derived for the general fractional model
based on the widely studied commensurate order fractional model
in [46]. The ﬁnal results show that the identiﬁed fractional model
structure is closer to the actual system, and the result is:
GsðÞ¼ 5594:32
s
2:75
þ259:08s
1:83
85950:3s
0:79
14240336:8:ð37Þ
Typically, the AMB model is close to the third order linear FOS
with the commensurate order
α
¼0.9, and a third order FADRC is
therefore adopted for the control of the plant. Parameters of
FADRC and ADRC are listed in Table 2.
Fig. 15 shows that thirdorder FADRC has faster adjustment
capability. Furthermore, FESO can get precise estimations when
the observer wplane bandwidth is increased to 1000. For ESO,
however, the precise estimations are not available until the
observer bandwidth is increased to 4000. This means that FADRC
does not require higher bandwidth or sampling rate and is more
suitable for practical engineering.
5.5. Robust analysis
A robustness assessment is carried out about the parameters of
Eq. (13). The following plant (38) with uncertain parameters
δ
and
ε
are considered:
y
1:8þ
ε
ðÞ
¼
5
8y
0:9þ
δ
ðÞ
10
8y
þ10
8u:ð38Þ
The normalized integral square error (ISE) trajectory tracking
performance indices are used as
ISE
v
tðÞ¼R
t
0
ytðÞv
1
tðÞðÞ
2
dt
ISE
f
tðÞ¼R
t
0
ftðÞz
3
tðÞðÞ
2
dt :
8
<
:ð39Þ
Output y (at 90% rated speed)
Control signal (at 90% rated speed)
Output y (at 93% rated speed)
00.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
1.2
t
output
transition process
FADRC
ADRC
0 0.5 1 1.5 2
0
0.004
0.008
0.012
t
control signal
FADRC
ADRC
00.5 11.5 2
0
0.2
0.4
0.6
0.8
1.0
1.2
t
output
transition process
FADRC
ADRC
Control signal (at 93% rated speed)
0 0.5 1 1.5 2
0
0.004
0.008
0.012
t
control signal
FADRC
control signal ADRC
Fig. 13. Response comparisons between FADRC and ADRC on gasturbine model.
(a) Output y (at 90% rated speed). (b) Control signal (at 90% rated speed). (c) Output
y (at 93% rated speed). (d) Control signal (at 93% rated speed).
Output y (ωo=300)
Output y (ωo=30)
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1.0
1.2
t
output
transition process
FADRC
ADRC
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1.0
1.2
t
output
transition process
FADRC
ADRC
Fig. 14. Response comparison between FADRC and ADRC on heatsolid model.
(a) Output y (ω
o
¼300). (b) Output y (ω
o
¼30).
D. Li et al. / ISA Transactions 62 (2016) 109–119 117
where v
1
tðÞis the transition process of the setpoint and ftðÞis the
total disturbance. Considering the dynamic characteristic of the
system and the accuracy of the identiﬁcation method, the varia
tion of uncertain parameter
ε
is set between 0.25 and 0.25. As
the second item produces small effects on the system, and para
meter
δ
is set between 0.5 and 0.5. Figs. 16 and 17 show the
trajectories of ISE
v;
δ
tðÞand ISE
f;
δ
tðÞfor different uncertain para
meters of
δ
in step of 2
ζ
, when
ε
¼0isﬁxed and ζ¼ﬃﬃﬃ
3
pﬃﬃﬃ
2
p
=6.
Fig. 17 and Fig. 18 show the trajectories ISE
v;
ε
tðÞand ISE
f;
ε
tðÞfor
different uncertain parameters of
ε
in step of
ζ
, when
δ
¼0isﬁxed.
From Fig. 16 to Fig. 19, the following conclusions can be
obtained. 1). For different uncertain parameters
δ
,Figs. 16 and 17
show that FADRC owns smaller ISE value in both the output and
disturbance trajectories. This means that FADRC could track tra
jectory (setpoint or total disturbance) more quickly and precisely.
The constant steady states of these loci indicate that the corre
sponding outputs are ultimately close to the desired reference
trajectories. 2). When the uncertain parameter
ε
changes, the
advantages of FADRC are distinct. Figs. 18 and 19 show that FADRC
obtains better performance when the ﬁrst item 1.8þ
ε
is changed,
while FESO is affected by the modiﬁcation of
ε
. When 1.8þ
ε
is
closer to 2, the ESO achieve a better performance. This is a good
example to prove that FADRC is necessary for a system with an
integerdistancing fractional order. 3). The highest order of FOS
plays an important role in FADRC designing for different system
because the nohighest items are treated as the total disturbance
and rejected by the controller. 4). More importantly, FADRC could
track the trajectory quickly and precisely even if there exist a large
change of uncertain parameter
δ
and
ε
, which means the FADRC is
also appropriate for incommensurate linear FOS.
6. Conclusion
The traditional ADRC solution for FOS is improved in this paper,
where the ESO and the NLSEF are replaced by FESO and PD
α
Fig. 16. Behavior of the output ISE trajectories for different values of the uncer
tainty order δ, when ε¼0isﬁxed (Red full line indicates FADRC, blue dotted line
indicates ADRC and the step is 2ζ. For interpretation of the references to color in
this ﬁgure legend, the reader is referred to the web version of this article.)
Fig. 17. Behavior of the disturbance ISE trajectories for different values of uncer
tainty order δ, when ε¼0isﬁxed (Red full line indicates FADRC, blue dotted line
indicates ADRC and the step is 2ζ. For interpretation of the references to color in
this ﬁgure legend, the reader is referred to the web version of this article.)
Output y
Control signal
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
1.2
t
output
transition process
FADRC
ADRC
0 0.5 1 1.5 2
3000
2000
1000
0
1000
t
control signal
FADRC
ADRC
Fig. 15. Response comparison between FADRC and ADRC on solidcore active
magnetic bearing. (a) Output y. (b) Control signal.
Fig. 19. Behavior of the disturbance ISE trajectories for different values of uncer
tainty order ε, when δ¼0isﬁxed (Red full line indicates FADRC, blue dotted line
indicates ADRC and the step is ζ. For interpretation of the references to color in this
ﬁgure legend, the reader is referred to the web version of this article.)
Fig. 18. Behavior of the output ISE trajectories for different values of the uncer
tainty order ε, when δ¼0isﬁxed (Red full line indicates FADRC, blue dotted line
indicates ADRC and the step is ζ. For interpretation of the references to color in this
ﬁgure legend, the reader is referred to the web version of this article.)
D. Li et al. / ISA Transactions 62 (2016) 109–119118
controller respectively. Linear bandwidthparameterization method
is applied to simplify the parameters tuning. In addition, stability
and frequencydomain characteristics of FADRC for FOS are also
analyzed. Numerical simulations show the superiority and effec
tiveness of the proposed scheme over the existing ADRC solution.
Moreover, robustness analysis shows that FADRC is also appropriate
for incommensurate FOS control. Furthermore, it is believed that
further improvements can be obtained by employing the nonlinear
gains in FADRC but it would require a thorough theoretical study of
necessity.
Acknowledgments
This research has been supported by the National Natural Sci
ence Foundation of P.R. China (Grant No. 61573052).
Appendix: A
The transfer functions of the blocks in Fig. 7 are
G
r1
sðÞ¼ k
d
s
α
þk
p
s
2
α
þβ
1
þk
d
s
α
þβ
1
k
d
þβ
2
þk
p
A1
G
r2
sðÞ¼k
d
s
3
α
þk
p
þk
d
β
1
s
2
α
þk
p
β
1
þk
d
β
2
s
α
þk
p
β
2
s
2
α
þβ
1
þk
d
s
α
þβ
1
k
d
þβ
2
þk
p
A2
G
f1
sðÞ¼ k
p
β
1
þk
d
β
2
s
α
þk
p
β
2
s
2
α
þβ
1
þk
d
s
α
þβ
1
k
d
þβ
2
þk
p
A3
G
f2
sðÞ¼ s
2
α
þk
d
s
α
þk
p
s
2
α
þβ
1
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