Article
Fuzzy observer–based disturbance
rejection control for nonlinear fractionalorder systems with time-varying delay
Journal of Vibration and Control
2021, Vol. 0(0) 1–10
© The Author(s) 2021
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DOI: 10.1177/10775463211006958
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Parvin Mahmoudabadi and Mahsan Tavakoli-Kakhki
Abstract
In this article, a Takagi–Sugeno fuzzy model is applied to deal with the problem of observer-based control design for
nonlinear time-delayed systems with fractional-order α 2 ð0; 1Þ. By applying the Lyapunov–Krasovskii method, a fuzzy
observer–based controller is established to stabilize the time-delayed fractional-order Takagi–Sugeno fuzzy model. Also,
the problem of disturbance rejection for the addressed systems is studied via the state-feedback method in the form of
a parallel distributed compensation approach. Furthermore, sufficient conditions for the existence of state-feedback gains
and observer gains are achieved in the terms of linear matrix inequalities. Finally, two numerical examples are simulated for
the validation of the presented methods.
Keywords
Fractional-order systems, T-S fuzzy model, time-varying delay, observer-based control, disturbance rejection
1. Introduction
Fractional-order systems are a natural generalization of
classical integer-order systems (Behinfaraz et al., 2019;
Zhang et al., 2019). The advantages of using fractionalorder for nonlinear systems are having an unlimited
memory and describing a real object more accurately than
the classical integer calculus methods (Yi et al., 2019; Jafari
et al., 2017a). Motivated by these prominent advantages,
fractional-order systems have received noticeable attention
in control engineering (Binazadeh and Yousefi, 2018;
Moradi et al., 2019). Moreover, nonlinearities in dynamic
systems cause some difficulties in control engineering
problems. One of the approaches to overcome these difficulties is using the T-S fuzzy model (Takagi and Sugeno,
1985). The T-S fuzzy model that is an efficient approach to
describe nonlinear systems has been presented by Takagi
and Sugeno in 1985 (Takagi and Sugeno, 1985). Fuzzy
sector nonlinearity which is a method to convert a nonlinear
system into the T-S fuzzy model can exactly describe
a nonlinear system as a set of local linear subsystems
blended by fuzzy membership functions (Tanaka and Wang,
2004). The T-S fuzzy model has been applied for solving
many control problems (Jafari et al., 2017b). Also, T-S
fuzzy control has been firstly extended to control fractionalorder systems in Zheng et al. (2010).
On the other hand, state estimation of dynamic systems is
very important in various control problems, such as fault
detection and tracking and state-based control (Li et al.,
2020; Tian et al., 2020). In a significant number of research
works, the problem of observer-based control design for
nonlinear integer-order systems (Majumder and Patre,
2019, Yu et al., 2020) and nonlinear fractional-order systems has been investigated (Selvaraj et al., 2019; Tian et al.,
2020). Furthermore, many practical systems include time
delay such as hydraulic systems, chemical processes, and
communication channels (Mahmoudabadi et al., 2017;
Binazadeh and Yousefi, 2018; Moradi et al., 2019). Time
delay causes instability, oscillation, and poor performance.
Therefore, time-delayed systems have achieved a lot of
attention in recent years (He et al., 2021). A lot of effective
research has been presented dealing with the state estimation problem for nonlinear integer-order systems with
time delay (Pourdehi and Karimaghaee, 2020; Sun et al.,
2020). However, state estimation for nonlinear fractionalorder systems with time delay requires more research (Trinh
Faculty of Electrical Engineering, K.N. Toosi University of Technology, Iran
Received: 30 October 2020; accepted: 10 March 2021
Corresponding author:
Mahsan Tavakoli-Kakhki, Department of Electrical Engineering, KN Toosi
University of Technology, Shariati Street, Seyyed Khandan bridge, Tehran
1631714191, Iran.
Email: matavakoli@kntu.ac.ir
2
Journal of Vibration and Control 0(0)
The rest of this article is constructed as follows. In
et al., 2019). By using feedback control strategy, the
problem of disturbance rejection and dealing with the Section 2, the considered system, lemmas, and definitions
system uncertainties have been investigated for a class of are presented. Observer-based control design for fractionalfractional-order complex dynamical networks with time order T-S fuzzy systems with time-varying delay is indelay in Sakthivel et al. (2019a). In Phat et al. (2020), vestigated based on the state-feedback controller in Section
observer-based control design for nonlinear time-delayed 3. A PDC-disturbance rejection controller is proposed in
fractional-order systems has been investigated. A reduced- Section 4. In Section 5, numerical results are presented.
order state observer has been considered for time-delayed Finally, the article is concluded in Section 6.
fractional-order systems with Lipschitz nonlinearities in
Huong and Thuan (2018). The problem of functional ob2. Preliminaries and problem formulation
server design for nonlinear time-delayed fractional-order
systems has been studied in Thuan et al. (2019). In 2.1. Notations
Parvizian et al. (2020), a non-fragile adaptive sliding mode
Throughout this article, ℜn states the n-dimensional Euobserver design has been investigated for nonlinear timeclidean space, ℜn×m is the set of n × m real matrices, the
delayed fractional-order systems. In Bettayeb et al. (2017),
notation A > ð≥ÞB means that A B is positive-definite
a high gain observer has been introduced for the syn(positive semi-definite), I(0) is the identity (zero) matrix
chronization of fractional-order systems by considering
with appropriate dimension, AT means the transpose of A,
delay in measurements. Although observer-based control
kk denotes the Euclidean norm in ℜn , k:k2 denotes for the
design for nonlinear time-delayed fractional-order systems
l2 norm, and h∗i stands for the elements below the main
based on the T-S fuzzy model is an important subject, it only
diagonal of a symmetric block matrix.
has been studied in Sakthivel et al. (2019b).
In this article, the Caputo fractional differential operator
Stability criteria for time-delayed systems can be cateis applied because the initial conditions for fractional-order
gorized into two types: delay-dependent and delaydifferential equations based on the Caputo derivative are
independent. The delay-independent stability criteria
like the needed initial conditions for solving integer-order
ensure the stability of the system irrespective of the size of
differential equations.
the time delay, whereas the delay-dependent stability criteria are related to the size of delay term (Mahmoudabadi
Definition 1. (Fractional integral (Podlubny (1998)): The
et al., 2017). The delay-dependent stability criteria guarfractional integral is defined as follows
antee the stability of the system for any values of the delay
Z t
term which are less than a specific upper bound. But, delay1
α1
α
I
f
ðtÞ
¼
ðt ηÞ f ðηÞdη, α 2 ℜþ
t0 t
independent criteria are beneficial when time delay is large
ΓðαÞ t0
or its upper bound is unknown (Lin et al., 2007). The
problem of observer-based controller design for time- where Γ() is the gamma function.
delayed fractional-order T-S fuzzy systems based on
delay-dependent criteria has been studied in Sakthivel et al. Definition 2. (Caputo fractional-order derivative (Podlubny
(2019b). But the investigation of this problem based on (1998)): The definition of the Caputo fractional-order dedelay-independent criteria has not been considered in any rivative is introduced as follows
Z t
research work.
1
nα1 ðnÞ
C α
×
D
f
ðtÞ
¼
ðt ηÞ
f ðηÞdη
Based on the above discussion, this article discusses the
t0 t
Γðn αÞ
t0
problem of observer-based disturbance rejection controller
design for nonlinear time-delayed fractional-order systems where n is the first integer which is greater than α, that is,
in the framework of the T-S fuzzy modeling based on n 1 ≤ α ≤ n.
a delay-independent criterion which to the best of the auConsider a class of nonlinear time-delayed fractionalthor’s knowledge is a completely new research issue. The order system which can be represented as a T-S fuzzy model
main objectives of this study are as follows.
as Sakthivel et al. (2019b).
Rule i: IF θ1(t) is M1i, θ2(t) is M2i, …, θp(t) is Mpi, THEN
8C α
1. The T-S fuzzy model is used to deal with the problem
D xðtÞ ¼ Ai xðtÞ þ Adi xðt τðtÞÞ þ Bi uðtÞ
>
>
of controller design for nonlinear fractional-order
>
<
þ Di ωðtÞ, t ≥ 0,
systems with time-varying delay.
(1)
>
2. By applying a delay-independent criterion, observeryðtÞ
¼
C
xðtÞ,
i
>
>
:
based feedback stabilization for fractional-order T-S
xðsÞ ¼ φðsÞ, s 2 ½τ,0
fuzzy systems with time-varying delay is studied.
n
3. A PDC-disturbance rejection controller is proposed to where xðtÞ 2 ℜ is the pseudo-state vector which for simeffectively attenuate the external disturbance in the plicity it is called as state vector in the rest of the article. Ai,
Adi, Bi, Ci, and Di are known matrices, θp ðtÞ denotes premise
system.
Mahmoudabadi and Tavakoli-Kakhki
3
variables, r is the number of rules, p is the number of
premise variables, φ(s) is the continuous vector-valued
initial function on ½τ,0, Mli(l = 1, 2, …, p, i = 1, 2, …, r)
are fuzzy sets, and τ(t) is a time-varying functional
satisfying
0 ≤ τðtÞ ≤ τ
(2)
τ_ ðtÞ ≤ μ < 1
(3)
Using the center-average defuzzifier, the product inference, and the singleton fuzzifier, the following T-S fuzzy
model is obtained
8
r
X
>
>
hi ðθðtÞÞfAi xðtÞ þ Adi xðt τðtÞÞ
> C Dα xðtÞ ¼
>
>
>
i¼1
>
>
>
<
þ Bi uðtÞ þ Di ωðtÞg, t ≥ 0,
(4)
r
X
>
>
>
yðtÞ
¼
h
ðθðtÞÞfC
xðtÞg,
>
i
i
>
>
>
i¼1
>
>
:
xðsÞ ¼ φðsÞ, s 2 ½τ,0
where hi(θ(t)) for θðtÞ ¼ ðθ1 ðtÞ,…,θp ðtÞÞ are fuzzy
weighting functions that are described as
hi ðθðtÞÞ ¼
p
∏i¼1
r p
P
∏
i¼1 l¼1
Mli ðθðtÞÞ
,
i ¼ 1; 2,…,r
:
(5)
Mli ðθðtÞÞ
hi ðθðtÞÞ ¼ 1,
i¼1
hi ðθðtÞÞ ≥ 0,
(6)
i ¼ 1; 2,…,r
In this article, the fuzzy observer-based control design
based on the PDC approach is presented for time-delayed
fractional-order T-S fuzzy systems. An observer with timevarying delay is designed as
8C α
D bxðtÞ ¼
>
>
>
>
<
r
X
hi ðθðtÞÞfAibxðtÞ þAdibxðt τðtÞÞ
i¼1
þ Bi uðtÞ þ Li ðyðtÞ byðtÞÞg,
>
>
r
>
>
: byðtÞ ¼ X h ðθðtÞÞfC bxðtÞg
i
i
t ≥ 0,
(7)
where bxðtÞ 2 ℜ is the estimate of xðtÞ and Li is the observer
gain matrix to be designed. The fuzzy controller is given as
n
r
X
hi ðθðtÞÞKibxðtÞ
r X
r
X
Dα xðtÞ ¼
hi ðθðtÞÞhj ðθðtÞÞfAi xðtÞ
i¼1 j¼1
þ Bi Kj xðtÞ þ Adi xðt τðtÞÞ þ Bi Kj eðtÞ þ Di ωðtÞ ,
r X
r
X
C α
D eðtÞ ¼
hi ðθðtÞÞhj ðθðtÞÞ Ai eðtÞ Li Cj eðtÞ
i¼1 j¼1
þ Adi eðt τðtÞÞ Di ωðtÞ
(9)
where eðtÞ ¼ bxðtÞ xðtÞ. The closed-loop system (9) can be
presented as
r X
r
X
C α
D x¼
hi ðθðtÞÞhj ðθðtÞÞ Gij xðtÞ
i¼1 j¼1
þ Hij xðt τðtÞÞ þ Di ωðtÞ ,
r
X
yðtÞ ¼
hi ðθðtÞÞC i xðtÞ
(10)
i¼1
where
xðtÞ
,
Gij ¼
eðtÞ
Adi 0
Hij ¼
,
0 Adi
Ai þ Bi Kj
Bi Kj
0
Ai Li Cj
Di
Di ¼
, C i ¼ ½ Ci 0 Di
Lemma 1 (Duarte-Mermoud et al., 2015). Let zðtÞ 2 ℜn be
a vector of differentiable functions. Then, for any time
instant t > t0, the following inequality holds
T
1C α T
C α
t0 Dt z ðtÞRzðtÞ ≤ z ðtÞRt0 Dt zðtÞ,
2
"α 2 ð0; 1Þ
where R 2 ℜn×n is a constant, square, symmetric, and
positive-definite matrix.
Lemma 2 (Cao et al. 1998). Let U and V be any appropriate
dimension matrices. For any constant matrix W > 0 and
a scalar γ > 0, the inequality
UV þ V T U T ≤ γ1 UW 1 U T þ γV T WV
is always satisfied.
i¼1
uðtÞ ¼
C
xðtÞ ¼
where Mli(θ(t)) are the grades of the membership of θl(t) in
Mli(θ(t)), and for all t, the following conditions are satisfied
8X
r
<
in which Ki is the state-feedback gain matrix. Also, the
closed-loop system can be proposed as
(8)
Lemma 3 (Schur complement) (Boyd et al. 1994). For
given matrices Σ, N, and Z with appropriate dimensions
satisfying Σ = ΣT and N = NT
Σ ZT
T 1
Σ þ Z N Z < 05
<0
Z N
i¼1
is always satisfied.
4
Journal of Vibration and Control 0(0)
3. Observer-based controller design
This section discusses the observer-based control design for
the fractional-order augmented system (10) without disturbance. The feedback and observer gain matrices are
obtained by the following theorem.
Theorem 1. The closed-loop system (10) with observerbased control law (8) is asymptotically stable if there exist
the symmetric positive-definite matrices X1, X2, S1, and S2
and any matrices Yj and Fi with appropriate dimensions
such that the following LMIs hold
Δii < 0, i ¼ 1; 2,…,r,
Δij þ Δji < 0, i,j ¼ 1; 2,…,r, i < j,
Vii < 0, i ¼ 1; 2,…,r,
Vij þ Vji < 0, i,j ¼ 1; 2,…,r, i < j
where
2
6 Ai X1 þ
Δij ¼ 4
X1 ATi
þ Bi Yj þ
YjT BTi
S1
þ
1μ
(11)
3
(12)
where xt ¼ xðt þ θÞ and θ 2 ðτ,0 and P and Q are
positive-definite matrices. The derivative of V ðxt Þ along the
trajectory of system (10) is obtained as (13)
1
ð1 τ_ ðtÞÞxT ðt τðtÞÞQxðt τðtÞÞ
1μ
(13)
1 T
x ðtÞQxðtÞ
1μ
1
ð1 μÞxT ðt τðtÞÞQxðt τðtÞÞ
1μ
hi ðθðtÞÞhj ðθðtÞÞ 2xT ðtÞPGij xðtÞ
i¼1 j¼1
þ xT ðtÞPHij Q1 HijT PT xðtÞ þ xT ðt τðtÞÞQxðt τðtÞÞ
þ
1 T
x ðtÞQxðtÞ xT ðt τðtÞÞQxðt τðtÞÞ
1μ
(15)
r X
r
X
hi ðθðtÞÞhj ðθðtÞÞ 2PGij þ PHij Q1 HijT PT
(16)
1
Q <0
þ
1μ
where P is a nonsingular matrix. Also, suppose that X = P1
and Q = X1SX1. Then, by pre- and post-multiplying the
both hand sides of (16) by the matrix X and its transpose,
respectively, (17) is obtained
r X
r
X
hi ðθðtÞÞhj ðθðtÞÞ Gij X þ XGTij
i¼1 j¼1
(17)
1
S <0
þ Hij X S
þ
1μ
"
#
"
#
X1
S1
0
0
By defining X ¼
,S ¼
,
0 X21
0 X21 S2 X21
(18) and (19) can be concluded from (17)
r X
r
X
XHijT
hi ðθðtÞÞhj ðθðtÞÞ Ai X1 þ X1 ATi þ Bi Kj X1
i¼1 j¼1
(18)
S1
þ X1 KjT BTi þ Adi X1 S11 X1 ATdi þ
<0
1μ
r X
r
X
hi ðθðtÞÞhj ðθðtÞÞ Ai X21 þ X21 ATi Li Cj X21
i¼1 j¼1
From (3) and by applying Lemma 1, the inequality (14)
is obtained
V_ ðxt Þ ≤ 2xT ðtÞPC Dα xðtÞ þ
1
Proof. For any t ≥ 0, choose the following LKF candidate
Z t
1
1α T
V ðxt Þ ¼ I
x ðtÞPxðtÞ þ
xT ðζ ÞQxðζ Þdζ
1 μ tτðtÞ
r X
r
X
i¼1 j¼1
Yi ¼ Ki X1 ði ¼ 1; 2,…,rÞ, Fi ¼ X2 Li ði ¼ 1; 2,…,rÞ.
1 T
x ðtÞQxðtÞ
1μ
V_ ðxt Þ≤
It is obvious that if inequality (16) is hold, then V_ ðxt Þ < 0
Adi X1 7
5,
∗
S1
2
3
S2
T
T T
6 X2 Ai þ Ai X2 Fi Cj Cj Fi þ 1 μ X2 Adi 7
Vij ¼ 4
5
∗
S2
V_ ðxt Þ ¼ C Dα xT ðtÞPxðtÞ þ
By substituting the system equation (10) in (14) and
using Lemma 2, the inequality (15) is obtained
X21 CjT LTi þ Adi S21 ATdi þ
X21 S2 X21
<0
1μ
(19)
(14)
By pre- and post-multiplying the both hand sides of (19)
by, respectively, the matrix X2 and its transpose, (20) is
obtained
Mahmoudabadi and Tavakoli-Kakhki
r X
r
X
5
hi ðθðtÞÞhj ðθðtÞÞ X2 Ai þ ATi X2 X2 Li Cj
i¼1 j¼1
(20)
CjT LTi X2T þ X2 Adi S21 ATdi X2T þ
S2
<0
1μ
By applying Schur complement for (18) and (20), the
stabilization conditions (11) are obtained. This completes
the proof. ■
4. Observer-based disturbance rejection
control
In this section, sufficient conditions for disturbance rejection of the closed-loop T-S fuzzy system (10) are proposed based on the state-feedback PDC. The block diagram
of the observer-based feedback control for disturbance
rejection is shown in Figure 1. A desired disturbance rejection can be achieved by minimizing γ in the optimization
problem (21) (Tanaka and Wang, 2004)
sup
kωðtÞk2 ≠ 0
kyðtÞk2
≤γ
kωðtÞk2
(21)
The inequality
V_ ðtÞ þ yT ðtÞyðtÞ γ2 ωT ðtÞωðtÞ ≤ 0
(21) is satisfied, if there exist the symmetric positivedefinite matrices X1, X2, Q1 , and Q2 and any matrices Yj
and Fi with appropriate dimensions such that the following
LMIs hold
Πii < 0, i ¼ 1; 2,…,r,
Πij þ Πji < 0, i,j ¼ 1; 2,…,r,
where
2
Ai X1 þ X1 ATi
6
0
6 þB Y þ Y T BT þ 1 Q
i j
6
1
j
i
1
μ
6
6
6
6
X2 Ai þ ATi X2
6
6
∗
1
6
Q2
Fi Cj CjT FiT þ
6
1
μ
6
6
6
∗
∗
Πij ¼ 6
6
6
6
∗
∗
6
6
6
∗
∗
6
6
6
6
∗
∗
6
6
6
∗
∗
4
∗
(22)
implies (21). Because by integrating (22) from 0 to T and
assuming xð0Þ ¼ 0, the inequality (23) is obtained
Z T
yT ðtÞyðtÞ γ2 ωT ðtÞωðtÞdt ≤ 0
(23)
V ðxðT ÞÞ þ
0
Because V ðxðT ÞÞ ≥ 0, (21) is concluded from (23). The
inequality (22) is used in the proof of Theorem 2.
Theorem 2. The closed-loop system (10) with observerbased control law (8) is asymptotically stable and H∞ index
(24)
i<j
∗
3
Di
Adi X1
0
0
Bi Yj
X2 Di
γ2
0
0
X2 Adi
0
I
0
0
0
∗
∗
Q1
∗
0
Q2
0
0
0
0
∗
∗
∗
υ1 X1
0
∗
∗
υ1
1 X1
∗
∗
∗
∗
∗
∗
∗
X1 CiT 7
7
7
0 7
7
0 7
7
7
0 7
7
0 7
7
7
0 7
7
0 7
7
7
I 5
Yi ¼ Ki X1 ði ¼ 1; 2,…,rÞ, Fi ¼ X2 Li ði ¼ 1; 2,…,rÞ:
Proof. The first steps of the proof of Theorem 2 are similar
to what has been done for the proof of Theorem 1 (see (12)–
(14)). From (22) and (14), (25) is obtained
Figure 1. Block diagram of observer-based feedback control.
V_ ðxt Þ þ yT ðtÞyðtÞ γ2 ωT ðtÞωðtÞ ≤ 2xT ðtÞPC Dα xðtÞ
(25)
1 T
þ
x ðtÞQxðtÞ xT ðt τðtÞÞQxðt τðtÞÞ
1μ
6
Journal of Vibration and Control 0(0)
By substituting the system equation (10) in (25) and by
applying Lemma 2, it is obtained that
V_ ðxt Þ þ yT ðtÞyðtÞ γ2 ωT ðtÞωðtÞ
r X
r
X
≤
hi ðθðtÞÞhj ðθðtÞÞ 2xT ðtÞPGij xðtÞ
It is clear that if inequality (28) is hold, then V_ ðxt Þ < 0
r X
r
X
hi ðθðtÞÞhj ðθðtÞÞΘij < 0
(28)
i¼1 j¼1
i¼1 j¼1
þ xT ðtÞPHij Q1 HijT PxðtÞ þ 2xT ðtÞPDi ωðtÞ
1 T
T
x ðtÞQxðtÞ þ xT ðtÞC i C j xðtÞ γ2 ωT ðtÞωðtÞ < 0
1μ
(26)
#
"
X11 0
Q1 0
and
, Q¼
By supposing P ¼
0 Q2
0
X2
substituting matrices Gij, Hij, Di , and C i , inequality (27) is
obtained
2
3T
xðtÞ
r
r
XX
6
7
hi ðθðtÞÞhj ðθðtÞÞ4 eðtÞ 5
i¼1 j¼1
ωðtÞ
3
2 1
T 1
X1 Ai þAi X1
7
6
7
6 þ X 1 Bi Kj
1
7
6
7
6
7
6 þ K T BT X 1
1
1 7
j i 1
6
X1 Bi Kj
X1 D 7
6
7
6 þ X 1 A Q1 AT X 1
7
6
di 1
1
di 1
7
6
7
6
1
7
6 þ
T
Q1 þCi Cj
7
6
1μ
7
6
7
6
7
6
7
6
T
X2 Ai þAi X2
7
6
7
6
7
6
7
6
X2 Li Cj
7
6
7
6
T T
7
6
Cj Li X2
6
∗
X2 D 7
7
6
7
6
1 T T
7
6
þ
X
A
Q
A
X
2
di
2
di 2
7
6
7
6
7
6
1
7
6
Q
þ
2
7
6
1μ
7
6
5
4
2
∗
∗
γ
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2
xðtÞ
3
Θij
By pre- and post-multiplying the both hand sides of (28)
by matrix diagfX ,I,Ig and its transpose, respectively, (29)
can be obtained
r X
r
X
hi ðθðtÞÞhj ðθðtÞÞ
i¼1 j¼1
8
>
>
3
2
>
>
>
Ai X1 þ X1 ATi þ Bi Yj
>
>
>
6
>
>
0
D 7
7
6
>
1
>
T T
7
6
>
þY
B
þ
Q
>
1
j
i
7
6
>
1μ
>
7
6
>
>
>
7
6
>
>
7
6
>
T
<6
7
X2 Ai þ Ai X2
7
6
7
6
T T
>
7
6
F
C
C
F
>
i j
j i
>
6
∗
X2 D 7
>
>6
7
>
>
7
6
1
>
>
7
6
Q
>
2
>
7
6
>
1μ
>
7
6
>
>
5
4
>
>
>
>
2
>
∗
∗
γ
>
>
:|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
2
Adi X1
3
2
Ξ
Adi X1
3T
2
0
3
2
0
3T
6
7 1 6
7
7
7
6
6
þ 4 0 5Q1 4 0 5 þ 4 X2 Adi 5Q1
2 4 X2 Adi 5
0
0
2
3
3T
0 2 Bi Yj 3T 2 Bi Yj 3 0
6
7
7
6
7
76
76
7
6
6
þ 6 X11 74 0 5 þ 4 0 56 X11 7
4
5
5
4
0
0
0
0
32
3T 9
2
>
T
T
>
X1 Ci
X1 Ci
>
>
76
7 =
6
76
7
6
< 0:
þ 6 0 76 0 7
54
5 >
4
>
>
>
;
0
0
2
0
0
(29)
6
7
4 eðtÞ 5 <0
ωðtÞ
(27)
From (29) and by applying Lemma 2, inequality (30) is
obtained
Mahmoudabadi and Tavakoli-Kakhki
7
8
3
2
Adi X1
>
<
6
7
hi ðθðtÞÞhj ðθðtÞÞ Ξ þ 4 0 5
>
:
i¼1 j¼1
0
3
3T
2
2
3T 2
Adi X1
0
0
1 6
7
7
6
7
6
Q1 4 0 5 þ 4 X2 Adi 5Q1
2 4 X2 Adi 5
Dα x1 ðtÞ ¼ x2 ðtÞ 0:05x1 ðt τðtÞÞ 0:1x2 ðtÞðt τðtÞÞ,
1
C α
ðg sinðx1 ðtÞÞ
D x2 ðtÞ ¼
4ðl=3Þ amlcos2 ðx1 ðtÞÞ
amlx22 ðtÞsinðð2x1 ðtÞÞ=2Þ a cosðx1 ðtÞÞuðtÞÞ
C
r X
r
X
0
3
2
0
2
0
3T
0
0:05x2 ðt τðtÞÞ
(31)
0
2
3
Bi Yj
6
7
7
6
6 1 7 1 6 1 7
6
7
1
6
7
7
þ6
6 X1 7 υ1 X1 6 X1 7 þ 4 0 5 υ1 X1
4
5
5
4
0
0
0
2
32
3T 9
T
T
>
2
3T
>
X
X
C
C
1
1
Bi Yj
i 76
i 7 >
>
6
6
76
7 =
6
7
6
7
6
7
<0
4 0 5 þ 6 0 76 0 7
>
4
54
5 >
>
>
0
;
0
0
(30)
From (30) and according to Lemma 3, the sufficient
conditions for disturbance rejection of system (10) is obtained in terms of the LMIs (24). This completes the proof.
■
Remark 1. It is worth mentioning that by considering the
inequality (21), the controller is designed such that the
effect of disturbance on the system output is minimized or
decreased. To achieve this goal, the inequality (22) which is
equivalent to (21) is used. Indeed, the parameter γ which
appears in the obtained LMI-based stability conditions (24)
is related to the problem of disturbance rejection which can
be determined by the designer or can be minimized to
achieve the best disturbance rejection performance.
Remark 2. The computational complexity of the LMIbased stability conditions in Theorem 2 is more than the
LMI-based conditions presented in Theorem 1, particularly
for systems with a large number of rules. However, the
number of variables in Theorem 2 is equal to that of
Theorem 1.
5. Numerical examples
In this section, the results of the article are used for state
estimation and stabilization of nonlinear time-delayed
fractional-order systems. Two numerical examples are investigated to show the validity of Theorem 1 and Theorem
2, respectively.
Example 1. Consider the unstable nonlinear time-delayed
fractional-order system (31)
where a = 0.1, l = 0.5, m = 2, g = 9.8, τ = 4, and μ = 0.8. To
model system (31) by a T-S fuzzy model, the membership
functions are obtained as h1 ðθðtÞÞ ¼ expð7ðx1 ðtÞðπ=
4ÞÞÞ=ð1þexpð7ðx1 ðtÞðπ=4ÞÞÞÞ×1=ð1þexpð7ðx1 ðtÞ
ðπ=4ÞÞÞÞ and h2(θ(t)) = 1 h1(θ(t)). Accordingly, the
matrices of the T-S fuzzy model (4) for the unstable nonlinear time-delayed fractional-order system (31) are obtained as follows
0
1
0
1
A1 ¼
, A2 ¼
,
0
17:2941 0
12:6305
0
0
, B2 ¼
,
B1 ¼
0:1765
0:0779
0:05 0:1
0:05 0:1
Ad1 ¼
, Ad2 ¼
,
0
0:05
0
0:05
C1 ¼ ½ 1 1 , C2 ¼ ½ 1 1 Based on Theorem 1, the matrices of the controller gain
(8) and the observer gain (7) are obtained as
K1 ¼ ½ 219:9092
62:6366
, K2 ¼ ½ 382:2705 125:9533,
2:5092
2:1104
, L2 ¼
L1 ¼
6:7316
5:4849
The initial conditions for real state variables and estimated state variables are chosen as φð0Þ ¼ ð 5 10 ÞT and
b
φð0Þ ¼ ð 20 10 ÞT , respectively. Figures 2 and 3 demonstrate the observer errors and also the state variables are
shown in Figures 4 and 5 for different values of fractionalorder α belongs to (0, 1). As it is seen from these figures, by
the designed controller and observer, the state estimation
and stabilization of the closed-loop system are done
efficiently.
Example 2. Consider the following continuous-time
fractional-order T-S fuzzy system (Sakthivel et al. 2019b)
2 0:1
1
1
1
A1 ¼
, A2 ¼
, B1 ¼
,
0 3 0:1 0:5 1 0:1 0
0:5
1
B2 ¼ B1 , Ad1 ¼
, Ad2 ¼
,
0
0:2
0:2 0:1
C1 ¼ ½ 50 , C2 ¼ C1 , D1 ¼ ½ 0:10:1 , D2 ¼ D1
Also, consider α = 0.85, τ = 0.2, μ = 0.1, h1 ðθðtÞÞ ¼
1=1 þ ex1 ðtÞþ0:5 , h2(θ(t)) = 1 h1(θ(t)), and the external
disturbance in the T-S fuzzy model as (32). This external
disturbance is depicted in Figure 6
8
Journal of Vibration and Control 0(0)
Figure 2. Observer error e1 ðtÞ in Example 1.
Figure 5. State variable x2 ðtÞ in Example 1.
Figure 3. Observer error e2 ðtÞ in Example 1.
Figure 6. Disturbance ωðtÞ in Example 2.
8
< 40:3ðsinð0:4πtÞ þ cosð0:2πtÞ
ωðtÞ ¼ þ sinð0:6πtÞ þ sinð0:5πtÞÞ, 10 ≤ t ≤ 37 (32)
:
0, otherwise
By applying Theorem 2 with ν1 = 0.01 and γ = 0.1, the
matrices of the controller gain (8) and the observer gain (7)
are obtained as follows
K1 ¼ ½86:9224 0:1597 ,
T
L1 ¼ ½ 21:8881 22:5101 ,
K2 ¼ ½93:8142 0:5676 ,
L2 ¼ ½ 13:6016
T
13:6803 :
Also, with ν1 = 0.001 and γ = 0.01, the matrices of the
controller gain (8) and the observer gain (7) are obtained as
follows
Figure 4. State variable x1 ðtÞ in Example 1.
K1 ¼ ½990:1903 0:2057 ,
K2 ¼ ½1000 0:0005 ,
Mahmoudabadi and Tavakoli-Kakhki
9
Figure 7. (a) Observer error e1 ðtÞðÞ and e2 ðtÞð:Þ for γ = 0.1
and (b) e1 ðtÞðÞ and e2 ðtÞð:Þ for γ = 0.01 in Example 2.
Figure 9. (a) Output of open-loop system and (b) output of
closed-loop system for γ = 0.1 () and output of closed-loop
system for γ = 0.01 (.) in Example 2.
6. Conclusions
In this article, by applying the T-S fuzzy model, some
sufficient conditions were presented for the stabilization and
state estimation of nonlinear time-delayed systems with
fractional-order α 2 ð0; 1Þ. A delay-independent Lyapunov–
Krasovskii functional was used to obtain observer-based
stabilization conditions in terms of LMIs. Two simulation
examples were provided to show the effectiveness of the
study results. For the future work, designing fault detection
observer for nonlinear time-delay fractional-order systems
based on the T-S fuzzy model can be considered as a noticeable research topic.
Declaration of conflicting interests
Figure 8. (a) State variable e1 ðtÞðÞ and x2 ðtÞð:Þ for γ = 0.1 and
(b) x1 ðtÞðÞ and x2 ðtÞð:Þ for γ = 0.01 in Example 2.
T
L1 ¼ 104 ½ 2:9546 2:9546 ,
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
T
L2 ¼ 104 ½ 1:7948 1:7948 :
Figures 7 and 8 show the simulation results for φð0Þ ¼
ð 0:4 0:9 ÞT and b
φð0Þ ¼ ð 0:2 0:1 ÞT as the initial conditions for real state variables and estimated state variables,
respectively. It can be seen that the observer error converges
to zero. Also, Figure 9 shows the output of open-loop and
closed-loop systems. It is obvious that the effect of external
disturbance is efficiently attenuated by the designed controller. As this figure depicts, decreasing the value of γ leads
to better disturbance rejection. This result confirms that the
inequality (21) is well satisfied by the designed controller.
Moreover, Theorem 2 can be applied for observer-based
disturbance rejection of time-delay fractional-order
T-S fuzzy system presented in Example 2 when the
fractional-order operator selects different values from 0 to 1.
Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.
ORCID iDs
Parvin Mahmoudabadi  https://orcid.org/0000-0001-9991-3413
Mahsan Tavakoli-Kakhki  https://orcid.org/0000-0001-68511614
References
Behinfaraz R, Ghaemi S and Khanmohammadi S (2019)
Adaptive synchronization of new fractional-order chaotic
systems with fractional adaption laws based on risk analysis.
Mathematical Methods in the Applied Sciences 42(6):
1772–1785.
10
Bettayeb M, Al-Saggaf UM and Djennoune S (2017) High gain
observer design for fractional-order non-linear systems with
delayed measurements: application to synchronisation of
fractional-order chaotic systems. IET Control Theory & Applications 11(17): 3171–3178.
Binazadeh T and Yousefi M (2018) Asymptotic stabilization of
a class of uncertain nonlinear time-delay fractional-order
systems via a robust delay-independent controller. Journal
of Vibration and Control 24(19): 4541–4550.
Boyd S, El Ghaoui L, Feron E, et al. (1994) Linear Matrix Inequalities
in System and Control Theory. Philadelphia, PA: SIAM.
Cao YY, Sun YX and Cheng C (1998) Delay-dependent robust
stabilization of uncertain systems with multiple state delays. IEEE Transactions on Automatic Control 43(11):
1608–1612.
Duarte-Mermoud MA, Aguila-Camacho N, Gallegos JA, et al.
(2015) Using general quadratic lyapunov functions to prove
lyapunov uniform stability for fractional order systems.
Communications in Nonlinear Science and Numerical Simulation 22(1–3): 650–659.
He BB, Zhou HC, Kou CH, et al. (2021) Stabilization of uncertain
fractional order system with time-varying delay using bmi
approach. Asian Journal of Control 23: 582–590.
Huong DC and Thuan MV (2018) Design of unknown-input
reduced-order observers for a class of nonlinear fractionalorder time-delay systems. International Journal of Adaptive
Control and Signal Processing 32(3): 412–423.
Jafari P, Teshnehlab M and Tavakoli-Kakhki M (2017a) Adaptive
type-2 fuzzy system for synchronisation and stabilisation of
chaotic non-linear fractional order systems. IET Control
Theory & Applications 12(2): 183–193.
Jafari P, Teshnehlab M and Tavakoli-Kakhki M (2017b) Synchronization and stabilization of fractional order nonlinear
systems with adaptive fuzzy controller and compensation
signal. Nonlinear Dynamics 90(2): 1037–1052.
Li S, Kang W and Ding DW (2020) Observer-based fuzzy faulttolerant control for nonlinear parabolic PDEs. International
Journal of Fuzzy Systems 22: 1–11.
Lin C, Wang G, Lee TH, et al. (2007) LMI Approach to Analysis
and Control of Takagi-Sugeno Fuzzy Systems with Time
Delay. New York: Springer Science & Business Media,
volume 351.
Mahmoudabadi P, Shasadeghi M and Zarei J (2017) New stability
and stabilization conditions for nonlinear systems with timevarying delay based on delay-partitioning approach. ISA
Transactions 70: 46–52.
Majumder K and Patre BM (2019) Adaptive sliding mode control
for asymptotic stabilization of underactuated mechanical systems via higher-order nonlinear disturbance observer. Journal
of Vibration and Control 25(17): 2340–2350.
Moradi L, Mohammadi F and Baleanu D (2019) A direct numerical solution of time-delay fractional optimal control
problems by using chelyshkov wavelets. Journal of Vibration
and Control 25(2): 310–324.
Parvizian M, Khandani K and Majd VJ (2020) An H∞ non-fragile
observer-based adaptive sliding mode controller design for
uncertain fractional-order nonlinear systems with time delay
and input nonlinearity. Asian Journal of Control 23 (1),
423–431.
Journal of Vibration and Control 0(0)
Phat V, Niamsup P and Thuan MV (2020) A new design method
for observer-based control of nonlinear fractional-order systems with time-variable delay. European Journal of Control 1:
1–18.
Podlubny I (1998) Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential
Equations, to Methods of Their Solution and Some of Their
Applications. San Diego: Elsevier.
Pourdehi S and Karimaghaee P (2020) Reset observer-based fault
tolerant control for a class of fuzzy nonlinear time-delay
systems. Journal of Process Control 85: 65–75.
Sakthivel R, Raajananthini K, Alzahrani F, et al. (2019a) Observerbased modified repetitive control for fractional-order nonlinear systems with unknown disturbances. IET Control Theory
& Applications 13(18): 3132–3138.
Sakthivel R, Raajananthini K, Kwon OM, et al. (2019b) Estimation and disturbance rejection performance for fractional
order fuzzy systems. ISA Transactions 92: 65–74.
Selvaraj P, Kwon OM and Sakthivel R (2019) Disturbance and
uncertainty rejection performance for fractional-order
complex dynamical networks. Neural Networks 112:
73–84.
Sun S, Wang Y, Zhang H, et al. (2020) Multiple intermittent fault
estimation and tolerant control for switched T-S fuzzy stochastic systems with multiple time-varying delays. Applied
Mathematics and Computation 377: 125114.
Takagi T and Sugeno M (1985) Fuzzy identification of systems and
its applications to modeling and control. IEEE Transactions on
Systems, Man, and Cybernetics SMC-15(1): 116–132.
Tanaka K and Wang HO (2004) Fuzzy Control Systems Design and
Analysis: A Linear Matrix Inequality Approach. New York:
John Wiley & Sons.
Thuan MV, Huong DC, Sau NH, et al. (2019) Unknown input
fractional-order functional observer design for one-side
lipschitz time-delay fractional-order systems. Transactions
of the Institute of Measurement and Control 41(15):
4311–4321.
Tian Y, Wang B, Chen P, et al. (2020) A state estimator–based
nonlinear predictive control for a fractional-order francis hydraulic turbine governing system. Journal of Vibration and
Control 26(11–12): 1068–1080.
Trinh H, Huong DC and Nahavandi S (2019) Observer design for
positive fractional-order interconnected time-delay systems.
Transactions of the Institute of Measurement and Control
41(2): 378–391.
Yi Y, Chen D and Xie Q (2019) Controllability of nonlinear
fractional order integrodifferential system with input delay.
Mathematical Methods in the Applied Sciences 42(11):
3799–3817.
Yu X, Liao F and Li L (2020) Observer-based decentralized
tracking control with preview action for a class of nonlinear
interconnected systems. Complexity, 2020: 1–18.
Zhang X, Li D and Zhang X (2019) Adaptive impulsive synchronization for a class of fractional order complex chaotic
systems. Journal of Vibration and Control 25(10): 1614–1628.
Zheng Y, Nian Y and Wang D (2010) Controlling fractional order
chaotic systems based on Takagi-Sugeno fuzzy model and
adaptive adjustment mechanism. Physics Letters A 375(2):
125–129.