1) The document analyzes the boundedness and domain of attraction of a fractional-order wireless power transfer (WPT) system. 2) It establishes a fractional-order piecewise affine model of the WPT system and derives sufficient conditions for boundedness using Lyapunov functions and inequality techniques. 3) The results provide a way to estimate the domain of attraction of the fractional-order WPT system and systems with periodically intermittent control.

1. Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
Contents lists available at ScienceDirect
Nonlinear Analysis: Hybrid Systems
journal homepage: www.elsevier.com/locate/nahs
Estimation of domain of attraction for the fractional-order
WPT system✩
Zhongming Yu, Yue Sun
∗
, Xin Dai
School of Automation, Chongqing University, Chongqing, 400030, PR China
Key Laboratory of Complex System Safety and Control, Ministry of Education, Chongqing University, Chongqing, 400030, PR China
a r t i c l e i n f o
Article history:
Received 8 September 2019
Received in revised form 18 May 2021
Accepted 25 May 2021
Available online 5 June 2021
MSC:
00-01
99-00
Keywords:
WPT system
Fractional-order
Domain of attraction
Boundedness
a b s t r a c t
This paper investigates the problem of domain of attraction of the fractional-order
wireless power transfer (WPT) system. As a fractional-order piecewise affine system,
firstly, the model of the fractional-order WPT system is established. Secondly, based
on the Lyapunov function approach and the inductive method, sufficient conditions of
the boundedness for the fractional-order WPT system and the fractional-order system
with periodically intermittent control are derived, respectively. In the meantime, the
relevant inequality technique is introduced so as to decrease the conservatism of the
results. The derived results can be used for estimating the domain of attraction of
the systems. Finally, several examples are given to demonstrate the obtained results.
Simulation shows that the conservatism of the results is indeed reduced in theory, and
the designed controller is effective.
© 2021 Elsevier Ltd. All rights reserved.
1. Introduction
In the past decade, wireless power transfer (WPT) technology based on the principle of electromagnetic induction has
received a lot of attention because of its many advantages such as flexible charging, high efficiency and no pollution to
the environment. Meantime, the technology has been successfully used in many fields such as household appliances [1],
electric vehicles [2,3] and biomedical implants [4]. Nevertheless, external disturbances such as load disturbance and
parameter variation are inevitable during the operation of the WPT system, which can cause the phenomenon of frequency
splitting [5,6] and further affect the domain of attraction of the system.
It is well known that the domain of attraction reflects the ability to resist the disturbance of the system, namely,
robustness. The larger the domain of attraction, the stronger the robustness of the system and vice versa. Besides, the
strength of the robustness has also a significant impact on energy transmission efficiency of the WPT system. As a result,
it is extremely important to research the domain of attraction of the WPT system.
In addition, fractional-order characteristics exist in many fields such as electronic circuits [7,8], electromagnetic
field [9], signal processing [10], electrochemical capacitor [11,12], power systems [13], energy fuels [14,15] and energy
balance [16–18]. Meanwhile, the model of fractional-order systems [19–21] can more accurately describe the system and
the physical phenomenon than the model of integer-order ones. In the WPT system, owing to it contains many fractional
✩ No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with
this work.
∗ Corresponding author at: School of Automation, Chongqing University, Chongqing, 400030, PR China.
E-mail addresses: yuzhgmin@sina.com (Z. Yu), syue@cqu.edu.cn (Y. Sun), toybear@vip.sina.com (X. Dai).
https://doi.org/10.1016/j.nahs.2021.101062
1751-570X/© 2021 Elsevier Ltd. All rights reserved.

2. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
components such as capacitor and inductor, the corresponding fractional-order model is more accurate than the integer-
order model in terms of modeling. Thus the fractional-order model of the system can more accurately describe parameters
of the system such as resonance frequency, inductance and capacitance, which is helpful to better analyze the frequency
drift phenomenon and estimate the domain of attraction. And recently, the effect of fractional orders on the transmission
power and efficiency of the fractional-order WPT system has been investigated [22]. Additionally, the WPT system also
belongs to a class of piecewise affine systems according to its working principle [23]. In the meanwhile, the problem
of the domain of attraction of the piecewise affine systems as one of the most challenging topics in the field of hybrid
systems has attracted great attention in the past few decades [24,25]. Stability regions for linear systems with saturating
control via circle and Popov criteria are estimated [24]. Results of global asymptotic stability of piecewise linear systems
based on impact maps and surface Lyapunov functions are given [25]. And compared with the paper [24], the results
in paper [25] show less conservatism, which leads to a larger stability region. Nonetheless, these works only focus on
the integer-order piecewise affine systems, so the obtained results cannot be applied to the fractional-order piecewise
affine systems. Besides, with regard to the fractional-order systems, the domain of attraction of the fractional-order linear
system is investigated [26]. However, up to now, to the best of our knowledge, the problem of the domain of attraction
for the fractional-order piecewise affine systems is still a gap.
The above mentioned works and observation inspire us to deal with the problem of the domain of attraction of the
fractional-order WPT system. First of all, the fractional-order model of the WPT system is built. Then, relevant results of
the fractional-order WPT system are derived by using Lyapunov function approach and inductive method, and further
extended to the fractional-order system with periodically intermittent control. The major contributions of this work are
as follows:
(a) To estimate the domain of attraction of the systems, sufficient conditions of the boundedness for the fractional-order
WPT system and the fractional-order system with periodically intermittent control are derived, respectively.
(b) Consider the conservatism of the results, the relevant inequality technique is used. And compared with the results
without using the inequality technique, the domains of attraction of the obtained results are larger. Besides, the
conservatism is also discussed from the practical point of view.
(c) The related results are also suitable for estimating the domain of attraction under the different τ.
(d) Estimation of the domain of attraction provides an important reference for the robustness evaluation of the WPT
system.
This paper is organized as follows. Section 2 introduces some relevant lemmas and definitions. Simultaneously, the
model of the fractional-order WPT system is described. In Section 3, the results of the boundedness of the fractional-
order WPT system and the fractional-order system with periodically intermittent control are demonstrated, respectively.
In Section 4, simulation is provided to verify the obtained results. Conclusions are drawn in Section 5.
2. Preliminaries and model of description
At first, some relevant definitions and lemmas are given as follows.
Definition 1 ([27]). For a continuous function f : [0, ∞) → R, the Caputo derivative of fractional order α is defined as
Dα
f (t) =
1
Γ (n − α)
∫ t
0
(t − s)n−α−1
f (n)
(s)ds (n − 1 < α < n, n = [α] + 1), (1)
where [α] denotes the integer part of the real number α. Γ (·) is the Gamma function.
Definition 2 ([28]). The Mittag-Leffler function with one parameter is defined as
Eα(z) = Eα,1(z) =
∞
∑
k=0
zk
Γ (kα + 1)
, (2)
where α > 0 and z ∈ C. The Mittag-Leffler function with two parameters appears most frequently and has the following
form:
Eα,β(z) =
∞
∑
k=0
zk
Γ (kα + β)
, (3)
where α > 0, β > 0, and z ∈ C. For β = 1 we obtain Eα,1(z) = Eα(z).
Lemma 1 ([29]). Let x(t) ∈ Rn
be a vector of differentiable functions. Then, for any time instant t ≥ t0, the following relationship
holds
1
2
C
t0
Dα
(xT
(t)Px(t)) ≤ xT
(t)P C
t0
Dα
x(t), ∀α ∈ (0, 1], ∀t ≥ t0, (4)
where P ∈ Rn×n
is a constant, square, symmetric and positive definite matrix.
2

3. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
Fig. 1. Typical WPT system.
Lemma 2 ([30]). For any matrices X and Y with appropriate dimensions, the following relationship holds
XT
Y + YT
X ≤ εXT
X +
1
ε
YT
Y, for any ε > 0. (5)
Lemma 3 ([31]). For any vectors a ∈ Rna , b ∈ Rnb , and any matrices N ∈ Rna×nb , X ∈ Rna×na , Y ∈ Rna×nb , Z ∈ Rnb×nb , if
[
X Y
YT
Z
]
≥ 0, then the following inequality holds
−2aT
Nb ≤
[
a
b
]T [
X Y − N
YT
− NT
Z
] [
a
b
]
(6)
Lemma 4 ([32]). Let H(t) be a continuous function on [0, ∞), if there exist k1 ∈ R and k2 ≥ 0 such that C
0 Dα
t H(t) ≤ k1H(t)+k2,
t ≥ 0, then
H(t) ≤ H(0)Eα(k1tα
) + k2tα
Eα,α+1(k1tα
), t ≥ 0. (7)
where 0 < α < 1, Eα(·) and Eα,α+1(·) are one-parameter Mittag-Leffler function and two-parameter Mittag-Leffler function,
respectively.
It is worth noting that k1 > 0 is required in work [32]. In fact, Lemma 3 still holds when k1 ∈ R. The proof is similar
to Lemma 3 of the work [32].
Lemma 5 ([33]). If α < 2, β is an arbitrary real number, η is such that πα
2
< η < min{π, πα} and m > 0 is a real constant,
then
| Eα,β(z) |≤
m
1+ | z |
, η ≤| arg(z) |≤ π, | z |≥ 0. (8)
The model of the typical WPT system is depicted in Fig. 1. E denotes DC power supply. The left LP CP resonant circuit is
called primary side and the right LS CS resonant circuit is called secondary side. LP , CP , RP and LS , CS , RS denote inductance,
capacitance and parasitic resistance of the primary side and the secondary side, respectively. RL denotes resistance of load.
MPS denotes mutual inductance. Besides, ip, up and is, us denote current and capacitance voltage of the primary side and
the secondary side, respectively.
Based on circuit theory, differential equations of the WPT system are listed as
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
s(t)E = up(t) + ip(t)Rp + Lp
dip
dt
+ Mps
dis
dt
0 = Ls
dis
dt
+ Mps
dip
dt
+ is(t)Rs + us + isRL
Cp
dup
dt
= ip(t)
Cs
dus
dt
= is(t)
(9)
Let vector x(t) = [ip, up, is, us]T
, then corresponding state space model of the WPT system is described as
ẋ(t) = Ax(t) + bi, i = 1, 2. (10)
3

4. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
where
A =
⎡
⎢
⎢
⎢
⎢
⎣
RpLs
M2
ps−LpLs
Ls
M2
ps−LpLs
(Rs+RL)Mps
LpLs−M2
ps
Mps
LpLs−M2
ps
1
Cp
0 0 0
RpMps
LpLs−M2
ps
Mps
LpLs−M2
ps
(Rs+RL)Lp
M2
ps−LpLs
Lp
M2
ps−LpLs
0 0 1
Cs
0
⎤
⎥
⎥
⎥
⎥
⎦
,
B =
[
ELs
LpLs−M2
ps
0
EMps
M2
ps−LpLs
0
]T
, s(t)=
{
1, nT ≤ t < nT + T
2
−1, nT + T
2
≤ t < (n + 1)T
,
bi = s(t)B =
{
b1 = B, nT ≤ t < nT + T
2
b2 = −B, nT + T
2
≤ t < (n + 1)T
.
T denotes the duty circle of the inverter. s(t) = 1 denotes that switch S1 and switch S4 are turned on while switch S2
and switch S3 are turned off. s(t) = −1 denotes that switch S1 and switch S4 are turned off while switch S2 and switch
S3 are turned on. n = 0, 1, 2, . . ..
On the basis of the analysis of the introduction part, one can see that the WPT system is the fractional-order piecewise
affine system. Hence the Caputo fractional-order model of the WPT system is established as
C
0 Dα
t x(t) = Ax(t) + bi, i = 1, 2. (11)
3. Main results
In this section, the boundedness and the domain of attraction with respect to the fractional-order WPT system and
the fractional-order system with periodically intermittent control are analyzed, respectively.
3.1. Boundedness analysis of the fractional-order WPT system
Theorem 1. The fractional-order WPT system (11) is bounded, and the trivial solution x(t) converges to the following compact
set
G = {x(t) ∈ R4
|∥ x(t) ∥≤
√
bT
1 b1m2
λmin(P)εγ1
} (12)
if there exist positive constants γ1, γ2, ε, a symmetric positive definite matrix P and matrices X1, X2, Y1, Y2, Z1 and Z2 with
appropriate dimensions satisfy the following conditions:
(i) (X1 + YT
1 + Y1 − NT
1 − N1 + Z1)/2 < 0, (13)
(ii) (X2 + YT
2 + Y2 − NT
2 − N2 + Z2)/2 < 0, (14)
(iii)
[
X1 Y1
YT
1 Z1
]
≥ 0,
[
X2 Y2
YT
2 Z2
]
≥ 0, (15)
where
N1 = −AT
P − PA − εP2
− γ1P, N2 = −AT
P − PA − εP2
− (γ1 + γ2)P.
Proof. Consider the following Lyapunov function
V = xT
(t)Px(t). (16)
Then, when t ∈ [nT, nT + T
2
), based on Lemma 1 we know that
C
0 Dα
t V ≤ (Ax + b1)T
Px + xT
P(Ax + b1)
= xT
(AT
P + PA)x + bT
1 Px + xT
Pb1.
(17)
Next, by Lemma 2 we can get that
C
0 Dα
t V ≤ xT
(AT
P + PA)x + εxT
P2
x +
1
ε
bT
1 b1
= −γ1V + xT
(AT
P + PA + εP2
+ γ1P)x +
1
ε
bT
1 b1
≤ −γ1V +
1
ε
bT
1 b1.
(18)
4

5. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
Then applying Lemma 3, and let N1 = −AT
P − PA − εP2
− γ1P, it yields
xT
(AT
P + PA + εP2
+ γ1P)x =
1
2
[−2xT
(−AT
P − PA − εP2
− γ1P)x]
=
1
2
(−2xT
N1x)
≤
1
2
[
x
x
]T [
X1 Y1 − N1
YT
1 − NT
1 Z1
] [
x
x
]
=
1
2
xT
[X1 + YT
1 + Y1 − NT
1 − N1 + Z1]x
(19)
Similarly, when t ∈ [nT + T
2
, (n + 1)T), we obtain that
C
0 Dα
t V ≤ xT
(AT
P + PA)x + bT
2 Px + xT
Pb2
= −(γ1 + γ2)V + xT
(AT
P + PA + εP2
+ (γ1 + γ2)P)x +
1
ε
bT
2 b2
≤ −(γ1 + γ2)V +
1
ε
bT
2 b2.
(20)
Let N2 = −AT
P − PA − εP2
− (γ1 + γ2)P, and by Lemma 3 we have
xT
(AT
P + PA + εP2
+ (γ1 + γ2)P)x =
1
2
[ − 2xT
( − AT
P − PA − εP2
− (γ1 + γ2)P)x]
=
1
2
(−2xT
N2x)
≤
1
2
[
x
x
]T [
X2 Y2 − N2
YT
2 − NT
2 Z2
] [
x
x
]
=
1
2
xT
[X2 + YT
2 + Y2 − NT
2 − N2 + Z2]x.
(21)
Then, when t ∈ [0, T
2
), by using Lemma 4 we have
V(t) ≤ V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
), (22)
and
V(
T
2
) ≤ V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
). (23)
Considering the memory property of fractional calculus, when t ∈ [T
2
, T), using Lemma 4, and combining (22), (23)
and the condition b1 = −b2, we have
V(t) ≤ V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) + V(
T
2
)Eα([−(γ1 + γ2) − (−γ1)]
· (t −
T
2
)α
) +
1
ε
(bT
2 b2 − bT
1 b1)(t −
T
2
)α
Eα,α+1([−(γ1 + γ2) − (−γ1)](t −
T
2
)α
)
≤ V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1
· (
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(t −
T
2
)α
),
(24)
and
V(T) ≤ V(0)Eα(−γ1Tα
) +
1
ε
bT
1 b1Tα
Eα,α+1(−γ1Tα
) + [V(0)Eα(−γ1(
T
2
)α
)
+
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(
T
2
)α
).
(25)
Similarly, when t ∈ [T, 3T
2
), applying Lemma 4, and combining (24), (25) and the condition b1 = −b2, it yields
V(t) ≤ V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1
5

6. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
· (
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(t −
T
2
)α
) + V(T)Eα([−γ1 − (−γ1 − γ2)]
· (t − T)α
) +
1
ε
(bT
1 b1 − bT
2 b2)(t − T)α
Eα,α+1([−γ1 − (−γ1 − γ2)](t − T)α
)
≤ V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1
· (
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(t −
T
2
)α
) + {V(0)Eα(−γ1Tα
) +
1
ε
bT
1 b1
· Tα
Eα,α+1(−γ1Tα
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]
· Eα(−γ2(
T
2
)α
)}Eα(γ2(t − T)α
),
(26)
and
V(
3T
2
) ≤ V(0)Eα(−γ1(
3T
2
)α
) +
1
ε
bT
1 b1(
3T
2
)α
Eα,α+1(−γ1(
3T
2
)α
) + [V(0)
· Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2Tα
) + {V(0)
· Eα(−γ1Tα
) +
1
ε
bT
1 b1Tα
Eα,α+1(−γ1Tα
) + [V(0)Eα(−γ1(
T
2
)α
)
+
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(
T
2
)α
)}Eα(γ2(
T
2
)α
).
(27)
When t ∈ [3T
2
, 2T), using Lemma 4, and combining (26), (27) and the condition b1 = −b2, we obtain that
V(t) ≤ V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1
· (
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(t −
T
2
)α
) + {V(0)Eα(−γ1Tα
) +
1
ε
bT
1 b1
· Tα
Eα,α+1(−γ1Tα
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]
· Eα(−γ2(
T
2
)α
)}Eα(γ2(t − T)α
) + V(
3T
2
)Eα((−(γ1 + γ2) − (−γ1))(t −
3T
2
)α
)
+
1
ε
(bT
2 b2 − bT
1 b1)(t −
3T
2
)α
Eα,α+1([−(γ1 + γ2) − (−γ1)](t −
3T
2
)α
)
≤ V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1
· (
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(t −
T
2
)α
) + {V(0)Eα(−γ1Tα
) +
1
ε
bT
1 b1
· Tα
Eα,α+1(−γ1Tα
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]
· Eα(−γ2(
T
2
)α
)}Eα(γ2(t − T)α
) + {V(0)Eα(−γ1(
3T
2
)α
) +
1
ε
bT
1 b1(
3T
2
)α
· Eα,α+1(−γ1(
3T
2
)α
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]
· Eα(−γ2Tα
) + {V(0)Eα(−γ1Tα
) +
1
ε
bT
1 b1Tα
Eα,α+1(−γ1Tα
) + [V(0)
· Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(
T
2
)α
)}Eα(γ2(
T
2
)α
)}
· Eα(−γ2(t −
3T
2
)α
),
(28)
6

7. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
and
V(2T) ≤ V(0)Eα(−γ1(2T)α
) +
1
ε
bT
1 b1(2T)α
Eα,α+1(−γ1(2T)α
) + [V(0)Eα(−γ1(
T
2
)α
)
+
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(
3T
2
)α
) + {V(0)Eα(−γ1Tα
) +
1
ε
bT
1 b1
· Tα
Eα,α+1(−γ1Tα
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]
· Eα(−γ2(
T
2
)α
)}Eα(γ2Tα
) + {V(0)Eα(−γ1(
3T
2
)α
) +
1
ε
bT
1 b1(
3T
2
)α
· Eα,α+1(−γ1(
3T
2
)α
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]
· Eα(−γ2Tα
) + {V(0)Eα(−γ1Tα
) +
1
ε
bT
1 b1Tα
Eα,α+1(−γ1Tα
) + [V(0)
· Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(
T
2
)α
)}Eα(γ2(
T
2
)α
)}
· Eα(−γ2(
T
2
)α
).
(29)
In the following, inductive method is used to analyze.
When t ∈ [nT, nT + T
2
), we have
V(t) ≤ V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) + V(
T
2
)Eα(−γ2(t −
T
2
)α
)
+ V(T)Eα(γ2(t − T)α
) + V(
3T
2
)Eα(−γ2(t −
3T
2
)α
) + V(2T)Eα(γ2(t − 2T)α
)
+ · · · + V(nT)Eα(γ2(t − nT)α
)
= V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) +
2n
∑
k=1
V(
kT
2
)Eα((−1)k
γ2(t −
kT
2
)α
).
(30)
When t ∈ [nT + T
2
, (n + 1)T), we can get that
V(t) ≤ V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) + V(
T
2
)Eα(−γ2(t −
T
2
)α
)
+ V(T)Eα(γ2(t − T)α
) + V(
3T
2
)Eα(−γ2(t −
3T
2
)α
) + V(2T)Eα(γ2(t − 2T)α
)
+ · · · + V(nT)Eα(γ2(t − nT)α
) + V(nT +
T
2
)Eα( − γ2(t − (nT +
T
2
))α
)
= V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) +
2n+1
∑
k=1
V(
kT
2
)Eα((−1)k
γ2(t −
kT
2
)α
).
(31)
Therefore, for any t ≥ 0, by Lemma 5, we can obtain that
∥ x(t) ∥ ≤
√
V(0) | Eα(−γ1tα) |
λmin(P)
+
√
1
ε
bT
1 b1tα | Eα,α+1(−γ1tα) |
λmin(P)
+
√∑2n+1
k=1 V(kT
2
) | Eα((−1)kγ2(t − kT
2
)α) |
λmin(P)
≤
√
V(0)m1
λmin(P)(1 + γ1tα)
+
√
1
ε
bT
1 b1tαm2
λmin(P)(1 + γ1tα)
+



√ 1
λmin(P)
·
2n+1
∑
k=1
V(kT
2
)m3
1 + γ2 | (t − kT
2
)α |
.
(32)
7

8. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
Besides, when t → ∞, we can know that
∥ x(t) ∥ ≤
√
bT
1 b1m2
λmin(P)εγ1
. (33)
Hence the system (11) is bounded. This completes the proof.
Remark 1. From (33), we can know that when time tends to infinity, V = xT
Px ≤
bT
1
b1m2
εγ1
. Thus the domain of attraction
of the system (11) can be estimated by this inequality.
Remark 2. During the derivation, some matrix variables by Lemma 3 are introduced. Conservatism of the obtained result
is improved in theory. This point can be seen in the analysis of later examples.
Especially, when γ1 = γ2 = γ , the following Corollary can be obtained.
Corollary 1. The fractional-order WPT system (11) is bounded, and the trivial solution x(t) converges to the following compact
set
G = {x(t) ∈ R4
|∥ x(t) ∥≤
√
bT
1 b1m2
λmin(P)εγ
} (34)
if there exist positive constants γ , ε, a symmetric positive definite matrix P and matrices X1, X2, Y1, Y2, Z1 and Z2 with
appropriate dimensions satisfy the following conditions:
(i) (X1 + YT
1 + Y1 − NT
1 − N1 + Z1)/2 < 0, (35)
(ii) (X2 + YT
2 + Y2 − NT
2 − N2 + Z2)/2 < 0, (36)
(iii)
[
X1 Y1
YT
1 Z1
]
≥ 0,
[
X2 Y2
YT
2 Z2
]
≥ 0, (37)
where
N1 = −AT
P − PA − εP2
− γ P, N2 = −AT
P − PA − εP2
− 2γ P.
3.2. Boundedness analysis of the fractional-order system with periodically intermittent control
To without loss of generality, consider the following fractional-order system with periodically intermittent control
C
0 Dα
t x(t) =
{
(A + K)x(t) + d, nT ≤ t < nT + τ,
Ax(t) + d, nT + τ ≤ t < (n + 1)T,
(38)
where d is an external bias vector. n = 0, 1, 2, . . ..
Theorem 2. The fractional-order system (38) is bounded, and the trivial solution x(t) converges to the following compact set
G = {x(t) ∈ Rl
|∥ x(t) ∥≤
√
dT dm2
λmin(P)εγ1
} (39)
if there exist positive constants γ1, γ2, ε, a symmetric positive definite matrix P, and matrices K, X1, X2, Y1, Y2, Z1 and Z2 with
appropriate dimensions satisfy the following conditions:
(i) (X1 + YT
1 + Y1 − NT
1 − N1 + Z1)/2 < 0, (40)
(ii) (X2 + YT
2 + Y2 − NT
2 − N2 + Z2)/2 < 0, (41)
(iii)
[
X1 Y1
YT
1 Z1
]
≥ 0,
[
X2 Y2
YT
2 Z2
]
≥ 0 (42)
where N1 = −PA − AT
P − Q − Q T
− εP2
+ γ1P, N2 = −AT
P − PA − εP2
+ (γ1 + γ2)P, Q = PK.
8

9. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
Proof. The process of the proof is similar to Theorem 1. Choosing the same Lyapunov function as Theorem 1. When
t ∈ [nT, nT + τ), by Lemmas 1 and 2 we get that
C
0 Dα
t V ≤ (xT
AT
+ xT
KT
+ dT
)Px + xT
P(Ax + Kx + d)
= xT
(PA + AT
P + PK + KT
P)x + dT
Px + xT
Pd
≤ xT
(PA + AT
P + PK + KT
P + εP2
)x +
1
ε
dT
d
= γ1V + xT
(PA + AT
P + PK + KT
P + εP2
− γ1P)x +
1
ε
dT
d
≤ γ1V +
1
ε
dT
d.
(43)
Then, let N1 = −PA − AT
P − PK − KT
P − εP2
+ γ1P, and by Lemma 3 we obtain that
xT
(PA + AT
P + PK + KT
P + εP2
− γ1P)x
=
1
2
[−2xT
(−PA − AT
P − PK − KT
P − εP2
+ γ1P)x]
=
1
2
(−2xT
N1x)
≤
1
2
[
x
x
]T [
X1 Y1 − N1
YT
1 − NT
1 Z1
] [
x
x
]
=
1
2
xT
[X1 + YT
1 + Y1 − NT
1 − N1 + Z1]x, (44)
when t ∈ [nT + τ, (n + 1)T), we have
C
0 Dα
t V ≤ (xT
AT
+ dT
)Px + xT
P(Ax + d)
= xT
(PA + AT
P)x + dT
Px + xT
Pd
≤ xT
(PA + AT
P + εP2
)x +
1
ε
dT
d
= (γ1 + γ2)V + xT
(PA + AT
P + εP2
− (γ1 + γ2)P)x +
1
ε
dT
d
≤ (γ1 + γ2)V +
1
ε
dT
d.
(45)
Similarly, let N2 = −AT
P − PA − εP2
+ (γ1 + γ2)P, and applying Lemma 3, it yields
xT
(AT
P + PA + εP2
− (γ1 + γ2)P)x =
1
2
[ − 2xT
( − AT
P − PA − εP2
+ (γ1 + γ2)P)x]
=
1
2
(−2xT
N2x)
≤
1
2
[
x
x
]T [
X2 Y2 − N2
YT
2 − NT
2 Z2
] [
x
x
]
=
1
2
xT
[X2 + YT
2 + Y2 − NT
2 − N2 + Z2]x.
(46)
Then, when n = 0, namely, t ∈ [0, T). For t ∈ [0, τ), using Lemma 4, we have
V(t) ≤ V(0)Eα(γ1tα
) +
1
ε
dT
dtα
Eα,α+1(γ1tα
), (47)
and
V(τ) ≤ V(0)Eα(γ1τα
) +
1
ε
dT
dτα
Eα,α+1(γ1τα
). (48)
When t ∈ [τ, T), utilizing Lemma 4 and combining and (47) and (48), we obtain that
V(t) ≤ V(0)Eα(γ1tα
) +
1
ε
dT
dtα
Eα,α+1(γ1tα
) + V(τ)Eα((γ1 + γ2 − γ1)(t − τ)α
)
+
1
ε
(dT
d − dT
d)(t − τ)α
Eα,α+1((γ1 + γ2 − γ1)(t − τ)α
)
≤ V(0)Eα(γ1tα
) +
1
ε
dT
dtα
Eα,α+1(γ1tα
) + (V(0)Eα(γ1τα
) +
1
ε
dT
dτα
· Eα,α+1(γ1τα
))Eα(γ2(t − τ)α
),
(49)
9

10. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
and
V(T) ≤ V(0)Eα(γ1Tα
) +
1
ε
dT
dTα
Eα,α+1(γ1Tα
) + (V(0)Eα(γ1τα
)
+
1
ε
dT
dτα
Eα,α+1(γ1τα
))Eα(γ2(T − τ)α
).
(50)
When t ∈ [T, T + τ), applying Lemma 4 and combining (49) and (50), we have
V(t) ≤ V(0)Eα(γ1tα
) +
1
ε
dT
dtα
Eα,α+1(γ1tα
) + (V(0)Eα(γ1τα
) +
1
ε
dT
dτα
· Eα,α+1(γ1τα
))Eα(γ2(t − τ)α
) + V(T)Eα((γ1 − (γ1 + γ2))(t − T)α
)
+
1
ε
(dT
d − dT
d)(t − T)α
Eα,α+1((γ1 − (γ1 + γ2))(t − T)α
)
≤ V(0)Eα(γ1tα
) +
1
ε
dT
dtα
Eα,α+1(γ1tα
) + (V(0)Eα(γ1τα
) +
1
ε
dT
dτα
· Eα,α+1(γ1τα
))Eα(γ2(t − τ)α
) + [V(0)Eα(γ1Tα
) +
1
ε
dT
dTα
· Eα,α+1(γ1Tα
) + (V(0)Eα(γ1τα
) +
1
ε
dT
dτα
Eα,α+1(γ1τα
))Eα(γ2(T
− τ)α
)]Eα(−γ2(t − T)α
),
(51)
and
V(T + τ) ≤ V(0)Eα(γ1(T + τ)α
) +
1
ε
dT
d(T + τ)α
Eα,α+1(γ1(T + τ)α
)
+ (V(0)Eα(γ1τα
) +
1
ε
dT
dτα
Eα,α+1(γ1τα
))Eα(γ2Tα
) + [V(0)
· Eα(γ1Tα
) +
1
ε
dT
dTα
Eα,α+1(γ1Tα
) + (V(0)Eα(γ1τα
)
+
1
ε
dT
dτα
Eα,α+1(γ1τα
))Eα(γ2(T − τ)α
)]Eα(−γ2τα
).
(52)
When t ∈ [T + τ, 2T), using Lemma 4 and combining (51) and (52), we have
V(t) ≤ V(0)Eα(γ1tα
) +
1
ε
dT
dtα
Eα,α+1(γ1tα
) + (V(0)Eα(γ1τα
) +
1
ε
dT
dτα
· Eα,α+1(γ1τα
))Eα(γ2(t − τ)α
) + [V(0)Eα(γ1Tα
) +
1
ε
dT
dTα
· Eα,α+1(γ1Tα
) + (V(0)Eα(γ1τα
) +
1
ε
dT
dτα
Eα,α+1(γ1τα
))Eα(γ2(T
− τ)α
)]Eα(−γ2(t − T)α
) + V(T + τ)Eα((γ1 + γ2 − γ1)(t − (T + τ))α
)
+
1
ε
(dT
d − dT
d)(t − (T + τ))α
Eα,α+1((γ1 + γ2 − γ1)(t − (T + τ))α
)
≤ V(0)Eα(γ1tα
) +
1
ε
dT
dtα
Eα,α+1(γ1tα
) + (V(0)Eα(γ1τα
) +
1
ε
dT
dτα
· Eα,α+1(γ1τα
))Eα(γ2(t − τ)α
) + [V(0)Eα(γ1Tα
) +
1
ε
dT
dTα
· Eα,α+1(γ1Tα
) + (V(0)Eα(γ1τα
) +
1
ε
dT
dτα
Eα,α+1(γ1τα
))Eα(γ2(T
− τ)α
)]Eα(−γ2(t − T)α
) + {V(0)Eα(γ1(T + τ)α
) +
1
ε
dT
d(T + τ)α
· Eα,α+1(γ1(T + τ)α
) + (V(0)Eα(γ1τα
) +
1
ε
dT
dτα
Eα,α+1(γ1τα
))
· Eα(γ2Tα
) + [V(0)Eα(γ1Tα
) +
1
ε
dT
dTα
Eα,α+1(γ1Tα
) + (V(0)
· Eα(γ1τα
) +
1
ε
dT
dτα
Eα,α+1(γ1τα
))Eα(γ2(T − τ)α
)]Eα(−γ2τα
)}
· Eα(γ2(t − (T + τ))α
),
(53)
10

11. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
and
V(2T) ≤ V(0)Eα(γ1(2T)α
) +
1
ε
dT
d(2T)α
Eα,α+1(γ1(2T)α
) + (V(0)Eα(γ1τα
)
+
1
ε
dT
dτα
Eα,α+1(γ1τα
))Eα(γ2(2T − τ)α
) + [V(0)Eα(γ1Tα
) +
1
ε
dT
dTα
· Eα,α+1(γ1Tα
) + (V(0)Eα(γ1τα
) +
1
ε
dT
dτα
Eα,α+1(γ1τα
))Eα(γ2(T
− τ)α
)]Eα(−γ2Tα
) + {V(0)Eα(γ1(T + τ)α
) +
1
ε
dT
d(T + τ)α
· Eα,α+1(γ1(T + τ)α
) + (V(0)Eα(γ1τα
) +
1
ε
dT
dτα
Eα,α+1(γ1τα
))
· Eα(γ2Tα
) + [V(0)Eα(γ1Tα
) +
1
ε
dT
dTα
Eα,α+1(γ1Tα
) + (V(0)
· Eα(γ1τα
) +
1
ε
dT
dτα
Eα,α+1(γ1τα
))Eα(γ2(T − τ)α
)]Eα(−γ2τα
)}
· Eα(γ2(T − τ)α
).
(54)
Therefore, when t ∈ [nT, nT + τ), by induction we can obtain that
V(t) ≤ V(0)Eα(γ1tα
) +
1
ε
dT
dtα
Eα,α+1(γ1tα
) + V(τ)Eα(γ2(t − τ)α
)
+ V(T)Eα(−γ2(t − T)α
) + V(T + τ)Eα(γ2(t − (T + τ))α
) + V(2T)
· Eα(−γ2(t − 2T)α
) + · · · + V(nT)Eα(−γ2(t − nT)α
)
= V(0)Eα(γ1tα
) +
1
ε
dT
dtα
Eα,α+1(γ1tα
) +
n−1
∑
k=0
V(kT + τ)Eα(γ2(t
− (kT + τ))α
) +
n
∑
k=1
V(kT)Eα(−γ2(t − kT)α
),
(55)
when t ∈ [nT + τ, (n + 1)T), we have
V(t) ≤ V(0)Eα(γ1tα
) +
1
ε
dT
dtα
Eα,α+1(γ1tα
) + V(τ)Eα(γ2(t − τ)α
)
+ V(T)Eα(−γ2(t − T)α
) + V(T + τ)Eα(γ2(t − (T + τ))α
) + V(2T)
· Eα(−γ2(t − 2T)α
) + · · · + V(nT)Eα(−γ2(t − nT)α
) + V(nT + τ)
· Eα(γ2(t − (nT + τ))α
)
= V(0)Eα(γ1tα
) +
1
ε
dT
dtα
Eα,α+1(γ1tα
) +
n
∑
k=0
V(kT + τ)Eα(γ2(t
− (kT + τ))α
) +
n
∑
k=1
V(kT)Eα(−γ2(t − kT)α
).
(56)
Similarly, for any t ≥ 0, by Lemma 5 we have
∥ x(t) ∥ ≤
√
V(0) | Eα(γ1tα) |
λmin(P)
+
√
1
ε
dT dtα | Eα,α+1(γ1tα) |
λmin(P)
+
√∑n
k=0 V(kT + τ) | Eα(γ2(t − (kT + τ))α) |
λmin(P)
+
√∑n
k=1 V(kT) | Eα(−γ2(t − kT)α) |
λmin(P)
11

12. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
≤
√
V(0)m1
λmin(P)(1 + γ1tα)
+
√
1
ε
dT dtαm2
λmin(P)(1 + γ1tα)
+



√ 1
λmin(P)
·
n
∑
k=0
V(kT + τ)m3
1 + γ2 | (t − (kT + τ))α |
+



√ 1
λmin(P)
·
n
∑
k=1
V(kT)m4
1 + γ2 | (t − kT)α |
.
(57)
Consequently, when t → ∞, we have
∥ x(t) ∥ ≤
√
dT dm2
λmin(P)εγ1
. (58)
Thus the system (38) is bounded. It completes the proof.
Remark 3. Similarly, the domain of attraction of the fractional-order system with periodically intermittent control can be
estimated via the compact set S = {x(t) ∈ Rn
| V = xT
Px ≤ dT dm2
εγ1
}. Furthermore, the Theorem is appropriate not only for
the case τ = T
2
, but for other cases τ ∈ (0, T).
Additionally, if d = 0, then we can obtain the following fractional-order system with periodically intermittent control.
C
0 Dα
t x(t) =
{
(A + K)x(t), nT ≤ t < nT + τ,
Ax(t), nT + τ ≤ t < (n + 1)T,
(59)
where n = 0, 1, 2, . . ..
Thereupon, from the Theorem 2, relevant stability result can be derived as follow.
Corollary 2. The fractional-order system (59) is stable if there exist positive constants γ1, γ2, a symmetric positive definite
matrix P, and matrices K, X1, X2, Y1, Y2, Z1 and Z2 with appropriate dimensions satisfy the following conditions:
(i) (X1 + YT
1 + Y1 − NT
1 − N1 + Z1)/2 < 0, (60)
(ii) (X2 + YT
2 + Y2 − NT
2 − N2 + Z2)/2 < 0, (61)
(iii)
[
X1 Y1
YT
1 Z1
]
≥ 0,
[
X2 Y2
YT
2 Z2
]
≥ 0 (62)
where N1 = −PA − AT
P − Q − Q T
+ γ1P, N2 = −PA − AT
P + (γ1 + γ2)P, Q = PK.
Remark 4. From the Corollary, the stability conditions of the fractional-order system with periodically intermittent control
are obtained. Moreover, the cost of time and control can be also saved by the control method.
4. Examples and numerical simulations
Example 1. In order to demonstrate the effectiveness of the Theorem 1, relevant parameters of the WPT system are
adopted as follows:
E = 10 V, Cp = Cs = 4 µF, Rp = Rs = 0.15 , Lp = Ls = 90 µH, Mps = 30 µH, RL = 25 , T = 0.00012 s. Besides,
taking α = 1.01, ε = 100, m2 = 1, γ1 = 200, γ2 = 300.
And then, using linear matrix inequality (LMI) tool box to solve conditions (i), (ii) and (iii) of the Theorem 1, we get a
feasible solution of the matrix P as
P =
⎡
⎢
⎣
3.9701 −0.0401 1.6476 1.7081
−0.0401 0.2497 −5.6865 −0.2294
1.6476 −5.6865 432.8121 10.7114
1.7081 −0.2294 10.7114 6.6787
⎤
⎥
⎦.
In addition, Fig. 2 shows the phase plot of the fractional-order WPT system, from which one can see that the WPT
system is bounded. In the meantime, x(t) converges to the compact set
G = {x(t) ∈ R4
|∥ x(t) ∥≤
√
bT
1 b1m2
λmin(P)εγ1
= 2235.496}. (63)
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13. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
Fig. 2. Phase plot of the fractional-order WPT system.
Fig. 3. Domain of attraction of the fractional-order WPT system.
Hence Theorem 1 is effective. Besides, from Remark 1, relevant two-dimensional projection of the domain of attraction
for the fractional-order WPT system is estimated and shown in Fig. 3. Consider the conservatism of the Theorem, Lemma 3
is used in this paper. From this picture, and by comparison, it can easily be seen that the domain of attraction obtained
by utilizing the inequality technique of Lemma 3 is larger. It indicates that the conservatism of the result is improved in
theory.
Remark 5. Via the above analysis, we know that the conservatism of the result is improved obviously. Nevertheless, from
the practical point of view, the conservatism of the system remains. Therefore, here we discuss it in two cases.
Case 1: The feasible solution exists in Theorem 1, which the domain of attraction can be estimated by this Theorem.
Thereby the robustness of the system can also be reflected though the domain of attraction.
Case 2: The feasible solution cannot be found by the Theorem, one can use state response to evaluate the robustness of the
system. First of all, performance indexes such as overshoot and regulation time can be obtained by the state response plot;
Secondly, construct a weight function about these performance indexes; Finally, evaluate the robustness of the system
by this function. Besides, one can utilize the robust control such as H∞ control and µ synthesis to increase the domain
of attraction and improve the robustness of the system.
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14. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
Fig. 4. The linear fractional transformation system with uncertainties and controller.
Fig. 5. Phase plot of the system (38) without control.
Remark 6. In practice, it is not comprehensive to evaluate the robustness of the system only from the perspective of the
domain of attraction. The performance indexes such as overshoot and regulation time should also be considered so as to
fully evaluate the robustness of the system. Besides, uncertainties exist in reality. Thus, one can evaluate the robustness
of the system by means of the structured singular value µ analysis and the state response. Main procedure are as follows:
Step 1. Obtain the linear fractional transformation (Fig. 4) representation Q via the closed-loop system model. In Fig. 4,
△ denotes the set of the model uncertainties; G denotes the transfer function; H denotes the controller;
Step 2. Calculate the maximum structured singular value ρ of Q by using structured singular value theory, and obtain
stability boundary 1
ρ
. This boundary is similar to boundary of the domain of attraction in this paper;
Step 3. Obtain overshoot δ and regulation time ts via the state response plot;
Step 4. Construct a weight function H( 1
ρ
, δ, ts), and evaluate the robustness of the system according to given conditions.
Note that we only list the main steps here due to this topic will be investigated in the next paper.
Example 2. For the system (38), we take
A =
[
0.6 −0.1
0.1 0.3
]
, d =
[
2
2
]
, x(0) =
[
0.1 0.1
]T
. Besides, choosing α = 0.8, T = 2.5s, τ = 2s, γ1 = 1, γ2 = 1.5,
ε = 0.1, m2 = 1. Via solving the linear matrix inequalities (40), (41) and (42) of the Theorem 2. The feasible solution of
the matrices P and K is obtained as P =
[
0.0053 −0.0001
−0.0001 0.0064
]
, K =
[
−0.8757 −2.1755
1.8113 −0.7940
]
, respectively.
Phase plot of the system without intermittent control and with intermittent control is depicted in Fig. 5 and Fig. 6,
respectively. The Fig. 5 shows that the system (38) without control is of instability. But from Fig. 6, we can easily see that
the system (38) with periodically intermittent control is stable, which indicates that the designed controller is effective.
14

15. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
Fig. 6. Phase plot of the system (38) with periodically intermittent control.
Fig. 7. Domain of attraction of the system (38) with periodically intermittent control.
Additionally, substituting relevant parameters into (39), we can get that
G = {x(t) ∈ R2
|∥ x(t) ∥≤
√
dT dm2
λmin(P)εγ1
= 89.8933}. (64)
Meantime, by Remark 3, the corresponding domain of attraction is estimated and presented in Fig. 7, from which it is not
difficult to see that the conservatism of the result is reduced because of the domain of attraction becomes larger.
Remark 7. From this example, it is easy to find that the derived results can not only analyze the boundedness of the
fractional-order piecewise affine system, but also estimate the domain of attraction of the system.
5. Conclusion
The problem of the domain of attraction for the fractional-order WPT system by employing the Lyapunov function
approach and the inductive method was studied in this paper. The relevant results were derived, and further verified by
15

16. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
a few examples, respectively. Simulation shows that the obtained results are valid, and less conservatism. Furthermore,
the designed controller also presents the good control effect. However, uncertainties are not considered in this paper, and
the issue of the robustness evaluation has not been fully addressed, which would be worthwhile for a further research.
CRediT authorship contribution statement
Zhongming Yu: Writing - original draft, Visualization, Investigation, Software, Formal analysis, Resources. Yue Sun:
Conceptualization, Methodology, Data curation. Xin Dai: Funding acquisition, Supervision, Validation, Writing - review &
editing, Project administration.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to thank the associate editor and the anonymous reviewers for their suggestions and comments.
This work was supported by National Natural Science Foundation of China under Grant 62073047.
References
[1] P. Raval, D. Kacprzak, A. Hu, A ICPT system for low power electronics charging applications, in: 2011 6th IEEE Conference on Industrial
Electronics and Applications, 2011, pp. 520–525.
[2] S. Lee, J. Huh, C. Park, et al., On-line electric vehicle using inductive power transfer system, in: Energy Conversion Congress and Exposition,
2010, pp. 1598–1601.
[3] C. Wang, O. Stielau, G. Covic, Design considerations for a contactless electric vehicle battery charger, IEEE Trans. Ind. Electron. 52 (5) (2005)
1308–1314.
[4] H. Jiang, J. Zhang, D. Lan, K. Chao, H. Shahnasser, A low-frequency versatile wireless power transfer technology for biomedical implants, IEEE
Trans. Biomed. Circuits Syst. 7 (4) (2013) 526–535.
[5] W. Niu, J. Chu, W. Gu, et al., Exact analysis of frequency splitting phenomena of contactless power transfer systems, IEEE Trans. Circuits Syst.
I. Regul. Pap. 60 (6) (2013) 1670–1677.
[6] Y. Zhang, Z. Zhao, K. Chen, Frequency-splitting analysis of four-coil resonant wireless power transfer, IEEE Trans. Ind. Appl. 50 (4) (2014)
2436–2445.
[7] A. Radwan, A. Emira, A. Abdelaty, Modeling and analysis of fractional order DC-DC converter, Isa Trans. 82 (2018) 184–199.
[8] A. Anis, T. Freeborn, A. Elwakil, Review of fractional-order electrical characterization of supercapacitors, J. Power Sources 400 (2018) 457–467.
[9] D. Baleanu, A. Golmankhaneh, A. Golmankhaneh, On electromagnetic field in fractional space, Nonlinear Anal. RWA 11 (1) (2010) 288–292.
[10] M. Ortigueira, C. Matos, M. Piedade, Fractional discrete-time signal processing: Scale conversion and linear prediction, Nonlinear Dynam. 29
(1–4) (2002) 173–190.
[11] V. Martynyuk, M. Ortigueira, Fractional model of an electrochemical capacitor, Signal Process. 107 (2014) 355–360.
[12] V. Martynyuk, M. Ortigueira, M. Fedula, O. Savenko, Methodology of electrochemical capacitor quality control with fractional order model,
AEU-Int. J. Electron. Commun. 91 (2018) 118–124.
[13] S. Majidabad, H. Shandiz, A. Hajizadeh, Nonlinear fractional-order power system stabilizer for multi-machine power systems based on sliding
mode technique, Internat. J. Robust Nonlinear Control 25 (2015) 1548–1568.
[14] T. Freeborn, B. Maundy, A. Elwakil, Fractional-order models of supercapacitors, batteries and fuel cells: a survey, Mater. Renew. Sustain. Energy
4 (9) (2015) 1–7.
[15] S. Westerlund, L. Ekstam, Capacitor. theory, Capacitor theory, IEEE Trans. Dielectr. Electr. Insul. 1 (1994) 826–839.
[16] T. Hartley, J. Trigeassou, C. Lorenzo, N. Maamri, Energy storage and loss in fractional-order systems, Circuits Devices Syst. Iet 9 (3) (2015)
227–235.
[17] J. Trigeassou, N. Maamri, A. Oustaloup, Lyapunov stability of commensurate fractional order systems: a physical interpretation, J. Comput.
Nonlinear Dyn. 11 (2016) 051007-1-051007-8.
[18] J. Trigeassou, M. Nezha, O. Alain, Lyapunov stability of noncommensurate fractional order systems: an energy balance approach, J. Comput.
Nonlinear Dyn. 11 (2016) 041007-1-041007-9.
[19] S. Wang, Y. Yu, G. Wen, Hybrid projective synchronization of time-delayed fractional order chaotic systems, Nonlinear Anal. Hybrid Syst. 11
(2014) 129–138.
[20] S. Zhang, Y. Yu, H. Wang, Mittag-Leffler stability of fractional-order Hopfield neural networks, Nonlinear Anal. Hybrid Syst. 16 (2015) 104–121.
[21] Y. Tan, M. Xiong, D. Du, S. Fei, Observer-based robust control for fractional-order nonlinear uncertain systems with input saturation and
measurement quantization, Nonlinear Anal. Hybrid Syst. 34 (2019) 45–57.
[22] X. Shu, B. Zhang, The effect of fractional orders on the transmission power and efficiency of fractional-order wireless power transmission
system, Energies 11 (7) (2018) 1–9.
[23] Z. Yu, Y. Sun, X. Dai, Z. Ye, Stability and control of uncertain ICPT system considering time-varying delay and stochastic disturbance, Internat.
J. Robust Nonlinear Control 29 (18) (2019) 6582–6604.
[24] C. Pittet, Stability regions for linear systems with saturating controls via circle and popov criteria, in: IEEE Conference on Decision and Control,
2015, pp. 4518–4523.
[25] J. Goncalves, A. Megretski, M. Dahleh, Global analysis of piecewise linear systems using impact maps and surface Lyapunov functions, IEEE
Trans. Automat. Control 48 (12) (2003) 2089–2106.
[26] Y. Lim, K. Oh, H. Ahn, Stability and stabilization of fractional-order linear systems subject to input saturation, IEEE Trans. Automat. Control 58
(4) (2013) 1062–1067.
[27] D. Baleanu, S. Rezapour, H. Mohammadi, Some existence results on nonlinear fractional differential equations, Phil. Trans. R. Soc. A 371 (1990)
(2013) 1–7.
16

17. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
[28] S.J. Sadati, D. Baleanu, A. Ranjbar, R. Ghaderi, T. Abdeljawad, Mittag-Leffler stability theorem for fractional nonlinear systems with delay, Abstr.
Appl. Anal. 2011 (12) (2014) 331–336.
[29] M. Duarte-Mermoud, N. Aguila-Camacho, J. Gallegos, R. Castro-Linares, Using general quadratic Lyapunov functions to prove Lyapunov uniform
stability for fractional order systems, Commun. Nonlinear Sci. Numer. Simul. 22 (1–3) (2015) 650–659.
[30] P. Khargonekar, I. Petersen, K. Zhou, Robust stabilization of uncertain linear systems: Quadratic stabilizability and H∞ control theory, IEEE
Trans. Automat. Control 35 (3) (1990) 356–361.
[31] Y. Moon, P. Park, W. Kwon, Y. Lee, Delay-dependent robust stabilization of uncertain state-delayed systems, Internat. J. Control 74 (14) (2001)
1447–1455.
[32] A. Wu, Z. Zeng, Boundedness, Mittag-Leffler stability and asymptotical ω-periodicity of fractional-order fuzzy neural networks, Neural Netw.
74 (2015) 73–84.
[33] X. Wen, Z. Wu, J. Lu, Stability analysis of a class of nonlinear fractional-order systems, IEEE Trans. Circuits Syst. II Express Briefs 55 (11) (2008)
1178–1182.
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