# Adaptive Inverse Optimal Control of a Novel Fractional-Order Four-Wing Hyperchaotic System with Uncertain Parameter and Circuitry Implementation

Authors: Chaojun Wu; Gangquan Si; Yanbin Zhang; Ningning Yang
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

An efficient approach of inverse optimal control and adaptive control is developed for global asymptotic stabilization of a novel fractional-order four-wing hyperchaotic system with uncertain parameter. Based on the inverse optimal control methodology and fractional-order stability theory, a control Lyapunov function (CLF) is constructed and an adaptive state feedback controller is designed to achieve inverse optimal control of a novel fractional-order hyperchaotic system with four-wing attractor. Then, an electronic oscillation circuit is designed to implement the dynamical behaviors of the fractional-order four-wing hyperchaotic system and verify the satisfactory performance of the controller. Comparing with other fractional-order chaos control methods which may have more than one nonlinear state feedback controller, the inverse optimal controller has the advantages of simple structure, high reliability, and less control effort that is required and can be implemented by electronic oscillation circuit.


---

## Body

## 1. Introduction

Fractional calculus is a classical mathematical theory with history of more than 300 years. Fractional-order differential equations can describe many systems in the real world more adequately, such as electrical circuits [1–3], polymer material [4], finance systems [5], and population models [6].Comparing with integer-order chaotic systems which exhibit complex nonlinear phenomena, fractional-order and multi-wing chaotic systems exhibit more complex and richer dynamical behaviors. It is expected that those chaotic systems will have a certain theoretical and practical significance for secure communication, control processing, and some other engineering applications. There exist some well-known fractional-order systems and multi-wing systems, such as the fractional-order Chua’s circuit [7], the fractional-order Rössler system [8], the fractional-order Chen system [9],the fractional-order Lu system [10], the first true four-wing attractor [11], a family of hyperchaotic systems with four-wing attractor [12], among many others [13–22].The applications of fractional-order differential equations in control processing developed rapidly in the last two decades. Fractional-order control methods and researches on the stability of fractional-order systems have become the frontier problem in modern nonlinear dynamics [23–25]. Podlubny and his colleagues proposed the fractional-order proportional-integral (PI) and proportional-integral-derivative (PID) controllers which are named P
I
λ and P
I
λ
D
μ controllers with the orders λ and μ [26] and designed analogue circuits to implement fractional-order controllers [27]. Hamamci presented a method to stabilize a given fractional dynamic system using fractional-order P
I
λ and P
I
λ
D
μ controllers [28]. In [29], the author proposed a solution scheme for a class of fractional optimal control problems. In [30], authors designed a fractional-order sliding mode controller to stabilize a fractional-order hyperchaotic system.Nonlinear controllers have been adopted in many fields in spite of having the complex structure and being not easy to obtain. Optimal control guarantees several desirable properties for the closed-loop system, including stability margins and robustness. To circumvent the task of solving a Hamilton-Jacobi-Bellman equation whose solution is nonexistent or nonunique in generally, the inverse optimal control technique based on input-to-state stability concept was developed [31].In this article, based on inverse optimal control methodology and fractional-order stability theory, we construct an adaptive state feedback controller to achieve the global asymptotic stabilization of a novel fractional-order four-wing hyperchaotic system. An electronic oscillation circuit is designed to realize the dynamical behaviors of the fractional-order system and verify the satisfactory performance of the controller. Comparing with other control methods, for example, feedback control [32], active control [33], impulsive control [34], back-stepping control [35], and sliding mode control [36], the inverse optimal controller has the advantages of simple structure, high reliability, and less control effort that is required.The rest of this paper is organized as follows. In Section2, the inverse optimal control methodology and fractional-order stability theory are introduced. In Section 3, firstly, a novel pseudo four-wing hyperchaotic system is analyzed. Then a derivative fractional-order four-wing hyperchaotic system is implemented by numerical simulations, and the adaptive inverse optimal control is applied to stabilize an unstable equilibrium point of the fractional-order four-wing hyperchaotic system. In Section 4, circuitry implementations are given for verifying the feasibility. Finally, the conclusion part summarizes the whole development process and presents some concluding remarks and comments.


## 2. Preliminaries

The inverse optimality approach of fractional-order four-wing hyperchaotic system used in this paper requires the knowledge of control Lyapunov function and fractional-order stability theory.


### 2.1. Inverse Optimal Control Approach

Consider the following nonlinear system:(1)
x
˙
=
f
x
+
g
x
u
,
where x
∈
R
n denotes the state vector and u
∈
R
m denotes the control vector, respectively. Moreover, f
:
R
n
→
R
n is vector function with f
(
0
)
=
0 and g
:
R
n
→
R
n
×
m is matrix-valued function.Definition 1 (see [37]).
V
(
x
) is a smooth, positive definite, and radially unbounded function. Taking the time derivative of V
(
x
), one obtains
(2)
V
˙
x
=
∂
V
x
∂
x
f
x
+
g
x
u
=
L
f
V
+
u
L
g
V
,
where
(3)
L
f
V
=
∂
V
x
∂
x
f
x
,
L
g
V
=
∂
V
(
x
)
∂
x
g
(
x
)
.
For all x
≠
0, V
(
x
) is a control Lyapunov function (CLF) for system (1), if it satisfies
(4)
L
g
V
=
0
⟹
L
f
V
<
0
.Lemma 2 (see [38]).
Suppose that the static state feedback control law(5)
u
=
k
x
≔
R
x
-
1
∂
V
∂
x
g
x
T
,
where R
:
R
n
→
R
n
×
m is a positive definite matrix-valued function, stabilizes system (1) with respect to a positive definite radially unbounded Lyapunov function V
(
x
). Then the control law
(6)
u
≔
-
β
k
(
x
)
=
-
β
R
-
1
(
x
)
∂
V
∂
x
g
x
T
is optimal with respect to the cost
(7)
J
u
=
∫
0
∞
l
x
+
u
T
R
x
u
d
t
,
where
(8)
l
x
=
-
2
β
∂
V
∂
x
(
f
(
x
)
+
g
(
x
)
u
)
+
β
β
-
2
∂
V
∂
x
g
x
R
-
1
x
∂
V
∂
x
g
x
T
.


### 2.2. The Fractional-Order Stability Theory

In the theory of fractional calculus,D
t
α
a represents an arbitrary order differ-integral operator. It is a notation for taking both fractional integrals and derivatives in one single expression and can be defined as
(9)
D
t
α
a
=
d
α
d
t
α
,
α
>
0
1
,
α
=
0
∫
a
t
d
τ
-
α
,
α
<
0
,
where α
∈
R is the order of the operation a and t are the bounds of the operation.There are some different definitions for fractional derivatives [39]. The most frequently used definitions are Riemann-Liouville definition and Caputo definition.Riemann-Liouville definition is given as(10)
d
α
f
(
t
)
d
t
α
=
1
Γ
(
n
-
α
)
d
n
d
t
n
∫
a
t
f
(
τ
)
t
-
τ
α
-
n
+
1
d
τ
,
where n
-
1
<
α
<
n, n is an integer number, and Γ
(
·
) is the Gamma function.If all the initial conditions are zero, the Laplace transform of Riemann-Liouville functional derivative is given as(11)
L
d
α
f
(
t
)
d
t
α
=
s
α
L
f
(
t
)
.Another alternative definition of Riemann-Liouville definition of fractional-order derivative was reported by Caputo as follows:(12)
d
α
f
(
t
)
d
t
α
=
1
Γ
(
n
-
α
)
∫
0
t
f
(
n
)
(
τ
)
t
-
τ
α
-
n
+
1
d
τ
.Since the Caputo definition is more convenient for initial conditions problems, in this paper, the operatorD
α denotes the α-order Caputo fractional derivative.Then some stability theorems for fractional-order systems are introduced.Consider the following autonomous system:(13)
d
x
d
t
=
A
x
x
.One can get the fractional-order form of system (13) as
(14)
d
α
x
d
t
α
=
A
x
x
,
where x
(
x
1
,
x
2
,
…
)
∈
R
n
(
n
∈
N
), 0
<
α
<
1.Lemma 3 (see [40]).
For fractional-order system (14), we have following. (i)
The system is asymptotically stable if and only ifarg
λ
i
[
A
(
x
)
]
>
α
π
/
2, (
i
=
1,2
,
…
,
n
), where arg
λ
i
[
A
(
x
)
] denotes the argument of the matrix A
(
x
)’s eigenvalue λ
i. In this case, components of the state decay toward 0 like t
-
α.
(ii)
The system is stable if and only if either it is asymptotically stable or those critical eigenvalues which satisfyarg
(
λ
i
(
A
)
)
=
α
π
/
2 have geometric multiplicity one.
The stability region for0
<
a
<
1 is illustrated in Figure 1.Figure 1
The stability region of fractional-order systems.Lemma 4 (see [41]).
If system (13) is asymptotically stable, in the range of state variable x (except the origin), all the real part of matrix A
(
x
)’s eigenvalues λ
i are not more than zero.Theorem 5.
If system (13) is asymptotically stable, fractional-order system (14) is also asymptotically stable.Proof.
According to Lemma4, assume that system (13) is asymptotically stable, one can get that all the real part of the matrix A
(
x
)’s eigenvalues λ
i are not more than zero. Then one can obtain arg
(
λ
)
>
π
/
2. It satisfies arg
(
λ
)
≥
a
π
/
2 for 0
<
α
<
1. According to Lemma 3, fractional-order system (14) is stable.


## 2.1. Inverse Optimal Control Approach

Consider the following nonlinear system:(1)
x
˙
=
f
x
+
g
x
u
,
where x
∈
R
n denotes the state vector and u
∈
R
m denotes the control vector, respectively. Moreover, f
:
R
n
→
R
n is vector function with f
(
0
)
=
0 and g
:
R
n
→
R
n
×
m is matrix-valued function.Definition 1 (see [37]).
V
(
x
) is a smooth, positive definite, and radially unbounded function. Taking the time derivative of V
(
x
), one obtains
(2)
V
˙
x
=
∂
V
x
∂
x
f
x
+
g
x
u
=
L
f
V
+
u
L
g
V
,
where
(3)
L
f
V
=
∂
V
x
∂
x
f
x
,
L
g
V
=
∂
V
(
x
)
∂
x
g
(
x
)
.
For all x
≠
0, V
(
x
) is a control Lyapunov function (CLF) for system (1), if it satisfies
(4)
L
g
V
=
0
⟹
L
f
V
<
0
.Lemma 2 (see [38]).
Suppose that the static state feedback control law(5)
u
=
k
x
≔
R
x
-
1
∂
V
∂
x
g
x
T
,
where R
:
R
n
→
R
n
×
m is a positive definite matrix-valued function, stabilizes system (1) with respect to a positive definite radially unbounded Lyapunov function V
(
x
). Then the control law
(6)
u
≔
-
β
k
(
x
)
=
-
β
R
-
1
(
x
)
∂
V
∂
x
g
x
T
is optimal with respect to the cost
(7)
J
u
=
∫
0
∞
l
x
+
u
T
R
x
u
d
t
,
where
(8)
l
x
=
-
2
β
∂
V
∂
x
(
f
(
x
)
+
g
(
x
)
u
)
+
β
β
-
2
∂
V
∂
x
g
x
R
-
1
x
∂
V
∂
x
g
x
T
.


## 2.2. The Fractional-Order Stability Theory

In the theory of fractional calculus,D
t
α
a represents an arbitrary order differ-integral operator. It is a notation for taking both fractional integrals and derivatives in one single expression and can be defined as
(9)
D
t
α
a
=
d
α
d
t
α
,
α
>
0
1
,
α
=
0
∫
a
t
d
τ
-
α
,
α
<
0
,
where α
∈
R is the order of the operation a and t are the bounds of the operation.There are some different definitions for fractional derivatives [39]. The most frequently used definitions are Riemann-Liouville definition and Caputo definition.Riemann-Liouville definition is given as(10)
d
α
f
(
t
)
d
t
α
=
1
Γ
(
n
-
α
)
d
n
d
t
n
∫
a
t
f
(
τ
)
t
-
τ
α
-
n
+
1
d
τ
,
where n
-
1
<
α
<
n, n is an integer number, and Γ
(
·
) is the Gamma function.If all the initial conditions are zero, the Laplace transform of Riemann-Liouville functional derivative is given as(11)
L
d
α
f
(
t
)
d
t
α
=
s
α
L
f
(
t
)
.Another alternative definition of Riemann-Liouville definition of fractional-order derivative was reported by Caputo as follows:(12)
d
α
f
(
t
)
d
t
α
=
1
Γ
(
n
-
α
)
∫
0
t
f
(
n
)
(
τ
)
t
-
τ
α
-
n
+
1
d
τ
.Since the Caputo definition is more convenient for initial conditions problems, in this paper, the operatorD
α denotes the α-order Caputo fractional derivative.Then some stability theorems for fractional-order systems are introduced.Consider the following autonomous system:(13)
d
x
d
t
=
A
x
x
.One can get the fractional-order form of system (13) as
(14)
d
α
x
d
t
α
=
A
x
x
,
where x
(
x
1
,
x
2
,
…
)
∈
R
n
(
n
∈
N
), 0
<
α
<
1.Lemma 3 (see [40]).
For fractional-order system (14), we have following. (i)
The system is asymptotically stable if and only ifarg
λ
i
[
A
(
x
)
]
>
α
π
/
2, (
i
=
1,2
,
…
,
n
), where arg
λ
i
[
A
(
x
)
] denotes the argument of the matrix A
(
x
)’s eigenvalue λ
i. In this case, components of the state decay toward 0 like t
-
α.
(ii)
The system is stable if and only if either it is asymptotically stable or those critical eigenvalues which satisfyarg
(
λ
i
(
A
)
)
=
α
π
/
2 have geometric multiplicity one.
The stability region for0
<
a
<
1 is illustrated in Figure 1.Figure 1
The stability region of fractional-order systems.Lemma 4 (see [41]).
If system (13) is asymptotically stable, in the range of state variable x (except the origin), all the real part of matrix A
(
x
)’s eigenvalues λ
i are not more than zero.Theorem 5.
If system (13) is asymptotically stable, fractional-order system (14) is also asymptotically stable.Proof.
According to Lemma4, assume that system (13) is asymptotically stable, one can get that all the real part of the matrix A
(
x
)’s eigenvalues λ
i are not more than zero. Then one can obtain arg
(
λ
)
>
π
/
2. It satisfies arg
(
λ
)
≥
a
π
/
2 for 0
<
α
<
1. According to Lemma 3, fractional-order system (14) is stable.


## 3. Adaptive Inverse Optimal Control of a Novel Fractional-Order Four-Wing Hyperchaotic System with Uncertain Parameter

### 3.1. Analysis of a Novel Pseudo Four-Wing Hyperchaotic System

We consider a hyperchaotic system described by(15)
d
x
1
d
t
=
a
-
x
1
+
x
2
x
3
,
d
x
2
d
t
=
b
x
2
-
h
x
1
x
3
+
c
x
4
,
d
x
3
d
t
=
k
x
1
x
2
-
g
x
3
,
d
x
4
d
t
=
-
m
x
2
+
n
x
3
,
where x
1, x
2, x
3, and x
4 are state variables and a, b, c, h, k, g, m, and n are positive real constants.Jacobian matrix is equal to(16)
J
=
-
a
-
h
x
3
k
x
2
0
a
x
3
b
k
x
1
-
m
a
x
2
-
h
x
1
-
g
-
n
0
c
0
0
.
With parameters a
=
10, b
=
2.5, c
=
1, h
=
1, k
=
4, g
=
4, m
=
1, and n
=
1
/
8, system (15) has three equilibrium points: P
1
(
0,0
,
0,0
)
,
P
2
(
-
8,1
,
-
8,61.5
), and P
3
(
-
8
,
-
1,8
,
-
61.5
).Then letJ
-
λ
I
=
0; eigenvalues of matrix J for equilibrium point P
1 are λ
1
p
1
=
0.5, λ
2
p
1
=
2, λ
3
p
1
=
-
10, and λ
4
p
1
=
-
4, where λ
1
p
1 and λ
2
p
1 are positive roots and λ
3
p
1 and λ
4
p
1 are negative roots. For equilibrium points P
2 and P
3, eigenvalues of matrix J are λ
1
p
2,3
=
-
11.8430, λ
2
p
2,3
=
0.007799, λ
3
p
2,3
=
0.1676
-
29.4284
i, and λ
4
p
2,3
=
0.1676
+
29.4284
i, where λ
1
p
2,3 is a negative root, λ
2
p
2,3 is a positive root, and λ
3
p
2,3 and λ
4
p
2,3 are a pair of conjugate complex roots with positive real part. Therefore, equilibrium points P
1, P
2, and P
3 are all unstable.The divergence is given by(17)
∇
V
=
∂
x
˙
∂
x
+
∂
y
˙
∂
y
+
∂
z
˙
∂
z
+
∂
v
˙
∂
v
=
-
11.5
<
0
.System (15) is dissipative.Figure2 shows Lyapunov exponents versus b. When b
=
2.5, Lyapunov exponents are as follows: λ
L
1
=
0.602, λ
L
2
=
0.187, λ
L
3
=
-
0.173, λ
L
4
=
-
12.166. λ
L
1
,
λ
L
2
>
0, and λ
L
3
,
λ
L
4
<
0, so system (15) is in hyperchaotic state.Figure 2
Lyapunov exponents versusb.Figure3 is the bifurcation diagram of x
3 versus b. The image shows complex and rich dynamical behaviors.Figure 3
Bifurcation diagram ofx
3 versus b.In the process of investigating this hyperchaotic system, we found this odd: different initial values would generate different area of attractor. With the aforementioned set of parameters and the initial values(
0.1,0.4,0.9,1
), (
0.1,0.4
,
-
0.9,1
), one can get the strange attractor as shown in Figure 4, where the upper one is indicated by a solid line and the lower one is indicated by a dotted line.Chaotic attractor of system (15).
(a)
Projection on thex
1-x
2 plane
(b)
Projection on thex
1-x
3 plane
(c)
Projection on thex
1-x
4 plane
(d)
Projection on thex
2-x
3 planeTo explain this weird phenomenon, detailed mathematical deduction and simulations will be given.Firstly, observing the first and third equations in system (15), one has
(18)
d
x
1
d
t
=
a
(
-
x
1
+
x
2
x
3
)
(19)
d
x
3
d
t
=
k
x
1
x
2
-
g
x
3
.Multiplying (18) with x
1 and (19) with x
3 yields
(20)
d
x
1
d
t
x
1
=
-
a
x
1
2
+
a
x
1
x
2
x
3
,
d
x
3
d
t
x
3
=
k
x
1
x
2
x
3
-
g
x
3
2
.Equation (20) can be rewritten as
(21)
1
a
d
x
1
d
t
x
1
=
-
x
1
2
+
x
1
x
2
x
3
(22)
1
k
d
x
3
d
t
x
3
=
x
1
x
2
x
3
-
g
k
x
3
2
.Subtracting (22) from (21), one obtains
(23)
1
k
d
x
3
d
t
x
3
-
1
a
d
x
1
d
t
x
1
=
x
1
2
-
g
k
x
3
2
.Equation (23) is equivalent to
(24)
d
(
a
x
3
2
-
k
x
1
2
)
d
t
=
2
a
k
x
1
2
-
a
g
x
3
2
.Equation (24) can be rewritten as
(25)
d
a
x
3
2
-
k
x
1
2
d
t
=
2
a
k
x
1
2
-
g
a
x
3
2
.By assumption,a
>
g
>
0, let a
=
μ
+
g, where μ is a positive real constant and μ
>
0. Equation (25) is equivalent to
(26)
d
a
x
3
2
-
k
x
1
2
d
t
=
-
2
g
a
x
3
2
-
k
x
1
2
+
2
μ
a
x
1
2
.Solving (26) yields
(27)
a
x
3
(
t
)
2
-
k
x
1
(
t
)
2
=
e
-
2
g
t
∫
0
t
2
μ
a
x
1
(
t
)
2
e
2
g
τ
d
τ
+
a
x
3
(
0
)
2
hhhhhhhhhhh
-
k
x
1
(
0
)
2
∫
0
t
2
μ
a
x
1
(
t
)
2
e
2
g
τ
.Sincee
-
2
g
t
>
0, 2
μ
a
x
1
(
t
)
2
e
2
g
τ
>
0; if the initial values of system (15) at time satisfy a
x
3
(
0
)
>
k
x
1
(
0
) or a
x
3
(
0
)
2
>
k
x
1
(
0
)
2, then (26) implies that a
x
3
(
t
)
2
-
k
x
1
(
t
)
2
>
0 for all t
≥
0.State variablex
3
(
t
) cannot cross the plane x
3
=
0 (Figure 10). The trajectories of system (15) are limited in the area of a
x
3
(
t
)
>
k
x
1
(
t
). If the trajectories of system (15) travel from the upper part to the lower part, there will be t
1
>
0 at x
3
(
t
1
)
=
0. That contradicts the fact that a
x
3
(
t
)
2
>
k
x
1
(
t
)
2 for all t
≥
0.For example, with the aforementioned set of parameters anda
>
g
>
0, the initial values are (
0.1,0.4,0.9,1
) and (
0.1,0.4
,
-
0.9,1
), respectively, which all belong to the area of a
x
3
(
0
)
>
k
x
1
(
0
), and one can find two coexisting attractors: the upper one with the initial value (
0.1,0.4,0.9,1
) and the lower one with the initial value (
0.1,0.4
,
-
0.9,1
), as shown in Figure 5. The trajectories of the attractor can travel only within the area a
x
3
(
t
)
>
k
x
1
(
t
) limited by the dashed line and cannot cross the plane x
3
=
0. That means that system (15) can only generate pseudo four-wing attractor.Figure 5
Two coexisting attractors ofx
1-x
3 phase plane.To produce a real four-wing attractor, we introduce a simple linear state feedbacke
x
4 to the third equation of system (15). Then one gets the following system:
(28)
d
x
1
d
t
=
a
-
x
1
+
x
2
x
3
,
d
x
2
d
t
=
b
x
2
-
h
x
1
x
3
+
c
x
4
,
d
x
3
d
t
=
k
x
1
x
2
-
g
x
3
+
e
x
4
,
d
x
4
d
t
=
-
m
x
2
+
n
x
3
.
With the aforementioned set of parameters and e
=
1, the real four-wing attractor is shown in Figure 6.The four-wing attractor of system (28).
(a)
Projection on thex
1-x
2 plane
(b)
Projection on thex
1-x
3 plane
(c)
Projection on thex
1-x
4 plane
(d)
Projection on thex
2-x
3 plane


### 3.2. Adaptive Inverse Optimal Control of the Fractional-Order Four-Wing Hyperchaotic System with Uncertain Parameter

The fractional-order hyperchaotic system with four-wing attractor can be constructed as the following form:(29)
d
α
x
1
d
t
α
=
a
-
x
1
+
x
2
x
3
,
d
α
x
2
d
t
α
=
b
x
2
-
h
x
1
x
3
+
c
x
4
,
d
α
x
3
d
t
α
=
k
x
1
x
2
-
g
x
3
+
e
x
4
,
d
α
x
4
d
t
α
=
-
m
x
2
+
n
x
3
.The predictor-corrector method is used to implement the fractional-order four-wing hyperchaotic system with the orderα
=
0.9. The fractional-order four-wing attractor of system (29) with the aforementioned set of parameters is shown in Figure 7.Fractional-order four-wing attractor of system (29) for α
=
0.9.
(a)
Projection on thex
1-x
2 plane
(b)
Projection on thex
1-x
3 plane
(c)
Projection on thex
1-x
4 plane
(d)
Projection on thex
2-x
3 planeThen the inverse optimal control methodology is developed to achieve the global asymptotic stabilization of the fractional-order four-wing hyperchaotic system with uncertain parameter. The closed-loop system with a controlleru is described by
(30)
d
α
x
1
d
t
α
=
a
-
x
1
+
x
2
x
3
,
d
α
x
2
d
t
α
=
b
t
x
2
-
h
x
1
x
3
+
c
x
4
+
u
,
d
α
x
3
d
t
α
=
k
x
1
x
2
-
g
x
3
+
e
x
4
,
d
α
x
4
d
t
α
=
-
m
x
2
+
n
x
3
.With the aforementioned set of parameters,b
(
t
) is an time-varying uncertain parameter described by b
(
t
)
=
b
+
Δ
b
(
t
), b is positive definite, and Δ
b
(
t
) is a bounded function which satisfied Δ
b
(
t
)
≤
ξ.Theorem 6.
The fractional-order hyperchaotic system with four-wing attractor can achieve global asymptotical stability by the following linear state feedback control law:(31)
u
=
-
(
b
1
+
λ
)
x
2
,
where b
1 is the estimate value of the unknown parameter b and λ
>
0. The parameter estimation update law b
˙
1 is
(32)
b
˙
1
=
4
a
x
2
2
h
.Proof.
According to Theorem5, one considers the integer-order dynamical system as follows:
(33)
d
x
1
d
t
=
a
(
-
x
1
+
x
2
x
3
)
d
x
2
d
t
=
b
(
t
)
x
2
-
h
x
1
x
3
+
c
x
4
+
u
d
x
3
d
t
=
k
x
1
x
2
-
g
x
3
+
e
x
4
d
x
4
d
t
=
-
m
x
2
+
n
x
3
.
Construct a Lyapunov function for system (33). Consider
(34)
V
=
x
1
2
+
2
a
h
x
2
2
+
a
k
x
3
2
+
2
a
x
4
2
+
1
2
b
~
2
,
where a
>
0, c
>
0, h
>
0, k
>
0, and b
~
=
b
-
b
1. Obviously, V
(
x
) is positive definite. The derivative of V
(
x
) along the time is
(35)
V
˙
x
=
2
x
1
x
˙
1
+
4
a
x
2
h
x
˙
2
+
2
a
x
3
k
x
˙
3
+
4
a
x
4
x
˙
4
-
b
~
b
˙
1
.
Substituting (32) and (33) into (35) yields
(36)
V
˙
x
=
2
a
(
-
x
1
+
x
2
x
3
)
x
1
+
4
a
h
b
t
x
2
-
h
x
1
x
3
+
c
x
4
+
u
x
2
+
2
a
k
(
k
x
1
x
2
-
g
x
3
+
e
x
4
)
x
3
-
4
a
m
x
2
+
n
x
3
x
4
-
b
-
b
1
4
a
x
2
2
h
.
Equation (36) can be rewritten as
(37)
V
˙
(
x
)
=
-
2
a
x
1
2
+
4
a
b
1
+
Δ
b
(
t
)
h
x
2
2
-
2
a
g
k
x
3
2
+
4
a
h
u
x
2
.
According to Definition1, one can obtain
(38)
L
f
V
=
-
2
a
x
1
2
+
4
a
b
1
+
Δ
b
(
t
)
h
x
2
2
-
2
a
g
k
x
3
2
,
L
g
V
=
4
a
h
x
2
.
It is easy to verify thatL
f
V
<
0 for L
g
V
=
0. Then V
(
x
) is indeed a control Lyapunov function (CLF).
Define a state feedback controller according to the results of Lemma2:
(39)
u
=
-
β
R
x
-
1
L
g
V
=
-
b
1
+
λ
x
2
,
where β is a positive constant and R
(
x
)
-
1 is a positive definite function of x. Consider
(40)
R
x
-
1
=
h
4
a
β
b
1
+
λ
.
Substituting (39) into (37) yields
(41)
V
˙
(
x
)
=
-
2
a
x
1
2
-
4
a
(
λ
-
Δ
b
(
t
)
)
h
x
2
2
-
2
a
g
k
x
3
2
.
According to the definition ofΔ
b
(
t
), if λ satisfies λ
≥
Δ
b
(
t
)
=
ξ, it can be proved that V
˙
(
x
) is negative definite for all x
≠
0 easily.
That means the close-loop system can achieve global asymptotical stabilization with the controller.
Define a cost functional as follows:(42)
J
u
=
lim
⁡
t
→
∞
2
β
V
x
+
∫
0
t
l
x
+
u
T
R
x
u
d
τ
,
where
(43)
l
(
x
)
=
-
β
L
f
V
+
β
2
R
x
-
1
L
g
V
2
.
Substituting (38) into (43) yields
(44)
l
x
=
-
β
-
2
a
x
1
2
-
2
a
g
k
x
3
2
+
4
a
b
1
+
Δ
b
t
h
x
2
2
+
4
a
h
β
(
b
1
+
λ
)
x
2
2
.
Then
(45)
l
x
=
2
a
β
x
1
2
+
2
a
g
β
k
x
3
2
+
4
a
β
h
λ
-
Δ
b
t
x
2
2
.
Equation (45) implies that l
(
x
) is positive definite for all x
≠
0. When x approaches infinity, l
(
x
) also approaches infinity. So function l
(
x
) is radially unbounded.
Substituting (39) into (2) yields
(46)
V
˙
(
x
)
=
L
f
V
-
β
R
-
1
(
x
)
L
g
V
2
.
Multiplying both sides of (46) with -
β, one obtains
(47)
-
β
V
˙
(
x
)
=
-
β
L
f
V
+
β
2
R
x
-
1
L
g
V
2
.
Substituting (43) into (47) yields
(48)
-
β
V
˙
x
=
l
x
.
According to (39), one obtains
(49)
u
T
R
(
x
)
u
=
β
2
R
x
-
1
L
g
V
2
.
Then(50)
l
x
+
u
T
R
x
u
=
-
β
V
˙
x
+
β
2
R
x
-
1
L
g
V
2
.
Substituting (50) into (42) yields
(51)
J
u
=
lim
⁡
t
→
∞
∫
0
t
-
β
V
˙
x
+
β
2
R
x
-
1
L
g
V
2
d
τ
2
β
V
(
x
)
hhhhihi
+
∫
0
t
-
β
V
˙
x
+
β
2
R
x
-
1
L
g
V
2
d
τ
.
In accordance with the optimal control law, the minimum of the cost function is(52)
J
u
min
⁡
=
2
β
V
x
0
.
Now, the linear state feedback control law (31) has been proved to be the optimal control law. Then integer-order system (33) is stable at zero equilibrium point with the controller u. According to Theorem 5, one can obtain that fractional-order four-wing hyperchaotic system (29) is also stable at the zero equilibrium point because all the real parts of matrix A
(
x
)’s eigenvalues are not more than zero.
The blocks diagram for adaptive inverse optimal control of the fractional-order four-wing hyperchaotic system is shown in Figure8.
The perturbation of parameterb
(
t
) is given by b
(
t
)
=
2.5
+
sin
(
7
t
). The state trajectories of the controlled fractional-order hyperchaotic system are displayed in Figure 9.
Whent
=
200
s, the controller begins to work. States converge to zero with the parameter perturbation which indicates the former uncertainty. The fractional-order four-wing hyperchaotic system is indeed globally asymptotically stabilized by the adaptive inverse optimal controller.Figure 8
Blocks diagram for adaptive inverse optimal control.Time waveform of state variables of the controlled fractional-order system with the controller started att
=
200
s.
(a)
x
1 state
(b)
x
2 state
(c)
x
3 state
(d)
x
4 stateFigure 10
The time evolutions of each state variable starting at timet
=
200
s.


## 3.1. Analysis of a Novel Pseudo Four-Wing Hyperchaotic System

We consider a hyperchaotic system described by(15)
d
x
1
d
t
=
a
-
x
1
+
x
2
x
3
,
d
x
2
d
t
=
b
x
2
-
h
x
1
x
3
+
c
x
4
,
d
x
3
d
t
=
k
x
1
x
2
-
g
x
3
,
d
x
4
d
t
=
-
m
x
2
+
n
x
3
,
where x
1, x
2, x
3, and x
4 are state variables and a, b, c, h, k, g, m, and n are positive real constants.Jacobian matrix is equal to(16)
J
=
-
a
-
h
x
3
k
x
2
0
a
x
3
b
k
x
1
-
m
a
x
2
-
h
x
1
-
g
-
n
0
c
0
0
.
With parameters a
=
10, b
=
2.5, c
=
1, h
=
1, k
=
4, g
=
4, m
=
1, and n
=
1
/
8, system (15) has three equilibrium points: P
1
(
0,0
,
0,0
)
,
P
2
(
-
8,1
,
-
8,61.5
), and P
3
(
-
8
,
-
1,8
,
-
61.5
).Then letJ
-
λ
I
=
0; eigenvalues of matrix J for equilibrium point P
1 are λ
1
p
1
=
0.5, λ
2
p
1
=
2, λ
3
p
1
=
-
10, and λ
4
p
1
=
-
4, where λ
1
p
1 and λ
2
p
1 are positive roots and λ
3
p
1 and λ
4
p
1 are negative roots. For equilibrium points P
2 and P
3, eigenvalues of matrix J are λ
1
p
2,3
=
-
11.8430, λ
2
p
2,3
=
0.007799, λ
3
p
2,3
=
0.1676
-
29.4284
i, and λ
4
p
2,3
=
0.1676
+
29.4284
i, where λ
1
p
2,3 is a negative root, λ
2
p
2,3 is a positive root, and λ
3
p
2,3 and λ
4
p
2,3 are a pair of conjugate complex roots with positive real part. Therefore, equilibrium points P
1, P
2, and P
3 are all unstable.The divergence is given by(17)
∇
V
=
∂
x
˙
∂
x
+
∂
y
˙
∂
y
+
∂
z
˙
∂
z
+
∂
v
˙
∂
v
=
-
11.5
<
0
.System (15) is dissipative.Figure2 shows Lyapunov exponents versus b. When b
=
2.5, Lyapunov exponents are as follows: λ
L
1
=
0.602, λ
L
2
=
0.187, λ
L
3
=
-
0.173, λ
L
4
=
-
12.166. λ
L
1
,
λ
L
2
>
0, and λ
L
3
,
λ
L
4
<
0, so system (15) is in hyperchaotic state.Figure 2
Lyapunov exponents versusb.Figure3 is the bifurcation diagram of x
3 versus b. The image shows complex and rich dynamical behaviors.Figure 3
Bifurcation diagram ofx
3 versus b.In the process of investigating this hyperchaotic system, we found this odd: different initial values would generate different area of attractor. With the aforementioned set of parameters and the initial values(
0.1,0.4,0.9,1
), (
0.1,0.4
,
-
0.9,1
), one can get the strange attractor as shown in Figure 4, where the upper one is indicated by a solid line and the lower one is indicated by a dotted line.Chaotic attractor of system (15).
(a)
Projection on thex
1-x
2 plane
(b)
Projection on thex
1-x
3 plane
(c)
Projection on thex
1-x
4 plane
(d)
Projection on thex
2-x
3 planeTo explain this weird phenomenon, detailed mathematical deduction and simulations will be given.Firstly, observing the first and third equations in system (15), one has
(18)
d
x
1
d
t
=
a
(
-
x
1
+
x
2
x
3
)
(19)
d
x
3
d
t
=
k
x
1
x
2
-
g
x
3
.Multiplying (18) with x
1 and (19) with x
3 yields
(20)
d
x
1
d
t
x
1
=
-
a
x
1
2
+
a
x
1
x
2
x
3
,
d
x
3
d
t
x
3
=
k
x
1
x
2
x
3
-
g
x
3
2
.Equation (20) can be rewritten as
(21)
1
a
d
x
1
d
t
x
1
=
-
x
1
2
+
x
1
x
2
x
3
(22)
1
k
d
x
3
d
t
x
3
=
x
1
x
2
x
3
-
g
k
x
3
2
.Subtracting (22) from (21), one obtains
(23)
1
k
d
x
3
d
t
x
3
-
1
a
d
x
1
d
t
x
1
=
x
1
2
-
g
k
x
3
2
.Equation (23) is equivalent to
(24)
d
(
a
x
3
2
-
k
x
1
2
)
d
t
=
2
a
k
x
1
2
-
a
g
x
3
2
.Equation (24) can be rewritten as
(25)
d
a
x
3
2
-
k
x
1
2
d
t
=
2
a
k
x
1
2
-
g
a
x
3
2
.By assumption,a
>
g
>
0, let a
=
μ
+
g, where μ is a positive real constant and μ
>
0. Equation (25) is equivalent to
(26)
d
a
x
3
2
-
k
x
1
2
d
t
=
-
2
g
a
x
3
2
-
k
x
1
2
+
2
μ
a
x
1
2
.Solving (26) yields
(27)
a
x
3
(
t
)
2
-
k
x
1
(
t
)
2
=
e
-
2
g
t
∫
0
t
2
μ
a
x
1
(
t
)
2
e
2
g
τ
d
τ
+
a
x
3
(
0
)
2
hhhhhhhhhhh
-
k
x
1
(
0
)
2
∫
0
t
2
μ
a
x
1
(
t
)
2
e
2
g
τ
.Sincee
-
2
g
t
>
0, 2
μ
a
x
1
(
t
)
2
e
2
g
τ
>
0; if the initial values of system (15) at time satisfy a
x
3
(
0
)
>
k
x
1
(
0
) or a
x
3
(
0
)
2
>
k
x
1
(
0
)
2, then (26) implies that a
x
3
(
t
)
2
-
k
x
1
(
t
)
2
>
0 for all t
≥
0.State variablex
3
(
t
) cannot cross the plane x
3
=
0 (Figure 10). The trajectories of system (15) are limited in the area of a
x
3
(
t
)
>
k
x
1
(
t
). If the trajectories of system (15) travel from the upper part to the lower part, there will be t
1
>
0 at x
3
(
t
1
)
=
0. That contradicts the fact that a
x
3
(
t
)
2
>
k
x
1
(
t
)
2 for all t
≥
0.For example, with the aforementioned set of parameters anda
>
g
>
0, the initial values are (
0.1,0.4,0.9,1
) and (
0.1,0.4
,
-
0.9,1
), respectively, which all belong to the area of a
x
3
(
0
)
>
k
x
1
(
0
), and one can find two coexisting attractors: the upper one with the initial value (
0.1,0.4,0.9,1
) and the lower one with the initial value (
0.1,0.4
,
-
0.9,1
), as shown in Figure 5. The trajectories of the attractor can travel only within the area a
x
3
(
t
)
>
k
x
1
(
t
) limited by the dashed line and cannot cross the plane x
3
=
0. That means that system (15) can only generate pseudo four-wing attractor.Figure 5
Two coexisting attractors ofx
1-x
3 phase plane.To produce a real four-wing attractor, we introduce a simple linear state feedbacke
x
4 to the third equation of system (15). Then one gets the following system:
(28)
d
x
1
d
t
=
a
-
x
1
+
x
2
x
3
,
d
x
2
d
t
=
b
x
2
-
h
x
1
x
3
+
c
x
4
,
d
x
3
d
t
=
k
x
1
x
2
-
g
x
3
+
e
x
4
,
d
x
4
d
t
=
-
m
x
2
+
n
x
3
.
With the aforementioned set of parameters and e
=
1, the real four-wing attractor is shown in Figure 6.The four-wing attractor of system (28).
(a)
Projection on thex
1-x
2 plane
(b)
Projection on thex
1-x
3 plane
(c)
Projection on thex
1-x
4 plane
(d)
Projection on thex
2-x
3 plane


## 3.2. Adaptive Inverse Optimal Control of the Fractional-Order Four-Wing Hyperchaotic System with Uncertain Parameter

The fractional-order hyperchaotic system with four-wing attractor can be constructed as the following form:(29)
d
α
x
1
d
t
α
=
a
-
x
1
+
x
2
x
3
,
d
α
x
2
d
t
α
=
b
x
2
-
h
x
1
x
3
+
c
x
4
,
d
α
x
3
d
t
α
=
k
x
1
x
2
-
g
x
3
+
e
x
4
,
d
α
x
4
d
t
α
=
-
m
x
2
+
n
x
3
.The predictor-corrector method is used to implement the fractional-order four-wing hyperchaotic system with the orderα
=
0.9. The fractional-order four-wing attractor of system (29) with the aforementioned set of parameters is shown in Figure 7.Fractional-order four-wing attractor of system (29) for α
=
0.9.
(a)
Projection on thex
1-x
2 plane
(b)
Projection on thex
1-x
3 plane
(c)
Projection on thex
1-x
4 plane
(d)
Projection on thex
2-x
3 planeThen the inverse optimal control methodology is developed to achieve the global asymptotic stabilization of the fractional-order four-wing hyperchaotic system with uncertain parameter. The closed-loop system with a controlleru is described by
(30)
d
α
x
1
d
t
α
=
a
-
x
1
+
x
2
x
3
,
d
α
x
2
d
t
α
=
b
t
x
2
-
h
x
1
x
3
+
c
x
4
+
u
,
d
α
x
3
d
t
α
=
k
x
1
x
2
-
g
x
3
+
e
x
4
,
d
α
x
4
d
t
α
=
-
m
x
2
+
n
x
3
.With the aforementioned set of parameters,b
(
t
) is an time-varying uncertain parameter described by b
(
t
)
=
b
+
Δ
b
(
t
), b is positive definite, and Δ
b
(
t
) is a bounded function which satisfied Δ
b
(
t
)
≤
ξ.Theorem 6.
The fractional-order hyperchaotic system with four-wing attractor can achieve global asymptotical stability by the following linear state feedback control law:(31)
u
=
-
(
b
1
+
λ
)
x
2
,
where b
1 is the estimate value of the unknown parameter b and λ
>
0. The parameter estimation update law b
˙
1 is
(32)
b
˙
1
=
4
a
x
2
2
h
.Proof.
According to Theorem5, one considers the integer-order dynamical system as follows:
(33)
d
x
1
d
t
=
a
(
-
x
1
+
x
2
x
3
)
d
x
2
d
t
=
b
(
t
)
x
2
-
h
x
1
x
3
+
c
x
4
+
u
d
x
3
d
t
=
k
x
1
x
2
-
g
x
3
+
e
x
4
d
x
4
d
t
=
-
m
x
2
+
n
x
3
.
Construct a Lyapunov function for system (33). Consider
(34)
V
=
x
1
2
+
2
a
h
x
2
2
+
a
k
x
3
2
+
2
a
x
4
2
+
1
2
b
~
2
,
where a
>
0, c
>
0, h
>
0, k
>
0, and b
~
=
b
-
b
1. Obviously, V
(
x
) is positive definite. The derivative of V
(
x
) along the time is
(35)
V
˙
x
=
2
x
1
x
˙
1
+
4
a
x
2
h
x
˙
2
+
2
a
x
3
k
x
˙
3
+
4
a
x
4
x
˙
4
-
b
~
b
˙
1
.
Substituting (32) and (33) into (35) yields
(36)
V
˙
x
=
2
a
(
-
x
1
+
x
2
x
3
)
x
1
+
4
a
h
b
t
x
2
-
h
x
1
x
3
+
c
x
4
+
u
x
2
+
2
a
k
(
k
x
1
x
2
-
g
x
3
+
e
x
4
)
x
3
-
4
a
m
x
2
+
n
x
3
x
4
-
b
-
b
1
4
a
x
2
2
h
.
Equation (36) can be rewritten as
(37)
V
˙
(
x
)
=
-
2
a
x
1
2
+
4
a
b
1
+
Δ
b
(
t
)
h
x
2
2
-
2
a
g
k
x
3
2
+
4
a
h
u
x
2
.
According to Definition1, one can obtain
(38)
L
f
V
=
-
2
a
x
1
2
+
4
a
b
1
+
Δ
b
(
t
)
h
x
2
2
-
2
a
g
k
x
3
2
,
L
g
V
=
4
a
h
x
2
.
It is easy to verify thatL
f
V
<
0 for L
g
V
=
0. Then V
(
x
) is indeed a control Lyapunov function (CLF).
Define a state feedback controller according to the results of Lemma2:
(39)
u
=
-
β
R
x
-
1
L
g
V
=
-
b
1
+
λ
x
2
,
where β is a positive constant and R
(
x
)
-
1 is a positive definite function of x. Consider
(40)
R
x
-
1
=
h
4
a
β
b
1
+
λ
.
Substituting (39) into (37) yields
(41)
V
˙
(
x
)
=
-
2
a
x
1
2
-
4
a
(
λ
-
Δ
b
(
t
)
)
h
x
2
2
-
2
a
g
k
x
3
2
.
According to the definition ofΔ
b
(
t
), if λ satisfies λ
≥
Δ
b
(
t
)
=
ξ, it can be proved that V
˙
(
x
) is negative definite for all x
≠
0 easily.
That means the close-loop system can achieve global asymptotical stabilization with the controller.
Define a cost functional as follows:(42)
J
u
=
lim
⁡
t
→
∞
2
β
V
x
+
∫
0
t
l
x
+
u
T
R
x
u
d
τ
,
where
(43)
l
(
x
)
=
-
β
L
f
V
+
β
2
R
x
-
1
L
g
V
2
.
Substituting (38) into (43) yields
(44)
l
x
=
-
β
-
2
a
x
1
2
-
2
a
g
k
x
3
2
+
4
a
b
1
+
Δ
b
t
h
x
2
2
+
4
a
h
β
(
b
1
+
λ
)
x
2
2
.
Then
(45)
l
x
=
2
a
β
x
1
2
+
2
a
g
β
k
x
3
2
+
4
a
β
h
λ
-
Δ
b
t
x
2
2
.
Equation (45) implies that l
(
x
) is positive definite for all x
≠
0. When x approaches infinity, l
(
x
) also approaches infinity. So function l
(
x
) is radially unbounded.
Substituting (39) into (2) yields
(46)
V
˙
(
x
)
=
L
f
V
-
β
R
-
1
(
x
)
L
g
V
2
.
Multiplying both sides of (46) with -
β, one obtains
(47)
-
β
V
˙
(
x
)
=
-
β
L
f
V
+
β
2
R
x
-
1
L
g
V
2
.
Substituting (43) into (47) yields
(48)
-
β
V
˙
x
=
l
x
.
According to (39), one obtains
(49)
u
T
R
(
x
)
u
=
β
2
R
x
-
1
L
g
V
2
.
Then(50)
l
x
+
u
T
R
x
u
=
-
β
V
˙
x
+
β
2
R
x
-
1
L
g
V
2
.
Substituting (50) into (42) yields
(51)
J
u
=
lim
⁡
t
→
∞
∫
0
t
-
β
V
˙
x
+
β
2
R
x
-
1
L
g
V
2
d
τ
2
β
V
(
x
)
hhhhihi
+
∫
0
t
-
β
V
˙
x
+
β
2
R
x
-
1
L
g
V
2
d
τ
.
In accordance with the optimal control law, the minimum of the cost function is(52)
J
u
min
⁡
=
2
β
V
x
0
.
Now, the linear state feedback control law (31) has been proved to be the optimal control law. Then integer-order system (33) is stable at zero equilibrium point with the controller u. According to Theorem 5, one can obtain that fractional-order four-wing hyperchaotic system (29) is also stable at the zero equilibrium point because all the real parts of matrix A
(
x
)’s eigenvalues are not more than zero.
The blocks diagram for adaptive inverse optimal control of the fractional-order four-wing hyperchaotic system is shown in Figure8.
The perturbation of parameterb
(
t
) is given by b
(
t
)
=
2.5
+
sin
(
7
t
). The state trajectories of the controlled fractional-order hyperchaotic system are displayed in Figure 9.
Whent
=
200
s, the controller begins to work. States converge to zero with the parameter perturbation which indicates the former uncertainty. The fractional-order four-wing hyperchaotic system is indeed globally asymptotically stabilized by the adaptive inverse optimal controller.Figure 8
Blocks diagram for adaptive inverse optimal control.Time waveform of state variables of the controlled fractional-order system with the controller started att
=
200
s.
(a)
x
1 state
(b)
x
2 state
(c)
x
3 state
(d)
x
4 stateFigure 10
The time evolutions of each state variable starting at timet
=
200
s.


## 4. Circuit Experimental Researches

In this part, an electronic oscillation circuit is designed to stabilize the equilibrium of the fractional-order four-wing hyperchaotic system based on inverse optimal control law.As shown in Figure11, the electronic oscillation circuit has two parts: one is the fractional-order four-wing hyperchaotic circuit and the other is an adaptive inverse optimal controller. The circuit consists of resistors, capacitors, operational amplifiers (AD741), and multipliers (AD633). The circuit parameters are as follows: R
1
=
R
3
=
R
9
=
R
14
=
100
k
Ω, R
2
=
R
4
=
R
8
=
10
k
Ω, R
5
=
R
6
=
R
12
=
1
k
Ω, R
13
=
R
18
=
R
19
=
1
k
Ω, R
7
=
R
16
=
40
k
Ω, R
10
=
R
11
=
R
15
=
10
k
Ω, R
17
=
R
20
=
R
22
=
10
k
Ω, R
21
=
2.5
k
Ω, R
23
=
80
k
Ω, R
24
=
400
k
Ω, R
25
=
R
27
=
R
29
=
10
k
Ω, R
26
=
400
k
Ω, R
28
=
2
k
Ω, R
30
=
20
k
Ω, R
31
=
100
k
Ω, and R
32
=
10
k
Ω
C
5
=
1
μ
F. State variables x
1, x
2, x
3, and x
4 are obtained from voltage outputs of fractional calculus unit, respectively. State feedback e
x
4 is R
24 in a dotted line zone, which is the key of the four-wing system. The perturbation of parameter b is generated by sine generator and multiplier.Figure 11
Analog electronic circuit implementations of the fractional-order four-wing hyperchaotic system and the adaptive inverse optimal control.To implement the electronic circuit of the fractional-order system, the frequency domain approximation method is adopted. In [42], an effective algorithm is developed to approximate fractional-order transfer functions. From Table 1 in [14], one can obtain an approximation of 1
/
s
0.9 with an error of about 2 dB as follows:
(53)
F
(
s
)
=
2.2675
(
s
+
1.292
)
(
s
+
215.4
)
(
s
+
0.01292
)
(
s
+
2.154
)
(
s
+
359.4
)
.The fractional calculus unit is shown in Figure12, which is the chain fractance.Figure 12
Chain fractance.One can get the transfer function between A and B. Consider(54)
G
(
s
)
=
R
a
1
+
R
a
C
a
s
+
R
b
1
+
R
b
C
b
s
+
R
c
1
+
R
c
C
c
s
.Using the method of undetermined coefficients and comparing (53) with (54), parameters of the fractional calculus unit are R
a
=
62.84
M
Ω, R
b
=
250
k
Ω, R
c
=
2.5
k
Ω, C
a
=
1.232
μ
F, C
b
=
1.835
μ
F, and C
c
=
1.1
μ
F [43].By circuit simulation, the real four-wing attractor without the controller is shown in Figure13.Phase portrait of the real four-wing attractor by Pspice.
(a)
Projection on thex
1-x
2 plane
(b)
Projection on thex
1-x
3 plane
(c)
Projection on thex
1-x
4 plane
(d)
Projection on thex
2-x
3 planeWhent
=
200
s, the controller starts to work. As shown in Figure 14, state variables are closer to zero. In other words, the fractional-order four-wing hyperchaotic system is stabilized to its unstable equilibrium point.Time waveform of the state variables of the controlled fractional-order system with the controller started att
=
200
s by Pspice.
(a)
x
1 state
(b)
x
2 state
(c)
x
3 state
(d)
x
4 stateThen an electronic oscillation circuit is constructed to implement the fractional-order four-wing attractor and the adaptive inverse optimal controller.As shown in Figure15, because the output of the circuit is limited precisely for using low-cost components, the experimental phase portraits of the fractional-order four-wing attractor are approximately in agreement with circuitry simulations.Phase portrait of the real four-wing attractor by circuitry implementation.
(a)
Projection on thex
1-x
2 plane
(b)
Projection on thex
1-x
3 plane
(c)
Projection on thex
1-x
4 plane
(d)
Projection on thex
2-x
3 planeThe waveforms of state variables in time domain are shown in Figure16. When the controller is added to the circuit, the waveforms are close to zero. It verifies the satisfactory performance of the proposed control law and proves the robustness of the system to some extent.Figure 16
Time waveform of state variables of the controlled fractional-order system by circuitry implementation.


## 5. Conclusions

In this paper, combining the adaptive inverse optimal control with the stability theory of fractional-order system, a linear state feedback controller is designed to make the fractional-order four-wing hyperchaotic system with uncertain parameter stable in the unstable equilibrium point. Through circuit simulations and circuit implementations, the method turned out to be workable. It is remarkably simple as comparing with other fractional chaos control methods which may have more than one nonlinear state feedback controller. This research has a certain theoretical and practical significance for the application of the adaptive inverse optimal control in nonlinear circuits, security communication, and many other engineering applications.



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