Chapter 9
Sliding-Mode Synchronization Control for Fractional-Order Chaotic Systems with Disturbance
9.1 Problem Statement
On the basis of the Caputo definition of the fractional derivative (2.17), the FOCS will be introduced. Considering the following FOCS as the drive system:
where the fractional order satisfies 
, 
denotes a constant matrix, 
is a state vector, and 
is the nonlinear function vector.
The response system is defined as
where 
is the state vector, 
is the nonlinear function vector, 
is the unknown bounded disturbance, and 
is the control input.
This chapter aims to develop a FODO-based adaptive sliding-mode synchronization control scheme, in which synchronization is realized between two identical fractional-order chaotic systems in the presence of unknown external disturbances. On the basis of the designed sliding-mode controller, the response system can well synchronize the drive system under the proper condition. To obtain the main results, the following assumption is introduced.
9.2 Design of Fractional-Order Disturbance Observer
In this section, a FODO will be designed to approximate the external disturbance in the response system (9.2). Without loss of generality, according to the response system (9.2), we have
where 
is the 
th element of 
, 
is the 
th element of 
, 
is the 
th element of 
, 
is the 
th element of 
, 
is the 
th element of 
, and 
.
Since 
in Equation (9.2) is unknown, 
cannot be applied to develop the synchronization control scheme for the drive system (9.1) and the response system (9.2). To overcome this problem, a fractional-order nonlinear disturbance observer is designed to estimate the disturbance 
.
To design the FODO, an auxiliary variable is introduced based on the design technique of the integer-order disturbance observer as follows [105]:
where 
is a constant to be determined.
Considering Equations (9.3) and (9.4), the Caputo fractional derivative of 
can be written as
To obtain the disturbance estimate, the estimate of the intermediate variable 
is described as
where 
is the estimate of 
.
According to Equation (9.4), the disturbance 
can be estimated as
Define 
. Considering Equations (9.4) and (9.7), we have
Consider Equations (9.5) and (9.6); the Caputo fractional derivative of 
can be written as
On the basis of these discussions, to analyze the convergence of disturbance estimation error 
, a Lyapunov function candidate can be chosen as
Invoking Equation (9.10) and Lemma 2.1, the Caputo fractional derivative of 
is described as
Substituting Equation (9.9) into Equation (9.11), we obtain
According to Equation (9.12) and Assumption 9.1, we have
where 
and 
. To ensure that the estimation error is bounded, the design parameter 
of the FODO should be chosen to make 
. The conclusion that the signals 
and 
are bounded can be drawn from Equation (9.13) and Lemma 2.3.
On the basis of Lemma 2.3 and Equation (9.13), we obtain
which means that
which demonstrates that the disturbance estimation error 
is upper bounded. For the external disturbance 
, 
, the disturbance approximation error 
satisfies 
, where 
is an unknown positive constant. In actual applications, the upper bound of 
is difficult to determine. Thus, the estimated value 
of 
is introduced, where 
.
This design procedure of FODO can be summarized in the following theorem.
On the basis of these analyses, Theorem 9.1 can be easily proven.
9.3 Disturbance-Observer-Based Synchronization Control of Fractional-Order Chaotic Systems
In this section, a FODO-based adaptive sliding-mode control scheme will be proposed to guarantee that the trajectories of drive system (9.1) and response system (9.2) are ultimately bounded synchronized. To design the synchronization control scheme, we first define the error variable 
. From Equations (9.1) and (9.2), the corresponding synchronization error system is given as
To investigate the stabilization of fractional-order synchronization error system (9.16), a simple sliding-mode surface is defined as
where 
.
From Equation (9.17), we have
where 
denotes the 
th row of 
, 
denotes the 
th element of 
.
Using the adaptive sliding control approach, the desired synchronization control input is designed as
where 
is the sign function and 
is a design constant. Meanwhile, the estimated value 
is updated by
where 
is a design constant.
If the synchronization control scheme is designed as Equation (9.19) for the fractional-order synchronization error system (9.16), the sliding-mode surface 
satisfies
where 
is an unknown constant.
From Equations (9.17) and (9.21), one obtains
According to this discussion, if the sliding surface 
is bounded, then the synchronization error 
is also bounded. Therefore, the FODO-based adaptive sliding-mode synchronization control scheme for fractional-order chaotic systems with unknown disturbances can be summarized in the following theorem, which will be proved using the fractional-order Lyapunov method.
9.4 Simulation Examples
9.4.1 Synchronization Control of Modified Fractional-Order Jerk System
Yu [232] investigated a new chaotic generator by constructing a three-segment piecewise-linear odd function with variable breakpoint. From the differential equation of chaotic generator [232], the modified fractional-order jerk system is given as follows:
where 
, 
, and 
are system state variables, the parameters 
and 
, and 
is a piecewise linear function defined by
where 
and 
, 
.
According to the system (9.38) and the piecewise linear function (9.39), the three equilibrium points of the modified fractional-order jerk system are given in Table 9.1. The Jacobian matrix for the system (9.38) can be written as
On the basis of Table 9.1, the corresponding eigenvalues for equilibrium point 
are 
and 
. For equilibrium points 
and 
, the eigenvalues are 
and 
. When the fractional order 
is chosen, we obtain the following characteristic equation of the equilibrium points 
and 
:
with unstable 
and 
, in which 
(
is the lowest common multiple of the fractional-order denominator). Thus, the modified fractional-order jerk system (9.38) has chaotic dynamic behaviors based on the literature [164]. When the initial values are chosen as 
and the fractional order 
, the fractional-order modified jerk system exhibits chaotic behaviors, as shown in Figure 9.1.
Table 9.1 Equilibrium points of the modified fractional-order jerk system
| Linear region |  |
Equilibrium points |
 |
 |
 |
 |
 |
 |
 |
 |
 |
Figure 9.1 Chaotic behaviors of modified fractional-order jerk system: (a) 
–
plane; (b) 
–
plane; (c) 
–
plane; (d) 
–
–
space.
In this section, to illustrate the effectiveness of the proposed synchronization controller, the synchronization of the modified fractional-order jerk system (9.38) is investigated. Consider the FOCS (9.38) as a drive system. From Equation (9.2), the response system is defined as follows:
where 
, 
, and 
are system state variables, 
, 
, and 
are unknown bounded disturbances, 
, 
, and 
are designed synchronization control inputs, and 
is defined as
According to Equations (9.38) and (9.42), the synchronization error system can be written as follows:
where 
, 
, and 
are synchronization error variables.
Referring to the designed controller (9.19), the synchronization controller can be written as follows:
Substituting Equation (9.45) into Equation (9.44), we have the following:
where 
, with 
and 
.
To demonstrate the effectiveness of the proposed FODO-based adaptive sliding-mode synchronization control scheme, numerical simulation results are presented for the modified fractional-order jerk system under the following conditions: initial conditions are chosen as 
, 
, 
, and 
, design parameters are chosen as 
, 
, 
, and 
. Disturbances are assumed as 
, 
, and 
. On the basis of the result of Ishteva [225], we have 
, where 
denotes the square root of minus one, and 
and 
are arbitrary numbers. In this simulation, the parameter 
and the fractional order 
. Thus, 
can be used to approximate 
. The comparison result is shown in Figure 9.2 for 
and 
. According to Figure 9.2, Assumption 9.1 is satisfied.
Figure 9.2 Comparison result of 
and 
.
Numerical results are shown in Figure 9.3 and Figure 9.4 under the proposed FODO-based adaptive sliding-mode control scheme. State synchronization results of the drive system (9.38) and the response system (9.42) are given in Figure 9.3a–c. It is shown that good synchronization performance is achieved. Figure 9.3d shows that the synchronization errors 
, 
, and 
are convergent. Furthermore, the disturbance observation performance of the proposed FODO (9.6) and (9.7) is presented in Figure 9.4. It is evident from Figure 9.4 that the disturbance observer is effective and feasible. According to the simulation results, the drive system (9.38) and the response system (9.42) are bounded synchronized under the designed sliding-mode controller (9.19) and the adaptive update law (9.20). Therefore, the proposed FODO-based adaptive sliding-mode synchronization control scheme is valid for fractional-order chaotic systems with external disturbance.
Figure 9.3 Synchronization control results of modified fractional-order jerk system: (a) 
and 
; (b) 
and 
; (c) 
and 
; (d) synchronization errors 
, 
, and 
.
Figure 9.4 Disturbance observer results of the modified fractional-order jerk system: (a) 
and 
; (b) 
and 
; (c) 
and 
; (d) observation errors 
, 
, and 
.
9.4.2 Synchronization Control of Fractional-Order Liu System
To further illustrate the effectiveness of the proposed synchronization controller, synchronization of the fractional-order Liu system [233] is studied in this section. The fractional-order Liu system is given as follows:
where 
is the fractional order, 
, 
, and 
are system state variables, and 
, 
, and 
are system parameters. The fractional order is chosen as 
, the system parameters are set as 
, 
, and 
and the initial conditions are chosen as 
. The simulation results of the fractional-order Liu system are shown in Figure 9.5.
Figure 9.5 Dynamic behaviors of fractional-order Liu system: (a) 
–
plane; (b) 
–
plane; (c) 
–
plane; (d) 
–
–
space.
To develop the synchronization control scheme, the fractional-order Liu system (9.47) is taken as the drive system, and the response system is constructed as follows:
where 
, 
, and 
are system state variables, 
, 
, and 
are unknown bounded disturbances, and 
, 
, and 
are designed synchronization control inputs.
According to Equations (9.47) and (9.48), the synchronization error system can be written as follows:
where 
, 
, and 
are synchronization error variables.
Invoking the designed controller (9.19), the synchronization controller is given by the following:
Substituting Equation (9.50) into Equation (9.49), we have the following:
where 
, with 
and 
.
For the numerical simulation, we choose the fractional order as 
; the disturbances are assumed as 
. The initial conditions are chosen as 
, 
, 
, and 
. The control parameters are designed as 
, 
, and 
.
According to these conditions and the proposed synchronization control scheme, numerical results are presented in Figure 9.6 and Figure 9.7. Good synchronization performance is shown in Figure 9.6a–c. Numerical results of the synchronization errors 
, 
, and 
are given in Figure 9.6d. Furthermore, the observation performance of the proposed FODO (9.6) and (9.7) is presented in Figure 9.7. It shows that the disturbance observer is effective based on the estimation performance of the designed FODO. On the basis of the simulation results, the drive system (9.47) can synchronize the response system (9.48) well based on the designed sliding-mode controller (9.19) and the adaptive update law (9.20). Thus the proposed adaptive sliding-mode synchronization control method is effective for fractional-order chaotic systems with external disturbance.
Figure 9.6 Synchronization control results of fractional-order Liu system: (a) 
and 
; (b) 
and 
; (c) 
and 
; (d) synchronization errors 
, 
, and 
.
Figure 9.7 Disturbance observer results of the fractional-order Liu system: (a) 
and 
; (b) 
and 
; (c) 
and 
; (d) observation errors 
, 
, and 
.
9.5 Conclusion
In this chapter, a FODO-based adaptive sliding-mode synchronization control scheme has been studied for fractional-order chaotic systems in the presence of external disturbance. A FODO has been developed to approximate the unknown disturbances. A sliding-mode synchronization controller has been designed based on the FODO for synchronization of fractional-order chaotic systems. Furthermore, two examples are given, i.e., synchronization between two modified fractional-order jerk systems and synchronization between two fractional-order Liu systems. Numerical simulations show the effectiveness of the proposed FODO-based adaptive sliding-mode synchronization control scheme.