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.

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 and controllers with the orders and [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 and 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 Section 2, 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: where denotes the state vector and denotes the control vector, respectively. Moreover, is vector function with and is matrix-valued function.

Definition 1 (see [37]). is a smooth, positive definite, and radially unbounded function. Taking the time derivative of , one obtains where For all , is a control Lyapunov function (CLF) for system (1), if it satisfies

Lemma 2 (see [38]). Suppose that the static state feedback control law where is a positive definite matrix-valued function, stabilizes system (1) with respect to a positive definite radially unbounded Lyapunov function . Then the control law is optimal with respect to the cost where

2.2. The Fractional-Order Stability Theory

In the theory of fractional calculus, 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 where is the order of the operation and 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 where , is an integer number, and is the Gamma function.

If all the initial conditions are zero, the Laplace transform of Riemann-Liouville functional derivative is given as

Another alternative definition of Riemann-Liouville definition of fractional-order derivative was reported by Caputo as follows:

Since the Caputo definition is more convenient for initial conditions problems, in this paper, the operator denotes the -order Caputo fractional derivative.

Then some stability theorems for fractional-order systems are introduced.

Consider the following autonomous system:

One can get the fractional-order form of system (13) as where , .

Lemma 3 (see [40]). For fractional-order system (14), we have following. (i)The system is asymptotically stable if and only if , , where denotes the argument of the matrix ’s eigenvalue . In this case, components of the state decay toward like .(ii)The system is stable if and only if either it is asymptotically stable or those critical eigenvalues which satisfy have geometric multiplicity one.
The stability region for 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 (except the origin), all the real part of matrix ’s eigenvalues are not more than zero.

Theorem 5. If system (13) is asymptotically stable, fractional-order system (14) is also asymptotically stable.

Proof. According to Lemma 4, assume that system (13) is asymptotically stable, one can get that all the real part of the matrix ’s eigenvalues are not more than zero. Then one can obtain . It satisfies for . 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 where , , , and are state variables and , , , , , , , and are positive real constants.

Jacobian matrix is equal to With parameters , , , , , , , and , system (15) has three equilibrium points: , and .

Then let ; eigenvalues of matrix for equilibrium point are , , , and , where and are positive roots and and are negative roots. For equilibrium points and , eigenvalues of matrix are , , , and , where is a negative root, is a positive root, and and are a pair of conjugate complex roots with positive real part. Therefore, equilibrium points , , and are all unstable.

The divergence is given by

System (15) is dissipative.

Figure 2 shows Lyapunov exponents versus . When , Lyapunov exponents are as follows: , , , . , and , so system (15) is in hyperchaotic state.

Figure 2 

Lyapunov exponents versus .

Figure 3 is the bifurcation diagram of versus . The image shows complex and rich dynamical behaviors.

Figure 3 

Bifurcation diagram of versus .

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 , , 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.

(a) Projection on the - plane

(b) Projection on the - plane

(c) Projection on the - plane

(d) Projection on the - plane

(a) Projection on the - plane
(b) Projection on the - plane
(c) Projection on the - plane
(d) Projection on the - plane

Figure 4 

Chaotic attractor of system (15).

To explain this weird phenomenon, detailed mathematical deduction and simulations will be given.

Firstly, observing the first and third equations in system (15), one has

Multiplying (18) with and (19) with yields

Equation (20) can be rewritten as

Subtracting (22) from (21), one obtains

Equation (23) is equivalent to

Equation (24) can be rewritten as

By assumption, , let , where is a positive real constant and . Equation (25) is equivalent to

Solving (26) yields

Since , ; if the initial values of system (15) at time satisfy or , then (26) implies that for all .

State variable cannot cross the plane (Figure 10). The trajectories of system (15) are limited in the area of . If the trajectories of system (15) travel from the upper part to the lower part, there will be at . That contradicts the fact that for all .

For example, with the aforementioned set of parameters and , the initial values are and , respectively, which all belong to the area of , and one can find two coexisting attractors: the upper one with the initial value and the lower one with the initial value , as shown in Figure 5. The trajectories of the attractor can travel only within the area limited by the dashed line and cannot cross the plane . That means that system (15) can only generate pseudo four-wing attractor.

Figure 5 

Two coexisting attractors of - phase plane.

To produce a real four-wing attractor, we introduce a simple linear state feedback to the third equation of system (15). Then one gets the following system: With the aforementioned set of parameters and , the real four-wing attractor is shown in Figure 6.

(a) Projection on the - plane

(b) Projection on the - plane

(c) Projection on the - plane

(d) Projection on the - plane

(a) Projection on the - plane
(b) Projection on the - plane
(c) Projection on the - plane
(d) Projection on the - plane

Figure 6 

The four-wing attractor of system (28).

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:

The predictor-corrector method is used to implement the fractional-order four-wing hyperchaotic system with the order . The fractional-order four-wing attractor of system (29) with the aforementioned set of parameters is shown in Figure 7.

(a) Projection on the - plane

(b) Projection on the - plane

(c) Projection on the - plane

(d) Projection on the - plane

(a) Projection on the - plane
(b) Projection on the - plane
(c) Projection on the - plane
(d) Projection on the - plane

Figure 7 

Fractional-order four-wing attractor of system (29) for .

Then 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 controller is described by

With the aforementioned set of parameters, is an time-varying uncertain parameter described by , is positive definite, and is a bounded function which satisfied .

Theorem 6. The fractional-order hyperchaotic system with four-wing attractor can achieve global asymptotical stability by the following linear state feedback control law: where is the estimate value of the unknown parameter and . The parameter estimation update law is

Proof. According to Theorem 5, one considers the integer-order dynamical system as follows:
Construct a Lyapunov function for system (33). Consider where , , , , and . Obviously, is positive definite. The derivative of along the time is
Substituting (32) and (33) into (35) yields
Equation (36) can be rewritten as
According to Definition 1, one can obtain
It is easy to verify that for . Then is indeed a control Lyapunov function (CLF).
Define a state feedback controller according to the results of Lemma 2: where is a positive constant and is a positive definite function of . Consider
Substituting (39) into (37) yields
According to the definition of , if satisfies , it can be proved that is negative definite for all easily.
That means the close-loop system can achieve global asymptotical stabilization with the controller.
Define a cost functional as follows: where
Substituting (38) into (43) yields Then
Equation (45) implies that is positive definite for all . When approaches infinity, also approaches infinity. So function is radially unbounded.
Substituting (39) into (2) yields
Multiplying both sides of (46) with , one obtains
Substituting (43) into (47) yields
According to (39), one obtains
Then
Substituting (50) into (42) yields
In accordance with the optimal control law, the minimum of the cost function is
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 . 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 ’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 Figure 8.
The perturbation of parameter is given by . The state trajectories of the controlled fractional-order hyperchaotic system are displayed in Figure 9.
When , 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.