Pure and Applied Mathematics Journal
Volume 6, Issue 1, February 2017, Pages: 5-13

Linear and Non-linear Feedback Control Strategies for a 4D Hyperchaotic System

Maysoon M. Aziz, Saad Fawzi AL-Azzawi*

Department of Mathematics, College of Computer Sciences and Mathematics, University of Mosul, Mosul, Iraq

(S. F. AL-Azzawi)

*Corresponding author

Maysoon M. Aziz, Saad Fawzi AL-Azzawi. Linear and Non-linear Feedback Control Strategies for a 4D Hyperchaotic System. Pure and Applied Mathematics Journal. Vol. 6, No. 1, 2017, pp. 5-13. doi: 10.11648/j.pamj.20170601.12

Received: January 1, 2017; Accepted: January 13, 2017; Published: February 16, 2017

Abstract: This paper investigates the stabilization of unstable equilibrium for a 4D hyperchaotic system. The linear, non-linear and speed feedback controls are used to suppress hyperchaos to this equilibrium. The Routh-Hurwitz theorem and Lyapunov's second methods are used to derive the conditions of the asymptotic stability of the controlled hyperchaotic system. Theoretical analysis, numerical simulation and illustrative examples are given to demonstrate the effectiveness of the proposed controllers.

Keywords: Chaos Control, Linear Feedback Control, Non-linear Feedback Control, Routh-Hurwitz Method, Lyapunov's Second Method

Contents

1. Introduction

Chaos is an important topic in nonlinear science. In many cases, chaos is sometimes undesirable, so we wish to avoid and eliminate such behaviors [1]. Chaos control and chaos synchronization were once believed to be impossible until the 1990s when Ott et al. developed the OGY method to suppress chaos. Recently, many different techniques and methods have been proposed to achieve chaos control such as impulsive control method, sliding method control, differential geometric method,  control, adaptive control method, and so on. Among them, the feedback control is especially attractive and has been commonly applied to practical implementation due to its simplicity in configuration and implementation [2]. Generally speaking, there are two main approaches for controlling chaos: linear feedback control and non- linear feedback control. The linear feedback control approach offers many advantages such as robustness and computational complexity over the non- feedback control method [2-7].

Many attempts have been made to control hyperchaos of hyperchaotic systems. Recently, Yan (2005)[8], Dou and Sun et al. (2009) [9], Zhu (2010)[10] and Aziz and AL-Azzawi (2015) [11] suppressed hyperchaotic systems to unstable equilibrium by using feedback control method. In 2017, the Ref. [12] present some problems of these strategies and how treatment. Our system is generated based on Pan system via a state feedback controller which is described by the following mathematical model:

(1)

where , and  are constant parameters. When parameters and , system (1) is hyperchaotic and has two Lyapunov exponents, i.e., [13], and hyperchaotic attractors and Lyapunov exponents spectrum are shown in Figure 1 and Figure 2 respectively. The system (1) has only one equilibrium and the equilibrium is an unstable under these parameters [13].

The aim of this paper is discuss three feedback control methods for a 4D hyperchaotic system. The rest of the paper is organized as follows. section 2 present helping results which inclined ordinary feedback control, dislocated feedback control, enhancing feedback control, and speed feedback control. main results are present in section 3. And section 4 present the flow chats for linear and non-linear feedback control. Finally, the conclusion are given in section 5. Moreover, numerical simulations and illustrative examples are applied to verify the effectiveness of chosen controllers.

Figure 1. The attractor of system (1) in x-y-z space.

Figure 2. The Lyapunov exponents of system (1).

2. Helping Results

To control the hyperchaotic system (1) to the unstable equilibrium , we use the feedback control approach to control it. Let us assume that the controlled hyperchaotic system (1) is given by

(2)

where  and  are external feedback control inputs which will be suitably derive the trajectory of the hyperchaotic system, specified by  to the equilibrium of uncontrolled system (i.e. .

Remark 1 (Ordinary feedback control method)

For the ordinary feedback control, the system variable is often multiplied by a coefficient as the feedback gain, and the feedback gain is added to the right -hand of the corresponding equation [3-5, 7, 10-12, 16].

Remark 2 (Dislocated feedback control method)

If a system variable multiplied by a coefficient is added to the right -hand of another equation, then this method is called dislocated feedback control [1, 3-5, 7, 10-12, 16].

Remark 3 (Enhancing feedback control method)

It is difficult for a complex system to be controlled by only one feedback variable, and in such cases the feedback gain is always very large. So we consider using multiple variables multiplied by a proper coefficient as the feedback gain. This method is called enhancing feedback control [1, 3-5, 7, 10-12, 16].

Remark 4.

The above three remarks are used in linear feedback control, also can use with nonlinear feedback control, but in nonlinear feedback the system variable multiplied by a quantity which contains coefficient and nonlinear term and added to the right -hand of equation according to each above three remarks.

Remark 5 (Speed feedback control method)

For the feedback control, the independent variable of a system function is often multiplied by a coefficient as the feedback gain, so the method is called displacement feedback control. Similarly, if the derivative of an independent variable is multiplied by a coefficient as the feedback gain, it is called speed feedback control [2-12, 16].

Remark 6 (Routh-Hurwitz method, [14]). All the roots of the indicated polynomial have negative real parts precisely when the given conditions are met.

Remark 7 (Barbalat Remark, [15, 16]).

If  and , then .

3. Main Results

Theorem 1 (Ordinary feedback control method).

Let the controlled hyperchaotic system (2) be

(3)

where  is the feedback coefficient. When

, system (3) will gradually converge to the unstable equilibrium.

Proof. The Jacobi matrix of system (3) is

(4)

The characteristic equation of matrix (4) is

(5)

where

According to the Routh-Hurwitz method, the Jacobian matrix of system (4) has four negative real part eigenvalues if and only if

Implying  since  and (given) and  is always positive and it is possible under the condition .

Thus system (3) will gradually converge to the unstable equilibrium . The proof is completed.

Illustrative Example 1. the hyperchaotic system (3) be

(6)

When , system (6) will gradually converge to the unstable equilibrium according to theorem 1.

Sol. The characteristic equation of system (6) is

(7)

Then the roots of equation (7) depended of the value of  as following:

when, the equation (7) became

and the all roots have negative real parts

, therefore the system (6) is asymptotically stable.

while , the equation (7) became

then not all roots have negative real parts

therefore the system (6) is unstable.

Also can justified the same result that obtain in theorem 1 by using MATLAB software where numerical simulations are used to investigate the controlled hyperchaotic system (3) using fourth-order Runge-Kutta scheme with time step . The initial values are taken as by . The feedback coefficients are given by , in Figure 3- a, b respectively. The behaviors of the states of the control hyperchaotic system (6) show converging to  when  (Figure 3,a.) and divergence to  when  (Figure 3,b.)

Figure 3. The difference of the state of the controlled system (3) with the control gain change.

Theorem 2 (Dislocated feedback control method).

Let , and the controlled hyperchaotic system (2) be

(8)

where is the feedback coefficient. When, system (8) will gradually converge to the unstable equilibrium .

Proof. The Jacobi matrix of system (8) is

(9)

The characteristic equation of matrix (9) is

(10)

where

By using Routh-Hurwitz method, the Jacobian matrix of system (9) has four negative real part eigenvalues if and only if

Implying  since a, c and d are positive (given) and k is always positive and , it is possible under the condition . Thus system (8) will gradually converge to the unstable equilibrium . The proof is completed.

Illustrative Example 2: Let the hyperchaotic system (8) be

(11)

When , system (11) will gradually converge to the unstable equilibrium  according to theorem 2.

Sol. The characteristic equation of system (11) is

(12)

Then the roots of equation (12) depended of the value of k as following:

when , the equation (12) became

and the all roots have negative real parts

, therefore the system (11) is asymptotically stable.

While , the equation (12) became

then not all roots have negative real parts

therefore the system (11) is unstable.

The feedback coefficients are given by, in Figure 4- a, b respectively. The behaviors of the states  of the control hyperchaotic system (11) show converging to . when (Figure 4,a.) and divergence to  when  in (Figure 4,b.).

Figure 4. The difference of the state of the controlled system (11) with the control gain change.

Theorem 3 (Enhancing feedback control method):

Let ,  and the controlled hyperchaotic system (2) be

(13)

where is the feedback coefficient.

When , system (13) will gradually converge to the unstable equilibrium .

Proof. Analogously as in proof of Theorem 1 and Theorem 2.

Illustrative Example 3: Let the hyperchaotic system (13) be

(14)

When , system (14) will gradually converge to the unstable equilibrium according to theorem 3.

Sol. The characteristic equation of system (14) is

(15)

Then the roots of equation (15) depended of the value of k as following:

when , the equation (15) became

and the all roots have negative real parts

therefore the system (14) is asymptotically stable.

while , the equation (15) became

then not all roots have negative real parts

therefore the system (14) is unstable.

The feedback coefficients are given by  in Figure 5- a, b respectively. The behaviors of the states  of the control hyperchaotic system (14) show converging to  when  (Figure 5,a.) and divergence to  when  (Figure 5,b.).

Figure 5. The difference of the state of the controlled system (14) with the control gain change.

Theorem 4 (Speed feedback control method).

Let , and the controlled hyperchaotic system be

(16)

where is the feedback coefficient. When, system (16) will gradually converge to the unstable equilibrium .

Proof. Analogously as in proof of Theorem 1 and Theorem 2.

Illustrative Example 4. Let the hyperchaotic system (16) be

(17)

When , system (17) will gradually converge to the unstable equilibrium  according to theorem 4.

Sol. The characteristic equation of system (17) is

(18)

Then the roots of equation (18) depended of the value of  as following:

when, the equation (18) became

and the all roots have negative real parts

therefore the system (17) is asymptotically stable.

while , the equation (18) became

then not all roots have negative real parts

therefore the system (17) is unstable.

The behaviors of the states  of the control hyperchaotic system (17) show converging to  when  (Figure 6,a) and divergence to  when  (Figure 6,b).

Figure 6. The difference of the state of the controlled system (17) with the control gain change.

Theorem 5 (Nonlinear feedback control method).

Consider the controlled hyperchaotic system (2) as follows:

(19)

Where  are a positive feedback gain, then system (19) will converge to the unstable equilibrium point .

Proof. Construct the Lyapunov function

where  is a parameter to be determined.

The derivation of  is:

To ensure the system (19) asymptotically stable, the symmetric matrix P should be positive definite, and when P satisfy the following conditions:

we can choose the constant  as

so that is positive definite and  is negative semi-definite. However, cannot get immediately the controlled system (19) is asymptotically stable at equilibrium  i.e. and , for this reason,  then, we have

where  is the smallest eigenvalue of the positive matrix P. and so as , according to the Barbalat Lemma, implies  as , therefore the controlled hyperchaotic system (19) is asymptotically stable to equilibrium point, This completes the proof.

4. Flow Chart of Linear and Non-linear Feedback Control Strategies

The following flow charts explains briefly the Linear and non-linear feedback control strategies respectively

Figure 7. Flow chart of Linear feedback control strategies.

Figure 8. Flow chart of non-Linear feedback control strategies.

5. Conclusions

In this paper, the control problem of a 4D hyperchaotic system is investigated: ordinary feedback control, dislocated feedback control, enhancing feedback control, speed feedback control are used to suppress hyperchaos to unstable equilibrium. The Routh-Hurwitz and Lyapunov's second methods are applied to derive the conditions of the asymptotic stability of the controlled hyperchaotic system. Theoretical analysis,numerical simulation and illustrative examples are given to demonstrate the effectiveness of the proposed controllers.

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