Advances in Mathematical Physics
Volume 2017 (2017), Article ID 1843179, 8 pages
https://doi.org/10.1155/2017/1843179

Research Article

Adaptive Modified Function Projective Lag Synchronization of Memristor-Based Five-Order Chaotic Circuit Systems

Qiaoping Li and Sanyang Liu

School of Mathematics and Statistics, Xidian University, Xi’an 710071, China

Correspondence should be addressed to Qiaoping Li; liqiaoping1981@126.com

Received 27 October 2016; Revised 22 February 2017; Accepted 8 March 2017; Published 23 March 2017

Academic Editor: Zhi-Yuan Sun

Copyright © 2017 Qiaoping Li and Sanyang Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The modified function projective lag synchronization of the memristor-based five-order chaotic circuit system with unknown bounded disturbances is investigated. Based on the LMI approach and Lyapunov stability theorem, an adaptive control law is established to make the states of two different memristor-based five-order chaotic circuit systems asymptotically synchronized up to a desired scaling function matrix, while the parameter controlling strength update law is designed to estimate the parameters well. Finally, the simulation is put forward to demonstrate the correctness and effectiveness of the proposed methods. The control method involved is simple and practical.

1. Introduction

The memristor, an abbreviation for memory resistor studied by Chua in 1971 [1], is described as the missing 4th passive fundamental circuit element along with resistors, capacitors, and inductors. The memristor is a two-terminal element, either a charge-controlled memristor or a flux-controlled memristor. More than forty years later, on the first day of May 2008, the Hewlett-Packard (HP) research team proudly announced their realization of a memristor prototype, with an official publication in Nature [2, 3]. This new circuit element shares many properties of resistors and shares the same unit of measurement, ohm. Much attention has been attracted to this novel device for its resistance upon turning off the power source; in other words, it depends on the integral of its entire past current waveform. At present, research on chaotic system based on memristor becomes a focal topic [3–10]. Itoh and Chua proposed the possible nonlinear circuits, with a memristor which replaces Chua’s diode in 2008 [4], showed a memristor-based four-order Chua’s circuit which is derived from Chua’s circuit using a PWL memristor. By replacing Chua’s diode with an active flux-controlled memristor circuit, a memristor-based five-order chaotic circuit is derived from four-order Chua’s oscillator By Bao et al. [10].

As is known to all, the synchronization of chaotic systems has been a subject of active research field due to its potential applications for secure communications and control. Up to now, many types of synchronization methods have been put forward in dynamical systems, such as complete synchronization (CS) [11, 12], antisynchronization (AS) [13], phase synchronization (PS) [14], lag synchronization (LS) [15], intermittent lag synchronization [16], generalized synchronization (GS) [17], intermittent generalized synchronization [18], time scale synchronization [19], projective synchronization [20, 21], modified projective synchronization (MPS) [22], and function projective synchronization (FPS) [23].

Recently a more general form of FPS called modified function projective synchronization (MFPS) [24–26] in which master and slave systems are synchronized up to a desired scaling function matrix has attracted attention of researchers as it can provide more security in secure communication. Therefore, the research on MFPS is more valuable in practice. Considering time-delays exist widely in engineering, recently, a general method called modified function projective lag synchronization (MFPLS) for chaotic systems has been proposed in [27].

To the best of our knowledge, the MFPLS of memristor-based five-order chaotic circuit system with unknown disturbances has not been reported yet. Motivated by the above discussion, we will give a comprehensive study on this topic in this article. Based on the parameter modulation, the adaptive control technique, and Lyapunov stability theorem, the adaptive control laws are designed to make the states of two different memristor-based five-order chaotic circuit systems asymptotically synchronized up to a desired scaling function matrix.

2. Memristor-Based Five-Order Chaotic Circuit System

By replacing Chua’s diode with an active flux-controlled memristor circuit, Bao derived a memristor-based five-order chaotic circuit from four-order Chua’s oscillator. This new chaotic circuit can be shown in Figure 1 and it can be described by the following nonlinear equations:

in which .

Figure 1: Memristor-based five-order chaotic circuit.

Denote

and then system (2) can be written as

If we set

and the initial value is chosen as

system (3) is chaotic and multiscroll attractor as shown in Figures 2–5.

Figure 2: Attractor of the memristor-based five-order chaotic circuit (a).

Figure 3: Attractor of the memristor-based five-order chaotic circuit (b).

Figure 4: Attractor of the memristor-based five-order chaotic circuit (c).

Figure 5: Attractor of the memristor-based five-order chaotic circuit (d).

3. MFPLS in Memristor-Based Five-Order Chaotic Circuit Systems

Taking into account the external disturbances, for the sake of convenience, we reexpress system (3) as

where

Taking system (6) as the drive system, the response system can be written as

where

Assumption 1. The unknown external time-varying disturbances and are bounded; in other words, there exist nonnegative constants and such that

Denoting , , we can further obtain that

where stands for the 1-norm.

Definition 2 (MFPLS, [27]). For the drive system (6) and the response system (8), it is said that these two systems are modified function projective lag synchronization (MFPLS), if there exist a delay time and a scaling function matrix such that

or

where is reversible and differentiable and is a bounded continuously differentiable function.

Remark 3. It is clear that (12) and (13) are equivalent in the form.

Remark 4. When ,  , or , MFPLS is simplified to MFPS, FPS, or complete synchronization, respectively.

Denote , , and then the diagonal matrix can be decomposed as

It is obvious that

in which is an unit matrix.

Noted that is a continuously differentiable function with bound, we further pose the following assumption.

Assumption 5. There exist three nonnegative constants , , and such that

The main purpose of this paper is to design an appropriate controller to ensure systems (6) and (8) are modified function projective lag synchronization.
Let us define the MFPLS error vector

Combining systems (6) and (8) with the MFPLS error (17), the following error dynamical system can be obtained:

where

Furthermore, we can obtain

It followed by designing an adaptive controller to achieve MFPLS of systems (6) and (8).

4. Design of the Adaptive Controller

4.1. Case  1

We start with a simple case in which the bounds and are known. In order to achieve the MFPLS, the control law is given by

where is the control gain matrix and .

Theorem 6. If there exists symmetric positive definite matrix , such that the following LMI holds:

then systems (6) and (8) are MFPLS.

Proof. Substituting the control law (21) into (20), we can obtain

Since is positive definite matrix, we design the following Lyapunov function:

Calculating the time derivative of along the trajectory of the error system (18), it can be found that

Utilizing Lyapunov stability theorem, we get

which means that systems (6) and (8) are MFPLS. This completes the proof.

4.2. Case  2

We now consider the general case in which the bounds and are unknown; denote and stands for the estimated value of . Based on the adaptive control method, the controller is designed as

while the parameter adaptive law is given by

where constant is the adjustable gain.

Theorem 7. If there exist a positive constant and a symmetric positive definite matrix such that the following LMIs hold:

then the two systems (6) and (8) are MFPLS.

Proof. Substituting the controller (27) and the adaptive update law (28) into (20), we can obtain

Choose the following Lyapunov function:

Taking the time derivative of along the error system leads to

Applying Lyapunov stability theorem, we can obtain

which means that systems (6) and (8) are MFPLS. Hence, the proof is completed.

4.3. Case  3

More generally, if all the bounds and are unknown, let us denote

with the vector standing for the estimated value of .

For this case, the adaptive control law and parameter update rule is chosen by

with

where is adjustable gain.

Theorem 8. If there exist positive constants and a symmetric positive definite matrix , such that the following LMIs hold:

then systems (6) and (8) are MFPLS.

Proof. Substituting the controller (35) and the adaptive update law (36) into (20), we obtain

The Lyapunov function is designed as

The time derivative of is given by

Substituting (38) into (40), it follows that

According to Lyapunov stability theorem, we can get

which means that the two systems (6) and (8) are MFPLS. Hence, the proof is completed.

5. Simulation

In this section, two different memristor-based five-order chaotic circuit systems with unknown bounded disturbances are considered as the master system and the slave system, respectively, which can be described by

where

The delay time is chosen as , and the scaling function matrix is given by

with the control gain

The drive system is initialized with

and the response system is started with

Using the control method proposed in Theorem 8, the MFPLS error state trajectories are depicted in Figure 6, which illustrate that the error can quickly approach zero while the controller is maintained in a reasonable range which is shown in Figure 7. At the meantime, as is shown in Figure 8, all of the unknown parameters can be tracked well under the parameter adaptive update law.

Figure 6: Time response of MFPLS error .

Figure 7: Time response of the input controller .

Figure 8: Time response of the estimated value of .

6. Conclusion

In this paper, the problem of MFPLS of memristor-based five-order chaotic circuit systems with unknown bounded disturbances has been addressed. Combining the LMI approach with Lyapunov stability theory, an adaptive control law is designed to make the states of two different memristor-based five-order chaotic circuit systems asymptotically synchronized up to a desired scaling function matrix and the unknown parameters can be estimated accurately. At the end of the paper, the corresponding numerical simulations have been given to verify the effectiveness of the proposed control techniques. The proposed method is also suitable for the MFPLS of other chaotic systems and has broad application in secure communication, image processing, and other fields.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (61373174 and 11301409), and thanks are due to all the references authors.

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