Abstract
This paper concerns the problem on the fuzzy synchronization for a kind of disturbed memristive chaotic system. First, based on fuzzy theory, the fuzzy model for a memristive chaotic system is presented; next, based on H-infinity technique, a multidimensional fuzzy controller and a single-dimensional fuzzy controller are designed to realize the synchronization of master-slave chaotic systems with disturbances. Finally, some typical examples are included to illuminate the correctness of the given control method.
1. Introduction
Since May 2008, by using nanotechnology physical techniques, HP laboratory research team successfully obtain the resistance with memory characteristic [1], which confirmed the concept of memristor proposed by Chua [2, 3]. As the fourth basic passive device, memristor establishes the relationship between the magnetic flux and the charge. It has been reported that memristor can be applied in the field of computer science [4], biological engineering [5], and electronic engineering [6]. Especially, memristor can be used to construct the chaotic circuits.
For chaotic circuits, the nonlinear device is the key component. In 2008, Itoh and Chua built the first memristor-based chaotic system by replacing the diode with a piecewise linear magnetron Chua’s memristor [7]. Compared with the conventional nonlinear-device-based chaotic circuits, the memristor-based circuit has two main characteristics: first, the memristor-based circuit can produce the complicated dynamical behavior, which is different from the general chaotic dynamical behavior; secondly, the memristor-based circuit is more suitable to generate the high-frequency chaotic signal and have potential applications in chaotic secure communication, signal generator, and image process [8–12]. Hence, up to now, a number of memristor-based chaotic circuit with different structures are proposed. For example, the chaotic circuit with one memristor is studied in [13, 14], the chaotic circuit with two memristor is concerned in [15, 16], the integer-order chaotic memristor circuit is investigated in [17, 18], and the fractional chaotic memristor circuit are researched in [19, 20].
Chaos synchronization is a common phenomenon and can be found in biological systems, chemical reactions, power converters, secure communication system, and so on. Fuzzy technique is a powerful tool [21–27] and especially suitable for the chaos synchronization in the case that disturbances exist. For general fuzzy control, the control input is multidimensional and requires all system state information. However, in practical engineering, it is not easy to get all system state information. The multidimensional control can not only increase the control cost but also result in disturbance input problem. Hence, it is meaningful to design a single-dimensional fuzzy controller which is just based on one system state variable. In addition, disturbance inputs exist in actual system widely, which should be considered in synchronization control. All these motivate our research.
The paper is schemed as follows: the fuzzy model for a memristor-based chaotic circuit is constructed and the preliminary knowledge will be given in Section 2; a multidimensional fuzzy controller and a single-dimensional fuzzy controller will be designed to achieve the chaos synchronization of the master-slave systems in Section 3; the typical simulation example will be included to validate the correctness of the scheme in Section 4; and finally, the paper will be concluded in Section 5.
Notations used in this paper are fairly standard. represents a block diagonal matrix, is the n-dimensional Euclidean space, denotes the set of real matrix, the superscript stands for matrix transposition, refers to the Euclidean vector norm or the induced matrix 2-norm, and represents the maximum eigenvalue.
2. System Description and Preliminaries
First, consider a memristor-based circuit as Figure 1.
One can get the equivalent dynamic system as
Define the state variable as
One can obtain the equivalent dynamical equation as with where is the state variable of the system and is the system parameter. The memristive system will possess the chaotic dynamical behavior when the system parameters are , , , , , , , , and .
Next, consider the fuzzy modeling of the memristive chaotic system.
For
Rule 1. If is , then where means , and define
Rule 2. If is , then where means , and define
Hence, the fuzzy model of the memristive chaotic system is defined as
Above model can be rewritten as where System (3) is supposed as the master system, and the slave system is constructed as where is the state variable vector of the slave system and is the disturbance input of the slave system.
Hence, the fuzzy model of the slave system can be represented as where is the synchronization fuzzy controller.
Define the synchronization error vector of the master-slave systems as where .
One can get the error dynamic system as
In this paper, the following lemmas are concerned:
Lemma 1 (see [28]). If and , one can get
Definition 1. For nonzero and under the assumption of zero initial condition, if there exists a positive scalar such that Then, the slave system will synchronize to the master system with norm bound .
3. Main Results
Based on fuzzy theory and Lyapunov theory, a controller is presented as follows.
Theorem 1. If there exist scalar , design the multidimensional fuzzy controller with following control regulation with Then, the slave system (12) can synchronize to the master system (3) with norm bound .
Proof 1. With (18), the error dynamic system can be transformed as
Choose the Lyapunov function candidate as
One can get the time derivative of as
With (19), one can conclude that
Consider the performance index as
For and , where
Consider (20), it can be concluded that . Based on Definition 1, slave system (12) can synchronize to master system (3) with norm bound .
Next, consider the design for the single-dimensional fuzzy synchronization controller.
Construct the slave system as where is the single-dimensional synchronization fuzzy controller.
Theorem 2. If there exist scalar design the single-dimensional fuzzy controller with following control regulation where Then, slave system (30) with any initial conditions can synchronize to master system (3) with norm bound .
Proof 2. With (31), the error dynamic system can be transformed as
Choose the Lyapunov function candidate as
One can get the time derivative of as
With (31), one can conclude that
where
Consider the performance index as
For and , where
Consider (32), it can be concluded that . Based on Definition 1, slave system (30) can synchronize to master system (3) with norm bound .
4. Example and Simulation
First, consider the dynamics of the memristive chaotic system, and the simulation result is shown in Figure 2.
Next, we study the synchronization control of the master-slave systems. In the simulation, the system initial values are and . The disturbance input is . Let , and based on Theorem 1, the control parameters for the multidimensional fuzzy controller are and ; the simulation result is shown in Figure 3. Then, based on Theorem 2, the control parameters for the single-dimensional fuzzy controller are and ; the simulation result is shown in Figure 4.
Remark 2. Figure 3 depicts the time response of the synchronization error variables of the memristive master-slave systems with the multidimensional fuzzy controller. Figure 4 depicts the time response of the synchronization error variables of the master-slave systems with the single-dimensional fuzzy controller. It can be seen that although there exists just one disturbance for single-dimensional fuzzy control, the disturbance has impact on all error synchronization variables. In addition, it can be seen that both controllers can be able to realize the synchronization of the master-slave systems; the multidimensional fuzzy controller has the better control performance and realizes the chaos synchronization during 2.0 seconds. However, the possession of the good control performance is at the cost of the acquirement of all system state information. In addition, multidimensional control may introduce more disturbance input. The single-dimensional synchronization controller has the general control performance but requires just one system state information, which can decrease the control cost and the disturbance input. Hence, two kinds of controllers are useful and recommended for the different applied cases.
Remark 3. For the nonlinear disturbed chaotic system, the fuzzy modeling technique is adopted to realize the exact linearization control, which can eliminate the constraint on the system nonlinear term, compared with the general nonlinear control method; in addition, H-infinity approach is introduced to deal with the case that disturbances exist.
5. Conclusion
This paper focuses on the fuzzy synchronization for a new memristive chaotic system with disturbances. Based on fuzzy theory and Lyapunov stability theory, we have built the fuzzy model for the memristive chaotic system. Then, by using H-infinity technique, we have presented two kinds of fuzzy controllers for the possible application in chaos synchronization of slave-master systems. Finally, we have included some example to demonstrate the effectiveness of the given fuzzy controllers. In addition, the proposed results can be extended to the memristive chaotic control system with daelay or event trigger, which is our future work.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is partially supported by the National Key Basic Research Program, China (2012CB215202), the Graduate Innovation Fund of Xihua University (ycjj2018015), the Open Research Fund of Key Laboratory of Fluid and Power Machinery of Ministry of Education, and the National Natural Science Foundation of China (51475453, 11472297).