Adaptive single input sliding mode control for hybrid-synchronization of uncertain hyperchaotic Lu systems
Introduction
In the last three decades, the problem of synchronization for chaotic systems has been attractive research area since the originating effort is given by Pecora and Carroll [1]. The chaotic system represents a nonlinear deterministic system that exhibits a random, noisy like and complex nature. The characteristic of chaotic behavior is sensitive to initial conditions and system parameter variations. Chaos synchronization has abundant promising uses specially in secure communication, information processing, chemical process, engineering fields, telecommunications, power converters, biomedical, etc. [2], [3], [4]. Especially, hyperchaotic systems have received more attention because of the availability of more than one Lyapunov exponent in comparison with the chaotic system. It is generally known that more complex dynamics provide more secure communication [5] among the chaotic and hyperchaotic systems discovered such as Lorenz [6], Lu [7], Chen systems [8].
Synchronization of hyperchaotic systems have been explored extensively using nonlinear control methods [9], [10]. In the existing research, various kinds of synchronization phenomenon have been found such as functional synchronization [11], generalized synchronization [12], complete synchronization (CS) [13], [14], [15], anti-synchronization (AS) [16], [17], robust synchronization [18], and projective synchronization [19], [20]. The most familiar synchronization phenomena are CS and AS, because of their easy implementation. The CS and AS of hyperchaotic systems are the specific cases of the hybrid synchronization, where the CS and AS phenomena exist simultaneously [21]. It is important to synchronize the system states even in the presence of undesired side effects such as disturbance or uncertain parameters. Therefore, both elimination and utilization in synchronization are essential depending on the particular application as in secure communication. Compared with the synchronization issue, the problem of anti-synchronization is not well-known [17], neither is the hybrid synchronization (HS) problem [22]. In HS systems [22], the CS is the drive part and the AS is the response part so that the CS and AS simultaneously exist in the same systems. The CS and AS co-existence are highly important for secure communication and encryption schemes.
Meanwhile, uncertainties appear in most real systems. In that case, it is tough to observe all the uncertain parameters in the system [23], [24], [25]. To solve the uncertain parameters problem for the hyperchaotic Lu system, the adaptive synchronization method was proposed [26]. This key idea of solving the problem has been applied in the single control input for hyperchaotic Lu systems [27]. This single control input idea is more and accessible to implement in practical system applications than the multi-control input. However, it was not minimized the adverse effect of uncertain parameters in [27], [28], [29]. Recently, SMC has been widely adopted as one of the distinguished control techniques due to the advantages of SMC such as the good transient accomplishment, fast response, and also insensitive to the parameters variation [30], [31], [32], [33]. Therefore, the sliding mode technique has been used for the uncertainty minimization of hyperchaotic systems [34].
Recently, some developments have been also involved in the SMC design issue for Markov jump systems [35], [36], [37]. In [35], SMC was designed such that system state trajectories can be drawn to the sliding manifold in the finite time via integral SMC. But in practice, some abrupt changes in their parameters and structures are also present in the system. For this reason, the SMC was designed for nonlinear Markovian jump systems [36]. Furthermore, the integral SMC concept has also been extended to the distributed system for the leader-follower [37]. By employing the Lyapunov approach, sufficient conditions were deduced to ensure the consensus tracking performance. Moreover, from the application point of view, Jerk and hyperjerk systems have been used in different synchronization with a single input SMC technique [38], [39], [40]. The circuit design of the chaotic communication system was implemented and tested for real applications [39], [40].
On the other hand, the linear feedback control technique was used to synchronize the chaotic system in various research works [41]. Due to the simplicity in configuration and without eliminating the nonlinear dynamics, the linear state feedback control is especially attractive and has been commonly adopted for practical implementations. In most existing works, the number of control inputs was assumed to be equal to the number of state variables based on the drive-response scheme. But, a smaller number of controllers and simpler forms of controllers are greatly practical. Therefore, control methods based on a single variable are more simple, efficient, and easy to implement in practical applications. Few more single variable control techniques are available in the literature for synchronization of a 4D chaotic system like linear state feedback [41], [42], nonlinear state feedback technique [43], adaptive control [44] and backstepping technique [45].
The aforementioned results only dealt with synchronization problems for hyperchaotic systems via multiple control input or single control input, not hybrid synchronization ones. A hybrid synchronization scheme is useful to maintain the vastly secured and secrecy in the area of secure communication by using the control theory approach. Most of the existing hybrid synchronization techniques dealing with a controller without considering the uncertain parameters [46], [47]. In the recent work [48], hybrid synchronization problems using multiple controllers and without considering the uncertain parameters are addressed, which is not a practical situation. Therefore, it is necessary to consider the uncertain term in the system parameter for the practical point of view. However, in our proposed method, the problem of hybrid synchronization is designed after consideration of both issues. To the best of the author’s knowledge, the stability analysis of the hybrid synchronization for hyperchaotic systems regarding the single control input design has not been addressed yet. In previous existing techniques, it was not available to provide a highly secured communication scheme, which is possible by the use of a hybrid synchronization technique. The proposed approach can handle the parameter uncertain systems with the single control input instead of four control inputs of the 4D hyperchaotic system and also providing more secured communication due to hybrid synchronization, which was not possible to existing techniques. Furthermore, the proposed approach is also extended for the uncertain environment, where adaptation laws are proposed to ensure the convergence of uncertain parameters to their original values. In this paper, the proportional-integral (PI) based SMC technique is used. This technique provides the robustness of the system and eliminates the phenomena of chattering. Actually, in the presence of uncertainty, the adaptive PI-based SMC offers a fast convergence of the sliding surface. Therefore, Adaptive PI-based SMC is more useful compared to the linear sliding surface method.
The main contributed contents can be summarized as
- 1.
A hybrid-synchronization scheme is proposed for hyperchaotic Lu systems by adaptive single input SMC.
- 2.
In the case of uncertain parameters, hybrid synchronization is also achieved with adaptation laws to estimate the uncertain parameters.
- 3.
Simulation results are presented to show the efficacy of the proposed strategy for the hyperchaotic Lu system.
The framework of the rest of the paper is arranged as follows: Hybrid synchronization problem formulation is provided for the hyperchaotic Lu system in Section 2. In Section 3, hybrid-synchronization results of the hyperchaotic system are presented. Sliding mode function and control input is designed without uncertain parameters case. In Section 3.2, represent the hybrid synchronization problem in the presence of uncertain parameters. Simulation results are provided to show the effectiveness of the proposed control technique in Section 4. In Section 5, conclusions and future scope of work are given.
Section snippets
Problem formulation
In this section, the hyperchaotic Lu system is described and the proportional integral sliding function technique is developed for the hybrid synchronization problem.
Consider the following nonlinear dynamical system aswhere known matrix is given. is considered as a drive state vector, and nonlinear vector function is given as . The dynamics of controlled response system is given bywhere is the response state vector and control signal
Main results
This section presents the single input SMC design method for the hybrid synchronization problem.
Numerical simulations
In this section, an example is provided to show the effectiveness of the proposed method with the absence and presence uncertain environment using the single input SMC. Nonlinear system solution is determined by the use of 4-order Runge–Kutta approach and the set the 0.1 step size of time. Initially parameter of systems are given as and . Reachable parameter is chosen as positive parameter suitably. Positive parameter is obtained as . The initial conditions of drive
Conclusions
The hybrid synchronization problem of hyperchaotic Lu systems without and with uncertain parameters via single input SMC has been proposed. The hybrid synchronization is achieved between the drive and response systems using the control input and the proper selection of sliding function. To estimate the uncertain parameters, adaptation laws have been proposed to ensure the convergence of the uncertain parameters to their original value. The obtained hybrid synchronization results are providing a
Declaration of Competing Interest
The authors declare that they have no conflict of interest.
Acknowledgments
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF ) funded by the Ministry of Education, Science and Technology (2019R1I1A3A01060151) and by the BK21 FOUR project funded by the Ministry of Education, Korea (4199990113966).
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