Robust Global Synchronization of a Hyperchaotic System with Wide Parameter Space via Integral Sliding Mode Control Technique

Chaotic systems have played a crucial role in the understanding of phenomena that governed the development of science and technology. During the last three decades, intensive research into chaos has resulted in a deeper understanding of the interactions between physical, biological, economic, physiological, and social sciences. The plethora of literature has convincingly demonstrated the applications of chaos in various disciplines, including ARTICLE INFO ABSTRACT

The inherent property of invariance to structural and parametric uncertainties in sliding mode control makes it an attractive control strategy for chaotic dynamics control. This property can effectively constrain the chaotic property of sensitive dependence on initial conditions. In this paper, the trajectories of two identical fourdimensional hyperchaotic systems with fully-known parameters are globally synchronized using the integral sliding mode control technique. Based on the exponential reaching law and the Lyapunov stability principle, the problem of synchronizing the trajectories of the two systems was reduced to the control objective of asymptotically stabilizing the synchronization error state dynamics of the coupled systems in the sense of Lyapunov. To verify the effectiveness of the control laws, the model was numerically tested on a hyperchaotic system with a wide parameter space in a master-slave configuration. The parameters of the hyperchaotic system were subsequently varied to evolve a topologically non-equivalent hyperchaotic system that was identically coupled. In both cases, the modeled ISM control laws globally synchronized the dynamics of the coupled systems after transient times, which sufficiently proved the invariance property of the ISMC. This study offers an elegant technique for the modeling of an ISMC for hyperchaotic coupling systems. As an open problem, this synchronization technique holds promises for applications in robot motion control, chaos-based secure communication system design, and other sensitive nonlinear system control. economics, medicine, finances, security studies, telecommunications. Chaos theory has been applied to robotics. As the understanding of chaos deepens, higher dimension chaotic systems have been evolved, even as the frontiers of applications and hypotheses expanded considerably over the last three decades. Essentially, in order to be useful, chaos must be controlled. Thus, various control techniques have been applied to the control and synchronization of chaos. These techniques include adaptive control, fuzzy control, hybrid feedback control, backstepping control, contraction control, among others. Synchronization is very useful in telecommunication science and occurs when two chaotic systems are coupled such that, in spite of the exponential divergence of their nearby trajectories, synchrony of the trajectories is still achieved as t → ∞, provided conditions related to the coupling strength, parameter region of the systems are satisfied, in addition to satisfying a necessary condition for master-slave synchronization which is that the non-driven slave subsystem must be asymptotically stable in the sense of Lyapunov [1].
Sliding mode control (SMC) has emerged as a robust control technique for systems constraint by uncertainties and unpredictability. SMC is a control technique that is based on the design of switching laws to drive system trajectories to a user-chosen hyperplane in the state space [2]. It is attractive due to its property of invariance to parametric and non-parametric uncertainties. The global response of SMC consists of two phases known as the reaching phase and the sliding phase. In the reaching phase, the system's states are constrained to reach a predetermined sliding surface in finite time. On this surface, the controlled system is adaptively altered to a sliding mode, resulting in the system sliding towards the origin along the sliding surface for a duration known as the sliding phase [3].
This research contributes to the application of the ISMC technique to synchronize the dynamics of wide parameter spaced hyperchaotic systems, which are traditionally more sensitive than other hyperchaotic systems due to their large Lyapunov exponents. The rest of the paper is organized as follows: Section 2 describes the selected hyperchaotic system. Section 3 presents the design of the switching surface and integral sliding mode controller. Section 4 unveiled the numerical simulation results, while the conclusion and future work is given in Section 5.

Methods
In this section, the architecture of the proposed ISMC system is presented. The algebraic structure of the hyperchaotic system and its 3-D phase portrait are presented.
The architecture of the ISMC system and algebraic structure of the hyperchaotic system The architecture of the proposed controller comprises the master and slave systems, synchronization error system, nonlinear functions (which provides the nonlinearity), and the sliding manifold on which the trajectories slide. The architecture is shown in Fig. 1.  Fig. 1. The architecture of the ISMC system The hyperchaotic system used in the study was first reported in [25]. The hyperchaotic system is well-suited for studying the robustness of integral sliding mode controllers due to its huge parameter space and unstable dynamics [26]- [29]. The algebraic structure of the system is represented by four-coupled ordinary differential equations of the form:

Integral sliding mode controller and sliding surface design and analysis
In this section, we summarized the results for the global synchronization of the two hyperchaotic systems. Specifically, we obtained parameter ranges that assure the global stability of the system dynamics in the presence of known-parameter variations. The results can also be applied to the synchronization of non-identical systems. Consider system (1) as the master system. Let the identical-parameter controlled slave system be represented in the form: ̇1 = − 1 ( 1 + 2 ) + 2 3 + 1 ̇2 = − 3 1 3 + 4 2 + 5 4 + 2 3 = 6 1 2 + 7 + 3 ̇4 = − 8 1 − 9 2 − 10 3 − 11 4 + 4 (2) Where 1 , 2 , 3 and 4 are the ISMC laws to be derived, and 1 , 2 , 3 and 4 are state variables. Let the synchronization error between the master and slave systems be of the form: Given that the initial conditions of the master and slave systems, i.e., (0) ≠ (0), the coupled systems can be synchronized such that lim ⟶∞ ‖ ( )‖ = 0, ∀ . The synchronization error system is given by Synchronization of the systems involves two known steps viz: the selection of an appropriate switching surface which can guarantee the convergence of the system dynamics such that the error state dynamics asymptotically stabilizes in the sense of Lyapunov. Secondly, the derivation of suitable control law guarantees the existence of the sliding mode ( ) = 0. We define the integral sliding surface of each error state variable is defined as follows: The system trajectories glide on the sliding manifold if it satisfies the condition ̇= 0, ( = 1,2, … ,4). Thus, differentiating (5) results in the following: The Hurwitz condition is satisfied if (1,2, … 4) are positive constants. We set the following exponential reaching laws [22] ̇1 = − 1 sgn( 1 ) − Φ 1 1 ̇1 = − 1 sgn( 1 ) − Φ 1 1 ̇2 = − 2 sgn( 2 ) − Φ 2 2 (7) ̇3 = − 3 sgn( 3 ) − Φ 3 3 ̇4 = − 4 sgn( 4 ) − Φ 4 4 Where > 0; Φ > 0 are positive constants to be determined. Comparing (6) and (7) gives

Robustness test via parameter variation
In the section, the parameters of the systems were varied to evolve a topologically noequivalent case. The resulting plots of the converged error state and controller dynamics are shown in Fig. 9 and Fig. 10.

Discussion
When the ISMC laws are applied according to the architecture in Fig. 1, the synchronization error dynamics converged asymptotically, as shown in Fig. 3. The ISMC laws asymptotically converged to the origin in the sense of Lyapunov. Furthermore, each of the state trajectories of the identical master and slave systems in Fig. 2 (a) -(d) were synchronized as shown in Fig. 5 -Fig. 8. The test of robustness based on known-parameters variation using the ISMC laws resulted in the error state trajectories depicted in Fig. 9 and converged ISMC laws depicted in Fig. 10. It can be seen from Fig. 9 and Fig. 10 that the controller is invariant in the presence of parameter variation. This fact can be extended to the case of parameter uncertainty of the slave system.

Conclusion and Future Work
In this paper, integral sliding mode control laws were derived from synchronizing the state trajectories of identical 4-D hyperchaotic systems based on the Lyapunov stability principle. The control laws were observed to be invariant to known-parameter variations of the system during robustness tests. As an open problem, the synchronization of chaotic systems continues to hold promises for applications in robotic motion control, electric drive system, and other traction control systems. The modeled ISMC laws can be improved upon so that the transient time that elapsed before coupling can be finite-time (i.e., precisely determined) for applications in timecritical dynamic systems.
Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflict of interest.