Synchronization of Chaotic Systems: A Generic Nonlinear Integrated Observer-Based Approach

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Introduction
In nature, most real systems are nonlinear. To better understand the performance of distinctive nonlinear systems, it is significant and interesting to study the synchronization between two systems. Synchronization, perceived as a procedure that normally happens, has a notable effect in different areas of science, design, and engineering, even in public activities. Nonlinear system synchronization is an interesting field amid specialisations in various trains of thought because of its various uses relating to design and innovation. Researchers stepped into the universe of nonlinear systems in 1988, and various papers were published on the subject [1,2]. Nonlinear systems do not obey the principle of superposition and their output is not directly proportional to their input. Pecora and Carroll were responsible for the earliest effective work on the subject, introducing an experiment for synchronization of nonlinear systems under various initial conditions. Pecora and Carroll published a seminal paper [3] in the field of nonlinear synchronization. In this study, they described that certain nonlinear chaotic systems can be made to synchronize by linking them with common signals. e criterion for this is the sign of the sub-Lyapunov exponents. We apply these ideas to a real set of synchronizing chaotic circuits. Subsequently, scientists have developed numerous nonlinear synchronization strategies. e process of synchronization is where the determined system (slave system) comes to be in parallel with the master system (driving system), meaning that the synchronized system moves in a specific way, following the direction of a synchronizing system [4,5]. e background of this article outlines that many different methodologies have been used, including the Runge-Kutta model-based nonlinear observer [6], linear feedback control (LFC) [7], and delay-range dependent methodologies [8,9]. Adaptive schemes using fuzzy disturbance observers [10], robust adaptive methodology [11,12], reduced-order and full-order output-related observers [13], synchronization with Huygens' coupling [14], adaptive generalized projective synchronization (GPS) [15], stepwise sliding mode observer techniques [16], evolutionary algorithms [17], backstepping techniques [18], and nonlinear synchronism of undefined inputs, as well as Takagi-Sugeno fuzzy [19], have all been implemented for the coordination of chaos systems. All of these defined methods of synchronization of nonlinear modules show their robustness to different technologies such as neural networks [20], biological systems [21], secure communication [22,23], robotics [24], optics and lasers [25], information science [26], and chemical reaction [27]. Observer-based synchronization methods are progressively pertinent to the condition, where the master and slave situations are unknown [28]. Research specialists are consistently investigating such methods with various kinds of observers for different applications, for example, synchronous chaos in coupled systems [29], comprehensive projective synchronization procedures dependent on state approximation of hyperchaotic modules without computing Lyapunov proponents, and nonlinear-based protective communication, using decreased-order and stepwise sliding state observers. Nevertheless, previous statements of observer-dependent synchronization methods do not explain the integrated chaotic synchronized (ICS) observer and integrated chaotic adaptive synchronized (ICAS) observerdependent control strategies shown in this article. e primary disadvantage of the strategies previously mentioned, as opposed to the ICS and ICAS observer-dependent control techniques, is their appropriateness for the lower degree of synchronization of the two nonlinear modules with inaccessible state vectors. An error concurrent observer-dependent synchronization method was suggested in a recent work [30]. However, the technique is only used in nonlinear modules for which the general error module is adaptable to a direct composition of several error parameters.
is is widely used in applications to secure communications.
e numerous forms of chaotic synchronization include synchronization of Lur'e master and slave system. e work behind this synchronization of the chaotic Lur'e system was controlled in different ways. e absolute stability theory and different circumstances have been established. e objective of this research paper is to synchronize the unbalanced master pendulum system and slave system using a robust feedback technique and the LMIbased method for the synchronization of the chaotic dynamical pendulum system and output feedback controller technique. e main contribution and the objectives of the paper are (i) development of robust adaptive feedback control for delay containing chaotic systems, (ii) the mitigation of the effect of the disturbances using novel integrated adaptive observers, and (iii) a sufficient condition for the existence of observer and controller gains for the synchronization of chaotic systems. e closed-loop error is minimized after very little time and the system becomes stable, so the disturbance input effect reduces. To validate our research results, we have considered the example of the phase-locked loop system.

System Description
Synchronization of nonlinear systems is a subject matter. It means that synchronization of the dynamics of those systems occur, containing nonlinearities in their dynamics. Mathematical representations of nonlinear systems, which will be synchronized, contain both types of nonlinearities mentioned. Following this discussion, it is necessary to consider the generalized model of nonlinear master and slave chaotic (nonlinear) system equations (1) and (2), defined by state space representation when disturbance and adaptation are zero. dm � 0; ds � 0.
where L m ∈ R n×m and L s ∈ R n×m are the observer gain matrices. With the help of equation (3), we can manage the model structure of the slave (S) observer given as where u(t) is a nonlinear element. Also, in addition, we define taking the derivative of both sides of the master, slave, and output error equation, we acquire

Synchronization Feedback Control
where u g (t) is the nonlinear part of the proposed control law, i.e., e assumption considers B T P m C ⊥ � 0, B T P s C ⊥ � 0, and B T P o C ⊥ � 0 , where C ⊥ stands for the orthogonal projection on the null of C. If the above assumption holds, solving Adaptive controller design is provided using ICAS observers.

Theorem
e given observer and controller are then able to gain matrices F ∈ R l×n , L m ∈ R n×m , and L s ∈ R n×m an appropriate state for synchronization of the (M) and (S) systems (1) and (2) with undefined dynamics θ m ∈ R p , θ s ∈ R p , θ m,d ∈ R p , and θ s,d ∈ R p which concern with the assumption, applying the control law and ICAS observers (4)-(5), together with the law of adaptation: ese are the adaptation rate and β 4 > 0 so that the inequality matrix is satisfied: where

Simulation and Results
Simulation of the suggested methods for the synchronization of the (M) and (S) systems with undefined parameters, as planned in eorem 1, is shown in the accompanying simulation outcomes for FHN (Fitz-Hugh-Nagumo) (M)-(S) designs. e suggested methodology was completed with the help of simulation work using MATLAB software and FHN numerical models. FHN is generally utilized in genetic systems, such as brain stimulation therapy, considering the performance of neurons in electricity. It helps in investigating symptoms and diseases of the brain, including tremors resulting from disorders of the brain's neurons.
is kind of infection occurs in various parts of the brain.
e FHN system is defined below: where I o is the current in the above equation, m � 0.099, ω � 2πf, and f � 0.128. Chaotic systems are sensitive to initial conditions. By changing the value of initial conditions, the phase portrait, i.e., behaviour of nonlinear chaotic systems, changes. e initial conditions for the (M) and (S) systems are X s1(0) � 0.399 and X si+1(0) � 0.099. e other parameters are B � 1.01, R 1 � 10.09, and R 2 � 9.89. e phasor picture and individual reactions to the nonlinear chaotic performance of the (M)-(S) FHN system are exposed in Figure 1. For the (M)-(S) systems, various initial conditions are used. Different error signals are designed between the master system with its observer and the slave system with its observer and introduced in Figures 2(a) and 2(b), for eorem 1, respectively. Figure 3 describes the error signal between the (M) observer state and the (S) observer state. e simulation results are given for a nonadaptive control strategy under three different conditions. First, neuron behaviour with the help of the FHN system is generalized. In neurons, membrane potential is not the same in all living beings. It can be standardized using methods for an alternative scaling factor, so this is pertinent for all types of neurons. Second, the numerical articulations of the FHN system are, for the most part, dependent on the ordinary membrane potential. ird, standardized potential usage gain matrices must be controlled to finally synchronize (M)-(S) systems. e FHN model is related to the matrices, as indicated by the nonlinear chaotic master and slave systems: By using the adaptive scheme, synchronizing the (M)-(S) systems according to eorem 1, Lm and Ls are the gain matrices for the observer's master and slave systems, respectively. By varying the values of these observers gain matrices Lm and Ls and control gain matrix F, the effectiveness and efficiency of proposed control methodology may vary. After some empirical analysis, the values of the observer gain matrices and controller gain matrix are chosen. ese Lm, Ls, and F values are chosen as follows:  Figure 4 represents the result of the controller. In equation (26), ξ is the controlling function, which controls the behaviour of the (M)-(S) system. eorem 1 illustrates this in Figure 1, which shows the standardized potential of the (M)-(S) system with its observers. In Figure 5, eorem 1 illustrates the observer recovery variables for the (M)-(S) systems. Figure 6 explains the error signals between the (M) system and its master observer and between the (S) system and its slave observer. Lm and Ls are the gain matrices for the observer's respective master and slave systems, potentially influencing the master error em(t) and slave error es(t). "F" can clearly affect eoi(t). In the unlikely event where we  change the gain matrices' standards, these straightforwardly influence the synchronizing time. In Figure 2, eorem 1 shows the error signals between master and slave states. Finally, Figure 3 describes the error signals between the (M) observer state and (S) observer states.
For the representation of the degree of synchronization statistically [31,32], error-based DOS criteria are defined as follows: where e 2 and e 2, max are the 2-norm of error e and the maximum value of the norm, respectively. Note that the minimum and maximum values of degree of synchronization (DOS(e)) are 0 and 1, respectively. e maxima occurs for the minimum synchronization error, that is, e 2 � 0. While the minima occurs for e 2 � e 2, max , when synchronization error is maximum. It is worth mentioning that the maximum value e 2,max can be achieved by selecting either L m � L s � 0 0 T for any particular value of F or by utilitarian of F � 0 0 with some fixed values of L m and L s . Degree of synchronization is calculated for nonadaptive case to show the effect of variations in L m , L s , and F. Tables 1 and  14 Complexity 2 demonstrate the effect of L m � L s and Fon the DOS, respectively. It can be concluded that increase in the entries of L m and F can increase the degree of synchronization errors e m1 (t) and e o1 (t), respectively.

Conclusion
Synchronization of the two nonlinear systems, as well as chaotic frameworks with time delay, uncertainties, and disturbance, are recognized in this research study. A controller is designed utilizing the robust adaptive input control hypothesis. Along with the laws of adaptation for the approximation of boundaries, the planned delay rate-dependent controller ensures the synchronization of chaos, bringing synchronization errors to zero. e simulations using MATLAB confirm the adequacy of the proposed strategy. is is despite the fact that the model considered is for complex nonlinear chaotic framework with time delays with undefined elements. e result is also significant for its moderately simple, nonlinear frameworks with defined elements and consistent delays. As far as the future work is concerned, the distributed systems' synchronization of nonlinear systems can be considered. New methodologies can be sought for the distributed nonlinear systems having network delays with varying parameters.

Data Availability
e data used to support the findings of this study are available within the article. e raw data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare no conflicts of interest.