Modeling and Inverse Complex Generalized Synchronization and Parameter Identification of Non-Identical Nonlinear Complex Systems Using Adaptive Integral Sliding Mode Control

This paper presents the Inverse Complex Generalized Synchronization (ICGS) of non-identical nonlinear complex systems with unknown parameters. Using the philosophy of adaptive integral sliding mode control, an adaptive controller and laws regarding parametric upgradation are designed to realize ICGS and parameter identification of two non-identical chaotic complex systems with respect to a given complex map vector. To employ the control, the error system is transformed into a unique structure containing a nominal part and some unknown terms, which are computed adaptively. Then, the error system is stabilized by using integral sliding mode control. The stabilizing controller for the error system is constructed, which consists of the fractional-order control plus some compensator control. To avoid the chattering phenomenon, smooth continuous compensator control is incorporated instead of traditional discontinuous control. The compensator controller and the adapted law are derived in such a way that the time derivative of a Lyapunov function becomes strictly negative. This scheme is applied to synchronize a Memristor-Based Hyperchaotic Complex (MBHC) Lu system and a Memristor-Based Chaotic Complex (MBCC) Lorenz system, a chaotic complex Chen system and a memristor-based chaotic complex Lorenz system with entirely unknown parameters. The effectiveness and feasibility of the proposed scheme is validated through computer simulation using MATLAB software package.


I. INTRODUCTION
Synchronization is a fundamental phenomenon that enables coherent behavior in coupled systems. Due to its high potential in theoretical and engineering applications, chaos synchronization has been extensively studied in recent years [1]- [4]. After the development of complex Lorenz system [5], modeling and analysis of complex systems' synchronization have received considerable attention from The associate editor coordinating the review of this manuscript and approving it for publication was Shafiqul Islam . researchers, mostly due to the fact that some of the phenomena and physical systems can be modeled accurately by complex systems [6]- [8]. The synchronization of such systems has many useful applications to a variety of fields ranging from physics, medicine, information and signal processing to system identification with principal advantages of its use to secure communication in order to achieve higher transmission rates and security [9]- [11].
To synchronize complex systems, many chaotic synchronization methods were proposed in the literature after Pecora and Carroll work [12]. These methods include but are not limited to the complete synchronization [13], [14], exponential synchronization [15], [16], lag synchronization [17], [18], projective lag synchronization [19], projective and modified projective synchronization [20]- [22] impulsive synchronization [23], combination synchronization [24], neural networkbased [25] and fractional order synchronization [26]- [28]. Currently, it has been reported that some synchronization schemes were also developed based on the original versions. A modified hybrid projective synchronization scheme with a complex transformation matrix was introduced in [29] to synchronize fractional-order complex systems with different dimensions. Since the fractional-order system possesses memory, and it was claimed that the developed scheme could be helpful in secure communication, encryption, and control process applications. In [30], a complex modified projective synchronization method was proposed to synchronize complex systems through the use of a complex scale matrix. In [31], [32], the same scheme was also employed to synchronize complex systems having a different dimension with uncertain parameters. Via this approach, the slave system could be asymptotically synchronized up to a non-identical or identical master system through a desired complex scaling matrix, and all of the unknown parameters in both systems were estimated by using complex update laws.
Based on the above-mentioned sophisticated synchronization techniques, generalized synchronization (GS) has been extensively studied in recent years. In [33], the GS of two identical fractional-order chaotic systems was presented. Note that in GS, the two chaotic or hyper-chaotic real systems are said to be synchronized if a functional relationship exists between the states of the drive and the response systems. In [34], the GS was achieved in three typical classes of complex dynamical networks, including scale-free networks, small-world networks, and interpolating networks. In [35], Adaptive Generalized Synchronization (AGS) between the Chen system and a multi-scroll chaotic system was investigated with unknown parameters. Then, by using Lyapunov stability theory, AGS errors and parameter estimation errors asymptotically converge to zero.
The GS can be extended from real space to sophisticated space, resulting in a new synchronization scheme called complex generalized synchronization (CGS). Depending upon Lyapunov stability theory, an adaptive controller and parameter update laws are designed to realize CGS [36]. CGS is one of the most widely studied synchronization types. It refers to the existence of a functional relationship between the drive states and the response states. Instead of the conventional definition of synchronization, which stipulates that the difference between the drive and response trajectories tends to zero as t→∞, e→∞, CGS forces the difference between the slave states and a function of the master states to zero.
On the other hand, studying the inverse problem of CGS, which produces a new synchronization type called inverse complex generalized synchronization (ICGS), is an attractive and vital idea. The importance of CGS and ICGS stems from the fact that they can enrich the behavior of chaotic systems.
Aforementioned strategies allow more flexibility and have proven useful in many applications, including secure communications. With the motivation from the above discussions, the present article devised for ICGS and parameter identification of different chaotic systems with unknown parameters. Since, in many practical situations, the values of some parameters are unknown in advance, the synchronization is greatly affected by these uncertainties. Therefore, uncertain nonlinear complex systems are selected as the research objects. Moreover, in our proposed work, the ICGS and parameter estimation scheme is devised by employing adaptive integral sliding mode control, which ensures robustness from the very beginning due to the absence of reaching phase, suppress chattering, ensure finite-time convergence via Lyapunov theory [37]- [39]. Also, two different examples with same-order and reduced-order depending upon the orders of the systems are presented to demonstrate the validity and feasibility of the proposed scheme.
The paper is organized as follows. Section II presents the design of ICGS of non-identical complex systems. The ICGS and parameter identification of an MBHC Lu system and an MBCC Lorenz system with the same orders are discussed and illustrated numerically in Section III. Further, a chaotic complex Chen system and a memristor-based chaotic complex Lorenz system via increased order are also investigated and illustrated with a numerical example. Finally, the conclusion is presented in Section IV, and references are listed after Section IV.

II. SOME PRELIMINARIES AND DEFINITION OF ICGS
The non-identical drive and response for a complex system with uncertain parameters are represented in the following form:ẋ where x = [x 1 x 2 · · · x n ] T is the complex state vectors of the drive system (1) and y = [y 1 y 2 · · · y m ] T is the complex state vectors of the response system (2), x k = x kr + jx ki , y l = y lr + jy li , k = 1, 2, . . . , n, l = 1, 2, . . . , m, j = √ −1, θ ∈ R p and ϑ ∈ R q are real vectors of uncertain parameters. Throughout this paper, the subscripts r and i denote the real and imaginary parts of complex variables (vectors, matrices, and functions). F 2 (x) ∈ C n×p and G 2 (y) ∈ C m×q are complex matrices, and F 2 (x) = F r (x) + jF i (x) and G 2 (y) = G r (y) + jG i (y). F 1 (x) ∈ C n and G 1 (y) ∈ C m are vectors of complex nonlinear functions, and F 1 (x) = f r (x) + jf i (x) and G 1 (y) = g r (y) + jg i (y). u (x, y) ∈ C m is the complex input vector, and u (x, y) = u r (x, y) + ju i (x, y). It can be observed that some complex nonlinear systems, for example [5], [40]- [42], etc., can be represented as a system (1). To synchronize such systems, the complex variables and functions can be decomposed into the real and imaginary parts.

GENERAL SCHEME FOR ICGS AND IDENTIFICATION OF PARAMETERS
We are considering the case for m = n. Define the complex ICGS error vector as where e = (e 1 , e 2 , . . . , e n ) T ∈ C n , e r = (e 1r , e 2r , . . . , e nr ) T ∈ R n , e i = (e 1i , e 2i , . . . , e ni ) T ∈ R n . Let J = ∂ ∂y φ(y) be the Jacobean of mapping φ (x) and J = J r + J i . Letθ andθ be the estimates of θ and ϑ, respectively. Supposeθ = θ − θ andθ = ϑ −θ are the errors in estimating the parameters θ and ϑ, respectively. Now, by taking the time derivative of (5), the dynamical ICGS error system takes the form: This can be written as: where The error system (6) can be expressed as: By choosing where e rr = e 2r e 3r . . . e nr e 1i T , and v is the new input. Now the error system takes the form as: e nṙ e 1i e 2i . . .
To employ the integral sliding mode control, choose the nominal system for (8) as: Define the Hurwitz sliding surfaces for nominal systems (9) as: 1 is chosen in such a way that σ 0 becomes Hurwitz polynomial, theṅ where k > 0, we haveσ 0 = −k σ 0 . Therefore, the nominal system (9) is asymptotically stable. Now, by determining the sliding surfaces for the system (8) as σ = σ 0 + z, where z is some integral term computed later. To avoid the reaching phase, choose z (0) such that where v 0 is the nominal input and v s is the compensator terms computed later, theṅ Using Lyapunov theorem, we chose a function: V = 1 2 σ 2 +

III. EXAMPLES
In this section, two examples are given to demonstrate the effectiveness of the proposed method.  In this example, we will discuss the ICGS of two nonidentical systems having the same order. Depending upon MBHC Lu system discussed in [43], let the drive system be an MBHC Lu system, which is given aṡ where x 1 = x 1r + jx 1i x 2 = x 2r + jx 2i are complex and x 3 = x 3r , x 4 = x 4r are real;x 1 andx 2 are conjugate of x 1 and x 2 respectively; a k , k = 1, 2, 3, 4 are real uncertain parameters; α 1 , β 1 > 0 are known constants. Concerning [36], the parameters in the drive system (12) are taken as  phase portrait of a memristor-based chaotic complex Lorenz system. Suppose a memristor-based chaotic complex Lorenz system, developed in [37], is taken as the response system.
., 4, respectively, then systems (12) and (13) can be expressed as:ẋ The synchronization errors. It is essential to see that the errors converge to zero quickly.
Here for simulations, the initial conditions are chosen as: 38960 VOLUME 8, 2020 The simulation results are obtained in Fig. 3 revels that the synchronization errors approach to zero in less than 15s. The identifying process of uncertain parameters is shown in Fig. 4. It is evident that the estimates approach exact value respectively, (â 1 ,â 2 ,â 3 ,â 4 ) → (a 1 , a 2 , a 3 , a 4 ) and  (b 1 ,b 2 ,b 3 ,b 4 ) → (b 1 , b 2 , b 3 , b 4 ) as time elapses. The ICGS process is illustrated in Fig. 5, which shows the synchronization of response and drive systems (12). It can be seen that synchronization among the drive and response system is very effective. Both systems are entirely synchronized in the seventh second. In the upcoming figures sliding surface and control effort are displayed. Figs. (6 and 7) presents the sliding surface and control input profile for synchronization of the drive and respond system.

B. EXAMPLE 2. ICGS OF A CHAOTIC COMPLEX CHEN SYSTEM AND AN MBCC LORENZ SYSTEM (n < m)
In this example, we have discussed the ICGS of two nonidentical complex systems having increased order. Let the drive system be a chaotic complex Chen system [44], which is expressed asẋ x 1 andx 2 are conjugate of x 1 and x 2 respectively; c 1 , c 2 , c 3 ∈ R is uncertain parameters. With respect to [30], the parameters in the drive (28) Fig. 8, which shows the system (28) operating in chaotic orbits.
The complexity of the Chen system is also demonstrated in Fig. 8 via of 3D phase portrait. Letĉ i , i = 1, 2, 3 be the estimates of c i , i = 1, 2, 3 andc i = c i −ĉ i , i = 1,2,3, be the errors in estimations of c i , i = 1, 2, 3, respectively, then system (28) can be rewritten as: The 3-D complex system (29) can be written into the 5-D real system as: We consider (30) and (31) as drive and response systems, respectively.
Here for simulations, the initial conditions of the drive (30) and response (31) systems respectively are, (x 1 (0) , x 2 (0) , x 3 (0)) = (−3 − 2j, −1 − 5j, −4), (y 1 (0) , y 2 (0) , y 3 (0) , y 4 (0)) = (2 − 2j, 1 − j, 6, 1). The values for parameters in (29)-(39) are as follows. The simulation results are displayed as Fig. 9. The errors approach zero in less than 10s, as evident from Fig. 9. It can be seen that synchronization among the drive and response system is very effective. Both systems are entirely synchronized in the twelfth second. Control effort and sliding surface are displayed in Figs. 12 and 13, respectively. The identification of unknown parameters is shown in Fig. 10, where one can see that the estimates converge to exact values, i.e., ĉ 1 ,ĉ 2 ,ĉ 3 → (c 1 , c 2 , c 3 ) and b 1 ,b 2 The ICGS process is demonstrated in Fig. 11, in which it is clear that the drive (30) and response (31) systems synchronize. However, the control input profile and sliding surface profile are demonstrated in Figs. 12 and 13, respectively. In the upcoming section, the conclusion is divined.

IV. CONCLUSION
Adaptive integral sliding mode control is employed in this article regarding synchronization and estimation of parameters of two non-identical complex nonlinear systems. The stability of the said technique is also proved profoundly, using Lyapunov. To make the system more responsive, an adaptive controller and unknown parameter estimator were designed to synchronize the drive system to the response system. Moreover, smooth continuous compensator control was employed instead of traditional discontinuous control to suppress chattering. The compensator controller and the adapted laws were derived in such a way that the time derivative of a Lyapunov function becomes strictly negative. The versatility of the proposed scheme was illustrated with the help of two examples. The numerical simulation results confirm the effectiveness of the designed synchronization scheme.
From the perspective of future work, fast integral terminalbased sliding mode control may be explored. WAQAR ASLAM received the M.Sc. degree in computer science from Quaid-i-Azam University, Islamabad, Pakistan, and the Ph.D. degree in computer science from the Eindhoven University of Technology, The Netherlands. He received the Overseas Scholarship, HEC, Pakistan, for his Ph.D. degree. He is currently an Assistant Professor of computer science & IT with The Islamia University of Bahawalpur, Pakistan. His research interests include performance modeling & QoS of wireless/computer networks, performance modeling of (distributed) software architectures, radio resource allocation, the Internet of Things, effort/time/cost estimation of software development in (distributed) agile setups, social network data analysis, and DNA/chaos-based information security. VOLUME 8, 2020