Parameter Identiﬁcation and the Finite-Time Combination–Combination Synchronization of Fractional-Order Chaotic Systems with Different Structures under Multiple Stochastic Disturbances

: This paper researches the issue of the ﬁnite-time combination-combination (C-C) synchronization (FTCCS) of fractional order (FO) chaotic systems under multiple stochastic disturbances (SD) utilizing the nonsingular terminal sliding mode control (NTSMC) technique. The systems we considered have different characteristics of the structures and the parameters are unknown. The stochastic disturbances are considered parameter uncertainties, nonlinear uncertainties and external disturbances. The bounds of the uncertainties and disturbances are unknown. Firstly, we are going to put forward a new FO sliding surface in terms of fractional calculus. Secondly, some suitable adaptive control laws (ACL) are found to assess the unknown parameters and examine the upper bound of stochastic disturbances. Finally, combining the ﬁnite-time Lyapunov stability theory and the sliding mode control (SMC) technique, we propose a fractional-order adaptive combination controller that can achieve the ﬁnite-time synchronization of drive-response (D-R) systems. In this paper, some of the synchronization methods, such as chaos control, complete synchronization, projection synchronization, anti-synchronization, and so forth, have become special cases of combination-combination synchronization. Examples are presented to verify the usefulness and validity of the proposed scheme via MATLAB.


Introduction
Chaos is not an accidental or individual event, but a universal existence in various macro and micro systems in the universe. It promotes and relies on other sciences, which derive many interdisciplinary subjects, such as chaotic meteorology, chaotic economics, chaotic mathematics, and so forth. Because chaos is ubiquitous in many systems, the research on chaotic systems has drawn widespread attention of scholars. Thanks to the nonlinear nature of the chaotic system and the sensitivity to the initial value, the control and synchronization to the chaotic system has become a very difficult problem. Up to now, many valid synchronization methods were researched, such as drive-response synchronization [1], projective synchronization [2,3], adaptive fuzzy control [4][5][6], neural network livered to the second drive system. The second way is to break down time into different intervals. Let the signals in different intervals load in different drive systems. It is clear to observe that the traditional master-slave synchronization schemes (one to one system) do not satisfy the above communication signals but can be transferred in our model. Thus, it is imperative to pay more attention to the synchronization research of multi-systems. Sun et al. [42] realized the parameter identification and C-C synchronization in a finite time. In [24], the authors handle a hybrid projective C-C synchronization scheme between four specific hyper-chaotic systems utilizing SMC. The idea of dual C-C multi switching synchronization adopted the eight chaotic systems was addressed in [45]. The global exponential multi switching combination synchronization was introduced in terms of three different chaotic systems, in [46]. There are also some papers here that also mention the issue of C-C synchronization [47][48][49][50]. However, the systems they consider are all integer order chaotic systems and some of them do not consider the SD.
In response to this situation, we are going to consider the finite-time combinationcombination (C-C) synchronization (FTCCS) of FO chaotic systems with different structures and unknown parameters under multiple SD via the NTSMC technique. The multiple SD are explained as parameter uncertainties, nonlinear uncertainties and external disturbances. In the light of finite-time Lyapunov stability theory and the SMC technique, we propose an FO adaptive combination controller and some appropriate ACL.
Compared with other references, there are four advantages of the proposed method: (1) The finite-time control theory is different from the traditional stability theory and its control structure can be regarded as closed-loop feedback control. The complexity of the finite-time controller is relatively high, which is reflected in the anti-interference ability to the outside world and the robustness to the uncertainty of the system itself; (2) This paper extends the traditional drive-response synchronization schemes (single drive-response system) to combination-combination synchronization schemes. Thus, when the specific parameter values are gained to the D-R systems, the corresponding system or systems' combination are chose. The controller does not need to be redesigned for two systems or systems' combinations for every application. This not only has a wider range of applications but also saves too much time and effort. This advantage is reflected in Corollaries 1-3 in the paper; (3) In communication theory, comparing the traditional transmission model with the combination-combination synchronization model, our method has stronger anti-attack ability and anti-translated capability; (4) The nonsingular terminal sliding mode control avoids the singularity problem effectively that terminal sliding mode control (TSMC) would have and retains the characteristic of the finite-time convergence. Besides, the NTSMC has higher control accuracy than linear sliding mode control (SMC); (5) Based on the nonsingular terminal sliding mode control (NTSMC) and adaptive control, the combination-combination drive-response systems with unknown parameters and multiple stochastic disturbances is considered. The controller and parameter updating laws are designed to make the state of drive-response system gradually stable within a finite time. Our controller has good robustness and anti-interference performance.
This article is organized as follows. In Section 2, some definitions, lemmas and stability theories that need to be used are introduced. In Section 3, problem statements and assumptions are given. In Section 4, sliding mode synchronization controller and adaptive control laws are designed. In Section 5, the numerical simulations proved that our method is effective. In Section 6, there is a conclusion.

Definitions and Lemmas of Fractional Derivative
Next, let us present the Riemann-Liouville (R-L) derivative and the Caputo derivative, which are equivalent if and only if the order α is a negative real number and a positive integer. The R-L definition is best suited for theoretical analysis and can simplify the computation of FO derivatives. The Caputo is more relevant to modern engineering and makes Laplace's transformation more concise. Thus, we only display the mathematical expression of the Caputo derivative with order α. Definition 1 ([51]). The mathematical expression of the fractional integral of the function f (t) is following: where Γ(α) indicates the Gamma function.

Stability Theories of Fractional Order System
It follows that, if most things around us are nonlinear, we write the FO nonlinear system to be: where α ∈ (0, 1), f = ( f 1 , f 2 , · · · , f n ) T , x(t) ∈ R n and f : [t 0 , ∞] × Ω → R n satisfies the requirements of Lipschitz conditions; the initial value is x(t 0 ) = x 0 , t 0 ≥ 0. The equilibrium point x * of (5) can be calculated from f (x * ) = 0.

Problem Description and Assumptions
In this chapter, since the the initial values have a great influence on the initial values, in practical application, it is inevitable that the orbit of the system will change dramatically due to some small disturbances. Therefore, it is reasonable to treat them as bounded. This will also make our theory easier to understand.

Remark 1.
The matrices A, B, C, D ∈ R n × R n C = 0, or D = 0 indicating in (22) are named as the scaling matrices. They can also have different meanings, either as constant matrices or as functions of state variables x 1 , x 2 , y 1 and y 2 .

Remark 2.
If C = D = I, A = B = λI, then it will be transformed into finite-time C-C complete synchronization with multiple SD for λ = 1; It will be transformed into finite-time C-C antisynchronization with multiple SD for λ = −1; What's more, if A = C = 0, D = I, B = λI, then it will be transformed into finite-time combination complete synchronization with multiple SD for λ = 1, the finite-time combination anti-synchronization with multiple SD for λ = −1.

Remark 5.
It is supposed that A = 0, B = 0, C = 0 or A = 0, B = 0, D = 0, then finite-time C-C synchronization with multiple SD will be transformed into the issue of chaos control with multiple SD in the finite time .

Remark 6.
Based on all the above synchronization methods, we can also consider ∆θ or all of the uncertainties and external disturbance equal to zero for i = 1, 2.

Remark 7.
Starting from Definition 3, the number of D-R systems can be extended to three or more equations. Furthermore, D-R systems of the C-C synchronization scheme can be the same It follows from the Equation (22) that the error system is rewritten as: where From the above discussion, we make the following assumptions to ensure that our conclusions are more realistic.

Assumption 1. Assume that uncertain nonlinear vectors
and the parameter uncertainties ∆θ i , ∆ϑ i for (i = 1, 2) all have a bounded norm. Namely, there are suitable positive constants h, l, q that satisfy: Remark 8. The parameter vectors of D-R systems θ i , ϑ i , (i = 1, 2) and the three constants h, l, q are all unknown. Later, the parameters adaptive laws will be selected to identify them.

Sliding Mode Synchronization Controller Design within Finite Time
The main feature of the sliding mode control is that it directs the system states from their initial states towards the appropriate sliding surface which is specified and then it keeps the states in the corresponding sliding surface for all subsequent times. Designing a sliding mode controller consists of the following two steps : (1) To select a sliding mode surface; (2) To design a controller to make sure that the system's state converges to the sliding surface.
The nonsingular terminal FO sliding mode surfaces are designed as: where γ > 0, 0 < α < 1 and 0 < ξ < 1 and its FO derivative with α satisfies: When the system is in the sliding mode surface, the following conditions should be satisfied: Thus, Remark 9. Now, the nonsingular terminal sliding mode control (NTSMC) technique is very popular in the study of stochastic disturbances of chaotic systems. This is a new technique. In addition, some the state-of-the-art methods have appeared in the study of the synchronization of chaotic systems, such as: based on the state decoupling strategy and the Lyapunov-based approach, the minimum-energy synchronization control for interconnected networks is addressed by Li et al. [55]. The synchronization of Henon maps using adaptive symmetry control has recently been proposed [56]. The finite-time and fixed-time synchronization analysis of shunting inhibitory memristive neural networks with time-varying delays is introduced via constructing Lyapunov functions and feedback control schemes [57]. Combining adaptive control theory with Lyapunov-Krasovskii theory, Yuan et al. [58] solved the problem of finite-time synchronization (FTS) for complex dynamical networks with time-varying delays and unknown internal coupling matrices. Furthermore, a novel decentralized non-integer order controller applied on nonlinear fractional-order composite system is addressed in [59]. Li et al. [60] explored the issue of network synchronization for an FO chaotic system based on an event-triggered mechanism for the first time.
Theorem 3. When Assumptions 1 and 2 are satisfied and assume that the error system (23) is controlled by following combination controller (30) and adaptive laws (31), then the state trajectory of the error systems (23) will arrive the sliding surface s(t) in the finite time given by: where k > ς > 0 and ρ 1 , ρ 2 , ρ 3 ∈ (0, 1).θ i ,θ i andĥ,l,q represent the estimations of θ i , ϑ i and h, l, q. Their errors defined asθ Proof. Adopting the Lyapunov function: The FO derivative is expressed as: +q T (γ s(t) ).
Substituting (30) into Equation (23), we obtain: Substituting (34) into Equation (33), we obtain: It follows from Assumption 1 that we get: 0 D α t V(t, x(t)) ≤ γ s (q + h + l − (ĥ +l +q)) +h T (γ s(t) ) +l T (γ s(t) ) +q T (γ s(t) ) − k s − s T (ς( θ 1 It follows from Assumption 2 that we get: According to the Lemma 2: Motivated by the Theorem 1, it is clear that the system (5) is Mittag-Leffler stable. Then, we can obtain that the combination drive-response systems (18)-(21) achieve finite-time synchronization. Additionally, where 0 < α < 1. Proof. Adopting the Lyapunov function: The FO derivative is illustrated as: Thus, the error system (23) is Mittag-Leffler stable in finite-time T 1 under the sliding mode dynamics (28), described by: Remark 10. According to Theorem 3, the FO error systems (23) can be driven to the sliding surface s(t) via the controller (30) in finite time T 1 , that is, the sliding mode surface has accessibility; when it is on the sliding mode surface, according to Theorem 4, the FO error system (23) converges to the equilibrium point in finite time T 2 . So Theorem 3 and Theorem 4 achieve combination-combination synchronization within time T ≤ T 1 + T 2 .
The following corollaries are successfully analyzed from Theorem 4 and their proofs are omitted here.

Corollary 1.
(i) Assume the matrix C = 0, then the drive systems (18), (19) achieve the finite-time combination synchronization (FTCS) with the response system (21) provided the following controller: and the adaptive updating laws, (ii) Assume the matrix D = 0, then the drive systems (18), (19) achieve the FTCS with the response system (20) provided the following controller: and the adaptive updating laws,

Corollary 2.
(i) Assume the matrices A = C = 0, D = I then the drive system (19) achieve the FTCS with the response system (21) provided the following controller: and the adaptive updating laws, (ii) Assume the matrices A = D = 0, C = I then the drive system (19) achieve the FTCS with the response system (20) provided the following controller: sgn(e(t)) e(t) ξ − ς( θ 2 + θ 1 + ρ 1 |ĥ| + ρ 2 |l| + ρ 3 |q| and the adaptive updating laws, (iii) Assume the matrices B = D = 0, C = I then the drive system (18) achieve the FTCS with the response system (20) provided the following controller: and the adaptive updating laws, (iv) Assume the matrices B = C = 0, D = I then the drive system (18) achieve the FTCS with the response system (21) provided the following controller: and the adaptive updating laws,

Corollary 3.
(i) Assume the matrices A = B = C = 0, D = I, then the equilibrium point (0, 0, 0, 0) of response system (21) is asymptotically stable provided the following controller: and the adaptive updating laws, (ii) Assume the matrices A = B = D = 0, C = I, then the equilibrium point (0, 0, 0, 0) of response system (20) is asymptotically stable provided the following controller: and the adaptive updating laws, Remark 11. The scaling matrices A, B, C, D ∈ R n × R n could be the diagonal matrices or the identity matrices, or some of them are zero. As described in Remark 2, when A = B = C = D = I ∈ R n × R n , then the topic will be transformed into finite-time C-C complete synchronization with multiple SD; the numerical simulation results are displayed in Section 5.

Remark 12.
The ranges of fractional order that make the FO hyper-chaotic Chen, Lorenz, L and Liu chaotic system appear hyper-chaotic are chose as 0.8 ≤ α < 1, 0.97 ≤ α < 1, 0.94 ≤ α ≤ 1, 0.96 ≤ α ≤ 1 respectively. If the drive-response systems are in hyper-chaotic, the influence of stochastic disturbances on the system can be better studied, and the effectiveness and robustness of the controller can be proved. It follows that the dynamic error has the same fractional-order as the drive and response systems that the fractional order α is chose as 0.97 ≤ α < 1 which can ensure that the all drive-response systems are hyper-chaotic. Thus, in the numerical simulation section, we also consider the α = 0.97 to validate the proposed method.
A comparison analysis between the proposed finite-time combination-combination (C-C) synchronization (FTCCS) scheme and the earlier published work is as follows. In Ref. [62], the author applied the adaptive control method to achieve C-C synchronization among four identical hyper-chaotic systems where it noted that the synchronization states happened at t = 5 (approx). In Ref. [61], the author used the sliding mode control scheme to address multiple chaotic systems with unknown parameters and disturbances in which the synchronization happened at t = 5 (approx). Besides, in Ref. [63], the author solved a new type of C-C synchronization for four identical or different chaotic systems via adaptive control, where the desired synchronization happened at t = 5.5 (approx). The combination synchronization of FO non-autonomous chaotic systems with different dimensions adopting a scaling matrix is studied in Ref. [64], where the error synchronization happened at t = 6 (approx). Furthermore, the phase synchronization of FO complex chaotic systems with different structures is discussed in Ref. [65]; in the process of C-C synchronization, the desired synchronization happened at t = 4.5 (approx). The nonsingular terminal sliding mode control to achieve the finite-time synchronization between two complex-variable chaotic systems with unknown parameters is adopted in Ref. [66]; here it has been found that the synchronization error converges to zero at t = 10 (approx). In addition to the above studies, we have investigated the FTCCS scheme among fractional order (FO) chaotic systems under multiple stochastic disturbances (SD), utilizing the nonsingular terminal sliding mode control (NTSMC) technique in which it has been recorded that the synchronization occurs at t = 3.1 (approx) as depicted in Figure 5. Therefore, comparing the synchronization times discussed above with those obtained by our proposed scheme, our method is dominant. This also illustrates the vitality and effectivity of the considered methodology.

Remark 14.
After calculation, the finite synchronization time satisfies T 1 ≤ 5.71, T 2 ≤ 7.34 theoretically. Thus, we have T ≤ T 1 + T 2 = 13.05. Comparing the numerical simulation results, we can see that our control scheme is effective.

Remark 15.
The dynamic error has the same fractional-order as the D-R systems in our paper. It is worth considering that the non-integer order in the derivative of error is different from the D-R. If we only consider this situation, there are many papers that have discussed it. In Ref. [67], the author proposed a modified adaptive sliding-mode control technique to investigate the reduced-order and increased-order synchronization. Ouannas et al. [68] investigated the inverse full state hybrid function projective synchronization (IFSHFPS) of non-identical systems characterized by different dimensions and different orders. Furthermore, the hybrid projective synchronization of different dimensional fractional order chaotic systems with time delay and different orders is discussed by [69]. More research results can be found in Ref. [70][71][72]. All the above literature about the non-integer order in the derivative of error is different from the drive-response systems. Our next step will consider this situation.

Conclusions
In this article, the FTCCS of FO chaotic systems among four systems with different structures and unknown parameters is solved. The most important point is that the conditions we consider are under multiple stochastic disturbances. Our thought for this topic is that under the action of the finite-time Lyapunov theory and the nonsingular terminal sliding mode control technique, we deduced a new FO sliding surface, adaptive combination controller and some parameter updating laws, which can achieve the combinationcombination synchronization of systems under multiple stochastic disturbances in finite time. The unknown parameters are identified precisely. Moreover, the combination drive systems and combination response systems that we introduced are very general. The expression of the synchronization error system makes many synchronization methods, such as chaos control, complete synchronization, projection synchronization, anti-synchronization and so forth, become special cases of combination-combination synchronization. From the numerical simulation results, it is obvious that the error variables of the D-R systems quickly converge to the origin point in the given time. Therefore, this controller and the updated parameter laws are effective. Next, for the multiple stochastic disturbances, we will study the fractional order multi switching synchronization of eight chaotic systems with time-delay in which the systems' parameters are still unknown.
Author Contributions: W.P. proposed the main the idea and M.S. prepared the manuscript initially. T.L. gave the numerical simulation of this paper. S.A. and L.P. revised the English grammar of this paper. All authors have read and agreed to the published version of the manuscript.

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Informed Consent Statement: Not applicable Data Availability Statement: The data used to support the findings of this study are available from the corresponding author upon request