Robust adaptive anti-synchronization control of multiple uncertain chaotic systems of different orders

The precise anti-synchronization control of uncertain chaotic systems has always remained an interesting problem. The anti-synchronization control of multiple different orders uncertain chaotic systems increases the complexity and enhances the security of the information signal in secure communications. Hence, it confines the hacking in digital communication systems. This paper proposes a novel adaptive control technique and studies the double combination anti-synchronization of multiple different orders uncertain chaotic systems. The proposed adaptive feedback control technique consists of three fundamental nonlinear components. Each component accomplishes a different objective; (i) stability of the closed-loop, (ii) smooth and fast convergence behaviour of the anti-synchronization error, and (iii) disturbance rejection. The theoretical analysis in (i) to (iii) uses the Lyapunov stability theory. This paper also provides parameters adaptation laws that stabilize the uncertain parameters to some constants. The paper discusses the simulation results of two representative examples of four different orders uncertain chaotic systems. These examples demonstrate anti-synchronization among hyperchaotic Lü, uncertain chaotic Shimizu Morioka, uncertain second-order nonlinear duffing, and uncertain parametrically excited second-order nonlinear pendulum systems. The computer-based simulation results certify the efficiency and performance of the proposed anti-synchronization control approach and compare them with peer works.


Introduction
Abbreviations. The following abbreviations will be used in this paper (Table 1).
In chaos theory, the synchronization phenomenon refers to a strong relationship between coupled chaotic systems. Synchronization behaviour is realized when the difference of output of state variables of the coupled chaotic systems in the master-slave (drive-response) system arrangement tends to zero after a transient time, that is, lim t→∞ e(t) = lim t→∞ y(t) − x(t) = 0, where x(t) and y(t) are the state vectors of the master and slave systems, respectively [1]. The synchronization of chaotic systems has been actively investigated in various areas of applied sciences. These include information process [2], reactions-diffusion systems [3], DC/AC inverter [4], neural networks [5], mechanical systems [6], secure communications [7], and power systems [8], etc. After the seminal work of [1], different types of synchronization have been reported in the relevant literature, including complete synchronization, generalized synchronization, phase synchronization, antiphase-synchronization (or anti-synchronization (AS)), fuzzy synchronization and lag synchronization [7][8][9][10], among others. Chaos AS constitutes an important type of synchronization, which is a classical feature of nonlinear dynamical systems collaborating through repulsive coupling [9]. By definition, AS is a process wherein the sum of the output of state variables vanishes and advances as symmetrical oscillators in a transient time, i.e. lim t→∞ e(t) = lim t→∞ y(t) + x(t) = 0. This notion of chaos AS phenomenon was first presented in [10]. In recent decades, AS of chaotic systems has received increasing interest and has been observed both theoretically and experimentally. Chaotic AS has shown important applications, especially in improving the performance of semiconductor chaotic lasers [11], secure encryptions [12], electronic circuits [13], and dynamical networks [14], and so forth. As compared to the chaos synchronization, the AS scheme for encryption and decryption of data in secrete communication systems is more reliable, secure, and provides the faster transformation of the digital message signal [12]. It has been investigated that the AS is more significant than synchronization in chaotic laser oscillators with negative couplings [11]. The repulsive interactions between the nodes have been realized in various natural networks [9]. The human brain is an example of a complex network having both in-phase (synchronization) and anti-phase (anti-synchronization) oscillations [12]. Similarly, ecological webs have both positive and negative connections between their components [11]. A recent study of the chaos AS suggests that it could be utilized as a method to study the properties of a chaotic satellite system evolving in a circular orbit [15].

Literature review
Fundamentally, research in the chaos theory reports two types of AS schemes; full order antisynchronization (FOAS) and reduced-order/increasing order anti-synchronization (ROAS/IOAS) in the master-slave system arrangement. FOAS is the realization of the AS among the similar orders chaotic systems. The ROAS/IOAS discusses the AS among the different orders chaotic systems. ROAS methodologies anti-synchronize a higher-order master system with a lower order slave system by comparing the states of the slave system with partial states of the master system. The main advantage of the ROAS is the reduced-order model in which all states of the slave system share information with some partial states of the response system. This attribute strengthens the security of the data in the transmission channel, eases extra communication loads, and simplifies the additional stability analysis. Several effective control techniques have been developed to address the FOAS and ROAS. These feedback control techniques have been utilized to accomplish either the AS between two coupled hyper(chaotic) systems (one-to-one system mode) or AS among multiple hyper(chaotic) system (one-to-many or many-to-one or many-to-many system configuration). The configuration in the latter case is termed as the CAS. In the one-to-one system model, the AS is studied between two coupled hyper(chaotic) systems, whereas in the CAS, the master and slave systems arrangement consists of more than two systems. In this line, researchers have proposed different control strategies to address the FOAS and ROAS. [16] investigates the AS of two identical hyperchaotic systems with known and uncertain parameters. Using the adaptive and active control strategies, the scheme in [17] realizes the AS between two non-identical chaotic systems. [18] proposes a nonlinear control technique to discuss the AS of two different hyperchaotic systems. Using the aggregation technique, [19] investigates the AS of two non-identical chaotic systems. The AS of different orders uncertain chaotic systems with parameters identification is an enhancement of the ASC strategy. To tackle the effect of uncertainties, the scheme [20] synthesizes a robust adaptive control approach to investigate the AS of two non-identical hyperchaotic systems. Based on the LST, [21] designs an ASMC technique that establishes the AS of uncertain chaotic systems. The proposed scheme in [22] investigates the adaptive lag projective AS of UC systems with bounded nonlinearity. [23] accomplishes the ASC of two dynamical networks with both equivalent and different topological structures. Using the adaptive feedback control strategy, [24] studies the dynamics, control, synchronization, and AS of a novel hyperchaotic system with circuit realization. The paper [25] investigates the chaos synchronization and AS of two identical fractional-order chaotic systems using the active control technique. [26] studies the synchronization and AS of two indistinguishable chaotic complex nonlinear frameworks. The works in [18][19][20][21][22][23][24][25][26][27] investigate the one-to-one system AS. The results in [28] are an improvement of the oneto-one system AS scheme to the CCAS of five chaotic systems with known parameters. The multi-switching CCAS of eight chaotic systems is another development of the CCAS [29]. Similar work can be seen in the AS scheme proposed in [30]. Most recently, [31] has proposed an AS scheme to study the compound difference AS of integer and fractional orders chaotic systems.

Motivations
In the past couple of decades, researchers have developed state-of-the-art ASC methodologies, however, technological advances emphasized the need for more secure communication systems. This need has initiated the requirement of complex ASC techniques. The design of such complex ASC methodologies is required to resolve the following challenging issues.
Items M(i) − M(v) present the motivations for this work.

M(i)
The reported ASC approaches [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] (among others) confine to the specific assumption that the required error information between the coupled hyper(chaotic) systems is known. These ASC techniques use the FL concept to develop feedback controllers for the AS. The FL approaches assume that nonlinear relationships of the AS errors between the coupled chaotic systems are known, and the controllers cancel these nonlinear terms of the system and form a closed-loop, which exhibits linear dynamic behaviour. This assumption compromises the complexity of ASC in practical applications M(ii) A key factor that affects the AS performance is the existence of uncertainties in chaotic systems. For instance, aging of apparatus, sudden variations in the system's dynamics due to faults and changes in the operating conditions, etc. The effect of these disturbances may shatter the ASC stability. The ASC approaches [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] either do not consider the effects of unknown SD and TD disturbances in the closed-loop or bounds of these disturbances are known in prior, which is hard to be determined in practice M(iii) The ASC scheme [20] (among others), use the signum 10650 function to tackle the effects of unknown SD and TD disturbances and to achieve the fast AS behaviour. The robust ASC approach [20] synthesis a large control effort that results in smaller tracking error and shorter transient response time. The switching behaviour of the sgn function leads to create unwanted large oscillations in the error and control input signals. However, this attribute causes drive system malfunctioning and actuator failure gives rise to aliasing problems and causes further degradation in the AS behaviour M(iv) The multi-systems ASC approaches [28][29][30][31] consider known systems. The AS of UC systems offers theoretical complications in designing the feedback controllers. Such ASC schemes [28][29][30][31] produce large oscillations in the closedloop, which further become prominent in uncertain systems [32] M(v) The rates of convergence in the AS schemes [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] are slow, which affects the performance of such ASC methods in practical applications, for example, cellular radio systems, multi-agent systems, and wireless networks, etc In light of the above discussion, it is important to investigate the double combination antisynchronization (DCAS) of multiple different orders UC systems that increase the complexity and improves the security of the digital message signal in secure communication systems. It is also essential to design an ASC strategy that should (i) be able to converge the AS error signals to the origin in a shorter time, (ii) be robust against unknown SD and TD disturbances, and (iii) suppress the undesirable oscillations in the AS error and control input signals.

Contributions
This paper introduces a new RAASC technique and studies the DCAS of UC systems. The chaotic systems under consideration have different orders, structures, and uncertain parameters. The bounds of the SD and TD disturbances are unknown. This AS structure significantly increases the complexity of the information signals in secure communication systems. The paper establishes the analytical expression of the proposed ASC scheme and adaptive laws for the adaptation of uncertain parameters. This work uses LST [32] to prove the asymptotic robust stability of the closed-loop. Computer-based simulations of two numerical examples are illustrated to evaluate the performance of AS among the uncertain hyperchaotic Lü [33], uncertain chaotic SM [34], uncertain SOND [35], and uncertain PESNP [36] systems. The simulation results endorse the theoretical findings. Comparative simulation studies provide evidence of better AS performance of the proposed RAASC technique.
Items C(i)-C(iv) describe the main contributions of this paper. C(i) Studies a new AS scheme, which is composed of different order multiple uncertain chaotic systems. The proposed DCAS is realized in considering two different uncertain chaotic systems as the master system (transmitter) and two different uncertain chaotic systems as the slave system (receiver) in the presence of unknown SD and TD disturbances. The two master systems have different orders, whereas the slave systems have similar orders. The parameters of the master and slave systems are different and uncertain. These systems exhibit different chaotic behaviour and topological properties. Similarly, the traces changes of all systems are different. The dynamic behaviour of the proposed mixed AS scheme is more complex and unpredictable, which will further enhance the security of the information signal in digital communication systems. Therefore, using the proposed AS methodology, it would be difficult for an intruder to hack the information signal during the communication process C(ii) Design of a novel RAASC technique. The proposed control algorithm • Increases the speed of AS and convergence rates • Reduces the transient oscillations in the AS error vectors and suppresses chattering in the control input signals C(iii) The computer-based simulation results are performed to validate the theoretical findings C(iv) This article also compares the performance of the proposed RAASC algorithm with different types of feedback controller approaches reported in [21,24,28] regarding the AS transient speed, The n−dimensional Euclidean space i ∈ Z + A positive integer for subscript m, n, p, r ∈ Z + Used as superscripts M 1 and M 2 The first and second master UC systems S 1 and S 2 The first and second slave UC systems n, m, p n represents the order of M 1 , m denotes the order of M 2 and p represents the order of S 1 and S 2 , respectively where n > m > p The state vector of M 1 The state vector of M 2 The state vector of S 1 The state vectors of S 2 θ ∈ R n×n Uncertain parameter vector of M 1 θ i An i th row of the matrix θ ∈ R n×n ψ ∈ R m×m Uncertain parameter vector of Continuous nonlinear vector functions in M 1 , M 2 , S 1 , and S 2 (1-4), respectively The unknown SD disturbances acting on M 1 , M 2 , S 1 , and S 2 (1-4), respectively The unknown TD disturbances acting on M 1 , M 2 , S 1 , and s and ν are any scalar and vector values, respectively |s| convergence rates, and the amplitude of the error and control input signals oscillations The remaining article is organized as follows: In Sec. 2, this work presents a model of the proposed DCAS scheme and designs RAASC strategy. Sec. 3 demonstrates two numerical examples and simulation results with a comparative study. The paper ends with the conclusions in Sec. 4.

Notations and symbols.
This article uses the following symbols and notations ( Table 2).

Robust adaptive double combination anti-synchronization control scheme
This section is divided into two subsections. Subsection 2.1 presents the proposed model for the ROAS of four different orders UC systems, while subsection 2.2 designs the proposed RAASC technique and provides the proof of global asymptotic stability of the closed-loop.

Problem formulation
The proposed DCAS model considers four different orders UC systems. The master system M is a combination of two different orders UC systems denoted by M 1 and M 2 (1-2). The slave system S consists of two same orders UC systems represented by S 1 and S 2 (3-4). Equations (1-4) describe the master and slave systems configuration.

3) & (4)
However, it is useful to recognize that the master system has two sub-systems, known as the projection and the remainder. In the project part (1a-2a), the ROAS problem plays the role of the master system, while the remaining part (1b-2b) becomes idle in the ROAS. The projection part: The rest of the systems (1-2) are written as follows: , and G r ∈ R r×1 . The orders p < n and r < n are chosen such that p + r = n. Therefore, the master and slave systems configuration for the proposed DCAS reduces to the ROAS between the M p (1a-2a) and S (3-4).

Definition 2.1: Assume xy(t) = x p (t) + y p (t) and zw(t) = z(t) + w(t), then define that
be a DCAS error system. The elements of e(t) are described as follows: Now, (6) represents the error dynamical system of ROAS between M p (1a-2a) and S (3)(4). where, According to the reported strategies [28][29][30], the proposed multi-systems ASC schemes [28][29][30] cancel the nonlinear terms (6), which reduce the complexity of the closed-loop system. Hence, the designed model-based ASC structures [28][29][30] are difficult to obtain in practice. To overcome such issues, let us consider the following restructuring of the nonlinear terms in the error dynamics (6) to facilitate the partial cancellation of the nonlinear terms by the proposed AS controller and to keep some complexity attributes in the closed-loop system.
is an element of the matrix L(xy p (t), zw(t)) ∈ R p×1 . The diagonal elements of Q(xy p (t), zw(t)) are associated with e i (t), whereas the matrix L(xy p (t), zw(t)) contains some linear and nonlinear residual terms. Therefore, using (8), the closed-loop (6) can be reformed as: where Objective 2.1: The error dynamical system (9) is global asymptotic stable at the origin if:

The proposed RAASC technique and the asymptotic stability analysis
Theorem 2.1: If the following RAASC function is designed as: where μ, ε > 0 are real constants and μ determines the steepness of the tanh function. Further, Equation (11) gives the design of the parameters adaptation laws.
alternatively and δ > 0 is a real constant. Then, the MMS arrangement (1a-2a, [3][4] realizes the asymptotic DCAS. For the proof of the above theorem, the following assumptions, and lemmas are necessary. Assumption 2.1: [37]. Since the chaotic attractor evolves in a bounded region, therefore the trajectories of hyper(chaotic) systems are also bounded. Hence, it is assumed that unknown SD disturbances f p i (x p (t)), g p i (y p (t)), h i (z(t)), and j i (w(t)) are bounded. i.e., Consequently, , and σ i are unknown positive real constants such that: Assumption 2.2: [38]. In general, it is assumed that the unknown TD disturbances are norm bounded in C 1 . There exist unknown real constants ρ M 1 where σ i and ρ i are upper bounds of unknown SD and TD disturbances in (12c), and (13b), respectively. Therefore, without loss of generality, it is assumed that σ i = ρ i .

Remark 2.2:
Based on (21a), the controller parameters α ii , β ii , and ε, and the constantδ provide complete control over the convergence rates. Larger values of α ii , β ii , δ and smaller value of ε provide faster AS convergence. Errors convergence behaviours in Figure 3(ad) and control input signals in Figure 4(a-d) illustrate this finding in the computer simulation results.

Remark 2.3:
For the algorithm of the proposed RAASC method, please refer to Appendix A.

Example 1. Anti-synchronization of uncertain hyperchaotic Lü, chaotic SM, chaotic SOND and chaotic PESNP systems
This subsection presents a numerical example dealing with the DCAS of different uncertain hyperchaotic Lü, chaotic SM, chaotic SOND, and chaotic PESNP systems. These chaotic systems have different orders and structures. The initial conditions and parameter values of these systems are different and uncertain. These systems exhibit different chaotic behaviour and topological properties. Similarly, the trace changes of the state variables of the considered systems are different as shown in Figure 1(a-d). Therefore, the secure communication system based on such an AS scheme enriches the security of the information signal.
The combination of the uncertain hyperchaotic Lü and chaotic SM systems represents the master system whereas the chaotic SOND and PESNP systems form the slave system. Equations (24)(25)(26)(27) describe the master and slave systems arrangement, where x(t) and y(t) are the state vectors of M 1 and M 2 (24)(25), while z(t) and w(t) are the state vectors of S 1 and S 2 (26-27), respectively.
(Master system) M : & (27) According to the proposed ROAS strategy, let us divide the master system M (24-25) into two subsystems known as the projection subsystem described in (24a-25b) and the remaining given in (24b-25b).
According to Theorem 2.1, the RAASC functions are designed as follows: The uncertain parameters are updated according to the following adaptation laws: = δy 2 (t)e 1 (t),ψ 2 (t) = δy 3 (t)e 2 (t),  (24)(25)(26)(27). In (30), the design parameters are set as α ii = 1, β ii = 1, μ = 0.9, ε = 0.5, and δ = 10, for i = 1, 2. In numerical simulations, the following SD and TD disturbances are applied to the M 1 M 2 S 1 S 2 (24-27), respectively. (SD disturbance signals) = 0.2 sin 2y 1 (t), g 2 (y 2 (t)) = 0.25 sin 2y 2 (t), and Figure 1(a-d) illustrate the 2-dimensional phase portraits of the hyperchaotic Lü, chaotic SM, chaotic SOND, and chaotic PESNP systems. Figure 2(a-d) illustrate the time series of the state trajectories of the hyperchaotic Lü, chaotic SM, chaotic SOND, and chaotic PESNP systems. From Figure 1(a-d) and Figure 2(a-d), it can be observed that these systems exhibit different chaotic behaviour, and the traces changes of all systems are different. Figure 3(a-b) depict the transient behaviour of the state variables of the AS among the uncertain hyperchaotic Lü, chaotic SM, SOND and PESNP systems (24)(25)(26)(27) under the feedback control technique (30). From these figures, it is evident that the projection part of the master system (24a-25a) is completely antisynchronized with the slave system (26)(27). Figure 4(a-d) and Figure 5(a-d) illustrate the convergence behaviour of the AS error vectors (29) and control input signals (30), respectively under the effects of the controller parameters α ii , β ii , δ, and ε. From Figure 4(ad), it can be seen that the AS time is shorter for larger values of the three coefficients α ii , β ii , δ, and a smaller value of ε. Figure 5(a-d) compare the activeness in the control signals for different values of α ii , β ii , δ, and ε. Figure 4(e) and 5(e) demonstrate the AS error vectors and control input signals when the sech and tanh functions are replaced with the sgn function in the RAASC approach (30), respectively. Activeness increases in the error signals using the large values of the designed parameters as shown in Figure 4(b-c). In Figure 4(a) and 4(e), the convergence time of the error vectors is almost the same. But, it can be observed that the control inputs (30) with sec h and tan h functions ( Figure 5(ad)) have lesser active than the control input (30) with the sgn function ( Figure 5(e)). Further, the control input signals (30) with sgn function produce chattering ( Figure 5(e)). Thus, the proposed RAASC strategy is successful in synthesizing chattering free control signals for reducing the AS errors transient oscillations and convergence time.

Example 2
This subsection discusses and compares the performance and efficiency of the RAASC technique (30) on the AS behaviour and feedback control strategies with the peer works [21,24,28]. The control signals u 1 (t) and u 2 (t) are synthesized by A(i) ASMC technique [21], A(ii) adaptive feedback control strategy [24], and A(iii) adaptive control approach [28] described in (34), (35), and (36), alternatively. Then, these control signals in the error system (29) establish the AS. The initial conditions, controller parameters, unknown SD and TD disturbances are set the same for the ASC schemes reported in [21,24,28] for the benchmarking.
A(iii) The adaptive control scheme [28]: According to [21,28], ii is the bound of unknown SD and TD disturbances, which is known in advance.
comparative study of the proposed control algorithm (30) in the presence of the disturbances with A(i) and A(ii). A comparison of the corresponding AS performance using the proposed RAASC strategy (30) with reported works [21,24,28] is illustrated in Tables 3-5 and Figures 6-8. Items S(i), andS(ii) provide an analysis of the results in Figure 6(a-d), and Figure 7(ad), respectively,while S(iii) demonstrates a comparison of the absolute value of the error (IAE), the integral of the time multiplied by the absolute value of the error (ITAE), and the integral of the square value (ISV) [40]. S(iv) discusses the AS rate of convergence.
Ref. [28] Eq.  [21], (c) Adaptive feedback control approach [24], and (d) ASC technique [28] (α ii = β ii = 1, δ = 10, ε = 0.5, μ = 0.9). Figure 6(a) illustrates that the AS errors converge to the origin by the proposed algorithm (30), while Figure 6(b-d) demonstrate the AS errors convergence by the reported schemes [21], [24], and [28], alternatively. In this work, the AS is accomplished in t = 0.5seconds, while by (34), (35) and (36), the AS error vectors converge to zero after t = 10seconds, t = 6seconds, and t = 5seconds, alternatively. As compared to the control approaches proposed in [21], [24], and [28], the AS time in the present work is shorter. The short time AS error convergence is crucial in practical applications. Furthermore, in the proposed AS approach, the transient phase has a lower amplitude and dies quickly. S(ii) Figure 7(a-d) depict the transient behaviour of the feedback terms in ASC strategies (30), (34), (35), and (37), alternatively. From these figures, one can observe that the control input signals in the proposed RAASC technique (30) are chattering free and yield lesser active than [21,24,28]. Table 4 summarizes the comparative analysis of the proposed method in the sense of IAE, ITAE, and ISV of Example 2. Table 4 verifies that the proposed ASC approach (30) has lower IAE, ITAE, and ISV than those of other ASC methods [21,24,28].

S(iii)
S(iv) From Table 5, we may write the following inequality. , (38) whereV(t) determines the rate of convergence.

Future research direction
Extension of this work is summarized as follows: (i) The model considered in this work is general and the theoretical results are encouraging. The proposed AS scheme can be used for encryption and decryption of an image in the secrete communication systems.The speed of the transportation of the information signal can be increased by selecting high feedback gains, but this attribute may give birth to signal saturation and the AS may lose its stability and the message signal may be interrupted during the communication process.The above issues might be tackled by the feature selection (FC) method [41,42]. The speed of AS could be further increased by reducing the dimensionality of the transmitted data using an FC method. The FC decreases computation cost, speeds up the classification process, and enhances the performance, modelling, and prediction.
(ii) The proposed ROAS will be further enhanced for the finite-time anti-synchronization of uncertain chaotic systems with time-delay.

Conclusions
This article proposes a new robust adaptive control technique and studies the double combination antisynchronization of multiple different orders uncertain chaotic systems in the presence of unknown statedependent and time-dependent disturbances. This controller accomplishes quick convergence. The rate of convergence decreases in the vicinity of the origin that causes the suppression of undesirable transient oscillations in the error signals and control inputs. Analysis based on the Lyapunov stability theory assures the convergence properties. The design of suitable adaptive laws converges the uncertain parameters to some constants. Computer-based simulation results confirm the performance and efficiency of the presented robust adaptive anti-synchronization control strategy. In comparison with reported anti-synchronization control strategies in the relevant literature, the proposed control technique shows better anti-synchronization performance.

Disclosure statement
No potential conflict of interest was reported by the author(s).