A hybrid adaptive synchronization protocol for nondeterministic perturbed fractional-order chaotic nonlinear systems

In this paper, we investigate hybrid adaptive synchronization issue for a class of perturbed fractional-order chaotic systems with nondeterministic nonlinear terms. On the basis of fractional-order extended version of Lyapunov stability criterion, a novel fuzzy adaptive synchronization control protocol coupled with backstepping-based method is constructed, ensuring that the synchronization errors converge to a sufficiently small region of the origin. In order to avert the occurrence of “explosion of complexity”, we take advantage of a fuzzy logic system to estimate the unknown systematic term approximately in every backstepping step. Finally, some numerical simulations are given to exemplify the effectiveness of the proposed approach.


Introduction
Fractional calculus [1], as a greatly ancient subject, is overwhelmingly superior than integer-order calculus in various applications. This is ascribed to that fractional calculus provides not only a powerful algorithmic tool to facilitate complex numerical computing, but also a comprehensive mathematical model of enormous practical problems [2]. In view of heredity and memristive feature, fractional-order calculus can be utilized to model most of complex dynamic behaviors or specific materials (such as chaos, anomalous diffusions, viscoelastic damping structures, neural networks, and so on, see [3][4][5][6][7][8]) more precisely, beyond the integer-order calculus in general. Due to this, the topic of synchronization protocol design for fractional-order nonlinear systems has dramatically stirred plenty of excitement in many research fields. The synchronization issues can be dealt with by applying abundant control methods, including resilient control [9,10], output-feedback control [11,12], sliding mode control [13][14][15][16][17], fuzzy control [18][19][20][21], dynamic surface control [22][23][24][25], etc.
Synchronization issue of nonlinear systems is widely considered due to its valuable significance in both theoretical and practical aspects. The goal of synchronization is to design an active controller to synchronize the so-called slave dynamical system with another diverse one, namely the master. Various synchronization protocols have been proposed, including lag synchronization [26], projective synchronization [27], fixed time synchronization [28], and chaos synchronization [29,30]. In essence, chaos synchronization generalizes chaos control [31], which enables the chaotic master-slave error dynamics trajectories to be asymptotically stable.
In real life, especially in control procedure, we are frequently confronted with a slew of information with ambiguity, randomness, incompatibility, incompleteness, and so forth. This led to the invention of many mathematical approaches (e.g., Zadeh's fuzzy set approach [32], backstepping approach [33][34][35][36]) to dispose of nondeterministic systematic parameters. As a kind of recursive control strategy, backstepping control has engaged attention because of its efficient performance in handling mismatched parametric uncertainties of integer-order nonlinear systems. Unfortunately, this control method has an inherent drawback, namely "explosion of complexity", which is triggered by iteratively differentiating virtual control inputs (see [22]). Additionally, it requires complicated analysis to compute a so-called "regression matrix" (see [37]). Dawson et al. [38] pointed out the fact that the size of the regression matrix displayed too large when backstepping technique was applied to manipulate DC motors in a conventional manner. Such a complexity might be augmented significantly for a fractional-order nonlinear system. An available remedy for relaxing the limitation of backstepping control is to incorporate fuzzy inference approach [39] into backstepping proceedings. For instance, Tong et al. [40] put forward an observer-based adaptive backstepping control protocol for nondeterministic stochastic strict-feedback integer-order systems via fuzzy inference approach, and they also developed a simplified control protocol. Liu et al. [41] introduced a robust fuzzy backstepping control method for fractional-order nonlinear systems with triangle structures. Shukla et al. [42] exploited a backstepping technique to synchronize the tracking signals of fractional-order chaotic systems with constant parameters. However, their works seldom took into account fractional-order chaotic systems with functional uncertainties and external perturbations.
Motivated by this, we aim to address the backstepping-based synchronization issue of a class of fractional-order chaotic master-slave nonlinear systems. Compared with the previous works, our problem model involves nondeterministic external perturbations and more complicated parametric uncertainties, which expands the scope of applications. In order to achieve this goal, we propose a hybrid adaptive control method combined with backstepping technique and fuzzy inference approach. The contributions of our synchronization protocol are outlined to be twofold: (1) An appropriate fuzzy logic system is adopted as an estimation function routinely for the nondeterministic nonlinear term in each backstepping step; (2) A reasonable fuzzy adaptive control strategy based on backstepping method is established to attenuate all estimation errors and realize the synchronization between master and slave systems. With the aid of the proposed protocol, the occurrence of the drawback of "explosion of complexity" will be denied in every backstepping step. The arrangement of this paper is listed as below. In Sect. 2, some fundamental notions and results involving with fractional calculus are recalled and a concrete description of model for the research issue is presented. In Sect. 3, we construct an adaptive backstepping-based controller via fuzzy inference approach, and analyze the systematic synchronization on the basis of our proposed synchronization scheme. The validity of this synchronization scheme is demonstrated by numerical simulation in Sect. 4. Finally, we summarize the research in this paper and present an outlook for our further research in Sect. 5.

Fractional calculus fundamental
In the full context, we denote the space of all real numbers (resp. complex numbers, ndimensional real vectors) by R (resp. C, R n ). For a vector x ∈ R n , x T denotes its transpose. A where f (t) is a time-dependent function with t ≥ 0, Γ denotes the Gamma function, that is, (2.1) In [1], the Laplace transform of Eq. (2.1) is represented as where L denotes the Laplace transform operator, F(s) = L{f (t); s}.
It is immediately seen that E 1,1 (z) = e z .

Lemma 2.2 ([43]) Let s be a variable of the Laplace domain and ν ∈ R. Then
where L -1 denotes the inverse Laplace transform operator, Re(s) means the real part of s.

Fuzzy logic systems
Let x x x(t) ∈ R n at time instant t ∈ [0, +∞), and R (1) , R (2) , . . . , R (m) be fuzzy rules, which can be interpreted as [18][19][20][21]39]: . . , n) and F k are fuzzy sets. A fuzzy logic system, with x x x(t) andf (x x x(t)) being the input-variable and the output-variable, respectively, is given bŷ where μ E k j is the fuzzy membership function of E k j defined from R to the interval [0, 1], and η k (t) (called the centroid of the fuzzy rule R (k) ) is a real number at which the fuzzy membership degree μ F k for F k is maximized, i.e., In general, we set μ F k (η k (t)) = 1 for simplicity.
A fuzzy basis function φ k : R n − → R based on the fuzzy rule R (k) is defined by Then the output-variable of the fuzzy logic system (2.3) can be simplified aŝ

Problem description
In this paper, we concentrate on chaos synchronization issue of a type of uncertain fractional-order master system, which is represented as . . , n) describe variables of master pseudo-states at time instant t ∈ [0, +∞), and f i denote nondeterministic smooth nonlinear functions.
The slave system coupled with system (3.1) is expressed by whereȳ i (t) = (y 1 (t), y 2 (t), . . . , y i (t)) T ∈ R i (1 ≤ i ≤ n) stand for variables of slave pseudostates, q i denote nondeterministic smooth nonlinear real-valued functions, d i (t) ∈ R represent external perturbations, and u(t) ∈ R is a synchronization controller which will be specified later. The relationship between systems (3.1) and (3.2) is that slave (3.2) subjected to the dynamic behaviors of master (3.1) is asymptotically synchronized with master (3.1) with the aid of controller u(t).
for some known positive constant d i .

Controller construction and stability analysis
The synchronization error e i (t) for the master variable x i (t) and its slave y i (t) is defined as for i = 1, 2, . . . , n -1, and h n ȳ n (t),x n (t) = q n ȳ n (t)f n x n (t) .
Next, let us focus on the design of a fuzzy adaptive backstepping-based control protocol step by step.
Step1. To approximate the nondeterministic continuous function h 1 , we construct the following fuzzy logic system, whose output-variable is determined by an estimation functionĥ 1 for h 1 : where η 1 (t) is an adjustable 1-dimensional parameter vector and φ 1 is a fuzzy basis function. Moreover, with respect to η 1 (t), Theorem 2.7 guarantees the existence of the optimal parameter η * 1 , which is given by Here, the employment of η * 1 just only facilitates the analysis of the systematic stability, but it is not mandatory in the control protocol.
On the other hand, it follows from Lemma 2.4 that As a consequence, for each > 0, there is a constant T 2 > 0 satisfying Select a 1n and a 2n such that a 2n a 1n ≤ 3 . Based on the preceding argument, we have From (4.33), it can be inferred that all state variables and estimation errors are bounded in the closed-loop system according to (4.29). Hence, by the arbitrariness of , every synchronization error e i (t) tends towards a sufficiently small region of the origin ultimately.
Remark 4.2 In order to realize the systematic synchronization, we should adjust a 2n /a 1n to be as small as possible by means of parameterizing the fuzzy logic system properly. For instance, we can enlarge ξ i and reduceξ i simultaneously.
Remark 4.4 It should be mentioned that Theorem 4.1 can be extended to analyze the stability of many other fractional-order nonlinear systems. Based on fractional Lyapunov stability criterion, it is not difficult to show that if there exist positive constants τ 1 , is a quadratic Lyapunov function, then e(t) ∈ R n is globally bounded and e(t) ≤ τ 2 τ 1 holds whenever the time instant t is sufficiently large.
Remark 4.5 Under the proposed fuzzy adaptive backstepping control protocol, it can be apparently seen that the superfluous terms which might appear by recurring fractional derivations on virtual control inputs are fully averted, which is also suitable for many other fractional-order nonlinear systems. For the details, the readers may refer to Appendix B of [35]. Suppose that a master system is formed by The relevant slave system is formulated by D α y 2 (t) = y 1 (t)y 3 1 (t) -0.15y 2 (t) + d 2 (t) + u(t).
Take two fuzzy logic systems into consideration in this simulation. Assume that y 1 (t) is the input of the first fuzzy logic system with the Gaussian membership functions, which are expressed by exp(-(xc i ) 2 /(2σ 2 i )) for i = 1, 2, 3, 4. Suppose they are uniformly distributed on [-3, 3], and consider the initial value η 1 (0) = (1, 1, 1, 1) T . The secondary one In view of the above-mentioned results, we observe that the synchronization errors reduce very rapidly and thereafter converge to a sufficiently small region as time elapses, which shows the outstanding performance of the fuzzy logic system in practical applications. Such results meet our expectation.

Figure 4
The fractional-order master system in Example 2

Figure 5
The fractional-order slave system in Example 2 The slave system defined in [48] is
The simulation involves three fuzzy systems. The first is based on the four fuzzy membership functions as defined in Example 1, viewing y 1 (t) as its input. Let the initial value be η 1 (0) = (1, 1, 1, 1) T . The second takes y 1 (t) and y 2 (t) as its inputs. For every input, the membership functions are defined similarly to that of the first fuzzy system with the initial value η 2 (0) = (1, . . . , 1) T ∈ R 16 . The last system regards y 1 (t), y 2 (t), and y 3 (t) as its inputs. With respect to y 3 (t), one defines five Gaussian membership functions uniformly distributed on [-4, 4]. The initial value is chosen as η 3 (0) = (1, . . . , 1) T ∈ R 80 .

Conclusion
This work provides a framework to study stabilization control of perturbed fractionalorder chaotic systems with nondeterministic terms based on extended Lyapunov stability criterion. It is demonstrated by numerical simulations that the proposed adaptive fuzzy backstepping-based control strategy not only overcomes the inherent drawback of "explosion of complexity", but also reflects the robust attribute for fractional-order chaotic systems consisting of parameter uncertainties and external perturbations. In the future, it is worth considering the synchronization issue of fractional-order systems with more sophisticated structures (for example, we can assume that the linear term coefficient is nondeterministic).