In this subsection, some basic preliminaries for fractional differential and integral equations are given (please refer to Ref.  [1] for more details). The lower limit of the fractional calculus is set to be 0 in this paper. The fractional-order integral with order α can be expressed as

where Γ (· ) represents the Euler Gamma function.

There are three definitions of fractional-order derivative which are often used in the literature, that is: Grunwald– Letnikov (G– L), Riemann– Liouville, and Caputo definitions. The Caputo derivative is frequently used in control engineering because the initial conditions of Caputo derivative take on the same form as those in integer-order systems. The Caputo fractional derivative with order α is given as

where n − 1 ≤ α < n. For convenience, in the rest of this paper we always assume that α ∈ (0, 1).

The following results will be used in this paper.

Lemma 1^[38] If x(t) ∈ C¹[0, T] for some T > 0, then

where α ₁, α ₂ > 0 and α ₁ + α ₂ ≤ 1.

Lemma 2^[1] If x(t) ∈ C¹[0, T] for some T > 0, and 0 < α ≤ 1, then the following equations hold:

Let the fractional-order drive system and response system respectively be

where x(t) = [x₁(t), … , x_n(t)]^T ∈ 𝓡 ⁿ and y(t) = [y₁(t), … , y_n(t)]^T ∈ 𝓡 ⁿ are the system state vectors which are assumed to have continuous derivatives, f(x(t)) = [f₁(x(t)), … , f_n(x(t))]^T ∈ 𝓡 ⁿ and g(y(t)) = [g₁(y(t)), … , g_n(y(t))]^T ∈ 𝓡 ⁿ are two continuous nonlinear functions, u(t) = [u₁(t), … , u_n(t)]^T ∈ 𝓡 ⁿ represents the control input, and d(t) = [d₁(t), … , d_n(t)]^T ∈ 𝓡 ⁿ corresponds to unknown external disturbance. The objective of this paper is to construct a proper controller so that the following two properties are guaranteed:

Property 1 The synchronization error e(t) = x(t) − y(t) = [e₁(t), … , e_n(t)]^T converges to origin asymptotically, and all of the signals in the closed-loop system remain bounded.

Property 2 The prescribed transient and steady state behavioral bounds on the tracking error e_i(t) are achieved.

Property 2 can be guaranteed with the help of the performance function which converts the prescribed performance characteristics into tracking error constraints.

Definition 1^{[23, 24]} A smooth function φ : R₊ → R₊ − {0} is called performance function if φ (t) is strictly decreasing and satisfies .

As a result, we can guarantee Property  2 by satisfying

for all t ≥ 0, i = 1, 2, … , n and φ _i(t) is a performance function associated with the synchronization error e(t). The decreasing rate of the performance function φ _i(t) gives a lower bound on the required speed of convergence of synchronization error e_i(t). The aforementioned properties can be seen in Fig.  1.

Fig.  1.

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	Fig.  1. Prescribed performance synchronization.

Remark 1 It should be mentioned that if inequality equation  (8) is satisfied, then the synchronization error will converge to zero asymptotically, with a convergence rate no less than a certain pre-specified value, exhibiting that the maximum overshoot is less than some sufficiently small preassigned constant. Prescribed performance synchronization can be well used in many fields, such as precise synchronization, finite-time synchronization, etc. In fact, if the prescribed performance function is chosen appropriately, then the finite-time synchronization (for example, see Refs.  [39]– [41]) turns into only a special case of the prescribed performance synchronization.

Remark 2 Traditional synchronization controller design for fractional-order systems, guarantees the convergence of the synchronization error to a residual set, whose size depends on control system design parameters and some bounded terms (see, Refs.  [13]– [22], etc). However, no systematic procedure exists to precisely estimate the required upper bounds, thus making an a priori choice of the aforementioned design parameters to satisfy some steady-state behaviors. Moreover, the performance of transient behaviors (i.e., overshoot and convergence rate) is difficult to analyze, even in the case of known nonlinearities. In this paper, the introduced performance function and the following synchronization error transformation will help us to solve this problem.

To meet the synchronization objective, we define an error transformation which can transform the original nonlinear system into an equivalent unconstrained system. We define

where φ _i(t) is the performance function, and z_i(t) is the transformed synchronization error. The introduced function s_i(· ) is smooth and strictly increasing, and satisfies the following conditions:

Remark 3 From the definition of the function s_i(· ), we know that it is an invertible function. In fact, there are many common functions satisfying the definition of s_i(· ). For example, tanh(z_i), 2arctan(z_i)/π , etc.

From the above discussion, we know that if z_i(t) can keep bounded, then we have − 1 < s_i(z_i(t)) < 1, thus – φ _i(t) < e_i(t) < φ _i(t). Noting that φ _i(t) converges to 0 asymptotically, to achieve Properties  1 and 2, we only need to design a proper controller such that z_i(t) keeps bounded. According to Remark 3 we know that the inverse transformation of z_i(t)

is well defined. Differentiating Eq.  (11) with respect to time gives

Let

then equation  (12) can be arranged as

From Lemma 1 we know

By substituting Eqs.  (6), (7), and (14) into Eq.  (13), we have

Now we consider synchronization of systems (6) and (7) without unknown external disturbances (i.e., d(t) ≡ 0). Then, we can design the following fractional synchronization controller:

where k_i is a positive design parameter.

From the above discussion, we have the following results.

Theorem 1 Consider the fractional-order drive system (6) and response system (7) on the assumption that d(t) ≡ 0. Then the proposed fractional-order controller, defined by Eq.  (16), guarantees the following properties: (i) All of the signals in the closed-loop system are bounded. (ii) The prescribed performance synchronization between two fractional-order chaotic systems is achieved.

Proof According to Lemma 2 we have

Substituting Eqs.  (16)– (18) into Eq.  (15) yields

Define the Lyapunov function candidate as , its derivative with respect to time will then be given as

As a result, z_i(t) is bounded for all t > 0. Because system  (6) represents a chaotic system, we have both x_i(t) and f_i(x(t)) that are bounded. Furthermore, we have | e_i(t)| ≤ φ _i(t), which means that the synchronization error converges to origin asymptotically and the system variables x_i(t) and y_i(t) are bounded. From the above discussion, we have that Properties  1 and 2 can be guaranteed. This completes the proof of Theorem  1.

Most practical systems suffer from several kinds of system uncertainties (such as the modeling errors, the plant parameter variations, and the unknown external disturbances) which can actually degrade the control performance if not well-disposed.^{[42– 44]} In this subsection, we consider the case where external disturbance d(t) cannot be neglected in chaos synchronization.

To proceed, the following assumption is needed.

Assumption 1 There exists some unknown positive constant c_i such that

Remark 4 Assumption  1 is not very restrictive, because the upper bound of fractional derivative of the external disturbance is assumed to be unknown. There are some frequently used external disturbance functions satisfying Assumption  1, such as sin(· ), cos(· ), etc. In fact, we denote

Let , then we will have

Similarly, we can obtain that G(3π ) ≥ 0 and G(4π ) ≤ 0. By using mathematical induction, we can easily conclude that G((2n – 1)π ) ≥ 0 and G(2nπ ) ≤ 0, . As a result, we have

Noting that

(by using the above method), we have

For all t > 0, there exists some such that 2nπ ≤ t < 2(n+ 1)π . Then, we have

which means that is bounded.

Remark 5 It should be pointed out that Assumption  1 is first proposed in this paper, and it may be helpful for tackling some periodic external disturbances in fractional-order systems.

We construct the following synchronization controller:

where b_i is a positive design parameter, and ĉ _i(t) represents the estimation of the unknown constant c_i. We denote the estimation error of c_i as

From the above discussion, we are now ready to give the following results.

Theorem 2 Consider the prescribed performance synchronization between two fractional-order chaotic systems  (6) and (7). Supposing that Assumption  1 is satisfied, then the proposed adaptive fractional-order controller (22) with the adaptation law:

where a_i and σ _i are positive design parameters, can guarantee that all signals in the closed-loop system will remain bounded and the prescribed performance synchronization between systems (6) and (7) can be obtained.

Proof Substituting the controller (22) into Eq.  (15) yields

By multiplying both sides of Eq.  (25) with z_i(t), and using Assumption  1, we have

Define the Lyapunov function candidate as

The derivative of V_1i(t) with respect to time is

Using Eqs.  (24) and (26), we have

where and ρ _i = min{2b_i, a_i} are two constants.

Following the same manipulators as those in Ref.  [45] (see, p.  14748, Eqs.  (48)– (50)), we know there exists some positive constant t_0i such that

for all t > t_0i. As a result, we know that both z_i(t) and c̃ _i(t) will keep bounded.

Thus, after the same discussion as that in the proof of Theorem  1, we know the prescribed performance synchronization between systems  (6) and (7) is achieved, and the proof of Theorem  2 is ended.

Remark 6 The two terms in Eq.  (24) have different applications. The σ _iγ _i(t)| z_i(t)| is used to cancel a term which is produced by differentiating a quadratic Lyapunov function, and the other is introduced for the parameter boundedness purpose.

Let d_i(t) = 0, i = 1, 2, 3. The controller design parameters are chosen as k₁ = k₂ = k₃ = 1. The simulation results are shown in Figs.  4 and 5. The evolution of the synchronization errors e₁(t), e₂(t), and e₃(t) are illustrated in Fig.  4. along with their corresponding performance bounds. The required control inputs u_i(t), i = 1, 2, 3 and the transformed synchronization errors z_i(t), i = 1, 2, 3 are depicted in Fig.  5. Obviously, synchronization with prescribed performance, by using the reasonable control inputs, is achieved as predicted by the theoretical analysis.

Fig.  4.

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	Fig.  4. Time responses of synchronization errors.

Fig.  5.

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	Fig.  5. Control inputs and zi(t), i = 1, 2, 3. (a) u1(t), (b) u2(t), (c) u3(t), and (d) zi(t), i = 1, 2, 3.

Let the external disturbances be d₁(t) = 0.4sin(t), d₂(t) = 0.4cos(t), and d₃(t) = − 0.4sin(t). The design parameters are chosen as σ _i = 5, a_i = 0.2, b_i = 1, and i = 1, 2, 3. Simulation results are given in Figs.  6– 8, from which we can see that the prescribed performance synchronization between systems  (32) and (31) is achieved, despite the unknown external disturbances. It should be pointed out that chattering phenomenon will not occur in this paper due to the fact that the fractional-order integration of the non-continuous term sign(· ) in the control input (22) is continuous.

Fig.  6.

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	Fig.  6. Simulation results with external disturbances. The inset is the enlarged figure in the range of 11  s– 15  s.

Fig.  7.

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	Fig.  7. Time responses of the estimation of ci.

Fig.  8.

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	Fig.  8. Control inputs and zi(t), i = 1, 2, 3. (a) u1(t), (b) u2(t), (c) u3(t), and (d) zi(t), i = 1, 2, 3.