Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems

Abstract In this paper, for multiple different chaotic systems with unknown bounded disturbances and fully unknown parameters, a more general synchronization method called modified function projective multi-lag combined synchronization is proposed. This new method covers almost all of the synchronization methods available. As an advantage of the new method, the drive system is a linear combination of multiple chaotic systems, which makes the signal hidden channels more abundant and the signal hidden methods more flexible. Based on the finite-time stability theory and the sliding mode variable structure control technique, a dual-stage adaptive variable structure control scheme is established to realize the finite-time synchronization and to tackle the parameters well. The detailed theoretical derivation and representative numerical simulation is put forward to demonstrate the correctness and effectiveness of the advanced scheme.


System description
In our drive-response type combination synchronization scheme, m different chaotic systems with unknown parameters and disturbance are considered as the drive systems. The lth drive system is given by (1) in which l D 1; 2; ; m.
At the meantime, the response system is described as: ; u n .t / T is the vector of control input.

Preliminary definition and lemmas
As the essence of finite-time synchronization, it means that the state trajectory of the response system can converge to the state trajectory of the drive system within a finite time. In this section, we introduce the precise definitions and several important lemmas, which are necessary for further study.
Assumption 2.1. The unknown parameters Â l and are bounded, in another word, there exist known constants N Â l 0 and N 0, such that where l D 1; 2; ; m; and k k stands for the 2-norm .
Assumption 2.2. The unknown external time-varying disturbances w l .t / and d i .t / are bounded, that is to say, there exist non-negative constants N w l i and d i satisfy where l D 1; 2; ; m and i D 1; 2; ; n. 40]). Assume that a continuous and positive-definite function V .t / satisfies the following differential inequality : where b 1 > 0; b 2 > 0 and 0 < # < 1 are constants.
, the following results are true: with T given by Lemma 2.4 ( [35]). Consider the system where the mapping function f W I ! R n is continuous. If there exists a continuous differential positive-definite function V W I ! R , real constants > 0; 0 < % < 1, satisfying then, the origin of system (5) is a locally finite-time stable equilibrium, the settling time T .x 0 / depends on the initial state x.0/ D x 0 , and the following inequality holds Lemma 2.5 ( [15]). Suppose a 1 ; a 2 ; ; a n and 0 < q < 2 are all real numbers, then the inequality below holds C C a n 2 / q 2 : Lemma 2.6. By choosing q D 1 in Lemma 2.5, we can obtain C C a n 2 / ; m/ and ƒ.t /, such that where A l D .a l ij / n n is constant matrix, ƒ.t / D d i ag f 1 .t /; ; n .t /g is a reversible function matrix whose elements are continuously differentiable nonzero function with bound.
then it is said that the group of the drive systems (1) and the response system (2) are finite-time modified function projective multi-lag combined synchronization. Remark 2.9. As is shown in Table 1, the proposed MFPMLCS is more general, and it concludes a large class of the previous synchronization methods. Selecting specific scaling matrix A l , ƒ.t / and specific delay times l ; l D 1; 2; ; m, the MFPMLCS will be simplified to specific synchronization. Here C S represents combined synchronization, CS means complete synchronization, ƒ D d i ag f 1 ; ; n g, I is a n n unit matrix.
Remark 2.10. As another advantage of the new method, the drive system is a linear combination of the multiple chaotic systems, which means the signal hidden channels are more diversified and the signal hidden methods are more flexible. The complexity of this new synchronization scheme improves, to a great degree, the abilities to anti attacking and anti decoding in the process of signal transmission.
Notice that i .t / ¤ 0 is a continuously differentiable function with bound, we can further put forward the following assumption.
Assumption 2.11. There exist positive constants p i and q i , i D 1; 2; ; n, i.e.
combining Assumption 2.2 with Assumption 2.11, we can obtain that i .t / is bounded.
To deal with the more general case in which the bound i > 0 is unknown, the following assumption is needed. ; n/ which are large enough, such that In order to solve the finite-time synchronization problem, we now define the MFPMLCS error vector that is to say from which, the corresponding error dynamic system below can be obtained: For convenience , let us denote Now, the error dynamics system (18) can be reduced as follows

Design of dual-stage finite-time control scheme
It is clear that the finite-time MFPMLCS problem is directly equivalent to the finite-time stabilization of the error system (20). In this section, we pay our attention to design an adaptive sliding mode variable structure control scheme to ensure the error trajectories converge to zero within a limited time. The finite-time control scheme is divided into the sliding mode stage and the sliding mode reaching stage. What is more, the time required for each stage is limited.

Sliding mode stage
In order to realize the desired finite-time sliding motion, let us establish a new nonsingular terminal sliding surface [41] as follows, where the constants 0 <˛i < 1, c iv > 0; v D 0; 1; 2; 3; i D 1; 2; ; n. which is proposed in [15], the terminal sliding surface (21) has the following advantage: the factor c i1 e i C c i 2 sgn.e i /je i j 2 ˛i plays a leading role to guarantee a fast convergence speed as je i .t /j is much larger than 1, while the factor c i3 sgn.e i /je i j˛i is the dominant one ensuring the finite-time convergence as je i .t /j is much less than 1.
According to the sliding mode control theory, when the state trajectories of the error system are located on the sliding surface, it is necessary and sufficient that with and Proof. Design the following Lyapunov function for the dynamics of the proposed nonsingular terminal sliding mode (22) Taking the time derivative of V 1i .t / , we obtain Applying the Lemma 2.3, we can directly deduce that during the sliding mode phase the error e i .t / converges to zero in the finite time T 1i given by (24). This yields that the error vector e.t / converges to e.t / D 0 in a finite time T 1 given by (23). Hence the proof is completed.

Sliding mode reaching stage
Until now, the suitable sliding surface is established and the finite-time convergence and stability in sliding mode stage has been proved. We now turn to design an adaptive controller to force the error trajectories move toward the sliding surface within a finite time and remain on it forever. In order to achieve the finite-time sliding mode reaching stage, the controller is given as follows: sgn.s i / js i j g i D 1; 2; ; n; with ; c n0 s n T , D min fc 10 k 1 ; c 20 k 2 ; ; c n0 k n g.
Meanwhile, the adaptive laws are given as follows to tackle the unknown parameters: in which, D minf ; &g.
Proof. Choose the following Lyapunov function candidate in which Taking the time derivative of V 21 .t / , we get sgn.e i /je i j 2 ˛i C c i3 sgn.e i /je i j˛i : Along the error system, P V 21 .t / can be described as Using the fact ; c n0 k n g ; we can derive The time derivative of V 22 .t / can be calculated as Combining (34) with (35), we can obtain According to Lemma 2.6, we get Applying Lemma 2.4, it follows that the error trajectory e.t / converges to the sliding surface s.t / D 0 in the finite time O T 2 and then remains on it forever, meanwhile the following inequality holds It is clear that O T 2 Ä T 2 in which T 2 is given by (31). This completes the proof.  ; c n 0k n ; &g. At the same time, the control input u i .t/ is proportional to 1 c i 0 , k i and & . Based on these relationships, the appropriate control gains above can be selected according to the specific requirements of designer. Remark 3.6. According to Eqs. (28) , the control input u i .t / contains the factor sgn.s i / js i j . In fact, during the sliding mode reaching phase, when the error trajectories e i .t / reach onto the sliding surfaces s i .t / D 0, it is obvious that sgn.s i / D s i D 0 , which means sgn.s i / js i j is singular. In order to overcome this disadvantage, the control law (28) is modified as follows where the switching gain ı is a sufficiently small positive constant which can be chosen according to the designer requirements.
Another effective approach is using the function sgn.s i / js i j C " ( " is a sufficiently small positive constant) to approximate sgn.s i / js i j , which is common in the sliding mode application.

Numerical simulation
In this section, we choose two famous chaotic systems: Lü system and Lorenz system with fully unknown parameters and unknown bounded disturbances as the drive systems. At the same time, another well-known chaotic system named Chen system is considered as the response system. They can be described as follows: Lü system: Lorenz system:  Figure 6 shows the sliding surface can rapidly converge to zero. The simulation results illustrate the effectiveness of the proposed method.

Conclusion
In this paper, we dealt with the problem of the finite-time modified function projective multi-lag combined synchronization (MFPMLCS) for a series of different chaotic systems with unknown bounded disturbances and fully unknown parameters. Based upon the sliding mode control technique and Lyapunov stability theory, we designed an adaptive dual-stage variable structure control scheme to realize the finite-time synchronization. The resulted systems are provided with fast convergence rate, strong robustness, small chattering and high accuracy. Finally, the numerical simulation demonstrated the correctness and effectiveness of the advanced scheme.