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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 15, Issue 1

Issues

Volume 13 (2015)

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

Qiaoping Li / Sanyang Liu
Published Online: 2017-08-19 | DOI: https://doi.org/10.1515/math-2017-0087

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.

Keywords: Finite-time adaptive control; Modified function projective multi-lag combined synchronization; Sliding mode variable structure control; Chaotic systems; Unknown parameter and disturbance

MSC 2010: 93C10

1 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 x˙1l(t)=F1l(xl(t))θl+f1l(xl(t))+w1l(t),x˙2l(t)=F2l(xl(t))θl+f2l(xl(t))+w2l(t),x˙nl(t)=Fnl(xl(t))θl+fnl(xl(t))+wnl(t),(1) in which l = 1, 2, ···, m.

At the meantime, the response system is described as: y˙1(t)=H1(y(t))ϕ+h1(y(t))+d1(t)+u1(t),y˙2(t)=H2(y(t))ϕ+h2(y(t))+d2(t)+u2(t),y˙n(t)=Hn(y(t))ϕ+hn(y(t))+dn(t)+un(t),(2) where xl=[x1l,x2l,,xnl]T,y(t)=[y1(t),y2(t),,yn(t)]TRn are the state vectors of the drive system and the response system respectively, fil(xl(t)), l = 1, 2, ···, m and hi (y(t)), i = 1, 2, ···, n are continuous nonlinear functions, Fil(xl(t)) and Hi (y(t)) are the ith row of the continuous linear function matrices Fl(xl(t)) and H(y(t)), respectively, θl=[θ1l,θ2l,,θnl]T and φ = [φ1, φ2, ···, φn]T are unknown parameter vectors, w(t) = [w1(t), w2(t), ···, wn(t)]T, d(t) = [d1(t), d2(t), ···, dn(t)]T and p(t) = [p1(t), p2(t), ···, pn(t)]T are unknown external time-varying disturbances, u(t) = [u1(t), u2(t), ···, un(t)]T is the vector of control input.

2 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 θl and φ are bounded, in another word, there exist known constants θ̄l ≥ 0 and φ̄ ≥ 0, such that θlθ¯l,ϕϕ¯, where l = 1, 2, ···, m, and ||·|| stands for the 2-norm.

Assumption 2.2

The unknown external time-varying disturbances wl (t) and di (t) are bounded, that is to say, there exist non-negative constants w¯il and di satisfy wil(t)w¯il,di(t)d¯i. where l = 1, 2, ···, m and i = 1, 2, ···, n.

Lemma 2.3

([40]). Assume that a continuous and positive-definite function V(t) satisfies the following differential inequality: V˙(t)b1Vϑ(t)b2V(t),tt0,V(t0)0,(3) where b1 > 0, b2 > 0 and 0 < ϑ < 1 are constants.

Then, when V1ϑ(t0)b1b2, the following results are true: V(t)eb2(tt0)[V1ϑ(t0)+b1b2b1b2eb2(1ϑ)(tt0)]1/(1ϑ),ift0t<T,V(t)=0,iftT. with T given by T=t0+1b2(1ϑ)ln(1+b2V1ϑ(t0)b1),(4)

Lemma 2.4

([35]). Consider the system x˙=f(x),f(0)=0,xRn(5) where the mapping function f : IRn is continuous. If there exists a continuous differential positive-definite function V : IR, real constants ζ > 0, 0 < ϱ < 1, satisfying V˙(x)ζVϱ(x),xI,(6) then, the origin of system (5) is a locally finite-time stable equilibrium, the settling time T (x0) depends on the initial state x(0) = x0, and the following inequality holds T(x0)V1ϱ(x0)ζ(1ϱ).(7)

Lemma 2.5

([15]). Suppose a1, a2, · · ·, an and 0 < q < 2 are all real numbers, then the inequality below holds |a1|q+|a2|q++|an|q(a12+a22++an2)q2.(8)

Lemma 2.6

By choosing q = 1 in Lemma 2.5, we can obtain |a1|+|a2|++|an|(a12+a22++an2)12.(9)

Definition 2.7

It is said that the group of the drive systems (1) and the response system (2) are modified function projective multi-lag combined synchronization (MFPMLCS), if there exist m different delay times τl and m + 1 scaling matrices Al(l = 1, 2, · · ·, m) and Λ(t), such that limtl=1mAlxl(tτl)Λ(t)y(t)=0,(10) or limtl=1mj=1naijlxjl(tτl)λi(t)yi(t)=0,i=1,2,,n,(11) where Al=(aijl)n×n is constant matrix, Λ(t) = diag {λ1(t), ···, λn(t)} is a reversible function matrix whose elements are continuously differentiable nonzero function with bound.

Definition 2.8

If there exist a constant T > 0, such that limtTl=1mAlxl(tτl)Λ(t)y(t)=0,(12) or limtTl=1mj=1naijlxjl(tτl)λi(t)yi(t)=0,i=1,2,,n,(13) and l=1mAlxl(tτl)Λ(t)y(t)=0iftT, 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 Al, Λ(t) and specific delay times τl, l = 1, 2, · · ·, m, the MFPMLCS will be simplified to specific synchronization. Here CS* represents combined synchronization, CS means complete synchronization, Λ = diag{λ1, · · ·, λn}, I is a n × n unit matrix.

Table 1

The special cases of MFPMLCS.

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 λi(t) ≠ 0 is a continuously differentiable function with bound, we can further put forward the following assumption.

Assumption 2.11

There exist positive constants pi and qi, i = 1, 2, ···, n, i.e. pi|λi(t)|qi. Let ρi(t)=l=1mj=1naijlwjl(tτl)λi(t)di(t),(14) combining Assumption 2.2 with Assumption 2.11, we can obtain that ρi(t) is bounded.

Denote ρ = [ρ1, ρ2, · · ·, ρn]T in which ρi = sup|ρi(t)|, i = 1, 2, · · ·, n. To deal with the more general case in which the bound ρi > 0 is unknown, the following assumption is needed.

Assumption 2.12

There exist definite positive constants ρ̄i (i = 1, 2, · · ·, n) which are large enough, such that ρi<ρ¯i.(15) In order to solve the finite-time synchronization problem, we now define the MFPMLCS error vector e(t)=l=1mAlxl(tτl)Λ(t)y(t),(16) that is to say ei(t)=l=1mj=1naijlxjl(tτl)λi(t)yi(t),i=1,2,,n.(17) from which, the corresponding error dynamic system below can be obtained: e˙i(t)=l=1mj=1naijlx˙jl(tτl)λi(t)y˙i(t)λ˙i(t)yi(t)=[l=1mj=1naijlfjl(x(tτl))λi(t)hi(y(t))λ˙i(t)yi(t)]+[l=1mj=1naijlFjl(xl(tτl))θlλi(t)Hi(y(t))ϕ]+[l=1mj=1naijlwjl(tτl)λi(t)di(t)]λi(t)ui(t).(18) For convenience, let us denote Ωi=l=1mj=1naijlfjl(xl(tτl))λi(t)hi(y(t))λ˙i(t)yi(t),u¯i(t)=λi(t)ui(t).(19) Now, the error dynamics system (18) can be reduced as follows e˙i(t)=Ωi+l=1mj=1naijlFjl(xl(tτl))θlλi(t)Hi(y(t))ϕ+ρi(t)u¯i(t).(20)

3 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.

3.1 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, si(t)=ci0ei(t)+0t(ci1ei+ci2sgn(ei(σ))|ei(σ)|2αi+ci3sgn(ei(σ))|ei(σ)|αi)dσ,(21) where the constants 0 < αi < 1, c > 0, υ = 0, 1, 2, 3, i = 1, 2, ···, n.

Remark 3.1

Compared with the terminal sliding surface si(t)=ciei(t)+0tsgn(ei(σ))|ei(σ)|αidσ,i=1,2,,n, which is proposed in [15], the terminal sliding surface (21) has the following advantage: the factor ci1ei + ci2sgn (ei)|ei|2−αi plays a leading role to guarantee a fast convergence speed as |ei(t)| is much larger than 1, while the factor ci3sgn (ei)|ei|αi is the dominant one ensuring the finite-time convergence as |ei(t)| 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 si(t)s˙i(t)=0,i=1,2,,n, from which, we can obtain the following dynamics of sliding mode: e˙i(t)=1ci0(ci1ei(t)+ci2sgn(ei(t))|ei(t)|2αi+ci3sgn(ei(t))|ei(t)|αi),i=1,2,,n.(22)

Theorem 3.2

The error vector e(t) of the sliding mode is finite-time stable and its trajectory converges to the equilibriums e(t) = 0 within a finite time T1, T1=max{T11,T12,,T1n},(23) with T1i=1b¯i2(1ϑ¯i)ln(1+b¯i2V1ϑ¯i(0)b¯i1),i=1,2,,n,(24) and b¯i1=21+αi2ci3ci0,b¯i2=2ci1ci0,ϑ¯i=1+αi2.(25)

Proof

Design the following Lyapunov function for the dynamics of the proposed nonsingular terminal sliding mode (22) V1i(t)=12ei2(t).(26) Taking the time derivative of V1i(t), we obtain V1i(t)=ei(t)e˙i(t)=1ci0(ci1(ei(t))2+ci2|ei(t)|3αi+ci3|ei(t)|1+αi)=1ci0(2ci1V1i+23αi2ci2(V1i)3αi2+21+αi2ci3(V1i)1+αi2)2ci1ci0V1i21+αi2ci3ci0(V1i)1+αi2.(27) Applying the Lemma 2.3, we can directly deduce that during the sliding mode phase the error ei (t) converges to zero in the finite time T1i given by (24). This yields that the error vector e(t) converges to e(t) = 0 in a finite time T1 given by (23). Hence the proof is completed. □

3.2 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: ui(t)=1λi(t){Ωi+1ci0(ci1ei+ci2sgn(ei)|ei|2αi+ci3sgn(ei)|ei|αi)+(ki+ρ^i)sgn(si)+l=1mj=1naijlFjl(xl(tτl))θ^lλi(t)Hi(y(t))ϕ^+ςgnci0sgn(si)|si|}i=1,2,,n,(28) with g=ϕ^+ϕ¯+ρ^+ρ¯+l=1m(θ^l+θ¯l),(29) in which, the constants ς > 0 and ki > 0 are the control gains, which can be designed according to the demands of the designer. ρ̂ = [ρ̂1, · · ·, ρ̂n]T is the estimation of the upper bound constant vector ρ, θ̂l and φ̂ are the estimations of the parameters θl, φ respectively, and η = [c10s1, c20s2, ···, cn0sn]T, μ = min{c10k1, c20k2, ···, cn0kn}.

Meanwhile, the adaptive laws are given as follows to tackle the unknown parameters: ρi^˙=ci0|si|,ρ^i(0)=ρ^i0,θl^˙=[AlFl(xl(tτl))]Tη,θ^l(0)=θ^0l,l=1,2,,m,ϕ^˙=[Λ(t)H(y(t))]Tη,ϕ^(0)=ϕ^0.(30)

Theorem 3.3

Using the controller (28) and the adaptive control laws (30), the state of the MFPMLCS error system (22) will reach to the sliding surface s = 0 in a finite time T2, and remain on it forever. Meanwhile, the sliding mode reaching time T2 satisfies T2[||s(0)||2+||ρ^0||2+||ρ¯||2+||ϕ^0||2+ϕ¯2+l=1m(||θ^0l||2+(θ¯l)2)]12γ,(31) in which, γ = min{μ, ς}.

Proof

Choose the following Lyapunov function candidate V2(t)=V21(t)+V22(t),(32) in which V21(t)=12||s||2,V22(t)=12(||ρ^ρ||2+12||ϕ^ϕ||2+l=1m||θ^lθl||2).(33) Taking the time derivative of V21(t), we get V˙21(t)=sTs˙=i=1nsis˙i=i=1nsi[ci0e˙i+ci1ei+ci2sgn(ei)|ei|2αi+ci3sgn(ei)|ei|αi]. Along the error system, V̇21(t) can be described as V˙21(t)=i=1nsici0ςgnci0sgn(si)|si|i=1nci0ki|si|+i=1n[sici0ρi(t)ci0|si|ρ^i+l=1mi=1nj=1nsici0aijlFjl(xl(tτl))(θlθ^l)+i=1nsici0[λi(t)Hi(y(t))(ϕϕ^)]=i=1nsici0ςgnci0sgn(si)|si|i=1nci0ki|si|+i=1n[sici0ρi(t)ci0|si|ρ^i]+l=1m(θlθ^l)T[AlFl(xl(tτl))]Tη+(ϕϕ^)T[Λ(t)H(y(t))]Tη. Using the fact sici0ρi(t)|sici0ρi(t)|=ci0|si|ρi(t)|ci0|si|ρi,i=1n(sici01nci0sgn(si)|si|)=1,μ=min{c10k1,c20k2,,cn0kn}, we can derive V˙21(t)l=1m(θlθ^l)T[AlFl(xl(tτl))]Tη+(ϕϕ^)T[Λ(t)H(y(t))]Tηςgμi=1n|si|+i=1nci0|si|(ρiρ^i).(34) The time derivative of V22(t) can be calculated as V˙22(t)=i=1n(ρ^iρi)ρi^˙+l=1m(θ^lθl)Tθ^˙+(ϕ^ϕ)ϕ^˙=i=1nci0|si|(ρ^iρi)+l=1m(θ^lθl)T[AlFl(xl(tτl))]Tη+(ϕ^ϕ)[Λ(t)H(y(t))]Tη.(35) Combining (34) with (35), we can obtain V˙2(t)=V˙21(t)+V˙22(t)μi=1n|si|ςg=μnn|si|ς[||ρ^||+||ρ¯||+||ϕ^||+ϕ¯+l=1m(||θ^l||+θ¯l)]γ[i=1n|si|+||ρ^||+||ρ¯||+||ϕ^||+ϕ¯+l=1m(||θ^l||+θ¯l)]γ(i=1n|si|+||ρ^ρ||+||ϕ^ϕ||+l=1m||θ^lθl||).(36) According to Lemma 2.6, we get V˙2(t)γ(i=1nsi2+l=1mθ^lθl2+||ρ^ρ||2+||ϕ^ϕ||)12=γ(||s||2+||ρ^ρ||2+||ϕ^ϕ||+l=1m||θ^lθl||2)12=2γ(12||s||2+12||ρ^ρ||2+12||ϕ^ϕ||+12l=1mθ^lθl2)12=2γV212(t).(37) Applying Lemma 2.4, it follows that the error trajectory e(t) converges to the sliding surface s(t) = 0 in the finite time 2 and then remains on it forever, meanwhile the following inequality holds T^2[||s(0)||2+||ρ^0ρ||2+ϕ^0ϕ2+l=1mθ^0lθl2]12γ.(38) It is clear that 2T2 in which T2 is given by (31). This completes the proof. □

Remark 3.4

The results of Theorem 3.2 and Theorem 3.3 imply that the group of the drive systems (1) and the response system (2) are MFPMLCS in the finite time T1 + T2 under the action of the adaptive control law (28)-(30).

Remark 3.5

According to the previous discussion, the convergence times T1, T2 and the controller ui(t) are depended on the control gains C, ki and ς. On the one hand, T1 is proportional to the value of C, which means a smaller ci0 results in a shorter convergence times T1, on the other hand, the sliding mode reaching time T2 is inversely proportional to γ = min{μ, ς} = min{c10k1, c20k2, · · ·, cn0kn, ς}. At the same time, the control input ui(t) is proportional to 1ci0, ki and ς. 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 ui(t) contains the factor sgn(si)|si|. In fact, during the sliding mode reaching phase, when the error trajectories ei(t) reach onto the sliding surfaces si(t) = 0, it is obvious that sgn (si) = si = 0, which means sgn(si)|si|is singular. In order to overcome this disadvantage, the control law (28) is modified as follows ui(t)=1λi(t){Ωi+1ci0(ci1ei+ci2sgn(ei)|ei|2αi+ci3sgn(ei)|ei|αi)+kisgn(si)+ρ^isgn(si)+l=1mj=1naijlFjl(xl(tτl))θ^lλi(t)Hi(y(t))ϕ^+ςgnci0Δ}i=1,2,,n,(39) with Δ=sgn(si)|si|,ifi=1n|si|δ,0,ifi=1n|si|<δ,(40) where the switching gain δ is a sufficiently small positive constant which can be chosen according to the designer requirements.

Another effective approach is using the function sgn(si)|si|+ε (ɛ is a sufficiently small positive constant) to approximate sgn(si)|si|, which is common in the sliding mode application.

4 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: (x˙11x˙21x˙31)=(0x11x314x12)f1(x1(t))+(x22x11000x11000x31)F1(x1(t))(10402.5)θ1+(0.5sint2sin(2t)2cost)w1(t). Lorenz system: (x˙12x˙22x˙32)=(0x12x32x22x12x22)f2(x2(t))+(x22x12000x12000x32)F2(x2(t))(10288/3)θ2+(cos2tsin3tcost)w2(t). Chen system (y˙1y˙2y˙3)=(0y1y3y1y2)h(y(t))+(y2y100y1y1+y2000y3)H(y(t))(35283)ϕ+(cos2tsin3tcost)d(t)+(u1(t)u2(t)u3(t))u(t). In the simulation, the drive systems are started with x1(0) = (2, 2, 2) and x2(0) = (3, 3, 3), and the response system is initialized with y(0) = (-6, -6, -6), the control gains are selected as k = (100, 80, 80), αi = 0.1, ci0 = 2, ci1 = 10, ci2 = 30, ci3 = 50 (i = 1, 2, 3) and ς = 0.1, it yields μ = 40, γ = 0.1. The bound vectors are chosen as ||ρ̄|| = 15, θ̄l = φ̄ = 55. Choosing delay times τ1 = 1, τ2 = 2 and the following scaling matrices A1=(100020001),A2=(200010001),Λ(t)=(2+sint0001+0.5cost00010.5sint) Using the modified controller (39)-(40) and the adaptive control law (30) with δ = 0.1, the MFPMLCS errors are revealed in Figure 1. It is observed that the MFPMLCS errors convergence to ei(t) = 0 within a very short time. The time responses of the adaptive parameter vectors ρ̂, θ̂l and φ̂, converge to the values ρ, θl and φ, respectively which can be shown in Figures 2-5. Meanwhile, Figure 6 shows the sliding surface can rapidly converge to zero. The simulation results illustrate the effectiveness of the proposed method.

Time response of MFPLS error e
Fig. 1

Time response of MFPLS error e

Time response of ρ̂
Fig. 2

Time response of ρ̂

Time response of θ̂1
Fig. 3

Time response of θ̂1

Time response of θ̂2
Fig. 4

Time response of θ̂2

Time response of φ̂
Fig. 5

Time response of φ̂

Time response of si(t)
Fig. 6

Time response of si(t)

5 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.

Acknowledgement

This paper is supported by the National Natural Science Foundation of China (61373174) and (11301409), thanks for all the references authors.

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About the article

Received: 2017-04-16

Accepted: 2017-06-19

Published Online: 2017-08-19


Competing interestsThe authors declare that there is no conflict of interests regarding the publication of this article.


Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 1035–1047, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0087.

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© 2017 Li and Liu. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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