Parameter identiﬁcation and the multi-switching sliding ...

Parameter identification and the multi-switching

sliding mode combination synchronization of

fractional order non-identical chaotic system under

stochastic disturbances

Weiqiu Pan, Tianzeng Li∗, Yu Wang

College of Mathematics and Statistics, Sichuan University of Science and Engineering,

Zigong 643000, China

South Sichuan Center for Applied Mathematics, Yibin 644000, China.

Abstract This paper deals with the issue of the multi-switching sliding modecombination synchronization (MSSMCS) of fractional order (FO) chaotic systems withdifferent structures and unknown parameters under double stochastic disturbances (S-D) utilizing the multi-switching synchronization method. The stochastic disturbancesare considered as nonlinear uncertainties and external disturbances. Our theoreticalpart is divided into two cases, namely, the dimension of the drive-response system aredifferent (or same). Firstly, a FO sliding surface was established in term of fraction-al calculus. Secondly, depended on the FO Lyapunov stability theory, the adaptivecontrol technology and sliding mode control technique, the multi-switching adaptivecontrollers (MSAC) and some suitable multi-switching adaptive updating laws (M-SAUL) are designed, so that the state variables of the drive systems are synchronizedwith the different state variables of the response systems. Simultaneously, the unknownparameters are assessed and the upper bound of stochastic disturbances are examined.Selecting the suitable scale matrices, the multi-switching projection synchronization,multi-switching complete synchronization, and multi-switching anti-synchronizationwill become special cases of MSSMCS. Finally, examples are displayed to certify theusefulness and validity of the demonstrated scheme via MATLAB.

Key words. Fractional-order chaotic system; Multi-switching combination synchro-nization (MSCS); Adaptive sliding mode control; Stochastic disturbance; Unknownparameters

1 Introduction

Chaos is an inherent characteristic of nonlinear dynamic systems and a commonphenomenon in real life. The chaotic phenomenon exhibited by the chaotic system is

∗Corresponding author. E-mail addresses: [email protected] (T.Li), [email protected](W.Pan), wangyu [email protected](Y.Wang)

1

uncertain, unrepeatable, and unpredictable. Therefore, experts in the fields of mathe-matics and control have carried out a series of researches on the control and synchro-nization of chaotic systems. So far, some effective synchronization control methodshave been proposed, such as drive-response synchronization [1], projection synchroniza-tion [2,3], adaptive fuzzy control [4–6], neural network synchronization [7,8], feedbacksynchronization [9], pulse synchronization [10, 11], sliding mode control [12, 13] etc.People apply chaotic synchronization to secure communication, signal processing andlife sciences, etc which has attracted great attention. Thus, chaos synchronization hasgradually become a core research topic in the field of control science. Because chaoticsystems are extremely sensitive to initial values, they are often subject to some SD.Whether it is the uncertainties within the system, such as parameter uncertainties,nonlinear uncertainties, or external disturbances, they cause an effect on the stabilityof the system. In the beginning, these synchronization methods are used by people toresearch the synchronization of single-drive system and single-response system [14–19].Gradually, some scholars have considered the influence of SD on this basis [20–27].Since Runzi et al [28] revealed the combination synchronization scheme, the synchro-nization of multi-drive and multi-response systems, multi-drive and single-responsesystems, single-drive and multi-response systems are suggested. Later, some new syn-chronization schemes appeared and have been developed by leaps and bounds, such asthe combination-combination [29–32], compound synchronization [33–35] and doublecompound synchronization [36, 37], etc. It can be said to be a major breakthrough inthe field of chaos synchronization. A major advantage of these new synchronizationschemes is that they can ensure the security of information. However, as the trans-mission of signals become more and more complex, how to strengthen the security ofinformation is a thought-provoking question.

In recent years, a new multi-switching combination synchronization (MSCS) schemewas analyzed by Vincent U et al [38] which means the state variables of the drive sys-tems are synchronized with the different state variables of the response systems, break-ing the conventional synchronization rules. Compared with the traditional synchroniza-tion or some extension of it, the MSCS scheme is very promising because it can providegreater security for the secure communication according to chaotic synchronization.The topic of dual combination-combination multi-switching synchronization in terms ofeight chaotic systems was solved in [39]. The global exponential MSCS was introducedin terms of three different chaotic systems [40]. Reference [41] solved the problem of M-SCS between three different integer-order chaotic systems. Adopting adaptive controltechnology, reference [42] investigated the multi-switching combination-combinationsynchronization of four integer-order chaotic systems which parameters are fully un-known. An further work of [42] has been developed by [43], which is indicated asinteger-order time-delay chaotic system. Shafiq M et al [44] proposed a robust adap-tive multi-switching technology to solve the issue of anti-synchronization for unknownhyper-chaotic systems under SD. The authors of references [45] and [46] considered themulti-switching synchronization of the single-drive and single-response system whichthe parameters are unknown. Their innovations lie in the order of the drive-responsesystem is different in [45] and the dimension is different in [46]. On the contrary, Chenet al [47] considered the synchronization of multiple chaotic systems with unknownparameters and disturbances. It’s a pity that the case of multi-switching is not con-sidered. Although reference [48] take multi-switching scheme into account in terms of

2

multiple chaotic systems, it does not consider the influence of unknown parametersand SD.

Hence, to address this limitation, we plan to solve the multi-switching slidingmode combination synchronization (MSSMCS) of fractional order (FO) chaotic sys-tems with different (or same) dimension under double stochastic disturbances (SD).Meanwhile, the parameters of two drive systems and one response system are fullyunknown. The double SD are considered as nonlinear uncertainties and external dis-turbance. In the light of Lyapunov stability theory, adaptive control technology andsliding mode control technique, we introduce two multi-switching adaptive controllers(MSAC) and multi-switching adaptive updating laws (MSAUL) to realise the multi-switching synchronization of D-R systems and assess the unknown parameters. Thereare two innovations points in this article. 1. Considering the combination D-R systemswith unknown parameters and double SD, the designed MSAC can make the state vari-ables of drive systems are synchronized with the different state variables of the responsesystem. Simultaneously, the designed MSAUL can identify the unknown parametersaccurately and bounded estimates are made for uncertainties and external disturbances.2. When the dimension of D-R systems are different (or same), scale matrices are choseas non-diagonal (or diagonal) matrices. If we adopt suitable scale matrices, the multi-switching projection synchronization, multi-switching complete synchronization, andmulti-switching anti-synchronization will become the special cases of MSSMCS. Fi-nally, numerical simulations via MATLAB demonstrate the multi-switching controllerswe conducted have good robustness and anti-interference performance and this methodwe investigated can make the multi-switching error systems quickly converges to theequilibrium point. The rest of the paper is described as follows. In Section 2, somedefinitions, lemmas that need to be used are introduced. In Section 3, the problemstatement are given. In Section 4, the MSAC and MSAUL are designed about D-Rsystems with the same dimension . In Section 5, the MSAC and MSAUL are designedabout D-R systems with the different dimension. In Section 6, the numerical simula-tions conducted that our method is effective and dependable. In Section 7, there is aconclusion.

2 Preliminaries

The fractional calculus is an ancient and ”fresh” concept. As early as the estab-lishment of integer calculus, some scholars began to consider its meaning. Up till thepresent moment, there are some commonly used types of fractional derivatives, suchas the Riemann-Liouville (R-L), Caputo and Grunwald-Letnikov (G-L) derivative etc.Among these definitions, Caputo’s derivative definition is the most generally utilized.

Definition 1. [49] The mathematical expression of the fractional integral of the func-tion f(t) is following

Iαt f(t) =1

Γ(α)

∫ t

a

f(υ)

(t− υ)1−αdυ, (1)

where Γ(α) indicates the Gamma function.

3

Definition 2. [49] The mathematical expression of Caputo derivative with order α isgiven as

Ca D

αt f(t) =

1

Γ(p− α)

∫ t

a

(t− υ)p−α−1f (p)(υ)dυ, (2)

where p− 1 < α < p, p ∈ Z+.

Lemma 1. [18] When x(t) ∈ Rn has continuous first derivative, then

aDαt (

1

2xT (t)Qx(t)) ≤ xTQaD

αt x(t), (3)

where α ∈ (0, 1) and Q ∈ Rn ×Rn indicates a positive definite matrix.

3 Problem description

The FO D-R systems with SD are demonstrated as

0Dα

t x(t) = F (x(t))θ + f (x(t)) + ∆f (x(t)) + d(x, t), (4)

0Dα

t y(t) = G(y(t))β + g(y(t)) + ∆g(y(t)) +D(y, t), (5)

where the systems (4), (5) are regarded as drive systems. x(t) = (x1, x2, · · · , xm)T

and y(t) = (y1, y2, · · · , ym)T are the state variables. θ = (θ1, θ2, · · · , θm)

T and β =(β1, β2, · · · , βm)

T are the parameter vectors; Fk(x(t)), (k = 1, 2, · · · , n) is the k throw of n × m matrix F (x(t)) and Gj(y(t)), (j = 1, 2, · · · , n) is the j th row ofn ×m matrix G(y(t)) whose elements are continuous nonlinear functions; f (x(t)) =(f1, f2, · · · , fm)

T , g(y(t)) = (g1, g2, · · · , gm)T are the nonlinear continuous functions;

d(x(t)) = (d1, d2, · · · , dm)T , D(y(t)) = (D1, D2, · · · , Dm)

T are the external distur-bances; ∆f (x(t)) = (∆f1,∆f2, · · · ,∆fm)

T , ∆g(y(t)) = (∆g1,∆g2, · · · ,∆gm)T are the

nonlinear uncertainties; αi ∈ (0, 1) represents the fractional order.

0Dα

t z(t) = H(z(t))ϑ + h(z(t)) + ∆h(z(t)) + µ(z, t) + u(t), (6)

where the systems (6) is regarded as response system. z(t) = (z1, z2, · · · , zn)T and

ϑ = (ϑ1, ϑ2, · · · , ϑn)T are the state variables and parameter vectors; Hi(z(t)), (i =

1, 2, · · · , n) is the i th row of n × n matrix H(z(t)) whose elements are contin-uous nonlinear functions; h(z(t)) = (h1, h2, · · · , hn)

T are the nonlinear continuousfunctions; µ(z(t)) = (µ1, µ2, · · · , µn)

T are the external disturbances; ∆h(z(t)) =(∆h1,∆h2, · · · ,∆hn)

T are the nonlinear uncertainties. u(t) = (u1, u2, · · · , un)T repre-

sent the controller.

Remark 1. It is worth noting that x(t),y(t) ∈ Rm, z(t) ∈ R

n, m = n or m 6= n;And the parameters of the D-R systems are unknown. We use θ, β, ϑ to represent theestimation of parameter θ, β, ϑ.

4

Definition 3. If there exist three constant matrices A ∈ Rn×m, B ∈ R

n×m, C ∈R

n×n, C 6= 0 such that

limt→∞

‖e‖ = limt→∞

‖Cz − (By + Ax)‖ = 0, (7)

then the drive systems (4), (5) and response system (6) can be reach combinationsynchronization.

Remark 2. We can redefined the error system (7) as

limt→∞

(lwv)ep = limt→∞

[

n∑

i=1

(clizi) − {

m∑

j=1

(bwjyj) +

m∑

k=1

(xvkxk)}], (8)

where i, j, k, p ∈ (1, 2, · · · , n) represent p th error component of e, i th components ofz, j th components of y, and k th components of x respectively, l, w, v represent theswitching mode. Suppose that l = w = v, then the error variables are expressed in theform of definition 3; if l 6= w 6= v, definition 3 will no longer apply.

Definition 4. We redefine the error state in definition 3 as

limt→∞


[n

∑

i=1

(clizi) − {m∑

j=1

(bwjyj) +m∑

k=1

(xvkxk)}] = 0, (9)

then the drive systems (4), (5) and the response systems (6) realize the MSCS, wherel = w 6= v or l 6= w = v or l = w 6= v or l 6= w 6= v.

Remark 3. The matrices A ∈ Rn×m, B ∈ R

n×m, C ∈ Rn×n, C 6= 0 are hailed as the

scaling matrices. Furthermore, they can be either constant matrices or the functionswith regard to state variables x, y, z.

Remark 4. If A = B = C = I, the MSCS becomes the multi-switching combinationcomplete synchronization; If A = 0, B = C = I or B = 0, A = C = I, the MSCSbecomes the multi-switching complete synchronization.

Remark 5. If A = B = −I, C = I, the MSCS becomes the multi-switching combi-nation anti-synchronization; If A = 0, B = −I, C = I or B = 0, A = −I, C = I, theMSCS becomes the multi-switching anti-synchronization.

Remark 6. If A 6= I, B 6= I, C = I, the MSCS becomes the multi-switching combina-tion projective synchronization; If A = I, B 6= I, C = I or B = I, A 6= I, C = I, theMSCS becomes the multi-switching projective synchronization.

4 The synchronization of multi-switching FO chaot-

ic system with same dimensions

In Section 4, the MSSMCS of FO chaotic systems with same dimension is for-mulated which means the dimension of D-R systems (4), (5) and (6) satisfies m = n.Thus, the scaling matrices A,B,C are given as diagonal matrices. Firstly, we know

5

that even if a chaotic system is slightly disturbed, its state orbit will change drasticallyover time. Therefore, it is crucial to suppose them as bounded. Then, we designedappropriate multi-switching adaptive controllers (MSAC) and some multi-switchingadaptive updating laws (MSAUL) to realize the synchronization of the D-R systemwhich are proved in Theorem 1.

When we choose m = n and the diagonal matrices A = diag(a11, a22, · · · , ann),B = diag(b11, b22, · · · , bnn), C = diag(c11, c22, · · · , cnn), the error system (9) can bedescribed as

limt→∞

(ijk)ep = limt→∞

[ciizi − (bjjyj + akkxk)] = 0, (10)

where i, j, k, p ∈ (1, 2, · · · , n) represent p th error component of e, i th components ofz, j th components of y, and k th components of x.

Assumption 1. Assumed that the external disturbances dk, Dj, µi, uncertain non-linear vectors ∆fk, ∆gj , ∆hi all have a bounded norm. Namely, there are suitablepositive constants (ijk)rp,

(ijk)qp that satisfy

|ciiµi − (bjjDj + akkdk)| ≤(ijk)rp,

|cii∆fk − (bjj∆gj + akk∆hi)| ≤(ijk)qp.

(11)

where p = 1, 2, · · · , n.

Remark 7. The positive constants (ijk)rp,(ijk)qp are unknown. (ijk)rp,

(ijk)qp are usedto represent the estimation of parameter (ijk)rp,

(ijk)qp.

According to the definition of the error vector (10), we get the FO error system as

0Dαt {

(ijk)ep} = cii{0Dαt zi} − bjj{0D

αt yj} − akk{0D

αt xk}

= ciiHi(z(t))ϑ + ciihi(z(t)) + cii∆hi(z(t)) + ciiµi(z, t) + cii{(ijk)up(t)}

− bjjGj(y(t))β − bjjgj(y(t))− bjj∆gj(y(t))− bjjDj(y, t)

− akkFk(x(t))θ − akkfk(x(t))− akk∆fk(x(t))− akkdk(x, t).

(12)

For convenience, we define the errors of unknown parameters θ,β,ϑ, (ijk)rp,(ijk)qp as

θ = θ − θ, β = β − β, ϑ = ϑ − ϑ, (ijk)rp = (ijk)rp −(ijk)rp,

(ijk)qp = (ijk)qp −(ijk)qp.

When the sliding mode surface is designed as (ijk)sp = (ijk)λp((ijk)ep) which can be

rewritten as s = ((ijk)s1,(ijk)s2, · · · ,

(ijk)sn), we can get the following MSAC (13) andMSAUL (16) are designed:

(ijk)up =1

cii{−ciihi + bjjgj + akkfk + akkFk(x(t))θ + bjjGj(y(t))β

− ciiHi(z(t))ϑ− ((ijk)rp +(ijk)qp)sign(

(ijk)sp)− k1tanh((ijk)sp)},

(13)

where (ijk)λp is a constant.Substituting (13) into Eq. (12), we obtain

0Dαt {

(ijk)ep} = −ciiHi(z(t))ϑ+ bjjGj(y(t))β + akkFk(x(t))θ

+ {cii∆hi(z(t))− bjj∆gj(y(t))− xkk∆fk(x(t))} + {ciiµi(z, t)

− bjjDj(y, t)− akkdk(x, t)} − ((ijk)rp +(ijk)qp)sign(

(ijk)sp(t))

− k1tanh((ijk)sp).

(14)

6

Thus, it follows from (ijk)sp =(ijk)λp(

(ijk)ep) and Eq. (14) that we obtain the followingsummation result.

n∑

p=1

{(ijk)sp}0Dαt {

(ijk)sp}

=

n∑

i=1

(ijk)si((ijk)λi){−ciiHi(z(t))ϑ+ cii∆hi(z(t)) + ciiµi(z, t)}

+n

∑

j=1

(ijk)sj((ijk)λj){bjjGj(y(t))β − bjj∆gj(y(t))− bjjDj(y, t)} (15)

+

n∑

k=1

(ijk)sk((ijk)λp){akkFk(x(t))θ − akk∆fk(x(t))− akkdk(x, t)}

−n

∑

p=1

((ijk)λp)((ijk)rp +

(ijk)qp)sign((ijk)sp(t))−

n∑

p=1

((ijk)λp)k1tanh((ijk)sp).

The MSAUL with regard to unknown parameters θ,β,ϑ, (ijk)rp,(ijk)qp are selected

to

0Dαt θ = −F T (x(t))γ − ϕ1sign(θ)|(θ)|

w,

0Dαt β = −GT (y(t))ξ − ϕ2sign(β)|(β)|

w,

0Dαt ϑ = HT (z(t)) − ϕ3sign(ϑ)|(ϑ)|

w,

0Dαt {

(ijk)rp} = m1((ijk)λp)|

(ijk)sp(t)|,

0Dαt {

(ijk)qp} = m2((ijk)λp)|

(ijk)sp(t)|,

(16)

where

γ = {((ijk)λ1)a∗

11((ijk)s1), (

(ijk)λ2)a∗

22((ijk)s2), · · · , (

(ijk)λn)a∗

nn((ijk)sn)

T},

ξ = {((ijk)λ1)b∗

11((ijk)s1), (

(ijk)λ2)b∗

22((ijk)s2), · · · , (

(ijk)λn)b∗

nn((ijk)sn)

T},

= {((ijk)λ1)c∗

11((ijk)s1), (

(ijk)λ2)c∗

22((ijk)s2), · · · , (

(ijk)λn)c∗

nn((ijk)sn)

T},

m1, m2 are positive constants and the meaning of matrices A∗ = diag(a∗11, a∗

22, · · · , a∗

nn),B∗ = diag(b∗11, b

∗

22, · · · , b∗

nn), C∗ = diag(c∗11, c

∗

22, · · · , c∗

nn) is the rearrangement of theoriginal matrices A,B,C.

Theorem 1. For any given initial conditions x(0),y(0), z(0), if the Assumption 1is hold, the synchronization error system (10) will achieve the multi-switching slid-ing mode combination synchronization (MSSMCS) via the multi-switching adaptivecontroller (MSAC) (13) and multi-switching adaptive updating laws (MSAUL) (16).

Proof. Adopting the Lyapunov function as:

V =1

2

n∑

p=1

(ijk)s2p +1

2m1

n∑

p=1

{(ijk)rTp }{(ijk)rp}+

1

2m2

n∑

p=1

{(ijk)qTp }{(ijk)qp}

+1

2θTθ +

1

2βTβ +

1

2ϑTϑ.

(17)

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Taking the α derivative

0Dαt V (t,x(t)) ≤

n∑

p=1

{(ijk)sp}0Dαt {

(ijk)sp}+1

m1

n∑

p=1

{(ijk)rTp }0Dαt {

(ijk)rp}

+1

m2

n∑

p=1

{(ijk)qTp }0Dαt {

(ijk)qp}+ θT0D

αt θ + βT

0Dαt β + ϑT

0Dαt ϑ.

(18)

Then substituting the Eq. (15) and the MSAUL (16) into Eq. (18), we obtain


n∑

i=1

(ijk)si((ijk)λi)(−ciiHi(z(t))ϑ + cii∆hi(z(t)) + ciiµi(z, t))

+

n∑

j=1

(ijk)sj((ijk)λj)(bjjGj(y(t))β − bjj∆gj(y(t))− bjjDj(y, t))

+n

∑

k=1

(ijk)sk((ijk)λk)(akkFk(x(t))θ − akk∆fk(x(t))− akkdk(x, t))

−n

∑

p=1



n∑

p=1

((ijk)λp)k1tanh((ijk)sp)

+ θT (−F T (x(t))γ − ϕ1sign(θ)|(θ)|w)

+ βT (−GT (y(t))ξ − ϕ2sign(β)|(β)|w)

+ ϑT (HT (z(t)) − ϕ3sign(ϑ)|(ϑ)|w)

+

n∑

p=1

(ijk)rTp ((ijk)λp)(|

(ijk)sp|) +

n∑

p=1

(ijk)qTp ((ijk)λp)(|

(ijk)sp|). (19)

Since

n∑

j=1

(ijk)sj((ijk)λj)bjjGj(y(t))β = βTGT (y(t))ξ,

n∑

k=1

(ijk)sk((ijk)λk)akkFk(x(t))θ = θTF T (x(t))γ,

n∑

i=1

(ijk)si((ijk)λi)ciiHi(z(t))ϑ = ϑTHT (z(t)).

(20)

Thus


n∑

i=1

(ijk)si((ijk)λi)(cii∆hi(z(t)) + ciiµi(z, t))

+

n∑

j=1

(ijk)sj((ijk)λj)(−bjj∆gj(y(t))− bjjDj(y, t))

+n

∑

k=1

(ijk)sk((ijk)λk)(−akk∆fk(x(t))− akkdk(x, t))

8

− θT (ϕ1sign(θ)|(θ)|w)− βT (ϕ2sign(β)|(β)|

w)− ϑT (ϕ3sign(ϑ)|(ϑ)|w)

−

n∑

p=1



n∑

p=1

((ijk)λp)k1tanh((ijk)sp)

+n

∑

p=1

((ijk)λp)((ijk)rTp )(|

(ijk)sp|) +n

∑

p=1

((ijk)λp)((ijk)qTp )(|

(ijk)sp|)

≤n

∑

p=1

((ijk)λp)((ijk)rp)|

(ijk)sp|+n

∑

p=1

((ijk)λp)((ijk)qp)|

(ijk)sp| (21)

−

n∑

p=1


(ijk)qp)|(ijk)sp|+

n∑

p=1

((ijk)λp)((ijk)rTp )(|

(ijk)sp|)

+

n∑

p=1

((ijk)λp)((ijk)qTp )(|

(ijk)sp|)− θT (ϕ1sign(θ)|(θ)|w)

− βT (ϕ2sign(β)|(β)|w)− ϑT (ϕ3sign(ϑ)|(ϑ)|

w)

−n

∑

i=1

(ijk)sp((ijk)λp)k1tanh(

(ijk)sp)

= −θT (ϕ1sign(θ)|(θ)|w)− βT (ϕ2sign(β)|(β)|

w)

− ϑT (ϕ3sign(ϑ)|(ϑ)|w)−

p∑

i=1

(ijk)spk1tanh(λ(ijk)sp).

Which establishes that the 0Dαt V (t,x(t)) is negative definite. Thus, it follows from the

FO Lyapunov stability theory that limt→∞

(ijk)ep = 0 is achieved, i.e. we can say thatthe MSSMCS of the drive systems (4), (5) and response system (6)is accomplished interms of m = n.

5 The synchronization of multi-switching FO sys-

tem with different dimensions

In Section 5, the MSSMCS of FO chaotic systems with different dimension isformulated which means the dimension of D-R systems (4), (5) and (6) satisfies m 6= n.Thus, the scaling matrices A,B,C are given as non-diagonal matrices. We designedappropriate multi-switching adaptive controllers (MSAC) and some multi-switchingadaptive updating laws (MSAUL) to realize the synchronization of the D-R systemwhich are proved in Theorem 2.

When we choose m 6= n and the non-diagonal matrices A ∈ Rn×m, B ∈ R

n×m, C ∈R

n×n, C 6= 0, the form of the error system (9) can be explained as:

limt→∞


[

n∑

i=1

(clizi) − {

m∑

j=1

(bwjyj) +

m∑

k=1

(xvkxk)}] = 0. (22)

Assumption 2. Assumed that the external disturbances dk, Dj, µi, uncertain non-linear vectors ∆fk, ∆gj , ∆hi all have a bounded norm. Namely, there are suitable

9

positive constants (lwv)ρp,(lwv)p that satisfy

∣

∣

∣

∣

n∑

i=1

cliµi − (

m∑

j=1

bwjDj +

m∑

k=1

xvkdk)

∣

∣

∣

∣

≤ (lwv)ρp,

∣

∣

∣

∣

n∑

i=1

cli∆hi − (m∑

j=1

bwj∆gj +m∑

k=1

xvk∆fk)

∣

∣

∣

∣

≤ (lwv)p,

(23)

where p = 1, 2, · · · , n.

Remark 8. The positive constants (lwv)ρp,(lwv)p are unknown,

(lwv)ρp,(lwv) ˆp are used

to represent the estimation of parameter (lwv)ρp,(lwv)p.

According to the definition of the error vector (22), we get the FO error system as

0Dαt {

(lwv)ep} =n

∑

i=1

cli{0Dαt zi} −

m∑

j=1

bwj{0Dαt yj} −

m∑

k=1

xvk{0Dαt xk}

=

n∑

i=1

cli{Hi(z(t))ϑ + hi(z(t)) + ∆hi(z(t)) + µi(z, t) +(lwv)up(t)}

−m∑

j=1

bwj{Gj(y(t))β + gj(y(t)) + ∆gj(y(t)) +Dj(y, t)}

−

m∑

k=1

xvk{Fk(x(t))θ + fk(x(t)) + ∆fk(x(t)) + dk(x, t)}.

(24)

The errors of unknown parameters θ,β,ϑ have been defined in (12). For conve-nience, we define error of unknown constants (lwv)ρp,

(lwv)p as(lwv)ρp =

(lwv)ρp−(lwv)ρp,

(lwv) ˜p = (lwv) ˆp −(lwv)p. Thus, the sliding mode surface is designed as (lwv)sp =(lwv)

λp((lwv)ep). We can get the following multi-switching adaptive controller (MSAC) (25)

multi-switching adaptive updating laws (MSAUL) (28):

n∑

i=1

cli{(lwv)up} = −

n∑

i=1

clihi +

m∑

j=1

bwjgj +

m∑

k=1

xvkfk +

m∑

k=1

xvkFk(x(t))θ

+m∑

j=1

bwjGj(y(t))β −n

∑

i=1

cliHi(z(t))ϑ− ((lwv)ρp +(lwv) ˆp)sign(

(lwv)sp)

− k1tanh((lwv)sp),

(25)

where (lwv)λp is a constant. Substituting (25) into Eq. (24), we obtain

0Dαt {

(lwv)ep} =n

∑

i=1

cli{−Hi(z(t))ϑ +∆hi(z(t)) + µi(z, t)}

+

m∑

j=1

bwj{Gj(y(t))β +∆gj(y(t)) +Dj(y, t)} (26)

+m∑

k=1

xvk{Fk(x(t))θ +∆fk(x(t)) + dk(x, t)}

10

− ((lwv)ρp +(lwv) ˆp)sign(

(lwv)sp)− k1tanh((lwv)sp).

Thus, it follows from (lwv)sp =(lwv)λp(

(lwv)ep) and Eq. (26) that we obtain the followingsummation result.

n∑

p=1

{(lwv)sp}0Dαt {

(lwv)sp}

=n

∑

l=1

n∑

i=1

(lwv)si((lwv)λi)cli{−Hi(z(t))ϑ +∆hi(z(t)) + µi(z, t)}

+n

∑

w=1

m∑

j=1

(lwv)sj((lwv)λj)bwj{Gj(y(t))β +∆gj(y(t)) +Dj(y, t)} (27)

+

n∑

v=1

m∑

k=1

(lwv)sk((lwv)λk)xvk{Fk(x(t))θ +∆fk(x(t)) + dk(x, t)}

−n

∑

p=1

((lwv)λp)((lwv)ρp +

(lwv) ˆp)sign((lwv)sp(t))−

n∑

p=1

((lwv)λp)k1tanh((lwv)sp).

The MSAUL with regard to unknown parameters θ,β,ϑ, (lwv)ρp,(lwv)p are select-

ed as


w,


w,


w,

0Dαt {

(lwv)ρp} = m1((lwv)λp)|

(lwv)sp(t)|,

0Dαt {

(lwv) ˜p} = m2((lwv)λp)|

(lwv)sp(t)|,

(28)

where

γ = {

m∑

k=1

((lwv)λ1)xvk((lwv)s1),

m∑

k=1

((lwv)λ2)xvk((lwv)s2), · · · ,

m∑

k=1

((lwv)λn)xvk((lwv)sn)}

T ,

ξ = (m∑

j=1

((lwv)λ1)bwj((lwv)s1),

m∑

j=1

((lwv)λ2)bwj((lwv)s2), · · · ,

m∑

j=1

((lwv)λn)bwj((lwv)sn))

T ,

= (

n∑

i=1

((lwv)λ1)cli(lwv)s1,

n∑

i=1

((lwv)λ2)cli((lwv)s2), · · · ,

n∑

i=1

((lwv)λn)cli((lwv)sn))

T ,

m1, m2 are positive constants.

Theorem 2. For any given initial conditions x(0),y(0), z(0), if the Assumption 2is hold, the synchronization error system (22) will achieve the multi-switching slid-ing mode combination synchronization (MSSMCS) via the multi-switching adaptivecontroller (MSAC) (25) and multi-switching adaptive updating laws (MSAUL) (28).

11

Proof. Adopting the Lyapunov function as:

V =1

2

n∑

p=1

(lwv)s2p +1

2m1

n∑

p=1

{(lwv)ρTp }{(lwv)ρp}+

1

2m2

n∑

p=1

{(lwv) ˜Tp }{(lwv) ˜p}

+1

2θTθ +

1

2βTβ +

1

2ϑTϑ.

(29)

Taking the α derivative


n∑

p=1

{(lwv)sp}0Dαt {

(lwv)sp}+1

m1{

n∑

p=1

(lwv)ρTp }0Dαt {

(lwv)ρp}

+1

m2

n∑

p=1

{(lwv) ˜Tp }0Dαt {

(lwv) ˜p}+ θT0D

αt θ + βT

0Dαt β + ϑT

0Dαt ϑ.

(30)

Then substituting the (27) and MSAUL (28) into Eq. (30), we obtain


n∑

l=1

n∑

i=1

(lwv)si((lwv)λi)cli{−Hi(z(t))ϑ +∆hi(z(t)) + µi(z, t)}

+n

∑

w=1

m∑

j=1

(lwv)sj((lwv)λj)bwj{Gj(y(t))β +∆gj(y(t)) +Dj(y, t)}

+

n∑

v=1

m∑

k=1

(lwv)sk((lwv)λk)xvk{Fk(x(t))θ +∆fk(x(t)) + dk(x, t)}

−

n∑

p=1


(lwv) ˆp)sign((lwv)sp(t))

−

n∑

p=1

((lwv)λp)k1tanh((lwv)sp) + θT{−F T (x(t))γ − ϕ1sign(θ)|(θ)|

ω}

+ βT{−GT (y(t))ξ − ϕ2sign(β)|(β)|ω}

+ ϑT{HT (z(t)) − ϕ3sign(ϑ)|(ϑ)|ω}

+

n∑

p=1

(lwv)ρTp ((lwv)λp)(|

(lwv)sp|) +

n∑

p=1

(lwv) ˜Tp ((lwv)λp)(|

(lwv)sp|).

(31)

Sincen

∑

v=1

m∑

k=1

(lwv)sk((lwv)λk)xvk{Fk(x(t))θ} = θTF T (x(t))γ,

n∑

w=1

m∑

j=1

(lwv)sj((lwv)λj)bwj{Gj(y(t))β} = βTGT (y(t))ξ,

n∑

l=1

n∑

i=1

(lwv)si((lwv)λi)cli{Hi(z(t))ϑ} = ϑTHT (z(t)).

(32)

Thus


n∑

l=1

n∑

i=1

(lwv)si((lwv)λi)cli{∆hi(z(t)) + µi(z, t)}

12

+

n∑

w=1

m∑

j=1

(lwv)sj((lwv)λj)bwj{∆gj(y(t)) +Dj(y, t)}

+n

∑

v=1

m∑

k=1

(lwv)sk((lwv)λk)xvk{∆fk(x(t)) + dk(x, t)}

+ θT {−F T (x(t))γ − ϕ1sign(θ)|(θ)|ω}

+ βT {−GT (y(t))ξ − ϕ2sign(β)|(β)|ω}

+ ϑT{HT (z(t)) − ϕ3sign(ϑ)|(ϑ)|ω} −

n∑

p=1

((lwv)λp)((lwv)ρp

+ (lwv) ˆp)sign((lwv)sp(t)) +

n∑

p=1


(lwv)sp|)

+

n∑

p=1


(lwv)sp|)−

n∑

p=1

((lwv)λp)k1tanh((lwv)sp) (33)

≤

n∑

p=1

(lwv)ρp((lwv)λp)(|

(lwv)sp|) +

n∑

p=1

(lwv)p((lwv)λp)(|

(lwv)sp|)

−n

∑

p=1


(lwv) ˆp)sign((lwv)sp(t))

+n

∑

p=1


(lwv)sp|) +n

∑

p=1


(lwv)sp|)

− θT (ϕ1sign(θ)|(θ)|ω)− βT (ϕ2sign(β)|(β)|

ω)

− ϑT (ϕ3sign(ϑ)|(ϑ)|ω)−

n∑

p=1

((lwv)λp)k1tanh((lwv)sp)

= −θT (ϕ1sign(θ)|(θ)|ω)− βT (ϕ2sign(β)|(β)|

ω)

− ϑT (ϕ3sign(ϑ)|(ϑ)|ω)−

n∑

p=1

((lwv)λp)k1tanh((lwv)sp).

Which establishes that the 0Dαt V (t,x(t)) is negative definite. Thus, it follows from the

FO Lyapunov stability theory that limt→∞

(lwv)ep = 0 is achieved. On the other hand,we can say that the MSSMCS of the drive systems (4), (5) and response system (6) isaccomplished in terms of m 6= n.

The following corollaries are successfully analyzed from Theorem 2 and their proofsare omitted here. By the way, the Theorem 1 has the same theory, we are not goingto describe.

Corollary 1. If the matrices A 6= 0, B = 0, C 6= 0, then the drive systems (4) achievethe MSSMCS with the response systems (6) provided the following controller,

n∑

i=1

cli{(lwv)up} = −

n∑

i=1

clihi +m∑

k=1

xvkfk +m∑

k=1

xvkFk(x(t))θ −n

∑

i=1

cliHi(z(t))ϑ



13

And the adaptive updating laws,


w,


w,

0Dαt {


(lwv)sp(t)|,

0Dαt {


(lwv)sp(t)|,

where the explanation of γ, can be seen (28).

Corollary 2. If the matrices A = 0, B 6= 0, C 6= 0, then the drive systems (5) achievethe MSSMCS with the response systems (6) provided the following controller,

n∑

i=1

cli{(lwv)up} = −

n∑

i=1

clihi +m∑

j=1

bwjgj +m∑

j=1

bwjGj(y(t))β −n

∑

i=1

cliHi(z(t))ϑ





w,


w,

0Dαt {


(lwv)sp(t)|,

0Dαt {


(lwv)sp(t)|,

where the explanation of ξ, can be seen (28).

Corollary 3. If the matrices A = 0, B = 0, C 6= 0, then the equilibrium point (0, 0, 0, 0)of response systems (6) is asymptotically stable provided the following controller,

n∑

i=1

cli{(lwv)up} = −

n∑

i=1

clihi −n

∑

i=1

cliHi(z(t))ϑ


(lwv)sp)− k1tanh((lwv)sp),



w,

0Dαt {


(lwv)sp(t)|,

0Dαt {


(lwv)sp(t)|,

where the explanation of can be seen (28).

6 Numerical simulation

This section mainly demonstrates the reliability and validity of the suggestedmulti-switching sliding mode combination synchronization scheme. For the D-R sys-tems (4), (5) and (6) with same dimension, we selected two error states to elaboratethe method, namely, i 6= j 6= k and i 6= j = k. For the D-R systems (4), (5) and (6)with different dimensions, we selected l 6= w 6= v and l 6= w = v. In each case, we havegiven the specific form of controller and parameter adapting rate via the specific FOhyper-chaotic or chaotic systems. .

14

6.1 Numerical simulations for FO chaotic system with same

dimension

As an example we choose FO hyper-chaotic Lorenz, Chen systems as the drivesystems, and the FO hyper-chaotic Lu system as the response system. Adding SD tothe D-R systems, we obtain

0Dαt x1

0Dαt x2

0Dαt x3

0Dαt x4

=

x2 − x1 0 0 00 x1 0 00 0 −x3 00 0 0 x4

a1b1c1d1

+

x4 +∆f1−x1x3 − x2 +∆f2

x1x2 +∆f3−x2x3 +∆f4

+

d1d2d3d4

, (34)

0Dαt y1

0Dαt y2

0Dαt y3

0Dαt y4

=

y2 − y1 0 0 0 00 0 y2 y1 00 −y3 0 0 00 0 0 0 y4

a2b2c2d2r

+

y4 +∆g1−y1y3 +∆g2y1y2 +∆g3y2y3 +∆g4

+

D1

D2

D3

D4

, (35)

0Dαt z1

0Dαt z2

0Dαt z3

0Dαt z4

=

z2 − z1 0 0 00 z2 0 00 0 −z3 00 0 0 z4

a3b3c3d3

+

z4 +∆h1

−z1z3 +∆h2

z1z2 +∆h3

z1z3 +∆h4

+

µ1 + u1

µ2 + u2

µ3 + u3

µ4 + u4

. (36)

Choosing the parameters are a1 = 10, b1 = 28, c1 = 8/3, d1 = −1, a2 = 35, b2 = 3, c2 =12, d2 = 7, r = 0.5, a3 = 36, b3 = 20, c3 = 3, d3 = 0.5. The initial conditions take asx(0) = (2,−2, 1, 1), y(0) = (1, 1, 2, 2), z(0) = (−20, 3, 1, 3, ). When ∆g = 0, ∆f = 0,∆h = 0, d(t) = 0, D(t) = 0, µ(t) = 0 and α = 0.97, we can obtain the attractorgraphs of the FO hyper-chaotic Lorenz, Chen, and Lu system using the FO prediction-correction method which are presented in Fig.1 The multi-switching error state modesbetween the drive systems (34), (35) and the response system (36) are listed.

i 6= j 6= k

(123)e1,(132)e2,

(124)e3,(142)e4,

(134)e5,(143)e6,

(213)e7,(231)e8,

(214)e9,(241)e10,

(234)e11,(243)e12,

(321)e13,(312)e14,

(324)e15,(314)e16,

(341)e17,(342)e18,

(412)e19,(421)e20,

(413)e21,(423)e22,

(432)e23,(431)e24.

i 6= j = k

{

(122)e25,(133)e26,

(144)e27,(211)e28,

(233)e29,(244)e30,

(311)e31,(322)e32,

(344)e33,(411)e34,

(422)e35,(433)e36.

i = j 6= k

{

(112)e37,(113)e38,

(114)e39,(221)e40,

(223)e41,(224)e42,

(331)e43,(332)e44,

(334)e45,(441)e46,

(442)e47,(443)e48.

i = k 6= j

{

(121)e49,(131)e50,

(141)e51,(212)e52,

(232)e53,(242)e54,

(313)e55,(323)e56,

(343)e57,(414)e58,

(424)e59,(434)e60.

15

040

10

20

20

x 3

30

x2

0

40

3020

50

10-20

x1

0-10

-20-40 -30

(a)

040

10

20

20

30

x 3

40

x2

030

50

2010

x1

-20 0-10

-20-40 -30

(b)

040

10

20

20

x 3

30

x2

0

40

30-20 20

x1

100

-10-40 -20

(c)

Fig.1. The 3D phase plot of FO hyper-chaotic Lorenz, Chen, Lu systems indicating in

sub-pictures (a)-(c) respectively.

i = k = j

{

(111)e61,(222)e62,

(333)e63,(444)e64.

In this chapter, we will select the appropriate switching error variable for the situationof i 6= j 6= k, namely, e = ((124)e3,

(243) e12,(312) e14,

(431) e24)T , i 6= j = k namely, e =

((122)e25,(244) e30,

(311) e31,(433) e36)

T and obtain the following two switching modes fornumerical simulation:

Switch−1 :

(124)e3 = c11z1 − (b22y2 + x44x4)(243)e12 = c22z2 − (b44y4 + x33x3)(312)e14 = c33z3 − (b11y1 + x22x2)(431)e24 = c44z4 − (b33y3 + x11x1)

, (37)

16

Switch−2 :

(122)e25 = c11z1 − (b22y2 + x22x2)(244)e30 = c22z2 − (b44y4 + x44x4)(311)e31 = c33z3 − (b11y1 + x11x1)(433)e36 = c44z4 − (b33y3 + x33x3)

. (38)

In order to confirm the robustness and reliability of the investigated MSAC (13)and MSAUL (16), the nonlinear uncertainty and external disturbance are selected asfollows:

d(x(t)) = (−0.1cos(t),−0.2cos(2t), 0.3sin(3t), 0.4sin(4t))T ,

D(y(t)) = (−0.1sin(t),−0.2sin(2t), 0.3cos(3t), 0.4cos(4t))T ,

µ(z(t)) = (0.1cos(5t), 0.2cos(6t), 0.3sin(7t), 0.4sin(8t))T ,

∆f (x(t)) = (0.1cos(2tx1), 0.2cos(2tx2), 0.3sin(2tx3), 0.4sin(2tx4))T , (39)

∆g(y(t)) = (0.1sin(2πt), 0.2sin(sign(y2)), 0.3cos(2πy3), 0.4cos(4tx8))T ,

∆h(z(t)) = (0.1sin(2tsign(z1)), 0.2sin(2tsign(z2)), 0.3sin(3t), 0.4sin(4t)T .

6.1.1 Switch-1

It follows from switch-1 (37) that the FO error dynamic system is expressed as:

0Dαt {

(124)e3} =c11{0Dαt z1} − b22{0D

αt y2} − a44{0D

αt x4}

=c11{(z2 − z1)a3 + z4 +∆h1 + µ1 +(124) u3}

−b22{y2c2 + d2y1 − y1y3 +∆g2 +D2}

−a44{x4d1 − x2x3 +∆f4 + d4},

0Dαt {

(243)e12} =c22{0Dαt z2} − b44{0D

αt y4} − a33{0D

αt x3}

=c22{z2b3 − z1z3 +∆h2 + µ2 +(243) u12}

−b44{y4r + y2y3 +∆g4 +D4}

−a33{−x3c1 + x1x2 +∆f3 + d3},

0Dαt {

(312)e14} =c33{0Dαt z3} − b11{0D

αt y1} − a22{0D

αt x2}

=c33{−z3c3 + z1z2 +∆h3 + µ3 +(312) u14}

−b11{(y2 − y1)a2 + y4 +∆g1 +D1}

−a22{x1b1 − x1x3 − x2 +∆f2 + d2},

0Dαt {

(431)e24} =c44{0Dαt z4} − b33{0D

αt y3} − a11{0D

αt x1}

=c44{z4d3 + z1z3 +∆h4 + µ4 +(431) u24}

−b33{(−y3b2 + y1y2 +∆g3 +D3}

−a11{(x2 − x1)a1 + x4 +∆f1 + d1}.

It follows from the form of MSAC (13) and MSAUL (16) that the controller is designedas follows:

(124)u3 =− z4 − (z2 − z1)a3 +b22c11

(−y1y3 + y2c2 + d2y1) +a44c11

(−x2x3 + x4d1)

−((124)r3 +(124) q3)sign(

(124)s3)− k1tanh((124)s3),

17

(243)u12 =z1z3 − z2b3 +b44c22

(y2y3 + y4r) +a33c22

(x2x1 − x3c1)

−((243)r12 +(243) q12)sign(

(243)s12)− k1tanh((243)s12),

(312)u14 =− z1z2 + z3c3 +b11c33

(y4 + (y2 − y1)a2) +a22c33

(x1x3 + x2 + x1b1)

−((312)r14 +(312) q14)sign(

(312)s14)− k1tanh((312)s14),

(431)u24 =− z1z3 − z4d3 +b33c44

(y1y2 − y3b2) +a11c44

(x4 + (x2 − x1)a1)

−((431)r24 +(431) q24)sign(

(431)s24)− k1tanh((431)s24).

and the parameters updating laws are designed as follows:

a1 = −(x2 − x1){(431)s24} − k2sign(a1)|a1|

ω, b1 = −x1{(312)s14} − k2sign(b1)|b1|

ω,

c1 = x3{(243)s12} − k2sign(c1)|c1|

ω, d1 = −x4{(124)s3} − k2sign(d1)|d1|

ω,

a2 = −(y2 − y1){(312)s14} − k3sign(a2)|a2|

ω, b2 = y3{(431)s24} − k3sign(b2)|b2|

ω,

c2 = −y2{(124)s3} − k3sign(c2)|c2|

ω, d2 = −y1{(124)s3} − k3sign(d2)|d2|

ω,r = −y4{

(243)s12} − k3sign(r)|r|ω, a3 = (z2 − z1){

(124)s3} − k4sign(a3)|a3|ω,

b3 = z2{(243)s12} − k4sign(b3)|b3|

ω, c3 = −z3{(312)s14} − k4sign(c3)|c3|

ω,

d3 = z4{(431)s24} − k4sign(d3)|d3|

ω, (124)r3 = m1((124)λ3)|

(124)s3|,(243)r12 = m1{

(243)λ12}|(243)s12|,

(312)r14 = m1{(312)λ14}|

(312)s14|,(431)r24 = m1{

(431)λ24}|(431)s24|,

(124)q3 = m1{(124)λ3}|

(124)s3|,(243)q12 = m2{

(243)λ12}|(243)s12|,

(312)q14 = m2{(312)λ14}|

(312)s14|,(431)q24 = m2{

(431)λ24}|(431)s24|.

6.1.2 Switch-2


0Dαt {

(122)e25} =c11{0Dαt z1} − b22{0D

αt y2} − a220{D

αt x2}

=c11{(z2 − z1)a3 + z4 +∆h1 + µ1 +(122) u25}

−b22{y2c2 + d2y1 − y1y3 +∆g2 +D2}

−a22{x1b1 − x1x3 − x2 +∆f2 + d2}

0Dαt {

(244)e30} =c22{0Dαt z2} − b44{0D

αt y4} − a44{0D

αt x4}

=c22{z2b3 − z1z3 +∆h2 + µ2 +(244) u30}

−b44{y4r + y2y3 +∆g4 +D4}

−a44{x4d1 − x2x3 +∆f4 + d4}

0Dαt {

(311)e31} =c33{0Dαt z3} − b11{0D

αt y1} − a11{0D

αt x1}

=c33{−z3c3 + z1z2 +∆h3 + µ3 +(311) u31}

−b11{(y2 − y1)a2 + y4 +∆g1 +D1}

−a11{(x2 − x1)a1 + x4 +∆f1 + d1}

0Dαt {

(433)e36} =c44{0Dαt z4} − b33{0D

αt y3} − a33{0D

αt x3}

=c44{z4d3 + z1z3 +∆h4 + µ4 +(433) u36}

18

−b33{−y3b2 + y1y2 +∆g3 +D3}

−a33{−x3c1 + x1x2 +∆f3 + d3}


(122)u25 =− z4 − (z2 − z1)a3 +b22c11

(y1y3 + y2c2 + d2y1) +a22c11

(−x1x3 − x2 + x1b1)

−((122)r25 +(122) q25)sign(

(122)s25)− k1tanh((122)s25),

(244)u30 =z1z3 − z2b3 +b44c22

(y2y3 + y4r) +a44c22

(−x2x3 + x4d1)

−((244)r30 +(244) q30)sign(

(244)s30)− k1tanh((244)s30),

(311)u31 =− z1z2 + z3c3 +b11c33

(y4 + (y2 − y1)a2) +a11c33

(x4 + (x2 − x1)a1)

−((311)r31 +(311) q31)sign(

(311)s31)− k1tanh((311)s31),

(433)u36 =− z1z3 − z4d3 +b33c44

(y1y2 − y3b2) +a33c44

(x1x2 − x3c1)

−((433)r36 +(433) q36)sign(

(433)s36)− k1tanh((433)s36).


a1 = −(x2 − x1)((311)s31)− k2sign(a1)|a1|

ω, b1 = −x1((122)s25)− k2sign(b1)|b1|

ω,

c1 = x3((433)s36)− k2sign(c1)|c1|

ω, d1 = −x4(((244)s30))− k2sign(d1)|d1|

ω,

a2 = −(y2 − y1)((311)s31)− k3sign(a2)|a2|

ω, b2 = y3((433)s36)− k3sign(b2)|b2|

ω,

c2 = −y2((122)s25)− k3sign(c2)|c2|

ω, d2 = −y1((122)s25)− k3sign(d2)|d2|

ω,r = −y4(

(244)s30)− k3sign(r)|r|ω, a3 = (z2 − z1)(

(122)s25)− k4sign(a3)|a3|ω,

b3 = z2((244)s30)− k4sign(b3)|b3|

ω, c3 = −z3((311)s31)− k4sign(c3)|c3|

ω,

d3 = z4((433)s36)− k4sign(d3)|d3|

ω, (122)r25 = m1((122)λ25)|

(122)s25|,(244)r30 = m1{

(244)λ30}|(244)s30|,

(311)r31 = m1{(311)λ31}|

(311)s31|,(433)r36 = m1{

(433)λ36}|(433)s36|,

(122)q25 = m2((122)λ25)|

(122)s25|,(244)q30 = m2{

(244)λ30}|(244)s30|,

(311)q31 = m2{(311)λ31}|

(311)s31|,(433)q36 = m2{

(433)λ36}|(433)s36|.

In this two numerical simulations, we adopt the matrices A,B,C as identity matri-ces. The initial values of the drive systems (34), (35) and response system (36) areselected as x(0) = (2,−2, 1, 1), y(0) = (1, 1, 2, 2), z(0) = (−20, 3, 1, 3). The initialconditions of the unknown parameter in D-R system and upper bound of SD are s-elected as (a1(0), b1(0), c1(0), d1(0)) = (2, 1,−2, 1), (a2(0), b2(0), c2(0), d2(0), r(0)) =(1,−3, 1, 1,−5), (a3(0), b3(0), c3(0), d3(0)) = (1, 1,−2,−2), (r1(0), r2(0), r3(0), r4(0)) =(−2.5, 1,−3, 1), (q1(0), q2(0), q3(0), q4(0)) = (1,−2.5,−5, 1). The constants (ijk)λp =2, m1 = 10, m2 = 10, k1 = 10, k2 = 10, k3 = 10, k4 = 10, ω = 0.5. Thetrajectories about the error variables e = ((124)e3,

(243) e12,(312) e14,

(431) e24)T and e =

((122)e25,(244) e30,

(311) e31,(433) e36)

T are drawn in figures 2 and 6. The synchronizationfor the state trajectories of drive systems (34), (35) and response system (36) are drawnin figures 3 and 7. The trajectories of estimations θ, β, ϑ are drawn in figures 4 and 8,(ijk)rp, and

(ijk)qp are drawn in figures 5 and 9. Motivated by the numerical simulation

19

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-20

-15

-10

-5

0

5

10

15

20

erro

rs

(124)e3

(243)e12

(312)e14

(431)e24

Fig.2. The synchronization errors (124)e3,(243) e12,

(312) e14,(431) e24 change with time t .

results, it can reveal that the drive systems (4), (5) and response system (6) achieveMSSMCS. Therefore, the multi-switching adaptive controllers (MSAC) (13) and somesuitable multi-switching adaptive updating laws (MSAUL) (16) are effective.

6.2 Numerical simulations for FO chaotic system with d-

iffrenrt dimensions

As an example we choose FO Lorenz, Chen chaotic system as the drive system,and the FO hyper-chaotic Lu system as the response system. Adding SD to the D-Rsystems, we obtain

0Dαt x1

0Dαt x2

0Dαt x3

=

x2 − x1 0 00 x1 00 0 −x3

a1b1c1

+

0 + ∆f1−x1x3 − x2 +∆f2

x1x2 +∆f3

+

d1d2d3

, (40)

0Dαt y1

0Dαt y2

0Dαt y3

=

y2 − y1 0 0−y1 0 y1 + y20 −y3 0

a2b2c2

+

0 + ∆g1−y1y3 +∆g2y1y2 +∆g3

+

D1

D2

D3

, (41)

0Dαt z1

0Dαt z2

0Dαt z3

0Dαt z4

=

z2 − z1 0 0 00 z2 0 00 0 −z3 00 0 0 z4

a3b3c3d3

+

z4 +∆h1

−z1z3 +∆h2

z1z2 +∆h3

z1z3 +∆h4

+

µ1 + u1

µ2 + u2

µ3 + u3

µ4 + u4

, (42)

Choosing the parameters are a1 = 10, b1 = 28, c1 = 8/3, a2 = 35, b2 = 3, c2 = 28, a3 =36, b3 = 20, c3 = 3, d3 = 0.5. The initial conditions take as x(0) = (2,−2, 1), y(0) =

20

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-100

-50

0

50

100

150z

1

x4+y

2

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-80

-60

-40

-20

0

20

40

60

80

100

120z

2

x3+y

4

(b)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-20

-15

-10

-5

0

5

10

15

20

25

30z

3

x2+y

1

(c)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-10

-5

0

5

10

15

20

25

30

35z

4

x1+y

3

(d)

Fig.3. The synchronization for state variable z1 and x4 + y2, z2 and x3 + y4, z3 and x2 + y1,

z4 and x1+y3 of drive systems (34), (35) and response system (36) indicating in sub-pictures

(a)-(d) respectively.

(1,−1, 3), z(0) = (−2, 3, 1, 3). When ∆g = 0, ∆f = 0, ∆h = 0, d(t) = 0, D(t) = 0,µ(t) = 0 and α = 0.97, we can obtain the attractor graphs of the FO Lorenz, Chenchaotic system which are presented in figure 10. The attractor graphs of hyper-chaoticLu system can be seen figure 1. Assume

A = B =

1 0 12 1 00 1 21 0 1

, C =

1 0 0 00 1 0 00 0 1 00 0 0 1

. (43)

The following multi-switching error state modes between the drive systems (40),(41) and the response system (42) are listed.

21

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

0

5

10

15

20

25

30

a1

b1c1

d1

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-5

0

5

10

15

20

25

30

a2

b2c2

d2r

(b)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

0

5

10

15

20

25

30

35

40

45

50 a3

b3c3

d3

(c)

Fig.4. The estimation of parameters a1, b1, c1, d1 of drive system (34) (a), a2, b2, c2, d2, r of

drive system (35) (b), a3, b3, c3, d3 of response system (36) (c).

0 1 2 3 4 5 6 7 8 9 10

t/s

1

1.5

2

2.5

3

(124)r3(243)r12(312)r14(431)r24

(a)

0 1 2 3 4 5 6 7 8 9 10

t/s

0

0.5

1

1.5

2

2.5(124)q3(243)q12(312)q14(431)q24

(b)

Fig.5. The estimation of parameters (124)r3,(243)r12,

(312)r14,(431)r24 and (124)q3,

(243)q12,(312)q14,

(431)q24 shown in sub-pictures (a) and (b).

22

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-15

-10

-5

0

5

10

15

erro

rs

(122)e25

(244)e30

(311)e31

(433)e36


(311) e31,(433) e36 change with time t.

l 6= w 6= v

(123)e1,(132)e2,

(124)e3,(142)e4,

(134)e5,(143)e6,

(213)e7,(231)e8,

(214)e9,(241)e10,

(234)e11,(243)e12,

(321)e13,(312)e14,

(324)e15,(314)e16,

(341)e17,(342)e18,

(412)e19,(421)e20,

(413)e21,(423)e22,

(432)e23,(431)e24.

l 6= w = v

{

(122)e25,(133)e26,

(144)e27,(211)e28,

(233)e29,(244)e30,

(311)e31,(322)e32,

(344)e33,(411)e34,

(422)e35,(433)e36.

l = w 6= v

{

(112)e37,(113)e38,

(114)e39,(221)e40,

(223)e41,(224)e42,

(331)e43,(332)e44,

(334)e45,(441)e46,

(442)e47,(443)e48.

l = v 6= w

{

(121)e49,(131)e50,

(141)e51,(212)e52,

(232)e53,(242)e54,

(313)e55,(323)e56,

(343)e57,(414)e58,

(424)e59,(434)e60.

l = w = v

{

(111)e61,(222)e62,

(333)e63,(444)e64.

In this Section, we will select the appropriate switching error variable for the situa-tion of l 6= w 6= v, namely, e = ((142)e4,

(234) e12,(321) e13,

(413) e21)T , l 6= w = v, namely,

e = ((122)e25,(244) e30,

(311) e31,(433) e36)

T and obtain the following two switching modesfor numerical simulation:

23

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-20

-15

-10

-5

0

5

10

15

20

25

30z

1

x2+y

2

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-100

-50

0

50

100z

2

x4+y

4

(b)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-20

-15

-10

-5

0

5

10

15

20

25

30

z3

x1+y

1

(c)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

10

20

30

40

50

60 z4

x3+y

3

(d)

Fig.7. The synchronization for state variable z1 and x2 + y2, z2 and x4 + y4, z3 and x1 + y1,

z4 and x3+y3 of drive systems (34), (35) and response system (36) indicating in sub-pictures

(a)-(d) respectively.

Switch− 1:

(142)e4 = [4

∑

i=1

(c1izi)− {3

∑

j=1

(b4jyj) +3

∑

k=1

(x2kxk)}],

(234)e12 = [

4∑

i=1

(c2izi)− {

3∑

j=1

(b3jyj) +

3∑

k=1

(x4kxk)}],

(321)e13 = [

4∑

i=1

(c3izi)− {

3∑

j=1

(b2jyj) +

3∑

k=1

(x1kxk)}],

(413)e21 = [4

∑

i=1

(c4izi)− {3

∑

j=1

(b1jyj) +3

∑

k=1

(x3kxk)}],

(44)

24

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

0

5

10

15

20

25a1

b1c1

d1

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-5

0

5

10

15

20

25

30

a2

b2c2

d2r

(b)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

0

5

10

15

20

25

30

35

40

45a3

b3c3

d3

(c)

Fig.8. The estimation of parameters a1, b1, c1, d1 of drive system (34) (a), a2, b2, c2, d2, r of

drive system (35) (b), a3, b3, c3, d3 of response system (36) (c).

Switch−2 :

(122)e25 = [4

∑

i=1

(c1izi)− {3

∑

j=1

(b2jyj) +3

∑

k=1

(x2kxk)}],

(244)e30 = [

4∑

i=1

(c2izi)− {

3∑

j=1

(b4jyj) +

3∑

k=1

(x4kxk)}],

(311)e31 = [4

∑

i=1

(c3izi)− {3

∑

j=1

(b1jyj) +3

∑

k=1

(x1kxk)}],

(433)e36 = [4

∑

i=1

(c4izi)− {3

∑

j=1

(b3jyj) +3

∑

k=1

(x3kxk)}],

(45)

In order to exhibit the effectiveness of the designed MSAC (25) and the MSAUL (28),

25

0 1 2 3 4 5 6 7 8 9 10

t/s

-2

-1

0

1

2

3

4

(122)r25(244)r30(311)r31(433)r36

(a)

0 1 2 3 4 5 6 7 8 9 10

t/s

-2

0

2

4

6

8(122)q25(244)q30(311)q31(433)q36

(b)

Fig.9. The estimation of parameters (122)r25,(244)r30,

(311)r31,(433)r36 and (122)q25,

(244)q30,(311)q31,

(433)q36 shown in sub-pictures (a) and (b).

0

10

40

20

30

x 3

20

40

50

x2

0 2010

x1

-20 0-10-40 -20

(a)

040

10

20

30

20

y 3

40

50

y2

0

60

3020-20

y1

100-10-20-40 -30

(b)

Fig.10. The 3D phase plot of FO Lorenz chaotic system (a), FO Chen chaotic system (b).

the nonlinear uncertainty and external disturbance are assumed as follows:

d(x(t)) = (−0.1cos(t),−0.2cos(2t), 0.3sin(3t))T ,

D(y(t)) = (−0.1sin(t),−0.2sin(2t), 0.3cos(3t))T ,

µ(z(t)) = (0.1cos(5t), 0.2cos(6t), 0.3sin(7t), 0.4sin(8t))T ,

∆f (x(t)) = (0.1cos(2tx1), 0.2cos(2tx2), 0.3sin(2tx3))T ,

∆g(y(t)) = (0.1sin(2πt), 0.2sin(sign(y2)), 0.3cos(2πy3))T ,

∆h(z(t)) = (0.1sin(2tsign(z1)), 0.2sin(2tsign(z2)), 0.3sin(3t), 0.4sin(4t)T .

26

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-25

-20

-15

-10

-5

0

5

10

15

20

erro

rs

(142)e4

(234)e12

(321)e13

(413)e21


(321) e13,(413) e21 change with time t .

6.2.1 Switch-1


0Dαt {

(142)e4} =[

4∑

i=1

c1i(0Dαt zi)− {

3∑

j=1

b4j(0Dαt yj) +

3∑

k=1

x2k(0Dαt xk)}]

={(z2 − z1)a3 + z4 +∆h1 + µ1 +(142)u4}

−{(y2 − y1)a2 − y3b2 + y1y2 +∆g1 +∆g3 +D1 +D3}

−{2(x2 − x1)a1 + x1b1 − x1x3 − x2 + 2∆f1 +∆f2 + 2d1 + d2},

0Dαt {

(234)e12} =[4

∑

i=1

c2i(0Dαt zi)− {

3∑

j=1

b3j(0Dαt yj) +

3∑

k=1

x4k(0Dαt xk)}]

={z2b3 − z1z3 +∆h2 + µ2 +(234)u12} − {−y1a2 − 2y3b2

+(y2 + y1)c2 − y1y3 + 2y1y2 +∆g2 + 2∆g3 +D2 + 2D3}

−{(x2 − x1)a1 − x3c1 + x1x2 +∆f1 +∆f3 + d1 + d3},

0Dαt {

(321)e13} =[

4∑

i=1

c3i(0Dαt zi)− {

3∑

j=1

b2j(0Dαt yj) +

3∑

k=1

x1k(0Dαt xk)}]

={−z3c3 + z1z2 +∆h3 + µ3 +(321)u13} − {2(y2 − 2y1)a2

+(y1 + y2)c2 − y1a2 − y1y3 + 2∆g1 +∆g2 + 2D1 +D2}

−{(x2 − x1)a1 − x3c1 − x1x2 +∆f1 +∆f3 + d1 + d3},

0Dαt {

(413)e21} =[4

∑

i=1

c4i(0Dαt zi)− {

3∑

j=1

b1j(0Dαt yj) +

3∑

k=1

x3k(0Dαt xk)}]

27

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-20

0

20

40

60

80

100 z1

2x1+x

2+y

1+y

3

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

20

40

60

80

100

120 z2

x1+x

3+y

2+2y

3

(b)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-50

0

50

100z

3

x1+x

3+2y

1+y

2

(c)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

0

20

40

60

80

100

120

z4

x2+2x

3+y

1+y

3

(d)

Fig.12. The synchronization for state variable z1 and 2x1 + x2 + y1 + y3, z2 and x1 + x3 +

y2 +2y3, z3 and x1 + x3 +2y1 + y2, z4 and x2 +2x3 + y1 + y3 of drive systems (40), (41) and

response system (42) indicating in sub-pictures (a)-(d) respectively.

={z4d3 + z1z3 +∆h4 + µ4 +(413)u21}

−{(y2 − y1)a2 − y3b2 + y1y2 +∆g1 +∆g3 +D1 +D3}

−{x1b1 − 2x3c1 − x1x3 − x2 + 2x1x2 +∆f2 + 2∆f3 + d2 + 2d3}.


(142)u4 =− z4 + y1y2 − x1x3 − x2 − a3(z2 − z1) + a2(y2 − y1)− b1y3

+2a1(x2 − x1) + b1x1 − ((142)ρ4 +(142) ˆ4)sign(

(142)s4)− k1tanh((142)s4),

(234)u12 =z1z3 − y1y3 + 2y1y2 + x1x2 − b3z2 − a2y1 − 2b2y3 + c2(y1 + y2)

+a1(x2 − x1)− c1x3 − ((234)ρ12 +(234) ˆ12)sign(

(234)s12)− k1tanh((234)s12),

(321)u13 =− z1z2 − y1y3 + x1x2 + c3z3 − a2y3 + 2a2(y2 − 2y1) + c2(y1 + y2)

+a1(x2 − x1)− c1x3 − ((321)ρ13 +(321) ˆ13)sign(

(321)s13)− k1tanh((321)s13),

(413)u21 =− z1z3 + y1y2 − x1x3 − x2 + 2x1x2 − d3z4 + a2(y2 − y1)− b2y3

28

+b1x1 − 2c1x3 − ((413)ρ21 +(413) ˆ21)sign(

(413)s21)− k1tanh((413)s21),


b1 = −x1{((142)s4) + ((413)s21)} − k2sign(b1)(abs(b1)

ω),

c1 = x3{((234)s12) + ((321)s13) + 2((142)s4)} − k2sign(c1)(abs(c1)

ω),

b2 = y3{((142)s4) + 2((234)s12) + ((413)s21)} − k3sign(b2)(abs(b2)

ω),

c2 = −(y1 + y2){((234)s12) + ((321)s13)} − k3sign(c2)(abs(c2)

ω),

a3 = (z2 − z1)((142)s4)− k4sign(a3)(abs(a3)

ω),

b3 = z2((234)s12)− k4sign(b3)(abs(b3)

ω),

c3 = −z3((321)s13)− k4sign(c3)(abs(c3)

ω),

d3 = z4((413)s21)− k4sign(d3)(abs(d3)

ω),

a1 = −(x2 − x1){((321)s13) + ((234)s12) + 2((413)s21)} − k2sign(a1)(abs(a1)

ω),

a2 = −{(y2 − y1)(((142)s4) + ((413)s21) + 2((321)s13))− y3((

(234)s12) + ((321)s13))}

− k1sign(a2)(abs(a2)ω),

(142)ρ4 = m1((142)λ4)|

(142)s4|,(234)ρ12 = m1{

(234)λ12}|(234)s12|,

(321)ρ13 = m1{(321)λ13}|

(321)s13|,(413)ρ21 = m1{

(413)λ21}|(413)s21|,

(142) ˆ4 = m2((142)λ4)|

(142)s4|,(234) ˆ12 = m2(

(234)λ12)|(234)s12|,

(321) ˆ13 = m2((321)λ13)|

(321)s13|,(413) ˆ21 = m2(

(413)λ21)|(413)s21|.

6.2.2 Switch-2


0Dαt {

(122)e25} =[

4∑

i=1

c1i(0Dαt zi)− {

3∑

j=1

b2j(0Dαt yj) +

3∑

k=1

x2k(0Dαt xk)}]

={(z2 − z1)a3 + z4 +∆h1 + µ1 +(122)u25} − {2(y2 − y1)a2

−y1a2 + (y1 + y2)c2 − y1y3 + 2∆g1 +∆g2 + 2D1 +D2}

−{2(x2 − x1)a1 + x1b1 − x1x3 − x2 + 2∆f1 +∆f2 + 2d1 + d2},

0Dαt {

(244)e30} =[4

∑

i=1

c2i(0Dαt zi)− {

3∑

j=1

b4j(0Dαt yj) +

3∑

k=1

x4k(0Dαt xk)}]

={z2b3 − z1z3 +∆h2 + µ2 +(244)u30}

−{(y2 − y1)a2 − y3b2 + y1y2 +∆g1 +∆g3 +D1 +D3}

−{(x2 − x1)a1 − x3c1 + x1x2 +∆f1 +∆f3 + d1 + d3},

0Dαt {

(311)e31} =[

4∑

i=1

c3i(0(Dαt zi)− {

3∑

j=1

b1j(0Dαt yj) +

3∑

k=1

x1k0Dαt xk)}]

={−z3c3 + z1z2 +∆h3 + µ3 +(311)u31}

−{(y2 − y1)a2 − y3b2 + y1y2 +∆g1 +∆g3 +D1 +D3}

−{(x2 − x1)a1 − x3c1 + x1x2 +∆f1 +∆f3 + d1 + d3},

29

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

0

5

10

15

20

25a1

b1c1

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

0

5

10

15

20

25

30

35

40a2

b2c2

(b)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-15

-10

-5

0

5

10

15

20

25

30

35a3

b3c3

d3

(c)

Fig.13. The estimation of parameters a1, b1, c1 of drive system (40) (a), a2, b2, c2 of drive

system (41) (b), a3, b3, c3, d3 of drive system (42) (c).

0Dαt {

(433)e36} =[4

∑

i=1

c4i(0Dαt zi)− {

3∑

j=1

b3j(0Dαt yj) +

3∑

k=1

x3k(0Dαt xk)}]

={z4d3 + z1z3 +∆h4 + µ4 +(433)u36} − {−y1a2 − 2y3b2

+(y2 + y1)c2 − y1y3 + 2y1y2 +∆g2 + 2∆g3 +D2 + 2D3}

−{x1b1 − 2x3c1 − x1x3 − x2 + 2x1x2 +∆f2 + 2∆f3 + d2 + 2d3}.


(122)u25 =− z4 − y1y3 − x1x3 − x2 − a3(z2 − z1) + 2a2(y2 − 2y1) + c2(y1 + y2)− a2y1

+2a1(x2 − x1) + b1x1 − ((122)ρ25 +(122) ˆ25)sign(

(122)s25)− k1tanh((122)s25),

(244)u30 =z1z3 + y1y2 + x1x2 − b3z2 + a2(y2 − y1)− b1y3

+a1(x2 − x1)− c1x3 − ((244)ρ30 +(244) ˆ30)sign(

(244)s30)− k1tanh((244)s30),

(311)u31 =− z1z2 + y1y2 + x1x2 + c3z3 + a2(y2 − y1)− b2y3

30

0 1 2 3 4 5 6 7

t/s

-3

-2

-1

0

1

2

3

4

5 (142)ρ4

(234)ρ12

(321)ρ13

(413)ρ21

(a)

0 1 2 3 4 5 6 7

t/s

-3

-2

-1

0

1

2

3

4

5

6(142) ˆ4(234) ˆ12(321) ˆ13(413) ˆ21

(b)

Fig.14. The estimation of parameters (142)ρ4,(234)ρ12,

(321)ρ13,(413)ρ21 and (142) ˆ4,

(234) ˆ12,(321) ˆ13,

(413) ˆ21 shown in sub-pictures (a) and (b).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-30

-20

-10

0

10

20

30

40

50

erro

rs

(122)e25

(244)e30

(311)e31

(433)e36

Fig.15. The synchronization errors (122)e25,(244)e30,

(311)e31,(433)e36 change with time t.

+a1(x2 − x1)− c1x3 − ((311)ρ31 +(311) ˆ31)sign(

(311)s31)− k1tanh((311)s31),

(433)u36 =− z1z3 − y1y3 + 2y1y2 − x1x3 − x2 + 2x1x2 − d3z4 − a2y1 − 2b2y3 + c2(y1 + y2)

+b1x1 − 2c1x3 − ((433)ρ36 +(433) ˆ36)sign(

(433)s36)− k1tanh((433)s36).


(122)ρ25 = m1((122)λ25)|

(122)s25|,(244)ρ30 = m1(

(244)λ30)|(244)s30|,

(311)ρ31 = m1((311)λ31)|

(311)s31|,(433)ρ36 = m1(

(433)λ36)|(433)s36|,

(122) ˆ25 = m2((122)λ25)|

(122)s25|,(244) ˆ30 = m2(

(244)λ30)|(244)s30|,

(311) ˆ31 = m2((311)λ31)|

(311)s31|,(433) ˆ36 = m2(

(433)λ36)|(433)s36|,

31

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-100

-80

-60

-40

-20

0

20

40

60

80

100

z1

2x1+x

2+2y

1+y

2

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

10

20

30

40

50

60

70

80

90

100z

2

x1+x

3+y

1+y

3

(b)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

0

10

20

30

40

50

60

70

80

90

100z

3

x1+x

3+y

1+y

3

(c)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

0

50

100

150

200z

4

x2+2x

3+y

2+2y

3

(d)

Fig.16. The synchronization for state variable z1 and 2x1 + x2 + 2y1 + y2, z2 and x1 + x3 +

y1 + y3, z3 and x1 + x3 + y1 + y3, z4 and x2 + 2x3 + y2 + 2y3 of drive systems (40), (41) and

response system (42) indicating in sub-pictures (a)-(d) respectively.

a1 = −(x2 − x1){(311)s31 +

(244) s30 + 2((433)s36)} − k2sign(a1)(abs(a1)ω),

b1 = −x1{(122)s25 +

(433) s36} − k2sign(b1)(abs(b1)ω),

c1 = x3{(244)s30 +

(311) s31 + 2((122)s25)} − k1sign(c1)(abs(c1)ω),

a2 = −((y2 − y1){(244)s30 + 2((433)s36) +

(311) s31} − y1{(122)s25 +

(433) s36}) (46)

− k3sign(a2)(abs(a2)ω),

b2 = y3{(244)s30 + 2((122)s25) +

(311) s31} − k3sign(b2)(abs(b2)ω),

c2 = −(y1 + y2)((122)s25 +

(433) s36)− k3sign(c2)(abs(c2)ω),

a3 = (z2 − z1)((122)s25)− k4sign(a3)(abs(a3)

ω),

b3 = z2((244)s30)− k4sign(b3)(abs(b3)

ω),

c3 = −z3((311)s31)− k4sign(c3)(abs(c3)

ω),

32

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-15

-10

-5

0

5

10

15

20

25 a1

b1c1

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-20

-10

0

10

20

30

40a2

b2c2

(b)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

-50

-40

-30

-20

-10

0

10

20

30

40

50

a3

b3c3

d3

(c)

Fig.17. The estimation of parameters a1, b1, c1 of drive system (40) (a), a2, b2, c2 of drive

system (41) (b), a3, b3, c3, d3 of drive system (43) (c).

d3 = z4((433)s36)− k4sign(d3)(abs(d3)

ω).

In this two numerical simulations, the definition of matrices A,B,C are shown in(43). The initial conditions of the drive systems (40), (41) and response system (42)are selected as x(0) = (2,−2, 1, 1), y(0) = (1, 1, 2, 2), z(0) = (−20, 3, 1, 3). The ini-tial values of the unknown parameter in D-R system and upper bound of SD areselected as (a1(0), b1(0), c1(0), d1(0)) = (2, 1,−2, 1), (a2(0), b2(0), c2(0), d2(0), r(0)) =(1,−3, 1, 1,−5), (a3(0), b3(0), c3(0), d3(0)) = (1, 1,−2,−2), (r1(0), r2(0), r3(0), r4(0)) =(−2.5, 1,−3, 1), (q1(0), q2(0), q3(0), q4(0)) = (1,−2.5,−5, 1). The constants (lwv)λp =2, m1 = 10, m2 = 10, k1 = 10, k2 = 10, k3 = 10, k4 = 10, ω = 0.5. Thetrajectories about the error variables e = ((142)e4,

(234) e12,(321) e13,

(413) e21)T and e =

((122)e25,(244) e30,

(311) e31,(433) e36)

T are drawn in figures 11 and 15. The synchronizationfor the state trajectories drive systems (40), (41) and response system (42) are drawnin figures 12 and 16. The trajectories of estimations θ, β, ϑ are drawn in figures 13and 17, (lwv)ρp, and

(lwv)p are drawn in figures 14 and 18. Motivated by the numericalsimulation results, it can reveal that the drive systems (4), (5) and response system (6)

33

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

0

5

10

15

20

25

(122)ρ25

(244)ρ30

(311)ρ31

(433)ρ36

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t/s

0

5

10

15

20

25

30

(122) ˆ25(244) ˆ30(311) ˆ31(433) ˆ36

(b)

Fig.18. The estimation of parameters (122)ρ25,(244)ρ30,

(311)ρ31,(433)ρ36 and

(122) ˆ25,(244) ˆ30,

(311) ˆ31,(433) ˆ36 shown in sub-pictures (a) and (b).

achieve MSSMCS. Therefore, the multi-switching adaptive controllers (MSAC) (25)and some suitable multi-switching adaptive updating laws (MSAUL) (28) are effective.

7 Conclusion

In this article, we investigated the multi-switching sliding mode combination syn-chronization (MSSMCS) of fractional order (FO) non-identical chaotic systems withunknown parameters under double stochastic disturbances (SD). In the theoreticalparts, the FO chaotic systems with the different (or same )dimension are considered.Our idea for this topic is that with the help of the Lyapunov theory, adaptive controltechnology and sliding mode control technique, we put forward the fractional order s-liding surface, multi-switching adaptive controllers (MSAC) and multi-switching adap-tive updating laws (MSAUL) which can achieve the the state variables of the drivesystems are synchronized with the different state variables of the response systems.Meanwhile, the unknown parameters are identified and upper bound of stochastic dis-turbances are examined accurately. What’s more, the combination drive systems andsingle response system we chose are very general. The different description of the s-cale matrices can make the multi-switching projection synchronization, multi-switchingcomplete synchronization, multi-switching anti-synchronization etc, become the specialcases of MSSMCS. Motivated by the numerical simulation results, it is clear that thedifferent error variables quickly converges to the equilibrium point. Therefore, themulti-switching adaptive controllers (MSAC) are effective and have strong robustness.Next, we will concentrate on the fractional order multi-switching synchronization oftime-delay systems under multiple stochastic disturbances, and the parameters of sys-tem are still unknown.

Declarations

34

Funding: This work is partly supported by Sichuan Youth Science and Technol-ogy Foundation (Grant No. 2019YJ0456), Fund of Sichuan University of Science andEngineering (Grant No. 2020RC26, 2020RC42).

Competing interests: The authors declare that they have no competing inter-ests.

Availability of data and material: The data used to support the findings ofthis study are available from the corresponding author upon request.

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