Synchronization of Two Different Chaotic Systems with Parameter Uncertainties

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Journal of Applied Mathematics Volume 2012, Article ID 607491,16pages doi:10.1155/2012/607491

Research Article

Robust Adaptive Finite-Time

Synchronization of Two Different Chaotic Systems with Parameter Uncertainties

Yun-An Hu, Hai-Yan Li, Chun-Ping Zhang, and Liang Liu

Department of Control Engineering, Naval Aeronautical and Astronautical University, Yantai, Shandong 264001, China

Correspondence should be addressed to Yun-An Hu,[email protected] Received 10 April 2012; Revised 20 July 2012; Accepted 22 July 2012 Academic Editor: Laurent Gosse

Copyrightq2012 Yun-An Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is concerned with the finite-time synchronization problem for two diﬀerent chaotic systems with parameter uncertainties. Using finite-time control approach and robust control method, an adaptive synchronization scheme is proposed to make the synchronization errors of the systems with parameter uncertainties zero in a finite time. On the basis of Lyapunov stability theory, appropriate adaptive laws are derived to deal with the unknown parameters of the systems.

And the convergence of the parameter errors is guaranteed in a finite time. The proposed method can be applied to a variety of chaos systems. Numerical simulations are given to demonstrate the eﬃciency of the proposed control scheme.

1. Introduction

In the past few decades, chaos synchronization has gained much attention from various fields1–3, since Pecora and carroll4introduced a method to synchronize two identical chaotic systems with different initial conditions in 1990. Most of the works on chaos synchronization have focused on two identical chaotic systems 5–11. However, in many real world applications, there are no exactly two identical chaotic systems. Therefore, the problem of chaos synchronization between two different chaotic systems with uncertainties is an important research issue 12. Different synchronization control methods for two different chaotic systems, such as adaptive control13–21, nonlinear feedback control22, backstepping23,24, fuzzy technique25–27, and sliding mode control28–30, have been proposed to solve the synchronization problem.

Since some systems’ parameters cannot be exactly known in advance, many eﬀorts have been devoted to adaptive synchronization. In 18, 31, Huang discussed the

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synchronizations between Lorenz-Stenflo LSsystem and CYQY system, and between LS system and hyperchaotic Chen system with fully uncertain parameters. Wang et al. 15 designed a general adaptive robust controller and parameter update laws which made the drive-response systems with different structures asymptotically synchronized. In16, the sufficient conditions for achieving synchronization between generalized Henon-Heiles system and hyperchaotic Chen system with unknown parameters were derived based on Lyapunov stability theory. A new adaptive synchronization scheme by pragmatical asymptotically stability theorem was proposed for two different uncertain chaotic systems 17, but the unknown signals were used in the controller. Chaos synchronization between two different chaotic systems with uncertainties in both master and slave chaotic systems remains a challenging problem30.

Most methods only guarantee the asymptotic stability of the synchronization error dynamics, namely, the trajectories of the slave system approach the trajectories of the master system as t → ∞. From a practical point of view, however, it is more valuable that the synchronization objective is realized in a finite time28. In recent years, some researchers have applied finite-time control techniques, such as nonsingular terminal sliding mode control method32, CLF-based method33,34, sliding mode control method28–30, and the finite-time stability theory-based method28,35,36, to realize synchronization.

Compared with the existing results in the literature, there are three advantages which make our approach attractive. First, based on the finite-time control technique, adaptive control, and robust control, a new synchronization method is presented for a wide class of nonlinear systems. Second, it guarantees that all the errors are driven to zero in a finite time even for the systems with parameter uncertainties. Third, it guarantees that all the parameter errors converge to zero in a finite time.

In this paper, an adaptive finite-time synchronization scheme is proposed for a class of chaotic systems. The rest of the paper is organized as follows. InSection 2, we introduce the chaotic systems considered in this paper and preliminary lemmas. InSection 3, the proposed finite-time controller is designed to synchronize two diﬀerent chaotic systems. We give the simulation results and the conclusions in Sections4and5, respectively.

2. System Description

Consider the following master chaotic system:

˙

x A₁ ΔA1x B₁ ΔB1f1x, 2.1

wherex x1, x₂, . . . , x_n^T ∈ Rⁿ denotes a state vector,f₁is a nonlinear continuous vector function,A1andB1aren×nnominal coeﬃcient matrices,ΔA1andΔB1are unknown parts ofn×ncoeﬃcient matrices.

The slave system is given with

˙

y A₂ ΔA2y B₂ ΔB2f2

y

u, 2.2

wherey y1, y2, . . . , yn^T ∈ Rⁿdenotes a state vector,f2 is a nonlinear continuous vector function,A₂andB₂aren×nnominal coeﬃcient matrices,ΔA2andΔB2are unknown parts

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ofn×ncoeﬃcient matrices, andu u1t, u2t, . . . , unt^T ∈Rⁿis a control input vector to be designed.

Subtracting2.1from2.2yields the error dynamical system as follows:

˙

e A₂ ΔA2y B₂ ΔB2f2

y

−A1 ΔA1x−B1 ΔB1f1x u, 2.3

where e y−x. Note that only a part of elements of the coeﬃcient matrices unknown, without loss of generality, we assume that the number of the unknown elements of theith row ofΔA1 isN_A1i, that ofΔA2isN_A2i, that ofΔB1isN_B1i, and that ofΔB2 isN_B2i. Then 2.3can be rewritten as

˙

eA₂y B₂f₂ y

−A₁x−B₁f₁x u

⎡

⎢⎢

⎢⎣

NA21

i1

δa_21iy_1i ...

NA2n

i1

δa_2niy_ni

⎤

⎥⎥

⎥⎦

⎡

⎢⎢

⎢⎣

NB21

i1

δb_21if_21i ...

NB2n

i1

δb_2nif_2ni

⎤

⎥⎥

⎥⎦

−

⎡

⎢⎢

⎢⎣

NA11

i1

δa_11ix_1i ...

NA1n

i1

δa_1nix_ni

⎤

⎥⎥

⎥⎦

−

⎡

⎢⎢

⎢⎣

NB11

i1

δb_11if_11i ...

NB1n

i1

δb_1nif_1ni

⎤

⎥⎥

⎥⎦

, 2.4

whereδa_∗jiare nonzero elements of the jth row ofΔA_∗,y_jiare corresponding elements ofy, δb_∗jiare nonzero elements of the jth row ofΔB_∗ andf_∗jiare corresponding elements off_∗, j1, . . . , n.

Assumption 2.1. The unknown parameters are norm-bounded, that is,

δa_∗ji≤d_a_∗_ji, δb_∗ji≤d_b_∗_ji, 2.5

whereda_∗jianddb_∗jiare known positive constants.

Definition 2.2see28. Consider the master and slave chaotic systems described by2.1 and2.2, respectively. If there exists a constantTTe0>0, such that

tlim→Tet0 2.6

andet ≡0, ift≥T, then the chaos synchronization between the systems2.1and2.2is achieved in a finite time.

Lemma 2.3see28. Consider the system

˙

xfx, f0 0, x∈Rⁿ, 2.7

wheref :D → Rⁿis continuous on an open neighborhoodD∈Rⁿ.

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Suppose there exists a continuous diﬀerential positive-definite functionVx : D → R, real numbersp >0, 0< η <1, such that

V˙x pV^ηx≤0, ∀x∈D. 2.8

Then, the origin of system 2.7 is a locally finite-time stable equilibrium, and the settling time, depending on the initial state x0 x₀, satisfies

Tx0≤ V^1−ηx0 p

1−η. 2.9

In addition, ifD Rⁿ and Vx is also radially unbounded i.e.,Vx → ∞ as x → ∞, then the origin is a globally finite-time stable equilibrium of system2.7.

Lemma 2.4see28. Suppose a1,a₂, . . . ,an, and 0<q<2 are all real numbers, then the following inequality holds:

|a1|^q |a2|^q · · · |an|^q≥

a²₁ a²₂ · · · a²_n_q/2

. 2.10

3. Synchronization of Two Different Chaotic Systems with Parameter Uncertainties

Consider two diﬀerent chaotic systems2.1and2.2from diﬀerent initial states. The aim of controller design is to determine appropriateusuch that

t→limTe0. 3.1

Now we are ready to give the design steps.

Define Lyapunov function

V 1 2e^Te 1

2 n j1

NA2j

i1

1 k_dj₋₁

γ_a2ji⁻¹ δa²_2ji 1 2

n j1

NA1j

i1

1 k_dj₋₁

γ_a1ji⁻¹ δa²_1ji

1 2

n j1

NB2j

i1

1 k_dj₋₁

γ_b2ji⁻¹ δb_2ji² 1 2

n j1

NB1j

i1

1 k_dj₋₁

γ_b1ji⁻¹ δb²_1ji,

3.2

whereδa_2ji δa_2ji−δa_2ji,δa_1ji δa_1ji−δa_1ji,δb_2ji δb_2ji−δb_2ji,δb_1jiδb_1ji−δb_1ji, and δa2ji,δa1ji,δb2ji,δb1jiare estimation values ofδa2ji,δa1ji,δb2ji,δb1ji, respectively, andγ_a2ji⁻¹ , γ_a1ji⁻¹ ,γ_b2ji⁻¹ ,γ_b1ji⁻¹ ,k_djare constants greater than zero.

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Taking the time derivative of3.2gives

V˙ e^Te˙ n j1

NA2j

i1

1 kdj

₋₁

γ_a2ji⁻¹ δa2jiδa˙2ji

n j1

NA1j

i1

1 kdj

₋₁

n j1

NB2j

i1

1 k_dj₋₁

γ_b2ji⁻¹ δb_2jiδb˙_2ji n

j1 NB1j

i1

1 k_dj₋₁

γ_b1ji⁻¹ δb_1jiδb˙_1ji.

3.3

Design the control law as

u −Ke−KDe˙ αs−

c1sgne1|e1|^α, . . . , cnsgnen|en|^α_T

−μ

⎡

⎣ⁿ

j1 NA2j

i1

δa_2ji d_a2ji_{1 α} ⁿ

j1 NA1j

i1

δa_1ji d_a1ji_{1 α}

n j1

NB2j

i1

δb_2ji d_b2ji_{1 α} ⁿ

j1 NB1j

i1

δb_1ji d_b1ji_{1 α}⎤

⎦α_e

_N A11

i1

δa11iy_1i, . . . ,

NA1n

i1

δa1niy_ni T

− _N

B21

i1

δb21if_21i, . . . ,

NA1n

i1

δb2nif_2ni T

− _N

A21

i1

δa_21iy_1i, . . . ,

NA2n

i1

δa_2niy_ni

T _N B11

i1

δb_11if_11i, . . . ,

NB1n

i1

δb_1nif_1ni T

,

3.4

whereK diag{k1, k2, . . . , kn},ki > 0,KD diag{kd1, kd2, . . . , kdn},kdi > 0, andci > 0 are constants, 0< α <1 is a constant,αs α_s1 ...α_sn^Tandαsiis given in3.5

α_si

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−∂VA/∂eifAi K_Ai

∂VA/∂eifAi

2

∂V_A/∂ei⁴

∂VA/∂ei , if ∂V_A

∂ei /0 0,if ∂VA

∂e_i 0, i1, . . . , n,

3.5

whereV_A 1/2e^Te,f_A−A2y−B₂f₂y A₁x B₁f₁xandK_Ai>0. Andα_e ^α^e1^{... α}^en^T andαeiis given in

αei

⎧⎪

⎪⎨

⎪⎪

⎩ 1

e_i, if|ei| ≥eσ

sgnei

e_σ , if|ei|< e_σ,

3.6

wheree_σ >0 is a small constant.

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Substituting3.4into2.4gives

˙

e −KI K_D⁻¹e−

⎡

⎢⎢

⎢⎣

NA21

i1

1 kd1⁻¹δa21iy_1i ...

NA2n

i1

1 k_dn⁻¹δa_2niy_ni

⎤

⎥⎥

⎥⎦

−

⎡

⎢⎢

⎢⎣

NB21

i1

1 kd1⁻¹δb21if_21i ...

NB2n

i1

1 k_dn⁻¹δb_2nif_2ni

⎤

⎥⎥

⎥⎦

fA αs

⎡

⎢⎢

⎢⎣

NA11

i1

1 k_d1⁻¹δa_11ix_1i ...

NA1n

i1

1 k_dn⁻¹δa_1nix_ni

⎤

⎥⎥

⎥⎦

⎡

⎢⎢

⎢⎣

NB11

i1

1 k_d1⁻¹δb_11if_11i ...

NB1n

i1

1 k_dn⁻¹δb_1nif_1ni

⎤

⎥⎥

⎥⎦

−μI KD⁻¹

⎡

⎣ⁿ

j1 NA2j

i1

δa2ji da2ji

_{1 α} ⁿ

j1 NA1j

i1

δa1ji da1ji

_{1 α}

n j1

NB2j

i1

δb2ji db2ji

_{1 α} ⁿ

j1 NB1j

i1

δb1ji db1ji

_{1 α}⎤

⎦αe

−I KD⁻¹

c1sgne1|e1|^α, . . . , cnsgnen|en|^α_T .

3.7

Case 1|ei| ≥e_σ. Substituting3.7into3.3yields

V˙ ≤ −e^TKI K_D⁻¹e−ⁿ

i1

c_i1 k_di⁻¹|ei|^{1 α}

−μk_d

⎡

⎣ⁿ

j1 NA2j

i1

δa_2ji d_a2ji_{1 α} ⁿ

j1 NA1j

i1

δa_1ji d_a1ji_{1 α}

n j1

NB2j

i1

δb_2ji d_b2ji_{1 α} ⁿ

j1 NB1j

i1

δb_1ji d_b1ji_{1 α}⎤

⎦

−ⁿ

j1

e_j

NA2j

i1

1 k_dj₋₁ δa_2jiy_ji

n j1

e_j

NA1j

i1

1 k_dj₋₁ δa_1jix_ji

−ⁿ

j1

e_j

NB2j

i1

1 k_dj₋₁

δb_2jif_2ji n j1

e_j

NB1j

i1

1 k_dj₋₁

δb_1jif_1ji

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n j1

NA2j

i1

1 kdj

₋₁

n j1

NA1j

i1

1 kdj

₋₁

n j1

NB2j

i1

1 k_dj₋₁

γ_b2ji⁻¹ δb_2jiδb˙_2ji n

j1 NB1j

i1

1 k_dj₋₁

γ_b1ji⁻¹ δb_1jiδb˙_1ji,

3.8

wherekdmin{1 kd1⁻¹,1 kd2⁻¹, . . . ,1 kdn⁻¹}. Choosing the updating law as

δa˙_2ji

γ_a2jie_jy_ji, ifa_2ji< d_a2ji

0, otherwise,

δa˙_1ji

−γa1jiejxji, ifa1ji< da1ji

0, otherwise,

δb˙2ji

⎧⎨

⎩

γb2jiejf_2ji, ifb2ji< db2ji

0, otherwise,

δb˙1ji

⎧⎨

⎩

−γb1jiejf_1ji, if b1ji< db1ji

0, otherwise.

3.9

Substituting3.9into3.8yields

V˙ ≤ −e^TKI K_D⁻¹e−ⁿ

i1

c_i1 k_di⁻¹|ei|^{1 α}

−μkd

⎡

⎣ⁿ

j1 NA2j

i1

δa2ji da2ji

_{1 α} ⁿ

j1 NA1j

i1

δa1ji da1ji

_{1 α}

n j1

NB2j

i1

δb2ji db2ji

_{1 α} ⁿ

j1 NB1j

i1

δb1ji db1ji

_{1 α}⎤

⎦.

3.10

Since

δa_2ji−δa_2ji≤δa_2ji δa_2ji≤δa_2ji d_a2ji, δa1ji−δa1ji≤δa1ji δa1ji≤δa1ji da1ji, δb_2ji−δb_2ji≤δb_2ji δb_2ji≤δb_2ji d_b2ji, δb_1ji−δb_1ji≤δb_1ji δb_1ji≤δb_1ji d_b1ji

3.11

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hold, one can conclude that−|δa2ji| da2ji^{1 α}≤ −|δa2ji−δa2ji|^{1 α},

−δa1ji da1ji

_{1 α}

≤ −δa1ji−δa1ji^{1 α},

−δb2ji db2ji

_{1 α}

≤ −δb2ji−δb2ji^{1 α},

3.12

and−|δb_1ji| d_b1ji^{1 α} ≤ −|δb_1ji−δb_1ji|^{1 α}. Therefore, the inequality3.10can be rewritten as

V˙ ≤ −e^TKI KD⁻¹e−ⁿ

i1

ci1 kdi⁻¹|ei|^{1 α}

−μk_d

⎡

⎣ⁿ

j1 NA2j

i1

δa_2ji−δa_2ji^{1 α} ⁿ

j1 NA1j

i1

δa_1ji−δa_1ji^{1 α}

n j1

NB2j

i1

δb_2ji−δb_2ji^{1 α} ⁿ

j1 NB1j

i1

δb_1ji−δb_1ji^{1 α}

⎤

⎦

≤ −c_μ

⎡

⎣ⁿ

i1

|ei|² ⁿ

j1 NA2j

i1

δa_2ji−δa_2ji² ⁿ

j1 NA1j

i1

δa_1ji−δa_1ji²

n j1

NB2j

i1

δb2ji−δb2ji² ⁿ

j1 NB1j

i1

δb1ji−δb1ji²

⎤

⎦

1 α/2

≤ −c_μV^{1 α/2},

3.13

wherec_μ min{ci1 k_di⁻¹, k_i1 k_di⁻¹, μk_d, i 1, . . . , n}. According toLemma 2.3,e → Beσ in a finite time, whereBeσ {e||ei| ≤eσ, i1, . . . , n}.

Case 2|ei|< e_σ. Using3.3–3.7and3.9, it is easy to show that V˙ ≤ −e^TKI KD⁻¹e−ⁿ

i1

ci1 kdi⁻¹|ei|^{1 α} 3.14

holds. According to Barbalat’s lemma37, we can conclude thate → 0 ast → ∞.

From the discussion above, we have the following result.

Theorem 3.1. For the systems2.1and2.2, underAssumption 2.1, if the control law is designed as3.4, updating laws are chosen as3.9, then e will converge toB_e_σ in finite time, e → 0 ast → ∞, andδa2ji,δa1ji,δb2ji, andδb1jiremain bounded.

Remark 3.2. Since the control signal3.4contains the discontinuous sign functions, as a hard switcher, it may cause undesirable chattering. In order to avoid the chattering, the “sgn”

function can be replaced by a continuous functiontanhto remove discontinuity.

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4. Numerical Simulation

In this section, we present numerical results to verify the proposed synchronization approach.

Consider the following master chaotic system:

⎡

⎣x˙1

˙ x₂

˙ x₃

⎤

⎦

⎡

⎣ax2−x1 bx₁−cx₁x₃

−gx3 hx²₁

⎤

⎦

⎡

⎣−a a 0 b 0 0 0 0 −g

⎤

⎦

⎡

⎣x1

x₂ x₃

⎤

⎦

⎡

⎣0 0 0 0 −c 0 0 0 h

⎤

⎦

⎡

⎣ 0 x₁x₃

x²₁

⎤

⎦

⎡

⎣−a0 a₀ 0 b0 0 0 0 0 −g0

⎤

⎦

A1

⎡

⎣x₁ x2

x2

⎤

⎦

x

⎡

⎣−δa0 δa₀ 0 δb0 0 0 0 0 −δg0

⎤

⎦

ΔA1

⎡

⎣x₁ x2

x3

⎤

⎦

x

⎡

⎣0 0 0 0 −c0 0 0 0 h0

⎤

⎦

B1

⎡

⎣ 0 x₁x₃

x²₁

⎤

⎦

f1x

⎡

⎣0 0 0 0 −δc0 0 0 0 δh0

⎤

⎦

ΔB1

⎡

⎣ 0 x₁x₃

x²₁

⎤

⎦

f1x

,

4.1

whereaa0 δa0,bb0 δb0,cc0 δc0,gg0 δg0,hh0 hg0,a08,δa02,b035, δb₀5,c₀ 0.7,δc₀0.3,g₀2.0,δg₀0.5,h₀0.8, andδh₀0.2.

The slave system is given with

⎡

⎣y˙₁

˙ y₂

˙ y3

⎤

⎦

⎡

⎣a₁

y₂−y₁ y₂y₃ b₁y₂−c₁y₁y₃

g1y2−h1y3

⎤

⎦

⎡

⎣−a1 a₁ 0 0 b₁ 0 0 g1 −h1

⎤

⎦

⎡

⎣y₁ y₂ y3

⎤

⎦

⎡

⎣a₁ 0 0 0 −c1 0

0 0 0

⎤

⎦

⎡

⎣y₂y₃ y₁y₃ 0

⎤

⎦

⎡

⎣−a10 a₁₀ 0 0 b10 0 0 g₁₀ −h10

⎤

⎦

A2

⎡

⎣y₁ y2

y₃

⎤

⎦

y

⎡

⎣−δa10 δa₁₀ 0 0 δb10 0 0 δg₁₀ −δh10

⎤

⎦

ΔA2

⎡

⎣y₁ y2

y₃

⎤

⎦

y

⎡

⎣a10 0 0 0 −c10 0

0 0 0

⎤

⎦

B2

⎡

⎣y2y3

y₁y₃ 0

⎤

⎦

f2^y

⎡

⎣δa10 0 0 0 −δc10 0

0 0 0

⎤

⎦

ΔB2

⎡

⎣y2y3

y₁y₃ 0

⎤

⎦

f2^y

⎡

⎣u1

u₂ u₃

⎤

⎦,

4.2

wherea₁a₁₀ δa₁₀,b₁b₁₀ δb₁₀,c₁ c₁₀ δc₁₀,g₁g₁₀ δg₁₀,h₁h₁₀ hg₁₀,a₁₀0.8, δa₁₀ 0.2,b₁₀ 2.0,δb₁₀0.5,c₁₀ 0.7,δc₁₀ 0.3,g₁₀ 0.7,δg₁₀ 0.3,h₁₀ 3.0,δh₁₀ 1.0.

x0 1.8,y0 −1.2, andz0 1.5.

The initial states in master system4.1arex₁0 1.8,x₂0 −1.2,x₃0 1.5. The initial states in slave system4.2arey₁0 1.5,y₂0 1.2,y₃0 1.1. The initial parameter estimation values of the systems2.1and2.2areδa00,δb00,δc0 0,δg00,δh00, δa₁₀ 0,δb₁₀0,δc₁₀0,δg₁₀ 0, andδh₁₀ 0.

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0 50 100 150

−20 −15 −10 −5 0 5 10 15 20 25 x1

x3

Figure 1: Chaotic behavior of the master chaotic system under the proposed parameters.

According toRemark 3.2, the control law3.4is modified as follows:

u −Ke−K_De˙ α_s−C_A2−C_B2 C_A1 C_B1

−μ

2|δa₀| d_a0^{1 α} δb₀ d_b0_{1 α}

|δc₀| d_c0^{1 α} δg0 dg0

_{1 α} δh0 dh0

_{1 α}

3|δa10| da10^{1 α} δb10 db10

_{1 α}

|δc₁₀| d_c10^{1 α} δg₁₀ d_g10_{1 α} δh₁₀ d_h10_{1 α} α_e

−

c₁tanhεe1|e1|^α, . . . , c_ntanhεen|en|^α_T ,

4.3

where

C_A2 δa₁₀

y₂−y₁

, δb₁₀y₂, δg₁₀y₂−δh₁₀y₃!_T , C_B2

δa₁₀y₂y₃,−δc₁₀y₁y₃,0T

, C_A1 δa₀x2−x₁, δb₀x₁,−δg₀x₃!T

,

CB1 0,−δc0x1x3, δh0x₁²!_T

, Kdiag{70,54,30}, K_Ddiag{0.93,0.75,0.1}, μdiag{1,0.2,0.01},

K_Adiag{2.3,2.1,2.3}da02, d_b010, d_c0 1, d_g01, d_h0 2, da10 1, db102, dc102, dg10 2, dh10 2, ε40, c11, c23, c31.

4.4

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−5

−4

−3

−2

−1 0 1 2 3

y3

0 5 10 15 20 25 30

y1

Figure 2: Chaotic behavior of the slave chaotic system under the proposed parameters.

Choosing the updating law as

δa˙0

0.30e₁x1−x₂, if|a₀|< d_a0

0, otherwise , δb˙0

⎧⎨

⎩

−0.90e2x1, ifb0< db0

0, otherwise,

δc˙₀

−0.001e2x1x3, if|c0|< dc0

0, otherwise , δg˙₀

0.002e3x3, ifg0< dg0

0, otherwise,

δh˙₀

⎧⎨

⎩

−0.018e3x²₁, ifh₀< d_h0

0, otherwise , δa˙₁₀

0.0006e1

−y1 y2 y2y3

, if|a10|< da10

0, otherwise,

δb˙10

⎧⎨

⎩

−0.08e2y2, if b10< db10

0, otherwise , δc˙10

−0.0015e2y₁y₃, if |c₁₀|< d_c10

0, otherwise,

δg˙₀

0.1e3y2, if g10< dg10

0, otherwise , δh˙₁₀

⎧⎨

⎩

−0.017e3y₃, if h₁₀< d_h10

0, otherwise.

4.5

Chaotic behavior of the master chaotic system under the proposed parameters is shown in Figure 1. Chaotic behavior of the slave chaotic system under the proposed parameters is shown inFigure 2. From Figures1 and2, we know that the two systems are still chaotic under adopted uncertain parameters. The synchronization errors between two diﬀerent chaotic systems are illustrated in Figures3,4, and5, where the control inputs are activated att1s. One can see that the synchronization errors converge to the zero in a finite time, which implies that the chaos synchronization between the two diﬀerent chaotic systems

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−20

−15

−10

−5 5 0 10 15

e1

0 10 20 30 40 50 60

t

Figure 3: Synchronization errore1.

−30

−20

−10 0 10 20 30 40

e2

0 10 20 30 40 50 60

t

is realized. The time responses of parameter estimationsa0,b0,c0,g0, andh0are depicted in Figure 6. The time responses of parameter estimationsa₁₀,b₁₀,c₁₀,g₁₀, andh₁₀ are depicted inFigure 7.

According to the simulations, it has been shown that the proposed control algorithm provides stable behavior when using online adaptive laws. The control performance is satisfactory and the chattering phenomenon has been successfully improved by using tanh functions. In addition, it is easy to see that the parameter estimation values approach their real values in a finite time.

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−120

−100

−80

−60

−40

−20 0

10

0 20 30 40 50 60

e3

t

−2 0 2 4 6 8 10 12

Parameter estimations

0 10 20 30 40 50 60

t Estimation forδ a0

Estimation forδ b0

Estimation forδ c0

Estimation forδ g0

Estimation forδ h0

Figure 6: Curves of parameter estimations for the master system.

5. Conclusions

In this paper, we have studied chaos synchronization of two different chaotic systems with parameter uncertainties. The two different chaotic systems with parameter uncertainties are synchronized via robust adaptive control based on the Lyapunov stability theory and finite- time theory. The proposed method can be applied to a variety of chaos systems. It guarantees that all the error states are driven to zero in a finite time. Numerical simulations are given to show the proposed synchronization approach works well for synchronizing two different

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−4

−2 0 2 4 6 8

Parameter estimations

0 10 20 30 40 50 60

t Estimation forδ a10

Estimation forδ b10

Estimation forδ c10

Estimation forδ g10

Estimation forδ h10

Figure 7: Curves of parameter estimations for the slave system.

chaotic systems in a finite time, even when the parameters of both the master and slave systems are unknown.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant no. 60674090.

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Synchronization of Two Different Chaotic Systems with Parameter Uncertainties

Research Article

Robust Adaptive Finite-Time

Synchronization of Two Different Chaotic Systems with Parameter Uncertainties

Yun-An Hu, Hai-Yan Li, Chun-Ping Zhang, and Liang Liu

1. Introduction

2. System Description

3. Synchronization of Two Different Chaotic Systems with Parameter Uncertainties

4. Numerical Simulation

5. Conclusions

Acknowledgment

References

Submit your manuscripts at http://www.hindawi.com

Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific World Journal

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Decision Sciences

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Stochastic Analysis