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Arabian Journal for Science and Engineering

, Volume 44, Issue 8, pp 7335–7350 | Cite as

Multi-Switching Combination Anti-synchronization of Unknown Hyperchaotic Systems

  • Muhammad Shafiq
  • Israr AhmadEmail author
Research Article - Systems Engineering

Abstract

The security of the information signal in digital communication systems is an active research problem. This paper proposes a new robust adaptive anti-synchronization control (RAASC) technique and studies the multi-switching combination anti-synchronization (MSCAS) of different unknown hyperchaotic (UH) systems. This controller reduces the overshoots in the transient phase and establishes the anti-synchronization convergence quickly. The proposed MSCAS scheme anti-synchronizes two master UH systems with a slave UH system using the multi-switching mechanism that increases the complexity and enhances the security of the information signal in digital communications. The Lyapunov direct method assures the robust stability of the closed-loop dynamical system. This work provides adaptive laws that estimate the true values of unknown parameters. A numerical example of three different hyperchaotic Pang, Lu and Lorenz–Stenflo systems is simulated. The simulation results verify the theoretical findings. The proposed MSCAS approach is used for the secret communication of digital messages using the chaotic masking technique. The receiver retrieves the information signal with high security and good precession qualitatively, under the effect of different types of disturbances. This paper also discusses the implications of controller constituents of the proposed RAASC algorithm that establishes fast convergence. The article suggests some future research problems related to this work.

Keywords

Multi-switched combined anti-synchronization Robust adaptive controller Lyapunov stability theory Hyperchaotic systems Unknown parameters 

List of Symbols

\(R^{n}\)

n-dimensional Euclidean space

T

Transpose of a vector (or matrix)

\(i,j,k,r,l\in Z^{+}\)

Positive numbers for subscripts

\(M_1\)

The first master hyperchaotic system

\(M_2\)

The second master hyperchaotic system

S

Stands for the slave hyperchaotic system

\({{\varvec{u}}}=( {u_1 ,u_2 ,\ldots ,u_n } )^\mathrm{T}\in R^{n}\)

State variables of \(M_1\)

\({{\varvec{v}}}=\left( {v_1 ,v_2 ,\ldots ,v_n } \right) ^\mathrm{T}\in R^{n}\)

State variables of \(M_2\)

\({{\varvec{w}}}=\left( {w_1 ,w_2 ,\ldots ,w_n } \right) ^\mathrm{T}\in R^{n}\)

State variables of S

\(G_1 ,G_2 ,G_3 :R^{n}\rightarrow R^{n}\)

Continuous nonlinear vector functions in \(M_1,M_2\), and S, respectively

\(G_{ir} (.)\in R^{1\times n}\)

rth element of the vector \(G_i(.)\in R^{n\times 1} \, ({i=1,2,3})\)

\(a=( {a_1 ,a_2 ,\ldots ,a_n})^\mathrm{T}\in R^{n}\)

Unknown parameter vector of \(M_1\)

\(A_a\)

An \(n\times n\) matric containing the unknown parameter vector a of \(M_1\)

\(A_{ai}\)

An ith row of the matrix \(A_a \in R^{n\times n}\)

\(b=\left( {b_1 ,b_2 ,\ldots ,b_n } \right) ^\mathrm{T}\in R^{n}\)

Unknown parameter vector of \(M_2\)

\(B_b\)

\(n\times n\) matrix containing the unknown parameter vector b of \(M_2\)

\(B_{bj}\)

jth row of the matrix \(B_b \in R^{n\times n}\)

\(c=\left( {c_1 ,c_2 ,\ldots ,c_n } \right) ^\mathrm{T}\in R^{n}\)

Unknown parameter vector of S

\(C_c\)

\(n\times n\) matrix containing the unknown parameter vector c of S

\(C_{ck}\)

kth row of the matrix \(C_c \in R^{n\times n}\)

\(Q_i \in R^{n\times 1}\)

ith row of the vector \(Q\in R^{n\times n}\)

\(f({{\varvec{u}}})=( f_1 ( {u_1 }),f_2({u_2}),\ldots ,\)

Unknown model

   \(f_n ( {u_n }))^\mathrm{T}\in R^{n\times 1}\)

   uncertainty in \(M_1\)

\(g({{\varvec{v}}})=(g_1 ({v_1}),g_2({v_2}),\ldots ,\)

Unknown model

   \(g_n ({v_n}))^\mathrm{T}\in R^{n\times 1}\)

   uncertainty in \(M_2 \)

\(h({{\varvec{v}}})=( h_1 ( {w_1 } ),h_2( {w_2 } ),\ldots ,\)

Unknown model

   \(h_n ( {w_n } ))^\mathrm{T}\in R^{n\times 1}\)

   uncertainty in S

\({\varvec{\rho }} =( {\rho _1 ,\rho _2 ,\ldots ,\rho _n })^\mathrm{T}\in R^{n\times 1},\)

Unknown external disturbance vector in \(M_1\)

\({\varvec{\sigma }} =( {\sigma _1 ,\sigma _2 ,\ldots ,\sigma _n })^\mathrm{T}\in R^{n\times 1}\)

Unknown external disturbance vector in \(M_2\)

\({\varvec{\delta }} =( {\delta _1 ,\delta _2 ,\ldots ,\delta _n })^\mathrm{T}\in R^{n\times 1}\)

Unknown external disturbance vector in S

\({\varvec{\mathcal{F}}}{=}( {\mathcal{F}_1 ,\mathcal{F}_2 ,\ldots ,\mathcal{F}_n \,})^\mathrm{T}{\in } R^{n\times 1}\)

Control input vector

\({{\varvec{e}}}\in R^{n\times n\times n}\)

MSCAS error vector

\(e\in R^{n\times 1}\)

A column vector, whose elements are chosen arbitrarily from \({{\varvec{e}}}\in R^{n\times n\times n}\)

\({{\varvec{e}}}_{ijk} \in R\)

An ith element of \({{\varvec{e}}}\in R^{n\times n\times n}\), chosen arbitrarily

\(X{=}\mathrm{diag}{[} {\alpha _1 ,\alpha _2 ,\ldots ,\alpha _n }{]}{\in } R^{n\times n}\)

Scaling matrix

\(Y{=}\mathrm{diag}{[} {\beta _1 ,\beta _2 ,\ldots ,\beta _n }{]}{\in } R^{n\times n}\)

Scaling matrix

\(Z{=}\mathrm{diag}{[} {\gamma _1 ,\gamma _2 ,\ldots ,\gamma _n }{]}{\in } R^{n\times n}\)

Scaling matrix

I

\(n\times n\) identity matrix

\(H^{M_1 M_2 S}( {{{\varvec{u,v,w}}}} )\in R^{1\times n}\)

A residual matrix of the nonlinear terms associated with \(M_1 ,M_2\), and S

\(H_i^{M_1 M_2 S} ( {{{\varvec{u,v,w}}}} )\)

An ith element of   \(H^{M_1 M_2 S}( {{{\varvec{u,v,w}}}} )\in R^{1\times n}\)

exp

The exponential function

sgn

The signum function

\(\circ \)

Hadamard product operator

\(D_{\hat{d}e} \in R^{n\times n}\)

\(D_{\hat{d}e}\) is a diagonal matrix, whose diagonal elements are the elements of column vector \(\hat{d}\circ \mathrm{exp}\,({-p| e |} )\)

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Copyright information

© King Fahd University of Petroleum & Minerals 2019

Authors and Affiliations

  1. 1.Department of Electrical and computer EngineeringSultan Qaboos UniversityMasqatOman
  2. 2.Department of General RequirementsCollege of Applied SciencesNizwaOman

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