Abstract
In this paper, new control approaches for synchronization the master and the slave chaotic systems is established by means of novel coupled chaotic synchronous observers and coupled chaotic adaptive synchronous observers. The simultaneous estimation of the master and the slave systems’ states is accomplished, by means of the proposed observers for each of the master and the slave systems, to produce error signals between these estimated states. This estimated synchronization error signal and the state-estimation errors converge to the origin by means of a specific observers-based feedback control signal to ensure synchronization as well as state estimation. Using Lyapunov stability theory, nonadaptive and adaptive control laws and properties of nonlinearities, a convergence condition for the state-estimation errors and the estimated synchronization error is developed in the form of nonlinear matrix inequalities. Solution of the resulted inequality constraints using a two-step approach is presented, which provides the necessary and sufficient condition to obtain values of the observer gain and controller gain matrices. Further, a method requiring less computational efforts for solving the matrix inequalities for obtaining the observer and the controller gain matrices using decoupling technique is also proposed. Numerical simulation of the proposed synchronization technique for FitzHugh–Nagumo neuronal systems is illustrated to elaborate efficaciousness of the proposed observers-based control methodologies.
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This work was supported by Higher Education Commission (HEC) of Pakistan by supporting Ph.D. studies of the first author through indigenous Ph.D. scholarship program (phase II, batch II, 2013).
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Appendices
Appendix 1: Proof of Theorem 2
Using (10)–(12), CCAS observers (22)–(23) and systems (1)–(2) reveal the error systems as
Applying \(\tilde{\theta }_\mathrm{m} (t)=\theta _\mathrm{m} -\hat{{\theta }}_\mathrm{m} (t)\) and \(g_\mathrm{m} (x_\mathrm{m} (t))=Bg(x_\mathrm{m} (t))\theta _\mathrm{m} \) and, further, employing the mathematical fact
we obtain
Similarly, it is implicit to obtain
Using \(u_g =g(\hat{{x}}_\mathrm{m} (t))\hat{{\theta }}_\mathrm{m} -g\left( {\hat{{x}}_\mathrm{s} (t)} \right) \hat{{\theta }}_\mathrm{s} \), we have
Consider the Lyapunov function
Its time derivative along (39)–(41), by employing \(B^\mathrm{T}P_\mathrm{m} -R_\mathrm{m} C=0\) and \(B^\mathrm{T}P_\mathrm{s} -R_\mathrm{s} C=0\), is given by
Using Assumption 1 for positive scaling factors \(\beta _1 \) and \(\beta _2 \), we have
Employing the above inequalities, using (43), \(\dot{\tilde{\theta }}_\mathrm{m} (t)=-\dot{\hat{{\theta }}}_\mathrm{m} (t)\) and \(\dot{\tilde{\theta }}_\mathrm{s} (t)=-\dot{\hat{{\theta }}}_\mathrm{s} (t)\) and incorporating the adaptation laws (24)–(25) under Assumption 2, it implies that
which further reveals
If (26) is satisfied, the above inequality (44) implies \(\dot{V}(t) < 0 \). Hence, the errors \(e_\mathrm{m} (t), e_\mathrm{s} (t)\) and \(e_\mathrm{o} (t)\) converge to the origin, which entails synchronization of the master and the slave chaotic oscillators. \(\square \)
Appendix 2: Proof of Theorem 4
Applying the congruence transformation, that is, by pre- and post- multiplying (33) by \(\mathrm{diag} (I_n , I_n ,I_n ,\beta _1 I_n ,\alpha _1 I_n ,)\), where \(\beta _1 =1/{\eta _1 \bar{{\beta }}_1 }\) and \(\alpha _1 =1/{\eta _1 \bar{{\alpha }}_1 }\) for an appropriate number \(\eta _1 \), the resultant matrix inequality
is obtained. Employing Schur complement obtains
Similarly, by using \(\beta _2 =1/{\eta _2 \bar{{\beta }}_2 }\), and \(\alpha _2 =1/{\eta _2 \bar{{\alpha }}_2 }\) for a scalar \(\eta _2 \), and following the same procedure as above, the matrix inequality (34) can be modified as
By application of congruence transformation to (35) using \(\mathrm{diag}(I_n ,I_n ,\alpha _3 I_n ,\alpha _3 I_n ,)\), where \(\alpha _3 =1/{\bar{{\alpha }}_3 }\), the resultant inequality is obtained as
By applying Schur complement, we achieve
Since we have
Consequently, we obtain
By considering \(H_3 =F\bar{{P}}_\mathrm{o} \) and applying congruence transformation by \(\mathrm{diag}(P_\mathrm{o} , P_\mathrm{o} )\), it results into
By lumping together the linear matrix inequalities (46), (47), and (48) and, further, using \(H_1 =\tilde{P}_\mathrm{m} L_\mathrm{m} \) and \(H_2 =\tilde{P}_\mathrm{s} L_\mathrm{s} \), it produces
We can regenerate the matrix inequality (26), by pre- and post-multiplying (49) by \( [\mathbb {I}_1^\mathrm{T} , \mathbb {I}_4^\mathrm{T} , \mathbb {I}_7^\mathrm{T} , \mathbb {I}_2^\mathrm{T} , \mathbb {I}_5^\mathrm{T} , \mathbb {I}_8^\mathrm{T} , \mathbb {I}_3^\mathrm{T} , \mathbb {I}_6^\mathrm{T} ]^\mathrm{T}\) and its transpose, respectively, where \(\mathbb {I}_{\alpha } \) is the matrix generated by replacing the ith \(0_{n\times n} \) with \(I_n \) in \(0_{n\times 8n} \) matrix (for example \(\mathbb {I}_2 = [0_{n\times n} , I_n, 0_{n\times n} , 0_{n\times n} , 0_{n\times n}, 0_{n\times n},0_{n\times n} ,0_{n\times n} ])\) and substituting \(\tilde{P}_\mathrm{m} =\eta _1 ^{-1}P_\mathrm{m} \),\(\tilde{P}_\mathrm{s} \,{=}\,\eta _1 ^{-1}P_\mathrm{s} \), \(\bar{{P}}_\mathrm{o} \,{=}\,P_\mathrm{o}^{-1} \), \(\bar{{\alpha }}_1 \,{=}\,1/{(\eta _1 \alpha _1 )}\), \(\bar{{\alpha }}_2 =1/{(\eta _2 \alpha _2 )}\), \(\bar{{\alpha }}_3 \,{=}\,\alpha _3 ^{-1}\), \(\bar{{\beta }}_1 \,{=}\,1/{(\eta _1 \beta _1 })\), and \(\bar{{\beta }}_2 \,{=}\,1/{(\eta _2 \beta _2 })\).
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Siddique, M., Rehan, M. A concept of coupled chaotic synchronous observers for nonlinear and adaptive observers-based chaos synchronization. Nonlinear Dyn 84, 2251–2272 (2016). https://doi.org/10.1007/s11071-016-2643-2
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DOI: https://doi.org/10.1007/s11071-016-2643-2