Control of chaotic synchronization in composite systems and its applications of secure half-duplex communication systems (Q2725090)

The author presents a control design method for chaotic synchronization of some composite systems with several workstations. Suppose that \(S\) is the composite system consisting of the subsystems \(S_i\), NEWLINE\[NEWLINES_i:\dot x_i= Ax_i+ f(x_i)+ Bu_i+ g(x_1,x_2,\dots, x_N),\quad i= 1,2,\dots, N,\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINES:\dot x=\overline Ax+\overline f(x)+\overline B u+\overline g(x),\tag{2}NEWLINE\]NEWLINE where \(x_i\in \mathbb{R}^n\), \(u_i\in \mathbb{R}^m\), \(A\in \mathbb{R}^{n\times n}\), \(B\in\mathbb{R}^{n\times m}\), \(x= (x^T_1,\dots, x^T_N)^T\) and \(u= (u^T_1,\dots, u^T_N)^T\in \mathbb{R}^{mN}\) are the state-vector and input-vector of (2) respectively, and \(f: \mathbb{R}^n\to \mathbb{R}^n\), \(g: \mathbb{R}^{nN}\to \mathbb{R}^n\) are continuous functions.NEWLINENEWLINENEWLINEApplying the linear control theory and the Lyapunov function method, two sufficient conditions for the synchronization of all the subsystems (2) are given as follows:NEWLINENEWLINENEWLINE(i) \((A,B)\) is controllable, and the function \(h(x_i)\) satisfies a Lipschitz condition;NEWLINENEWLINENEWLINE(ii) \(A\equiv 0\) and there exists a smooth function \(h(x)\) with values in \(\mathbb{R}^m\) which satisfies: \(f(x)+ Bh(x)\) has the linearization \(Mx+ L\), where \(L\in \mathbb{R}^n\) and \(M\in \mathbb{R}^{n\times n}\) is a stable matrix.NEWLINENEWLINENEWLINESome results on the synchronization of all the subsystems (2) controlled by a chosen principal subsystem are also indicated. An application of the main results in secure half-duplex communication is discussed in detail by using \textit{Wu-Chua's} circuit [Int. J. Bifur. Chaos 4, 974-998 (1994)].