an:05563201
Zbl 1183.34072
Wang, Hua; Han, Zheng-Zhi; Xie, Qi-Yue; Zhang, Wei
Finite-time synchronization of uncertain unified chaotic systems based on CLF
EN
Nonlinear Anal., Real World Appl. 10, No. 5, 2842-2849 (2009).
00249623
2009
j
34D06 34C28 34H05
finite-time synchronization; unified chaotic systems; uncertain parameters; control Lyapunov function
Consider the master system
\[
\begin{aligned} \dot x_1 & = (25\alpha+ 10)(x_2- x_1),\\
\dot x_2 & = (28-35\alpha) x_1- x_1 x_3+ (29\alpha- 1)x_2,\\
\dot x_3 & = x_1 x_2- {(8+ \alpha)\over 3} x_3.\end{aligned}\tag{1}
\]
For \(\alpha\in [0,1]\) system (1) is chaotic, for certain \(\alpha\)-values it is related to the Lorenz, L?? and Chen system. Representing (1) in the form \(\dot x= f(x,\alpha)\), the authors consider together with (1) the slave system \(\dot y= f(y,\alpha)+ u\). The goal of the authors is to find a control \(u\) such that the slave system synchronizes the master system in finite time, that is, the corresponding error system
\[
\dot e=\widetilde f(e,y,\alpha)+ u\text{ with }e= y- x
\]
has the property that their solutions tend to zero in a finite time. Of course, this requires that the error system is not Lipschitzian in \(e\). The authors construct such a control by means of a control Lyapunov function. Moreover, they show that this control is robust against perturbations of some coefficients of (1).
Klaus R. Schneider (Berlin)