Global adaptive stabilization of cascade nonlinear systems
The paper deals with global adaptive stabilization for systems of the form \[ \begin{aligned} \dot x & = Fx+ H(\xi)x+ h^T(\xi)\theta+ g^T(x,\xi,y)y,\\ \dot\xi & = A\xi+ b^T(x,\xi,y)\theta y,\\ \dot y & = \varphi^T(x,\xi,y)\theta+ u\end{aligned} \] with \(H(0)= 0\), \(h(0)=0\); \(\theta\in\mathbb{R}^p\) is the vector of the unknown parameters. The authors use a Lyapunov function for the zero dynamics with respect to \(y\) and obtain adaptive stabilization by using a modified Lyapunov function of the form \[ V_\theta(x,\xi,y,\widetilde\theta)= W_\theta(x,\xi)+ \textstyle{{1\over 2}} y^2+ \textstyle{{1\over 2}} |\widetilde\theta|^2, \] where \(\widetilde\theta\) is the estimation error for the parameter \(\theta\) and \(W\) is the Lyapunov function associated to the stable zero dynamics with respect to \(y\).
scientific article; zbMATH DE number 1008768