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Robust adaptive tracking for a class of uncertain nonlinear systems with unmodeled dynamics. (Chinese. English summary) Zbl 1017.93052

The authors consider the robust adaptive tracking problem of a class of single-input and single-output uncertain nonlinear systems of the form \[ \begin{aligned} \dot x_j &= x_{j+1}+ f(x_1,x_2,\dots, x_j)+ \theta^T \varphi_j(x_1,x_2,\dots, x_j)+ \Delta_j(x, u,t),\quad j= 1,2,\dots, n-1,\\ \dot x_n &= u+f(x_1,x_2,\dots, x_n)+ \theta^T \varphi_n(x_1,x_2,\dots, x_n)+ \Delta_n(x, u,t),\quad y= x_1,\end{aligned} \] where \(x= (x_1,x_2,\dots, x_n)^T\in \mathbb{R}^n\), \(y\in \mathbb{R}\), and \(\theta\in \mathbb{R}^q\) being an unknown constant parameter, functions \(f_i\), \(\varphi_i\), \(\Delta_i\) \((i= 1,2,\dots, n)\) may be nonlinear and \(\Delta_i\) \((i= 1,2,\dots, n)\) are uncertain functions.
The assumptions used herein are: (1) \(f_i\), \(\varphi_i\in C^n\) and there exist functions \(\psi_i\in C^n\) such that \(|\Delta_i(x, u,t)|\leq p_i\psi_i(x_1,x_2,\dots, x_i)\), \(i= 1,2,\dots, n\), hold for some positive constant \(p_i\); and (2) the reference signal \(y_r\in C^n\), and \(y_r,y^{(1)}_r,\dots, y^{(n)}_r\) are uniformly bounded.
Adaptive controllers are obtained by the backstepping procedure. The boundedness of all signals and arbitrary small tacking error are guaranteed under the proposed control law.

MSC:

93C40 Adaptive control/observation systems
93D21 Adaptive or robust stabilization
93C10 Nonlinear systems in control theory
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