## Adaptive dynamic surface control: A simplified algorithm for adaptive backstepping control of nonlinear systems.(English)Zbl 0969.93037

The authors discuss nonlinear uncertain systems of the form \begin{aligned} x_j &= x_{j+1}+ a_j f_j(x_1,\dots, x_j),\quad j= 1,\dots, n-1,\\ x_n &= u+ a_nf_n(x_1,\dots, x_n),\quad y= x_1,\end{aligned} where $$a_i$$, $$i= 1,\dots, n$$, are the unknown constant parameters and $$f_i: \mathbb{R}^i\to \mathbb{R}$$ are $$C^1$$-functions with $$f_i(0,\dots, 0)= 0$$.
By extending the dynamic surface control technique due to D. Swaroop et al. [Proceedings of the 1997 American Control Conference, Albuquerque, NM (1997)], the authors propose a new algorithm for adaptive backstepping control of the above system. Since one adds first-order low pass filters, this algorithm can be implemented without differentiating any model nonlinearities. Using a singular perturbation theorem in H. K. Khalil [Nonlinear systems, New York, Macmillan (1992)], the combined adaptive backstepping first-order filter system is proven to be semi-globally stable for sufficiently fast filters. A detailed numerical example is also given.

### MSC:

 93D21 Adaptive or robust stabilization 93C40 Adaptive control/observation systems 93C70 Time-scale analysis and singular perturbations in control/observation systems 93B51 Design techniques (robust design, computer-aided design, etc.) 93C10 Nonlinear systems in control theory
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