On modular backstepping design with second order sliding modes (Q1592904)

The paper deals with systems in the parametric-strict feedback form \[ \begin{aligned} x_i'' &=x_{i+1}+\theta \Phi _i(x_1,\dots ,x_i),\quad i=1,\dots ,n-1, \\ x_n'' &=\Phi _0(x_1,\dots ,x_n)+\theta \Phi _n(x_1,\dots ,x_n)+\beta (x_1,\dots ,x_n)u,\end{aligned} \] where \(\theta \) is a vector of constant unknown parameters, \(\Phi _i\) and \( \beta \) are known smooth nonlinear functions. The control objective is to make \(\gamma =x_1(t)\) track a smooth reference trajectory \(\gamma _r(t).\) A modular procedure to design estimation-based second order sliding mode controllers for uncertain nonlinear systems expressible in parametric-strict feedback form is presented. It consists of \(n-2\) steps of modular backstepping to compute all the quantities needed to build a second order auxiliary system. In particular, the authors propose a desired algorithm which is a modification of the Bartolini algorithm [\textit{G. Bartolini, A. Ferrara} and \textit{E. Usai}, Int. J. Robust Nonlinear Control 7, 299-319 (1997)]. Convergence of the desired algorithm is proved. A numerical example is considered.