I would like to majorate some given vector of values, but I does not accomplish it. I will describe my problem below:
Consider a differential equation $\dot s = \dot s_r + f_s + M_s \, u$, $s \in \mathbb{R}^m$. Than, a vector to bring it to zero is $u = - M_s^{-1}( \dot{s}_r + f_s + K \, sgn(s))$ (Post-edited: K was with - signal), for a positive matrix K. Adopt e.g. a diagonal matrix diag$(k_{ii})$, $k_{ii} > 0$. Hence, For parameters $M_s$ and $f_s$ identical to those of differential equation, than $\dot{s} = - K sgn(s)$ and s converge to 0 in finite time.
In case the terms does not coincide (adopt the hat signal for approximate parameter), than, $\dot{s} = -\Delta (\dot{s}_r + \hat{f}_s)- (I + \Delta) K sgn(s)$, such that $M_s \hat{M}_s^{-1} = I + \Delta$. The variable $s_r$ notoriously does not depend on estimates as $f_s$ and $M_s$. The literature below recommends one to consider $\dot{s}_i = -\eta_i \, sgn(s_i)$. The equality signal might be changed by $\leq$. Therefore, it accomplishes certain form to obtain $K$ by means of the maximum module of $f_s - \hat{f}_s$ i.e. $\mid f_s - \hat{f}_s\mid \leq F_s$ and $\mid \Delta \mid \leq D$.
I am a humble man, so I try to deduce step by step to learn in the process. My steps follow below:
Substitute the term provided by Slotine in the original equation and isolate K in the equality. Than
\begin{equation} (I + \Delta) K sgn(s) = H sgn(s) -\Delta (\dot{s}_r + \hat{f}_s) \end{equation}
Obtain the module of both sides and obtain $\mid(I + \Delta) K sgn(s) \mid= \mid H \,\, sgn(s) -\Delta (\dot{s}_r + \hat{f}_s) \mid$ and apply Cauchy-Schwarz theorem (CHT). The RHS is lesser than $\eta + D \, \mid \dot{s}_r + \hat{f}_s \mid$. Than $\mid (I + \Delta) k \mid \leq \eta + D\mid \dot s_r + \hat{f}_s \mid$. My guess is that I should also apply CHT on the left side, but than I could violate the inequality since the value is greater than the original one. Any contribution is appreciated.
I thank in advance. Best regards.
Slotine, J. J. E., & Li, W. (1991). Applied nonlinear control (Vol. 199, No. 1). Englewood Cliffs, NJ: Prentice hall.