## 3 Main Results
** The proof easily follows by using the Lyapunov function \(V=\rho^{2}\) or by noting that in the interval \((\rho_{1},\ \rho_{2})\ \rho(t)\) is decreasing (see Figure 2, as well). \(\square\)  
It is worth noting that for sufficiently large \(a\) from Theorem 2 it is \(\rho_{1}\cong\gamma_{c}/a\). Hence, for sufficiently large \(a\), if \(\|x(0)\|_{P}\leq\rho_{1}\) then:  
\[\|e(t)\|\leq\frac{\rho_{1}}{a}\cong\frac{\gamma_{c}}{a^{2}},\|\dot{e}(t)\|\leq \sqrt{2}\rho_{1}\cong\frac{\sqrt{2}\gamma_{c}}{a},\forall t\geq 0 \tag{46}\]  
**Remark 3**. : _From (46) and Theorem 2 it follows that, for sufficiently large \(a\) and for sufficiently large \(t\), \(\forall\|x_{0}\|_{P}<\rho_{2}\) the error norm decreases with respect to parameter \(a\) with an almost quadratic behavior, and the derivative error norm decreases, always with respect to parameter \(a\), with an almost linear behavior._  
**Remark 4**. : _Note that the time constant \(\tau_{l}\) of the linearized model of the majorant system \(\dot{\rho}=\chi(\rho)\) (36) around the equilibrium point \(\rho_{1}\) for sufficiently large \(a\) is \(\tau_{l}\cong 1/a\)._  
By integrating the equation \(\dot{\rho}=\chi(\rho)\) with initial condition \(\rho_{2}^{-}<\rho_{2}\) it is possible to obtain the \"half time\" \(t_{50\%}\) and a \"settling time\" \(t_{5\%}\) for any \(\rho_{0}<\rho_{2}^{-}\) shown in Figure 3.  
Figure 3: Illustration of the times \(t_{50\%}\) and \(t_{5\%}\). **Remark 5**. : _Note that if \(F\) and \(f\) are independent of \(p\) and \(f(t,y,\dot{y},d)=\overline{f}(t,y,\dot{y})+d\) by using the linearizing control law:_  
\[u=F^{\dagger}(h2a^{2}le+h2 aI\dot{e}+\tilde{r}-\overline{f}), \tag{47}\]  
_where \(F^{\dagger}=F^{T}{(FF^{T})}^{-1}\) is the pseudo-inverse of \(F\), the closed-loop control system (1), (47) becomes:_  
\[\dot{x}=\left[ \begin{array}{cc}0&I\\ -h2a^{2}I&-h2 aI\end{array} \right]x+\left[ \begin{array}{cc}0\\ I\end{array} \right](-d),\ e=\left[ \begin{array}{cc}I&0\end{array} \right]x, \ \dot{e}=\left[ \begin{array}{cc}0&I\end{array} \right]x. \tag{48}\]  
It is easy to prove that the control system (48) has the following properties:  
(a): if \(h=1:\)  
- behavior of a Butterworth filter with angular frequency \(\omega_{b}=\sqrt{2}a\);  
- time constant \(\tau=1/a\);  
- overshoot equal to \(4.32\%\);  
- damping ratio \(\zeta=\sqrt{2}/2\);  
- forced responses:  
\[ \begin{array}{ll}\|e(t)\|<\int_{0}^{\infty}\left|e^{-at}\sin(at)dt\right| \cdot\max\|\tilde{r}\|\leq\frac{0.545}{a^{2}}\max\|d\|\\ \|\dot{e}(t)\|<\int_{0}^{\infty}\left|\frac{d}{dt}\big{(}e^{-at}\sin(at)\big{)} dt\right|\cdot\max\|d\|\leq\frac{0.674}{a}\max\|d\|.\end{array}  \tag{49}\]  
(b): if \(h>1\):  
- cutoff angular frequency \(\omega_{b}\in[a,\ \sqrt{2}a]\);  
- max time constant \(\tau_{\max}\in[1/2a,\ 1/a]\);  
- overshoot \(\leq 4.32\%\);  
- damping ratio \(\zeta\leq\sqrt{2}/2\);  
- forced responses:  
\[ \begin{array}{ll}\|e(t)\|\leq\frac{[0.5,\ 0.545]}{ha2}\max\|d\|\\ \|\dot{e}(t)\|\leq\frac{[0.674,\ 1]}{ha}\max\|d\|.\end{array}  \tag{50}\]  
The proposed control law is also robust with respect to a possible bounded measurement noise \(n_{\dot{y}}\) on \(\dot{y}\) as it is shown by the following theorem. **Theorem 3**. : _Consider the uncertain nonlinear dynamic system (1), satisfying conditions (1), (2), and controlled with the control law (18). If \(\dot{y}\) is affected by a bounded measurement noise \(n_{\dot{y}}\), the rate of \(\|e\|\) due to the noise \(n_{\dot{y}}\), for sufficiently large \(a\) is inversely proportional to \(a\)._  
Proof. : If \(\dot{y}\) is affected by a bounded measurement noise \(n_{\dot{y}}\) the term \(\|2 aHn_{\dot{y}}\|\) must be added to \(\gamma_{c}\). From (46), for sufficiently large \(a\), \(\|e\|\) is inversely proportional to \(a^{2}\). Hence, the rate of \(\|e\|\) due to the noise \(n_{\dot{y}}\), for sufficiently large \(a\) is inversely proportional to \(a\). **Remark 6**. : _To design the controller it is essential to determine a matrix \(G\) satisfying condition (2).