## 3 Main Results  
This section presents the main theorems and results, providing a general approach to designing smooth robust controllers for a system (1) to track a sufficiently smooth reference signal with error norm smaller than a prescribed value. **Theorem 1**. : _Consider the uncertain nonlinear dynamic system (1), satisfying conditions (1), (2) and controlled with the control law (18), i.e., the following:_  
\[ \begin{array}{l}\dot{x}=\left[\begin{array}{cc}0&I\\ -2a^{2}H&-2 aH\end{array}\right]x+\left[\begin{array}{cc}0\\ I\end{array}\right]w=Ax+Bw\\ e=\left[\begin{array}{cc}I&0\end{array}\right]x=C_{1}x,\ \dot{e}=\left[ \begin{array}{cc}0&I\end{array}\right]x=C_{2}x\end{array}  \tag{34}\]  
_where:_  
\[H=FG,\ w=\tilde{r}-f-Fu_{c} \tag{35}\]  
_Then, if \(\tilde{r}\) is bounded and the possible compensation signal \(u_{c}\) is such that \(\|\tilde{r}-f-Fu_{c}\|\leq\|\tilde{r}-f\|\) a majorant system of the closed loop system (34) is:_  
\[ \begin{array}{l}\dot{\rho}=-a\rho+\psi_{\gamma_{c}}(\rho)=\chi(\rho)\\ \|e\|=\frac{\rho}{a},\ \|\dot{e}\|=\sqrt{2}\rho\end{array}  \tag{36}\]  
_where \(\rho=\|x\|_{P}\), with:_  
\[P=\left[ \begin{array}{cc}2a^{2}I&aI\\ aI&I\end{array} \right],\ a>0 \tag{37}\]  
_and \(\psi_{\gamma_{c}}\) is a function of class \(K_{\gamma_{c}}\), with \(\gamma_{c}=\max\|\tilde{r}\|+\varphi_{\gamma_{0}}\bigg{(}\max\|\left[r^{T}\ \dot{r}^{T}\right]^{T}\|_{P}/\lambda_{\min}(P)\bigg{)}\)._  
**Proof. ** By choosing as \"Lyapunov function\" the quadratic form \(x^{T}Px=\left\|x\right\|_{P}^{2}=\rho^{2}\), for \(x\) belonging to the generic hyper-circumference \(C_{P,\rho}\), its derivative satisfies the following inequalities:  
\[2\rho\dot{\rho}\leq\max\left(x^{T}Qx+2x^{T}PBw\right)\leq\max\left(x^{T}Qx \right)+\max\left(2x^{T}PBw\right),\;Q=A^{T}P+PA \tag{38}\]  
from which, for (26) and (27) of Lemma 1, it follows that:  
\[\dot{\rho}\leq\max\lambda(QP^{-1})\rho/2+\sqrt{\lambda_{\max}(B^{T}PB)}\max \left\|w\right\| \tag{39}\]  
Since:  
\[P^{-1}=\left[ \begin{array}{cc}I/a^{2}&-I/a\\ -I/a&2I\end{array} \right] \tag{40}\]  
if \(\lambda_{\min}\big{(}H+H^{T}\big{)}\geq 2\), after certain mathematical steps, then:  
\[\max\lambda(QP^{-1})=\max\lambda\left[A^{T}+PAP^{-1}\right]=\max\lambda\left( \left[ \begin{array}{cc}-2 aI&2a^{2}(2I-H-H^{T})\\ 0&-2a(H+H^{T}-I)\end{array} \right]\right)=-2a \tag{41}\]  
Moreover, it is easy to verify that:  
\[\sqrt{\lambda_{max}(B^{T}PB)}=1,\;\sqrt{\lambda_{max}(C_{1}P^{-1}C_{1}^{T})}= \frac{1}{a},\;\sqrt{\lambda_{max}(C_{2}P^{-1}C_{2}^{T})}=\sqrt{2} \tag{42}\]  
hence:  
\[\dot{\rho}\leq-a\rho+\max\|w\| \tag{43}\]  
The proof follows from (43), from (32) of Lemma 5 and by noting that, for \(x\) belonging to the generic hyper-circumference \(C_{P,\rho}\) from (28) of Lemma 1 and (42) it is:  
\[\|e\|\leq\frac{\rho}{a},\|\dot{e}\|\leq\sqrt{2}\rho \tag{44}\]  
\(\square\)  
Theorem 1 allows one to establish the following significant theorems. **Theorem 2. **_Consider the system (34). If hypotheses of Theorem 1 are satisfied then, for \(\forall a\geq\overset{\sim}{a}\), the following inequalities hold:_  
\[ \begin{array}{l}\rho(t)\leq\rho_{1},\forall\rho_{0}\leq\rho_{1};\underset{t \rightarrow\infty}{\lim}\rho(t)\leq\rho_{1},\forall\rho_{0}\in[\rho_{1},\rho_ {2})\\ \|e(t)\|\leq\frac{\rho_{1}}{a},\forall\rho_{0}\leq\rho_{1};\underset{t \rightarrow\infty}{\lim}\|e(t)\|\leq\frac{\rho_{1}}{a},\forall\rho_{0}\in[ \rho_{1},\rho_{2})\\ \|\dot{e}(t)\|\leq\sqrt{2}\rho_{1},\forall\rho_{0}\leq\rho_{1};\underset{t \rightarrow\infty}{\lim}\|\dot{e}(t)\|\leq\sqrt{2}\rho_{1},\forall\rho_{0}\in[ \rho_{1},\rho_{2})\end{array}  \tag{45}\]  
_where \(\overset{\sim}{a}\) is the value of a for which the line \(\dot{\rho}=a\rho\) is tangent to the curve \(\dot{\rho}=\psi_{\gamma_{c}}(\rho)\) and \(\rho_{1},\;\rho_{2},\;\rho_{1}<\rho_{2}\) are the smallest equilibrium points of the majorant system (36) (see Figure 2)._  
**Proof.