## 2 Problem Formulation and Preliminaries
A possible control law, to force plant (1) to track a reference signal \(r(t)\in R^{m}\) with bounded second derivative, is:  
\[u(t)=G(t,y,\dot{y})\left(2a^{2}e(t)+2a\dot{e}(t)\right)+u_{c} \tag{18}\]  
where \(e(t)=r(t)-y(t)\), \(G\in R^{r\times m}\), \(a\in R^{+}\) and \(u_{c}\) is a possible compensation signal, e.g., \(u_{c}=F^{\dagger}(t,\dot{\zeta},\dot{p})\Big{(}\ddot{r}-f(t,\dot{\zeta},\dot{ d},\dot{p})\Big{)}\) where \(\zeta=\left[ \begin{array}{cc}y^{T}&\dot{y}^{T}\end{array} \right]^{T}\) or \(\zeta=\left[ \begin{array}{cc}r^{T}&\dot{r}^{T}\end{array} \right]^{T}\), \(F^{\dagger}=F^{T}(FF^{T})^{-1}\) is the pseudo-inverse of \(F\) and \(\dot{p}\), \(\dot{d}\) are the nominal values of \(p\), \(d\). **Remark 1**. : _A reference signal \(r(t)\) with bounded second derivative can be obtained by interpolating a set of given points \((t_{k},\;r_{k}),\;k=0,\;1,\;\ldots,\;n\) with quadratic or cubic splines or by filtering any piecewise constant or piecewise linear signal \(\ddot{r}(t)\) with the following MIMO filter:_  
\[\dot{\zeta}=\left[ \begin{array}{ccc}0&I&0\\ 0&0&I\\ -f_{1}I&-f_{2}I&-f_{3}I\end{array} \right]\zeta+\left[ \begin{array}{c}0\\ 0\\ 0\\ f_{1}I\end{array} \right]\left[ \begin{array}{c}r\\ \dot{r}\\ \ddot{r}\end{array} \right]=\left[ \begin{array}{ccc}I&0&0\\ 0&I&0\\ 0&0&I\end{array} \right]\zeta \tag{19}\]  
_In this way, since \(\dot{r}(t)\) is available, the derivative action of the controller can be avoided, if \(\dot{y}(t)\) is measurable. Moreover, by suitably choosing the initial conditions of the filter and its cutoff frequency, it is possible to reduce the control action, above all during the transient phase.__It is worth noting that if the filter is a Bessel one with cutoff angular frequency \(\omega_{b}\) larger or equal to the angular frequency of \(\widetilde{r}(t)\), then \(r(t)\cong\widetilde{r}(t-t_{r})\), \(t_{r}=3\pi/(4\omega_{b})\) ([34; 35])._  
By setting:  
\[e=r-y,x_{1}=e,x_{2}=\dot{e},x=\left[x_{1}^{T}\ x_{2}^{T}\right]^{T} \tag{20}\]  
the plant (1) with the control law (18) is described by the following equations:  
\[ \begin{array}{l}\dot{x}=\left[\begin{array}{cc}0&I\\ -2a^{2}H&-2 aH\end{array}\right]x+\left[\begin{array}{cc}0\\ I\end{array}\right]w=Ax+Bw,\ x=\left[\begin{array}{c}x_{1}\\ x_{2}\end{array}\right]=\left[\begin{array}{c}e\\ \dot{e}\end{array}\right]\\ e=\left[\begin{array}{cc}I&0\end{array}\right]x=C_{1}x,\dot{e}=\left[ \begin{array}{cc}0&I\end{array}\right]x=C_{2}x,\end{array}  \tag{21}\]  
where:  
\[H=FG,w=\tilde{r}-f-Fu_{c} \tag{22}\]  
To design the control law (18) in order to force plant (1) to track a reference signal \(r(t)\in R^{m}\) with bounded second derivative, with tracking error norm smaller that a prescribed value, despite the presence of parametric and/or structural uncertainties and bounded disturbances, the following preliminary notations, definitions and lemmas are introduced. \[ \begin{array}{l}\|x\|_{p}=\sqrt{x^{T}Px},\ \|x\|=\|x\|_{I}=\sqrt{x^{T}x},\ S_{p, \ \rho}=\{x:\|x\|_{p}\leq\rho\},\ \rho\geq 0\\ C_{p,\ \rho}=\{x:\|x\|_{p}=\rho\},\ \hat{C}_{p,\ \rho}\supseteq C_{p,\ \rho} \end{array}  \tag{23}\]  
where \(P\in R^{n\times n}\) is a symmetric and positive definite (\(p.d.\)) matrix, \(x^{T}\) is the transpose of \(x\in R^{n}\), \(\hat{C}_{p,\ \rho}\) is a compact set, \(\lambda_{\max}(W)\ (\lambda_{\min}(W))\) is the maximum (minimum) eigenvalue of a square matrix \(W\in R^{n\times n}\) with real eigenvalues. **Definition 1**. : _Given the system:_  
\[ \begin{array}{l}\dot{x}=f(t,x,u,p),\ y=Cx\\ t\in\mathcal{T}=[0\,,\ t_{f}]\subseteq R,\ x\in R^{n},\ u\in R^{r}\\ p\in\wp\subset R^{n},\ y\in R^{m},\ C\in R^{m\times n}\end{array}