## Appendix A
If \(L=0.1,\ H=1,\ g=R\) it is:  
\[\ddot{y}+K\dot{y}=u+d,\ K\in[0.50,\ 3.50]\]0  
If \(R\gg L\), i.e., under the hypothesis of an almost ideal actuator, then \(u\cong v\) and, hence, for \(R/L\rightarrow\infty\), it is:  
\[\ddot{y}+K\dot{y}=u+d,\ K\in[0.50,\ 3.50]\]1  
By using the control law:  
\[\ddot{y}+K\dot{y}=u+d,\ K\in[0.50,\ 3.50]\]2  
the dynamical model of the tracking error, in the hypothesis that \(K_{p}=2a^{2},\ K_{d}=2a\) is:  
\[\ddot{y}+K\dot{y}=u+d,\ K\in[0.50,\ 3.50]\]3  
For \(e(0)=\dot{e}(0)=0\) and \(\forall d(t):|d(t)|\leq\delta\), applying the first of (49), it is \(|e(t)|\leq 0.545\delta/a^{2}\). From (101), (103) and (105) the exact model turns out to be:  
\[ \begin{array}{l}\ddot{e}=-\dot{e}-u+\ddot{r}+\dot{r}-d\\ \dot{u}=10R2a^{2}e+10(R2a-R+1)\dot{e}-10 Ru+10(R\ddot{r}+(R-1)\dot{r})\end{array}  \tag{107}\]  
from which, by setting \(x=\left[ \begin{array}{ccc}e&\dot{e}&u\end{array} \right]^{T}\), it is:  
\[\dot{x}=\left[ \begin{array}{ccc}0&1&0\\ 0&-1&-1\\ 10R2a^{2}&10(R2a-R+1)&-10R\end{array} \right]x+\left[ \begin{array}{c}0\\ \ddot{r}+\dot{r}-d\\ 10(R\ddot{r}+(R-1)\dot{r})\end{array} \right]=Ax+\sigma \tag{108}\]  
It is easy to verify that for \(R<0.102a-0.30,\ a\in[5,\ 50]\), the control system is unstable, while for \(R>0.102a-0.15,\ a\in[5,\ 50]\), the control system is stable (see Figure 11). For \(R=2a,\ a\in[5,\ 50]\), it is: \(\lambda(A)\cong\{-a\pm ja,\ -18a\}\), i.e., if the time constant of the actuator \(\tau_{a}\) is equal to \(\frac{1}{20a}\) in the hypothesis of real actuator (A11), the dynamic matrix A has a couple of dominant eigenvalues equal to the poles of the closed-loop system (A14). Hence, it can be said that the control techniques of the mechanical systems, especially for robots, which make the hypothesis of ideal actuators (torque and/or force actuators), are not reliable! **Example A4**. : _Consider the system (A2):_  
\[\ddot{y}+K\dot{y}=u+d,\ K\in\left[ \begin{array}{cc}0.50&3.50\end{array} \right]\] (A17)  
that can be rewritten as follows:  
\[ \begin{array}{l}\dot{x}=Ax+Bu+Ed,\ y=Cx\\ x=\left[\begin{array}{c}y\\ \dot{y}\end{array}\right],\ A=\left[\begin{array}{cc}0&1\\ 0&-K\end{array}\right],\ B=E=\left[\begin{array}{c}0\\ 1\end{array}\right],\ C=\left[\begin{array}{cc}1&0\end{array}\right]\end{array} \] (A18)  
To control the system (A17) with the MPC technique (e.g., [37; 43]), consider the following system as the nominal model in order to predict its evolution:  
\[ \begin{array}{l}\dot{\widetilde{x}}=\widetilde{A}x+\widetilde{B}u,\ \widetilde{y}= \widetilde{C}\widetilde{x}\\ \widetilde{A}=\left[\begin{array}{cc}0&1\\ 0&-2\end{array}\right],\ \widetilde{B}=\left[\begin{array}{c}0\\ 1\end{array}\right],\ \widetilde{C}=\left[\begin{array}{cc}1&0\end{array}\right]\end{array} \] (A19)  
Fixed the sampling time \(T\), let:  
\[x_{k+1}=A_{d}x_{k}+B_{d}u_{k}+E_{d}d_{k},\ y_{k}=C_{d}x_{k}\] (A20)  
\[\widetilde{x}_{k+1}=\widetilde{A}_{d}x_{k}+\widetilde{B}_{d}u_{k},\ \widetilde{y}_{k}= \widetilde{C}_{d}\widetilde{x}_{k}\] (A21)  
be the discrete time models of the systems (A18) and (A19), respectively. Said \(r_{k}\) the reference signal to track, it is:  
\[e_{k+1}=r_{k+1}-\widetilde{y}_{k+1}=r_{k+1}-(\widetilde{C}_{d}\widetilde{A}_ {d}x_{k}+\widetilde{C}_{d}\widetilde{B}_{d}u_{k})\] (A22)  
As control \(u_{k}\) consider the one minimizing the performance index:  
\[\min_{u_{k}\in[u^{-},\ u^{+}]}\left(e_{k+1}^{2}\right)\] (A23)  
Figure A2: Stability and instability regions (in red lower and upper bounds of separation between the two regions). Suppose that:  
\[T=0.10,\ K=1,\ x_{0}=\left[0.50\ 0\right]^{T},\ u^{-}=-5,\ u^{+}=5,\ d=1,\ r=\sin t \tag{10}\]  
In Figure 11 the time histories of \(u,r,\) and \(y\) are reported. If the velocity \(\dot{y}\) is affected by a measurement noise \(n_{\dot{y}}\) uniformly distributed in the interval \([-0.10,\ 0.10],\) some time histories of u, r and y are reported in Figures 12 and 13. From Figures A4 and A5 it clearly emerges that in general an MPC controller is very sensitive to measurement errors.