## Appendix A  
In the following six easy examples are provided. The first five ones clearly show how numerous control techniques available in the literature for significant classes of linear and nonlinear systems can produce unstable control system or can significantly reduce the performance of the control system, under the following realistic hypotheses:  
1. parametric uncertainties;
2. real actuators;
3. measurement noise;
4. finite online computation time of the control signal. The sixth example shows that the above mentioned disadvantages can be eliminated or reduced with the results proposed in the paper. **Example A1**. : _Consider the position control of an AGV. A model of the simple AGV in Figure A1 is:_  
\[M\ddot{y}+K\dot{y}+Mg(y)=u+d\] (A1)  
where \(y\) is the position, \(u\) is the control force, \(d\) is the disturbance force, \(M\) is the mass of the AGV and of its load, \(K\) is the friction coefficient, and \(Mg(y)\) is the action of the gravity force dependent on the slope of the road. Suppose the slope of the road constant and unknown. Including the gravity action in d, in the hypothesis that \(M=1\) (\([q=\text{quint}]\)) and \(K\in[0.50,\ 3.50]\) the model is represented as:  
\[\ddot{y}+K\dot{y}=u+d,\ K\in[0.50,\ 3.50]\] (A2)  
By using the control law:  
\[u=\hat{K}\dot{y}+1\cdot\ddot{r}+1\cdot\left(K_{p}e+K_{i}\!\int ed\tau+K_{d}e \right),\ e=r-y\] (A3)  
based on the inverse dynamic control technique, the closed-loop control system turns out to be:  
\[\dddot{e}+(K_{d}+K-\hat{K})\ddot{e}+K_{p}\dot{e}+K_{i}e=(K-\hat{K})\ddot{r}- \dot{d}\] (A4)  
If \(\hat{K}=K_{p}=K_{d}=K_{i}=2\) the dynamical model of the tracking error (A4) is:  
\[\dddot{e}+K\ddot{e}+2\dot{e}+2=(K-2)\ddot{r}-\dot{d}\] (A5)  
For \(K=\hat{K}=2\) and \(e(0)=\dot{e}(0)=\ddot{e}(0)=0\) in the presence of any disturbance with bounded derivative, applying the method proposed in [34; 35], the tracking error is \(|e(t)|\leq 1.357\delta_{p}\), where \(\delta_{p}=\max\abs{\dot{d}(t)}\). For \(K<1\), instead, the tracking error is not acceptable since system (A4) is unstable. Therefore, the inverse dynamic control technique, under the hypothesis of parametric uncertainties, is not reliable! **Example A2**. : _Suppose that \(M\in[0.50,\ 1.50]\), \(K=1\) and that the slope of the road is such that \(g(y)=5\sin(y)\). In this case the model is:_  
\[M\ddot{y}+\dot{y}+M5\sin y=u+d,\ M\in[0.50,\ 1.50]\] (A6)  
By using the control law:  
\[u=\hat{M}5\sin y+K_{p}e+K_{d}\dot{e},\ e=r-y,\ \hat{M}=K_{p}=K_{d}=1\] (A7)  
Figure A1: AGV. based on the feedback linearization technique, the dynamical model of the tracking error is:  
\[M\ddot{e}+2\dot{e}+e=(M-1)5\sin(r-e)+M\ddot{r}+\dot{r}-d \tag{100}\]  
For \(M=\hat{M}=1\) and \(e(0)=\dot{e}(0)=0\), in the presence of any bounded disturbance \(d(t)\), applying the method proposed in [34; 35], the tracking error of any reference with bounded second derivative is \(|e(t)|\leq 1\cdot\delta\), where \(\delta=\max\lvert\ddot{r}+\dot{r}-d\rvert\). Instead, for \(M<4/5\) the tracking error is not always acceptable, since system (101) can be unstable for \(r=0\). Hence, the control method based on the feedback linearization technique, under the hypothesis of parametric uncertainties, is not always reliable, especially for robots! **Example A3. **_Consider the system (102) for \(K=1\)_:  
\[\ddot{y}+\dot{y}=u+d \tag{101}\]  
activated by a DC motor supplied with an amplifier of gain \(g\):  
\[\dot{u}=-\frac{R}{L}u-\frac{H^{2}}{L}\dot{y}+\frac{Hg}{L}v \tag{102}\]  
in which \(v\) is the amplifier supply voltage.