## 2 An Approximated Derivative Structure
The numerator constraint is the following:  
\[\frac{\mathrm{d}\hat{r}(t)}{\mathrm{d}t}=-\mathcal{M}\Big{(}\hat{r}(t)-\frac{ \mathrm{d}y(t)}{\mathrm{d}t}\Big{)}, \tag{7}\]4  
Considering the conditions of the denominator, we obtain  
\[\frac{\mathrm{d}\hat{r}(t)}{\mathrm{d}t}=-\mathcal{M}\Big{(}\hat{r}(t)-\frac{ \mathrm{d}y(t)}{\mathrm{d}t}\Big{)}, \tag{7}\]5  
This yields  
\[\frac{\mathrm{d}\hat{r}(t)}{\mathrm{d}t}=-\mathcal{M}\Big{(}\hat{r}(t)-\frac{ \mathrm{d}y(t)}{\mathrm{d}t}\Big{)}, \tag{7}\]6  
\(k_{\text{app}}\) being, in our context, a function of time \(k_{\text{app}}=k_{\text{app}}(t)\), it is possible to write in \(\mathcal{Z}\)-domain as follows:  
\[\frac{\mathrm{d}\hat{r}(t)}{\mathrm{d}t}=-\mathcal{M}\Big{(}\hat{r}(t)-\frac{ \mathrm{d}y(t)}{\mathrm{d}t}\Big{)}, \tag{7}\]7  
and thus consider the back \(\mathcal{Z}\)-transform  
\[\frac{\mathrm{d}\hat{r}(t)}{\mathrm{d}t}=-\mathcal{M}\Big{(}\hat{r}(t)-\frac{ \mathrm{d}y(t)}{\mathrm{d}t}\Big{)}, \tag{7}\]8  
If \(t_{\text{s}}\) is small enough, \(k_{\text{app}}(k)\approx k_{\text{app}}(k-1)\). This implies  
\[\frac{\mathrm{d}\hat{r}(t)}{\mathrm{d}t}=-\mathcal{M}\Big{(}\hat{r}(t)-\frac{ \mathrm{d}y(t)}{\mathrm{d}t}\Big{)}, \tag{7}\]9  
\(\square\)  
**Remark 1**. : _Conditions (57), (59), and (60) guarantee that signal \(\hat{r}(t)\) equals \(y(t)\). Nevertheless, obtaining the rejection of the noise coefficients \(b_{1}\), \(b_{2}\), and \(c_{1}\) should be adaptively calculated using LSM. In our tests, in order to reduce the calculation load, the following conditions are considered: \(c_{1}=b_{1}=b_{2}=0\)._  
**3. Using an Adaptive Kalman Filter to Approximate a Derivative Controller**  
Assuming that the polynomial that approximates the derivative of the signal is of the first order as follows:  
\[e_{r}(t)=r(t)-\hat{r}(t), \tag{8}\]0  
in which \(\hat{y}(t)\) represents the polynomial approximation of the signal \(y(t)\). The following adaptive Kalman filter (KF) can be implemented in which at each sampling time, constant parameters \(p_{0}\) and \(p_{1}\) should be calculated. The following state representation is obtained:  
\[ \begin{bmatrix}\dot{\hat{y}}(t)\\ \hat{r}(t)\end{bmatrix} = \begin{bmatrix}0&1\\ 0&0\end{bmatrix}\begin{bmatrix}\hat{y}(t)\\ \hat{r}(t)\end{bmatrix} . \tag{62}\]  
It should be noted that \(\dot{\hat{y}}(t)=\hat{r}(t)\) represents the approximation of the derivative of the signal as proposed in Proposition 1. In this sense, according to the following general notation, \(\mathbf{x}(t)\in\mathbb{R}^{2}\), \(u(t)\in\mathbb{R}\), and \(y(t)\in\mathbb{R}\), then:  
\[\left\{ \begin{array}{l}\dot{\mathbf{x}}(t)=\mathbf{Ax}(t)\\ y(t)=\mathbf{Hx}(t),\end{array} \right. \tag{63}\]where  
\[\dot{\mathbf{x}}(t)= \begin{bmatrix}\dot{y}(t)\\ \dot{t}(t)\end{bmatrix} \text{and}\ \ \mathbf{A}= \begin{bmatrix}0&1\\ 0&0\end{bmatrix} \, \tag{64}\]  
and \(\mathbf{H}=[1\ \ \ 0]\). Considering the discretization of system (63), the following discrete system is obtained:  
\[\left\{ \begin{array}{l}\dot{\mathbf{x}}(k/k-1)=\mathbf{A}_{d}\mathbf{x}(k-1/ k-1)+\mathbf{Q}_{w}\\ y(k/k)=\mathbf{H}\mathbf{x}(k/k)+\zeta,\end{array} \right. \tag{65}\]  
where \(\mathbf{Q}_{w}\) is the process noise covariance matrix, and \(\zeta\) is the measurement noise covariance. The discrete forms of matrix \(\mathbf{A}\) of (64) are represented by \(\mathbf{A}_{\text{d}}\), respectively. If the forward Euler method with the sampling time \(t_{\text{s}}\) is applied, the following matrices are obtained:  
\[\mathbf{A}_{\text{d}}= \begin{bmatrix}1&t_{\text{s}}\\ 0&1\end{bmatrix} .