## 4 Results and Discussion  
The following section presents the results using a derivative realized through the adaptive KF and the proposed algorithm. The results are compared with the exact mathematical derivative. In order to reduce the calculation load in the case study, the following conditions were considered: \(c_{1}=b_{1}=b_{2}=0\). This approach allows us to also compare the results with a polynomial adaptive KF method. Figure 2 shows an ideal measured position \(y(t)\) signal. Therefore neither noise nor faults are present in the measured data. Figure 3 shows the approximated derivative of the measured sine function, and the result of this is shown in detail in Figure 4. With this graphical representation of the result, it is visible that the adaptive KF, compared with the proposed algorithm structure, shows a better performance. Figure 3: Graphical representation of the resulting velocity and its estimations from Figure 2.  
Figure 2: Graphical representation of the position \(y(t)\). Figure 5 shows the position signal used, \(y_{\text{noise}}(t)\), in which noise in superposition is added, and the graphical representation of the resulting velocity is shown in Figure 6. The signal used, \(y_{\text{noise}}(t)\), is represented in Figure 7 together with a fault in its measurement. Figure 8 shows the approximation derivative of the measured signal, and details of this result are shown in Figure 9. Also in the case of the presence of faults and using a more appropriate adaption of the adaptive KF, the two methods offer similar results. Figure 4: Detailed representation of Figure 3 and its estimations. Figure 5: Graphical representation of the position \(y(t)\) with noise. Table 1 presents the results of different input signals and the estimation errors of both adaptive estimators based on the Euclidean norm. The adaptive derivative estimator shows worse results with respect to the results obtained by the derivative obtained through the adaptive KF in the case of missing noise and faults. Figure 6: Detailed representation of Figure 5 and its estimations. Figure 7: Graphical representation of the position \(y(t)\) with noise and faults. In case of the presence of noise and using a more appropriate adaption of the adaptive KF, the two methods offer similar results. The proposed algorithm structure shows better results in the presence of faults. Figure 10 shows that a part of a chirp input signal starts from a frequency of 200 Hz and in 0.3 s reaches to 2 kHz. Because of the presence of the derivative, the signal to be approximated changes its  
Figure 3:0 Graphical representation of the resulting velocity from Figure 7 and its estimations with noise and faults. Figure 3:1 Detailed representation of Figure 7 with its estimations. amplitude linearly as a function of the frequency over time. The low-frequency part of the chirp input signal is presented in Figure 11, and the high-frequency part is shown in Figure 3:2 It also shows the resulting tracking of the polynomial adaptive KF and the proposed algorithm structure. Figure 3:3 Detailed representation of the low-frequency part of the chirp input signal and its estimations from Figure 3:4  
Figure 3:5 Graphical representation of chirp input signal with a starting frequency of 200 Hz. A fault is defined as an abrupt change in the amplitude over time, and in this sense is characterized by a high amplitude and high frequencies. In this context, it is clear that the proposed algorithm can better localize the faults because it is tuned through the choice of \(t_{\text{s}}\) on the desired signal. The proposed algorithm structure does not need to be tuned or, in other words, is tuned using the sample time, which in general is fixed in terms of upper bound by the Shannon Theorem. Now \(k_{\text{app}}\) represents the time constant of our approximating derivative, which is calculated adaptively through the least squares method.