# Parameter Observer Useable for the Condition Monitoring of a Capacitor
## 2 Materials and Methods
For the case in Figure 5, when the observer \(S_{0}\) is a discrete-time system, we considered that the tracking subsystem results from the replacement in Equations (6), (8) and (9) of the integration and derivation operations according to [PERSON]'s method and the backward difference, respectively [30]. Thus, for a constant sampling period \(h\), we obtain, respectively:  
\[\hat{z}[k]=\hat{z}[k-1]+h\cdot\hat{c}[k],\;\;\hat{z}[0]=\hat{z}_{0}, \tag{22}\]  
\[\varepsilon[k]=z[k]-\hat{z}[k], \tag{23}\]  
\[\hat{c}[k]=K_{p}(\varepsilon[k]-\varepsilon[k-1])+h\cdot K_{i}\cdot\varepsilon [k-1]+\hat{c}[k-1],\;\;\;\hat{c}[0]=\hat{c}_{0}\;. \tag{24}\]  
In (23) we have  
\[z[k]=\ln(y_{I}[k]). \tag{25}\]  
Equation (22) corresponds to block \(M\) in Figure 6, and Equation (24) to block \(T\)C.  
Figure 5: The block diagram of the observation of the parameter \(T\). Finally, instead of Equation (7), we will operate with Equation (26) and the average value (27):  
\[\hat{T}[k]=-\frac{1}{\hat{c}[k]}, \tag{26}\]  
\[y(t)=10\cdot e^{-\frac{t}{\omega_{0}\omega_{0}}}\cdot\left[2\cdot\left(\frac{ t}{0.007}\right)^{2}-2\cdot\left(\frac{t}{0.007}\right)+1\right]. \tag{19}\]0  
Here, \(k_{0}\) corresponds to the time \(t_{0}\) from which the average value is calculated. It is chosen so that the average value is not calculated over the time interval when the mentioned transient regime occurs. The input signal (19) sampled with \(h=5\cdot 10^{-6}\) s was used to validate the discrete-time PO. The results are shown in \[y(t)=10\cdot e^{-\frac{t}{\omega_{0}\omega_{0}}}\cdot\left[2\cdot\left(\frac{ t}{0.007}\right)^{2}-2\cdot\left(\frac{t}{0.007}\right)+1\right]. \tag{19}\]1 On the left are the input signal and the corresponding estimated signal. After a short-term transient regime, the follow-up is carried out without issue. According to the error \(\left\{\varepsilon[k]\right\}\) represented in Figure 7b, the transient regime lasts about \(2\cdot 10^{-3}\) s. On the right is illustrated \(\left\{\hat{T}_{m}[k]\right\}\). Ignoring the transient regime, the result practically coincides with that in Figure 4c. In (27) \(k_{0}=0\) corresponds to \(t_{0}=10^{-3}\) s.  
Additionally, for the validation of the discrete-time PO, we used the input signal (28) with a time constant which for \(t=4\cdot 10^{-3}\) s has a jump from the value \(T_{1}=5.8\cdot 10^{-3}\) s to the value \(T_{2}=2.9\cdot 10^{-3}\) s.  
\[y(t)=10\cdot e^{-\frac{t}{\omega_{0}\omega_{0}}}\cdot\left[2\cdot\left(\frac{ t}{0.007}\right)^{2}-2\cdot\left(\frac{t}{0.007}\right)+1\right]. \tag{19}\]2  
The signal's sampling period was \(h=5\cdot 10^{-6}\) s. The results are shown \[y(t)=10\cdot e^{-\frac{t}{\omega_{0}\omega_{0}}}\cdot\left[2\cdot\left(\frac{ t}{0.007}\right)^{2}-2\cdot\left(\frac{t}{0.007}\right)+1\right]. \tag{19}\]3 Figure 8 illustrates the signal \(y(t)\) and the estimated signal \(\left\{\hat{y}[k]\right\}\), and Figure 8b the estimated variation \(\left\{\hat{T}[k]\right\}\). Note that the tracking is performed very well after going through a transient regime with an approximate duration of \(0.25\cdot 10^{-3}\) s.  
\[y(t)=10\cdot e^{-\frac{t}{\omega_{0}\omega_{0}}}\cdot\left[2\cdot\left(\frac{ t}{0.007}\right)^{2}-2\cdot\left(\frac{t}{0.007}\right)+1\right]. \tag{19}\]4 Block diagram of the discrete-time tracking subsystem.