# Parameter Observer Useable for the Condition Monitoring of a Capacitor
## 2 Materials and Methods
In (14), \(\hat{c}_{f}(t)\) is the forced response of the tracking loop to the input signal \(z(t)\), i.e., the original of the expression \(\frac{\omega_{0}\cdot(2\cdot s+\omega_{0})}{(s+\omega_{0})^{2}}\cdot z(s)\). From (14) results the equations:  
\[T(t)\dot{z}(t)+1=0\quad,z(0)=z_{0}=\ln(y_{0}), \tag{2}\]6  
According to Equations (14) and (15), the output of the tracking subsystem shows the deviation \(\hat{c}_{0}-2\cdot\omega_{0}\cdot\hat{2}_{0}\), from the desired value, which tends asymptotically to 0. Practically, the attenuation is obtained for \(\omega_{0}\cdot t\geq 10\). If \(\tau\) is the chosen approximate value of the time interval after which the tracking mode must be installed, we derive the Formula (16) for adopting the value of \(\omega_{0}\):  
\[T(t)\dot{z}(t)+1=0\quad,z(0)=z_{0}=\ln(y_{0}), \tag{2}\]7  
The state Equation (17) corresponds to the tracking subsystem described by (6), (8), (9) and (13):  
\[\left\{ \begin{array}{l}\left[\begin{array}{c}\dot{x}_{1}(t)\\ \dot{x}_{2}(t)\\ y_{1}(t)\\ y_{2}(t)\end{array}\right]=\left[\begin{array}{cc}-2\cdot\omega_{0}&1\\ -\omega_{0}^{2}&0\\ 1&0\\ -2\cdot\omega_{0}&1\end{array}\right]\cdot\left[\begin{array}{c}x_{1}(t)\\ x_{2}(t)\\ x_{1}(t)\\ x_{2}(t)\end{array}\right]+\left[\begin{array}{c}2\cdot\omega_{0}\\ \omega_{0}^{2}\\ 0\\ 2\cdot\omega_{0}\end{array}\right]\cdot\left[z(t)\right],\ \ \ \left[\begin{array}{c}x_{1}\\ x_{2}\end{array}\right]=\left[\begin{array}{c}\hat{c}_{0}\\ \hat{c}_{0}\\ 0\\ 2\cdot\omega_{0}\end{array}\right]\quad,\end{array} \right. \tag{17}\]  
here:  
\[x_{1}(t)=\hat{z}(t),\ x_{2}(t)=K_{i}\int_{0}^{t}\varepsilon(\tau)d\tau,\ \ y_{1}(t)=\hat{z}(t),\ \ y_{2}(t)=\hat{c}(t). \tag{18}\]  
Figures 3 and 4 show two examples of how the observation process works. In the scenario of Figure 3, the observer's task is to estimate the time constant of the exponential signal \(y(t)=10e^{-t}\) represented in Figure 3a, i.e., the value \(T=1\) s. Figure 3b shows that the transient regime caused by the chosen initial conditions is attenuated in less than 0.1 s. As a result, Formula (16) proves to be sufficiently wide in scope. Figure 3c illustrates the fair identification of the value of \(T\); the estimated value is denoted as \(T^{\star}\). The scenario in Figure 4 refers to the signal (19) in Figure 4a, for which \(T\) varies and has values of a few milliseconds:  
\[y(t)=10\cdot e^{-\frac{t}{\omega_{0}\omega_{0}}}\cdot\left[2\cdot\left(\frac{ t}{0.007}\right)^{2}-2\cdot\left(\frac{t}{0.007}\right)+1\right]. \tag{19}\]  
The calculation of the value of \(T\) in Figure 4b used the formula:  
\[T(t)=-\frac{y(t)}{\dot{y}(t)}. \tag{20}\]Note too that in this case, except for the initial portion corresponding to the transient caused by the initial conditions, the estimated variation \(\hat{T}(t)\) and the calculated variation \(T(t)\) present a high degree of overlap (Figure 3(c)). Discrete-Time Observer for Determining the Variation in the Parameters of a First-Order Continuous-Time System from Its Sampled Free Regime  
In this section, we refer to the situation when the output of system \(S\) in Figure 1 is not available as a continuous-time signal \(y(t)\), but as a discrete-time signal \(\left\{y[k]\right\}_{k\in N}\) according to Equation (21) and Figure 5, where \(h\) is a constant sampling time:  
\[\left\{y[k]\right\}_{k\in N},\;\;y_{k}=y(k\cdot h). \tag{21}\]  
Two possibilities for approximating the variation in \(T\) may be considered:  
* Interpolation of the signal (21) followed by the processing of the interpolated signal with the PO from Figure 2
* Replacing the continuous-time PO with a homologous PO in discrete-time. Due to the great computational effort associated with interpolation with continual smooth functions, only the second approach is of practical interest.