# Dynamic Observer Modeling and Minimum-Variance Self-Tuning Control of EDM Interelectrode Gap
## 2 EDM Interelectrode Gap Model Based on Plate Capacitor
### Verification of EDM Interelectrode Gap Model  
The established model was verified by using an EDM single-pulse discharge test. The voltage of the pulsed power supply was 100 V, the pulse width was 100 \(\upmu\)s, and the interval pulse was 400 \(\upmu\)s. All of these parameters were kept constant. The interelectrode gap ranged from 10 to 80 \(\upmu\)m, increasing by 10 \(\upmu\)m each time. The time constant \(\tau\) was introduced into the EDM interelectrode gap model  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Actual Gap (\(\upmu\)m)** & \(\tau_{1}\) (\(\upmu\)s) & \(\beta_{1}\) & \(\tau_{2}\) (\(\upmu\)s) & \(\beta_{2}\) & \(\tau_{3}\) (\(\upmu\)s) & \(\beta_{3}\) & \(\overline{\beta}\) \\ \hline
10 & 1  
by Equation (9), from which the interelectrode gap \(d\) can be calculated. Each group was conducted 30 times. According to error theory, the 3\(\sigma\)-test method was used to analyze the test data. \(\sigma\) is the standard deviation, which is expressed below as  
\[\sigma=\sqrt{\sum\begin{smallmatrix}i=1\\ i=1\end{smallmatrix}\begin{smallmatrix}v_{i}^{2}/(n-1)\end{smallmatrix}} \tag{10}\]  
where, when \(n\geq 25\) for a certain observation data \(x_{i}\), its residual \(v_{i}\) is satisfied as below:  
\[v_{i}=\left|x_{i}-\overline{x}\right|\geq 3\sigma,i=1,2,\cdots\cdots,n \tag{11}\]  
The observation data \(x_{i}\) is the gross error, where \(\overline{x}\) is the mean of observation data \(x_{i}\):  
\[\overline{x}=\frac{1}{n}\!\!\sum\begin{smallmatrix}n\\ i=1\end{smallmatrix}\begin{smallmatrix}x_{i}\end{smallmatrix} \tag{12}\]  
Table 3 shows the actual gap value as well as the minimum, maximum, and average values of the model-calculated gap of the 30 sets of tests. The values of the maximum absolute error \(\delta_{\text{max}}\) under each gap are also given.  
Using a rectangular coordinate system, the horizontal axis is the gap \(d\) calculated by the model and the vertical axis is the actual gap. The minimum, maximum, and average values of the gap calculated by the model are plotted in Figure 6. It can be seen from the figure that the minimum, maximum, and average gap values calculated by the model are within \(\pm 3\sigma\), where \(\sigma\) is the minimum value 0.9165 in Table 3. Therefore, there is no gross error in the gap \(d\) calculated by the model. When the actual gap is 80 \(\upmu\)m, the maximum absolute error \(\delta_{\text{max}}\) between the calculated gap and the actual gap is only 2.31 \(\upmu\)m, which indicates that the established interelectrode gap model is correct and reliable. As the actual gap increases, the absolute error between the calculated gap and the actual gap of the model also increases. This is because the fitting error between \(\beta\) and the interelectrode gap \(d\) increases as the interelectrode gap increases. It can be seen in Figure 5 that the degree of dispersion of several fitting curves reaches the maximum at 80 \(\upmu\)m.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Actual Gap (\(\upmu\)m)** & **Model-Calculated Gap (\(\upmu\)m)** & **Model-Calculated Mean Gap \(\overline{x}\)** & \(\sigma\) & \(\delta_{\text{max}}\) \\ \hline
10 & 9.1562 & 10.2138 & 1.2168 & 1.10 \\
20 & 11.1022 & & & \\
20 & 18.5918 & & & \\
20 & 20.7725 & 19.8764 & 1.1365 & 1.41 \\
30 & 29.4844 & & & \\
30 & 31.6051 & 30.3362 & 1.0687 & 1.61 \\
40 & 39.0474 & & & \\
40 & 41.1400 & & & \\
50 & 48.8590 & & & \\
51.5708 & & 50.2152 & 0.9862 & 1.57 \\
60 & 58.3669 & & & \\
61.7297 & & 59.8568 & 0.9676 & 1.73 \\
70 & 68.1800 & & & \\
71.1806 & & 69.7634 & 1.2043 & 1.82 \\
80 & 78.7784 & & & \\
82.3050 & & 80.2652 & 1.1862 & 2.31 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Actual gap and model-calculated gap.