# Dynamic Observer Modeling and Minimum-Variance Self-Tuning Control of EDM Interelectrode Gap
## 1 Introduction  
Single-crystal Si is a difficult-to-machine material with high brittleness and high hardness [1; 2]. Electric discharge machining (EDM) [3; 4; 5] technology for high-efficiency, the high-quality processing of single-crystal Si has become a research topic of great interest [6; 7; 8; 9]. EDM is a typical nonlinear multi-parameter time-varying system [10]. Mechanical parameters and power parameters affect the system discharge stability and discharge state, determine the workpiece removal efficiency, surface quality, and electrode loss. According to different gaps between the electrode and the workpiece, the EDM discharge state can be divided into five types: The no-load state, normal spark discharge state, transitional arc state, stable arc state, and short-circuit state [4; 11]. Each discharge state corresponds to different voltage and current waveforms. Therefore, EDM equipment should have a perfect interelectrode gap detection and control system. The discharge mechanism of EDM is very complex and is often affected by many factors, such as adhesion, cavitation, and short-circuit phenomena [12; 13], that make it difficult to detect and control the gap between electrodes. [PERSON] proposed to identify the EDM discharge state by comparing the detected gap voltage with the preset voltage [14]. In Reference [15], the discharge state was distinguished by monitoring the interelectrode characteristic voltage value and the known voltage threshold. According to [PERSON] and [PERSON] et al., although the gap voltage threshold comparison method is simple, the determination of the voltage thresholdrequires a large number of tests as a support. It is difficult to determine the thresholds for stable arc discharge and transitional arc discharge, and the determined voltage threshold is not universal. The corresponding voltage threshold of different processing conditions needs to be determined according to specific conditions. [PERSON] et al. (2016) proposed a method based on a transistor constant pulse generator to monitor the gap by using average breakdown delay time. Other methods for identifying EDM gap distance include fuzzy identification ([PERSON] et al., 2017) and neural network identification ([PERSON] et al., 2018). None of the above methods can accurately detect the interelectrode gap, and only the artificial intelligence method can be used to estimate the interelectrode gap to control the electrode feed based on the experimental data. The literature ([PERSON] et al., 2018; [PERSON] et al., 2018) has established an EDM stochastic model using the spark frequency as the feedback signal of the discharge gap to adjust the servo feed rate, which can reduce the change of the control process and improve the machining efficiency. [PERSON] et al. (2017) used the breakdown delay time and fall time as the feedback signal of the adaptive control system to control the interelectrode gap to ensure that the discharge was in a stable state. [PERSON] et al. (2018, 2020) used the spark deviation rate and the average gap voltage as the inputs of the fuzzy controller. The output was the electrode feed rate and feed direction, and the performance of the paste controller was evaluated by the gain control controller. [PERSON] et al. (2018) adopted arc rate and short-circuit rate as the inputs of the self-adjusting fuzzy control system and controlled the knife-lifting movement of the electrode to increase the machining speed and the processing depth. [PERSON] et al. (2018) used the discharge current, pulse width, and ultrasonic vibration of the electrode as input parameters for fuzzy control. The control system can simultaneously consider the material removal rate, surface roughness, and electrode loss. Because there is no accurate mathematical model, the gap can only be estimated by relying on data obtained from trial and error, with large errors and low-control accuracy. In view of the above problems, in this paper, we used p-type single-crystal silicon EDM processing as an example and introduced a flat-plate capacitance model.