# Principles and Methods of Servomotor Control: Comparative Analysis and Applications
## 4 Control Methods of Servomotors
### Field-Oriented Control  
Field-oriented control (FOC), based on the vector control principle, is a method of controlling current and voltage in a servomotor, considering the direction of the magnetic field rotation. FOC supports system control dynamics and efficiency, which is suitable for high-performance systems and mechanisms with high precision. Examples of applications may include CNC machines and laser cutting machines [72].  
The field-oriented control method involves transforming the magnetic field into a rotating coordinate system known as the \"d-q\" system. The d-axis is aligned with the magnetic flux, while the q-axis is perpendicular to the d-axis. This transformation aligns the magnetic flux inside the motor along one axis, and the variable current is aligned along the other axis, which is the torque axis. Therefore, control is carried out based on the variable current [71; 87].  
The mathematical equation for Park transformation is:  
\[\begin{bmatrix}\mathrm{i}_{\alpha}\\ \mathrm{i}_{\beta}\end{bmatrix}=\begin{bmatrix}\cos(\theta)&-\sin(\theta)\\ \sin(\theta)&\cos(\theta)\end{bmatrix}\begin{bmatrix}\mathrm{i}_{\mathrm{d}} \\ \mathrm{i}_{\mathrm{q}}\end{bmatrix}, \tag{9}\]where \(i_{\alpha}\) and \(i_{\beta}\)--stator currents in fixed axes, \(i_{d}\) and \(i_{q}\)--stator currents in rotated axes, and \(\theta\)--magnetic flux rotation angle.  
The mathematical equation for Clark transformation is:  
\[\begin{bmatrix}i_{d}\\ i_{q}\end{bmatrix}=\begin{bmatrix}\cos(\theta)&\sin(\theta)\\ -\sin(\theta)&\cos(\theta)\end{bmatrix}\begin{bmatrix}i_{\alpha}\\ i_{\beta}\end{bmatrix}, \tag{10}\]  
The feedback system measures the magnetic flux and then, using Park and Clarke transformations, converts them from the stator coordinate system to the rotating d-q coordinate system. The use of a PI controller in a field-oriented control system allows maintaining the specified flux for each axis. The Clarke transformation is used to create d-q coordinates, and the inverse Park transformation is used to transform the currents back into the stator coordinate system [71; 87; 88].  
The equations for motor voltage and torque are as follows:  
\[T_{e}=\frac{3}{2}P\big{(}\psi i_{q}+(\psi_{m}-\psi)i_{d}\big{)}, \tag{11}\]  
\[V_{e}=R_{s}i_{s}+L_{s}\frac{di_{s}}{dt}+e_{s}, \tag{12}\]  
where \(V_{s}\)--stator voltage, \(R_{s}\)--stator resistance, \(L_{s}\)--stator inductance, \(i_{s}\)--stator current, \(p\)--number of pole pairs, \(\psi\)--rotor flux, \(\psi_{m}\)--maximum rotor flux, \(T_{e}\)--electromagnetic torque, \(e_{s}\)--back EMF.  
Advantages of the field-oriented control method [71; 87; 88]:  
1. Precision control. This method allows for increased precision in controlling both the torque and speed of the servomotor.
2. Low noise and vibration levels. FOC reduces mechanical and electrical noise in the operation of the servomotor.
3. High energy efficiency. The method reduces losses and allows for increased energy utilization efficiency.  
The development of FOC faces two of its most significant drawbacks: complexity of implementation and sensitivity to motor parameters. The more complex the mechanism control system, the more difficult the algorithm implementation and the more precise the behavior model should be. In turn, changes in motor parameters lead to the revision of the entire control algorithm [71; 87; 88].