## 2 Proposed PMSM-SCS Based on DTC Strategy  
The DTC-type control strategy, together with the FOC-type control strategy, are the most commonly used PMSM control strategies. Each of these control strategies has a number of advantages and disadvantages when compared directly with each other. Therefore, this section presents the DTC strategy based on the PMSM operating equations in the \(d\)-\(q\) rotating reference-frame coordinate system, and Figure 1 depicts the block diagram of the DTC strategy. The operating equations are given in Equations (1)-(4) [8; 9; 24].  
\[\begin{bmatrix}\psi_{d}\\ \psi_{q}\end{bmatrix}=\begin{bmatrix}L_{d}&0\\ 0&L_{q}\end{bmatrix}\begin{bmatrix}i_{d}\\ i_{d}\end{bmatrix}+\begin{bmatrix}\lambda_{0}\\ 0\end{bmatrix} \tag{1}\]  
\[\begin{bmatrix}u_{d}\\ u_{q}\end{bmatrix}=R_{s}\begin{bmatrix}i_{d}\\ i_{q}\end{bmatrix}+\begin{bmatrix}L_{d}&0\\ 0&L_{q}\end{bmatrix}\frac{d}{dt}\begin{bmatrix}i_{d}\\ i_{q}\end{bmatrix}+n_{p}\omega&\begin{bmatrix}0&-L_{d}\\ L_{d}&0\end{bmatrix}\begin{bmatrix}i_{d}\\ i_{q}\end{bmatrix}+n_{p}\omega\begin{bmatrix}0\\ \lambda_{0}\end{bmatrix} \tag{2}\]  
\[T_{e}=\frac{3}{2}n_{p}\big{(}\big{(}L_{d}-L_{q}\big{)}i_{d}i_{q}+\lambda_{0}l_ {q}\big{)} \tag{3}\]  
\[J\dot{\omega}=T_{e}-T_{L}-B\omega \tag{4}\]  
In the above equations, the usual notations are used for the inductance \(L\), the stator resistance \(R_{s}\), the stator flux \(\psi\), the currents \(i\) and the voltages \(u\) in the coordinate system of the \(d\)-\(q\) rotating reference frame, plus the mechanical angular speed of the PMSM rotor denoted by \(\omega\), the flux linkage denoted by \(\lambda_{0}\), the viscous friction coefficient denoted by \(B\), and the rotor inertia combined with the load inertia denoted by \(J\). \(T_{e}\) is the electromagnetic torque, \(T_{L}\) is the load torque and the number of PMSM pole pairs is given by \(n_{p}\).  
Figure 1: Flowchart of the proposed control method.  
The DTC strategy contains a two-loop control, where the outer loop implements speed control and the inner loop implements torque and flux control via two control loops, based on hysteresis ON-OFF controllers [8; 9].  
In the case of sensorless control, it is also necessary to synthesise an observer that provides the estimated value of the stator flux, the estimated value of the PMSM rotor speed and the load torque. The main feature of this DTC strategy is the use of quick-response controllers with reduced design simplicity for the inner loops, while classical PI-type controllers generally ensure good performance for the outer loop, maintaining a reduced complexity of real-time implementation.  
Based on these aspects, this section presents implementations of control structures that provide improved performance without significantly burdening the hardware/software structure for real-time implementation. Therefore, these types of outer loop controllers are presented when the reference stator flux is constant. For cases where the reference stator flux is variable, a separate section presents the implementation of SMC and FOSMC-type control structures.  
The implementation of a flux, speed and torque load observer is also presented in a separate section. In order to compare the performance for the proposed PMSM-SCS presented, the FD of the control signals, in this case the PMSM speed signal, is compared in addition to the classic comparison elements such as response time, settling time and PMSM speed-signal ripple. These can be summarized in a flowchart, as in Figure 1.