# Overview of Predictive Control Technology for Permanent Magnet Synchronous Motor Systems
## Composition of a PMSM Drive System Based on Predictive Control  
### Structure of a PMSM System  
Traditional predictive control is based on a discrete mathematical model of the drive motor. Additionally, it is predicted according to the characteristics of the switching state of the voltage source inverter. Take the most common three-phase PMSM as an example. Figure 2 is a topology diagram of the most common two-level three-phase voltage inverter in PMSM. The three-phase full bridge used consists of six IGBT power devices. According to the principle that the upper and lower switching devices of the inverter (VSI) cannot be turned on at the same time, eight switch combinations are formed. The conduction of the upper bridge arm of the inverter is 1 and the conduction of the lower bridge arm is 0. For different switch states, 8 basic voltage vectors can be obtained, and each vector can be expressed by Equation (1). \[U_{out}=\frac{2}{3}U_{dc}(s_{A}+s_{B}e^{j\frac{2}{8}\pi}+s_{C}e^{-j\frac{2}{8} \pi}) \tag{1}\]  
where \(U_{dc}\) is the DC bus voltage, \(j\) is an imaginary number. Map the basic spatial voltage vectors of 8 combinations onto the complex plane, as shown in Figure 2b.The red arrow from Figure 2a indicates the space voltage vector corresponding to the inverter switching signal (100). Figure 1: The main research direction and classification of predictive control. The realization of the PC algorithm relies on the discrete mathematical model. Based on the above-mentioned three--phase PMSM inverter topology as an example, the mathematical model of the three-phase PMSM is introduced. For the convenience of control, the motor equation of PMSM has been transformed by [PERSON] and [PERSON], which can be written as:  
\[\left[ \begin{array}{c}u_{d}\\ u_{q}\end{array} \right]=R_{s}\left[ \begin{array}{c}i_{d}\\ i_{q}\end{array} \right]+\frac{d}{dt}\left[ \begin{array}{c}\psi_{d}\\ \psi_{q}\end{array} \right]+\omega_{e}\left[ \begin{array}{c}-\psi_{q}\\ \psi_{d}\end{array} \right] \tag{2}\]  
where \(u_{d}\), \(i_{d}\), and \(\psi_{d}\) represent voltage, current, and magnetic flux on the \(d\)-axis, respectively; \(i_{q}\), \(u_{q}\), and \(\psi_{q}\) are represent current, voltage, and magnetic flux on the \(q\)-axis, respectively; and \(\omega_{e}\) is the electrical angular velocity. \(\psi_{q}\) and \(\psi_{d}\) can be expressed as:  
\[\left[ \begin{array}{c}\psi_{d}\\ \psi_{q}\end{array} \right]=\left[ \begin{array}{c}L_{d}\\ L_{q}\end{array} \right]\left[ \begin{array}{c}i_{d}\\ i_{q}\end{array} \right]+\left[ \begin{array}{c}\psi_{f}\\ 0\end{array} \right] \tag{3}\]  
where \(L_{d}\) and \(L_{d}\) are the stator inductance of \(d\) and \(q\) axis. \(\psi_{f}\) is the permanent magnet flux linkage. The electromagnetic torque equation of the motor is expressed as Equation (4), and the mechanical equation can be expressed as Equation (5). \[T_{e}=\frac{3}{2}p_{n}i_{q}\Big{[}(L_{d}-L_{q})i_{d}+\psi_{f}\Big{]} \tag{4}\]  
\[J\frac{d\omega_{m}}{dt}=T_{e}-T_{L}-B\omega_{m} \tag{5}\]  
where \(p_{n}\) is the number of pole pairs. \(J\) is the rotational inertia, \(B\) is the damping coefficient, \(\omega_{m}\) is the mechanical angular velocity, and \(T_{e}\) and \(T_{L}\) are the electromagnetic torque and the load torque, respectively. The mathematical model of the time invariant model of PMSM in a synchronous rotating reference frame can be expressed as:  
\[\left\{ \begin{array}{c}pi_{d}(t)=-\frac{R_{s}}{L_{q}}i_{q}(t)-\frac{L_{d} \omega_{e}(t)}{L_{q}}i_{d}+\frac{1}{L_{q}}u_{q}(t)-\frac{\psi_{d}\omega_{e}(t) }{L_{q}}\\ pi_{d}(t)=-\frac{R_{s}}{L_{d}}i_{d}(t)+\frac{L_{q}\omega_{e}(t)}{L_{d}}i_{q}+ \frac{1}{L_{d}}u_{d}(t)\end{array} \right. \tag{6}\]  
where \(pi_{d}(t)\) and \(pi_{q}(t)\) represent the reciprocal of the d-axis and q-axis currents, respectively. Based on the predictive current control, the motor equation is generally discretized through  
Figure 2: Topology of voltage source inverter and spatial distribution diagram of voltage vectors.