# MFC/IMC Control Algorithm for Reduction of Load Torque Disturbance in PMSM Servo Drive Systems
## 2 Mathematical Model of Surface-Mounted PMSM  
One of the most common control schemes implemented in the servo drives systems is a cascade structure with a field-oriented control (FOC) algorithm [9; 10]. It consists of two loops, each controlling different physical quantity: Electric current (inner loop) and velocity (outer loop). The former loop has faster cycle times (i.e., is executed more frequently by the control system) and regulates the currents in the \(d\)-\(q\) rotating reference frame. The latter one generates the \(q\)-axis reference current that serves as a set point for the inner loop. This control scheme is usually exploited to maintain a constant velocity.  
The dynamical model of PMSM can be expressed by a set of voltage and current equations [11; 12; 13]. In order to reduce the complexity of the original three-phase model description, most often, [PERSON] and [PERSON]'s transformations are used to convert three-phase quantities (ABC reference frame) into two-phase reference frame quantities (\(d\)-\(q\) reference frame). This operation allows to decouple state variables and solve time-dependent equations in a common frame of reference.  
In this paper, a PMSM with surface-mounted magnets is considered. The dynamical model in \(d\)-\(q\) reference frame assumes that:  
* The machine is symmetrical.
* The induced EMF is sinusoidal.
* Eddy currents and hysteresis are neglected.
* Resistance and inductance of motor's windings are constant.
* Magnetic circuits are linear,
* Cogging torque effect is neglected.
* Static friction model with Stribeck's effect is adopted.  
Basing on these assumptions a set of differential equations can be given in the explicit form (1):  
\[\begin{array}{l}\frac{d\dot{t}_{d}(t)}{dt}=-\frac{R_{s}}{L_{d}}\dot{t}_{d}(t)+ \frac{L_{q}P}{L_{d}}\omega(t)\dot{t}_{q}(t)+\frac{1}{L_{d}}u_{d}(t)\\ \frac{\dot{u}_{q}(t)}{dt}=-\frac{R_{s}}{L_{q}}\dot{t}_{q}(t)-\frac{L_{q}P}{L_{ d}}\omega(t)\dot{t}_{d}(t)-\frac{P\dot{P}_{f}}{L_{q}}\omega(t)+\frac{1}{L_{q}}u_{q}(t) \\ \frac{d\omega(t)}{dt}=\frac{1}{f}\Big{[}\frac{3P}{2}\Big{(}\psi\dot{f}i_{q}(t) +\big{(}L_{d}-L_{q}\big{)}i_{d}(t)\dot{t}_{q}(t)\Big{)}-T_{F}(t)-\tau_{load}(t )\Big{]}\\ T_{F}(t)=T_{p}\omega(t)-T_{c}\text{sgn}(\omega(t))-(T_{s}-T_{c})e^{-\frac{g \left[\omega(t)\right]}{\omega_{s}}}\text{sgn}(\omega(t))\end{array} \tag{1}\]  
where \(\big{[}i_{d}(t),i_{q}(t),\omega(t)\big{]}^{T}\) represents the state vector of the output currents in direct and quadrature axes, and rotor mechanical velocity respectively. Stator voltages in \(d\)-axis and \(q\)-axis are denoted by \(u_{d}(t)\) and \(u_{q}(t)\). Lumped friction torque expressed by \(T_{F}\) involves static friction model with Stribeck's effect. Parameter \(R_{s}\) is the stator's winding resistance, \(L_{d}=L_{q}\) is the stator's winding inductance, \(\psi_{f}\) is the permanent magnet flux, \(p\) is the number of pole pairs, \(J\) is the rotor's moment of inertia, \(T_{v}\) is the coefficient of viscous friction, \(T_{c}\) is the coefficient of Coulomb friction, \(T_{s}\) is the coefficient of static friction, \(\omega_{s}\) is the Stribeck velocity, \(\delta\) is an additional empirical parameter in range from 0.5 to 1 and \(\tau_{load}(t)\) is the load torque.