## 1 Introduction  
Key applications for the Permanent Magnet Synchronous Motor (PMSM) include industrial applications, aerospace applications, electric drives and electric vehicles, computer peripherals and robotics. Common applications of the PMSM include AC drives, automated guided vehicles, heating, ventilation and air conditioning (HVAC), robots, servo drives and home appliances. The growing interest in the widespread use of PMSMs in a variety of applications can be explained both by the fact that PMSMs have a number of advantageous design features, and by the fact that PMSM control systems can be characterised by an extended range of control algorithms with increased flexibility and performance, ranging from their mathematical description to their implementation in embedded systems [1; 2; 3; 4; 5]. In terms of overall PMSM control strategies, two strategies can be distinguished, namely the Field-Oriented Control (FOC)-type strategy [6; 7] and the Direct Torque Control (DTC)-type strategy [8; 9]. The FOC-type strategy in the classical description contains two tandem control loops, the outer speed loop and the inner current loop, and the controllers are of the classical Proportional Integrator (PI) controller type. When using the DTC strategy, there are two tandem control loops, the outer one for speed control and the inner one for torque and flow control. As far as the controllers are concerned, in the classical description the speed controller is of the PI-type, while the flux and torque controllers are of the hysteresis ON-OFF type. At first glance, the advantages and disadvantages of using the FOC-type strategy vs. the DTC strategy in terms of control algorithm complexity vs. control algorithm computation times are clearly antagonistic. Obviously, the use of hysteresis ON-OFF controllers, which require very short computing times, will inherently introduce small oscillations that must be cancelled out by the outer speed loop controller. Typically, the classical PI-type controller for the outer speed control loop can be an acceptable compromise between the required computation time when carried out in an embedded system and the overall performance of the control system, represented by response time, settling time, overshoot, steady-state error and speed ripple. In order to improve this global performance of PMSM control systems, given that PI-type controllers provide good performance around global operating points even for relatively small variations in external parameters and disturbances caused in particular by load torque variations, a number of more complex controllers of the following types can be used: adaptive [10], predictive [11], robust [12], fuzzy [13], and neural [14]. Although the performance of these controllers is superior to that of classical PI-type controllers, care must be taken when implementing them in real time in terms of the trade-off between the computing power of the embedded system and the overall cost of implementation. A special type of controller can be obtained by using fractional elements. Thus, a number of advanced integer-order controllers can be described by fractional means, with the following improvements in control system performance [15; 16; 17; 18; 19; 20]. A convergent approach to that represented by the control system synthesis is the use of velocity observers to estimate the PMSM rotor speed and possibly other control system characteristic parameters, in the case of the DTC strategy, torque and flux. The use of these observers ensures the sensorless nature of the control system and also contributes to the overall reliability of the PMSM control system through the use of sensors, particularly those using mechanical elements in motion [21; 22; 23; 24]. In addition to the performance indicators described above, an additional one can be introduced, namely the fractal dimension (FD) of the controlled signal, in this case the FD of the PMSM speed signal.