International Journal of Applied Sciences and Smart Technologies Volume 4, Issue 1, pages 59-74 p-ISSN 2655-8564, e-ISSN 2685-9432 59 The Effect of Motor Parameters on the Induction Motor Speed Sensorless Control System using Luenberger Observer Bernadeta Wuri Harini1,* 1 Department of Electrical Engineering, Sanata Dharma University, Yogyakarta, Indonesia *Corresponding Author: wuribernard@usd.ac.id (Received 08-04-2022; Revised 27-05-2022; Accepted 29-05-2022) Abstract The sensorless control system is a control system without a controlled variable sensor. The controlled variable is estimated using an observer. In this investigation, the sensorless control system is used to control induction motor speed. The observer that is used is the Luenberger observer. One of the drawbacks of the sensorless control system is precision motor parameter values. In this research, the effect of induction motor parameters in a speed sensorless control system, i.e. resistance and inductance motor, will be investigated. The differences in induction motor parameters between the controller and the actual value affect the system response. The value differences of Rr and Rs that can be applied are a maximum of 50%. However, the small differences in the inductance value greatly affect the system response. To get a good response, the value differences of Ls and Lr are between -5% to +5%, while the difference in the value of Lm is between -3% to +3%. This work is licensed under a Creative Commons Attribution 4.0 International License International Journal of Applied Sciences and Smart Technologies Volume 4, Issue 1, pages 59-74 p-ISSN 2655-8564, e-ISSN 2685-9432 60 Keywords: inductance, induction motor, Luenberger observer, resistance, sensorless 1 Introduction A control system without a controlled variable sensor, often known as "sensorless control," is one controller that is still being researched. Sensorless control systems evolved to overcome the challenges of sensor installation that sensor-based control systems faced. Sensor-based control systems are widely used by researchers, such as those of Z. Alpholicy X., et al [1] and Y. E. Loho, et al [2]. Sensors will drive up prices and complicate installation [3]. The controlled variable in this system is approximated from the plant’s current input using an observer rather than being measured directly by a sensor [4]. The stator current is used to estimate the motor speed using an observer. Sensorless control will be used to control the speed of the Induction Motor in this investigation. The induction motor is one of the Alternating Current (AC) motors. The phase angle, as well as the modulo current (current vector), must be controlled while driving an AC motor [4]. It is not the same as a DC motor. The torque and flux that produce the AC motor current are decoupled in vector control so that they can be controlled independently. Precision motor parameter values are one of the drawbacks of the sensorless control approach for controlling motor speed. For this sensorless speed control to work properly, parameter values must be clearly understood. As a result, a variety of approaches for determining induction motor parameter values have been offered by different researchers [5][6]. The importance of induction motor parameters is also underlined in the paper[7]. The disparity in parameter values causes inaccuracies in motor speed, according to this article. However, it is not indicated in these trials how much variances in motor parameter values will affect the speed controller. A motor speed error will occur if the motor parameters deviate from the real parameter [8]. In that research, it is used MRAS observer to estimate the Permanent Magnet Synchronous Motor (PMSM). International Journal of Applied Sciences and Smart Technologies Volume 4, Issue 1, pages 59-74 p-ISSN 2655-8564, e-ISSN 2685-9432 61 In this research, the effect of induction motor parameters in a speed sensorless control system, i.e. resistance and inductance motor, will be investigated. To estimate the motor speed, it is used the Luenberger observer. 2 Research Methodology This section provides the research methodology that we use in this work. 2.1. Induction Motor Sensorless Control System The block diagram of the system is shown in Figure 1. Each part is explained below. Figure 1. Block diagram of the system 2.1.1. Induction Motor Mathematic Model Using Clarke and Park transforms, the three-phase mathematical model of the induction motor will be transformed into a two-phase mathematical model. The Clarke transformation converts balanced three-phase values (Vsa,sb,sc) into a two-phase stationary reference frame (α,β,0) using equation [9]:                           − −− =           sc sb sa s s v v v v v v 2 1 2 1 2 1 2 3 2 3 0 2 1 2 1 1 3 2 0   (1) where Vsα and Vsβ are the stator voltage in α,β reference frame. M Induction InverterPWMDecoupling Speed Controller IP Luenberger Current Controller dq abc dq abc isd* isq*ωr* usd usq isd isq vsq vsd ia va vb vc ib ic r   r ̂ s 1 International Journal of Applied Sciences and Smart Technologies Volume 4, Issue 1, pages 59-74 p-ISSN 2655-8564, e-ISSN 2685-9432 62 The Park transformation transforms a stationary reference frame into a rotating reference (d, q, 0) frame using the equation             − =          s s ee ee sq sd v v v v cossin sincos (2) where e is the electric angle of the motor, while Vsd and Vsq are the stator voltage in d-q reference frame. The induction motor mathematical model in d-q frame [10] is 𝑑 𝑑𝑡 𝑖𝑠𝑑 = 1 𝜎𝐿𝑠 𝑉𝑠𝑑 − ( 𝑅𝑠 𝜎𝐿𝑠 + (1−𝜎) 𝜎𝜏𝑟 ) 𝑖𝑠𝑑 + (1−𝜎) 𝜎𝜏𝑟 𝑖𝑟𝑑 + (1−𝜎)𝑁𝑝𝜔𝑟 𝜎 𝑖𝑟𝑞 + 𝜔𝑒 𝑖𝑠𝑞 (3) 𝑑 𝑑𝑡 𝑖𝑠𝑞 = 1 𝜎𝐿𝑠 𝑉𝑠𝑞 − ( 𝑅𝑠 𝜎𝐿𝑠 + (1−𝜎) 𝜎𝜏𝑟 ) 𝑖𝑠𝑞 + (1−𝜎) 𝜎𝜏𝑟 𝑖𝑟𝑞 + (1−𝜎)𝑁𝑝𝜔𝑟 𝜎 𝑖𝑟𝑑 + 𝜔𝑒 𝑖𝑠𝑑 (4) 𝑑 𝑑𝑡 𝑖𝑟𝑑 = − 𝑅𝑟 𝐿𝑟 𝑖𝑟𝑑 + 𝑅𝑟 𝐿𝑟 𝑖𝑠𝑑 + (𝜔𝑒 − 𝑁𝑝𝜔𝑟)𝑖𝑟𝑞 (5) 𝑑 𝑑𝑡 𝑖𝑟𝑞 = − 𝑅𝑟 𝐿𝑟 𝑖𝑟𝑞 + 𝑅𝑟 𝐿𝑟 𝑖𝑠𝑞 − (𝜔𝑒 − 𝑁𝑝𝜔𝑟)𝑖𝑟𝑑 (6) 𝑑 𝑑𝑡 𝜃𝑒 = 𝑁𝑝𝜔𝑟 + 𝑖𝑠𝑞 𝜏𝑟𝑖𝑚𝑟 (7) 𝑑 𝑑𝑡 𝜔𝑟 = 1 𝐽 (𝑇𝑒 − 𝑇𝐿 − 𝐵. 𝜔𝑟 ) (8) 𝑑 𝑑𝑡 𝜃𝑟 = 𝜔𝑟 (9) Where 𝑖𝑠𝑑 is stator current in d-frame, 𝑖𝑠𝑞 is stator current in q-frame, 𝑖𝑟𝑑 is rotor current in d-frame, 𝑖𝑟𝑞 is rotor current in q-frame, 𝜃𝑒 is voltage vector angle, and 𝜔𝑟 is rotor speed. Table 1 shows the parameter values of the induction motor that is used in this paper. The parameters are shown in Figure 2 [11]. Table 1. Parameter Values of Induction Motor Symbol Description Values Unit Np Pole pairs 2 pairs Rr Rotor resistance 2.9 Ω Rs Stator resistance 2.76 Ω Ls Stator inductance 0.2349 H Lr Rotor inductance 0.2349 H Lm Mutual inductance 0.2279 H International Journal of Applied Sciences and Smart Technologies Volume 4, Issue 1, pages 59-74 p-ISSN 2655-8564, e-ISSN 2685-9432 63 Figure 2. Equivalent Circuit in d-q frame 2.1.2. Observer Luenberger Luenberger observer is one of observer that uses adaptive method to estimate the controlled variable [12]. The equations of the estimation [10] are 𝑑 𝑑𝑡 𝑖̂𝑠𝑑 = − ( 𝑅𝑠 𝜎𝐿𝑠 + (1−𝜎) 𝜎𝜏𝑟 ) 𝑖̂𝑠𝑑 + 𝜔𝑒 𝑖̂𝑠𝑞 + 𝐿𝑚 𝜎𝐿𝑠𝐿𝑟𝜏𝑟 �̂�𝑟𝑑 + 𝐿𝑚𝑁𝑝𝜔𝑟 𝜎𝐿𝑠𝐿𝑟 �̂�𝑟𝑞 + 1 𝜎𝐿𝑠 𝑣𝑠𝑑 + 𝑔1(𝑖𝑠𝑑 − 𝑖̂𝑠𝑑) − 𝑔2(𝑖𝑠𝑞 − 𝑖̂𝑠𝑞 ) (10) 𝑑 𝑑𝑡 𝑖̂𝑠𝑞 = −𝜔𝑒 𝑖̂𝑠𝑑 + 1 𝜎𝐿𝑠 (−𝑅𝑠 − 𝐿𝑚 2 𝜏𝑟𝐿𝑟 ) 𝑖̂𝑠𝑞 − 𝐿𝑚𝑁𝑝𝜔𝑟 𝜎𝐿𝑠𝐿𝑟 �̂�𝑟𝑑 + 𝐿𝑚 𝜎𝐿𝑠𝐿𝑟𝜏𝑟 �̂�𝑟𝑞 + 1 𝜎𝐿𝑠 𝑣𝑠𝑞 + 𝑔2(𝑖𝑠𝑑 − 𝑖̂𝑠𝑑) + 𝑔1(𝑖𝑠𝑞 − 𝑖̂𝑠𝑞 ) (11) 𝑑 𝑑𝑡 �̂�𝑟𝑑 = 𝑅𝑟 𝐿𝑟 𝐿𝑚𝑖̂𝑠𝑑 − 1 𝜏𝑟 �̂�𝑟𝑑 + (𝜔𝑒 − 𝑁𝑝𝜔𝑟 )�̂�𝑟𝑞 + 𝑔3(𝑖𝑠𝑑 − 𝑖̂𝑠𝑑 ) − 𝑔4(𝑖𝑠𝑞 − 𝑖̂𝑠𝑞 ) (12) 𝑑 𝑑𝑡 �̂�𝑟𝑞 = 𝐿𝑚 𝜏𝑟 𝑖̂𝑠𝑞 − (𝜔𝑒 − 𝑁𝑝𝜔𝑟)�̂�𝑟𝑑 + 1 𝜏𝑟 �̂�𝑟𝑞 + 𝑔4(𝑖𝑠𝑑 − 𝑖̂𝑠𝑑) + 𝑔3(𝑖𝑠𝑞 − 𝑖̂𝑠𝑞 ) (13) where 𝑔1 = (𝑘−1) 𝑘 (− 𝑅𝑠 𝜎𝐿𝑠 − 𝑅𝑟 𝜎𝐿𝑟 ) (14) 𝑔2 = − (𝑘−1) 𝑘 𝑁𝑝𝜔𝑟 (15) 𝑔3 = (𝑘−1) 𝑘(𝜏𝑟 2𝑁𝑝 2�̂�𝑟 2+1) ( 𝑅𝑠𝑅𝑟𝜏𝑟+𝐿𝑠𝑅𝑟−𝜎𝜏𝑟𝐿𝑠𝐿𝑟𝑁𝑝 2�̂�𝑟 2 𝐿𝑚 ) (16) 𝑔4 = (𝑘−1) 𝑘(𝜏𝑟 2𝑁𝑝 2�̂�𝑟 2+1) ( (𝑅𝑠𝐿𝑟𝜏𝑟+𝐿𝑠𝑅𝑟𝜏𝑟−𝜎𝐿𝑠𝐿𝑟)𝑁𝑝 2�̂�𝑟 𝐿𝑚 ) (17) International Journal of Applied Sciences and Smart Technologies Volume 4, Issue 1, pages 59-74 p-ISSN 2655-8564, e-ISSN 2685-9432 64 The estimation speed (�̂�𝑟 ) is then calculated using equation �̂�𝑟 = 𝐾𝑝(�̂�𝑟𝑞 𝑒𝑖𝑠𝑑 − �̂�𝑟𝑑 𝑒𝑖𝑠𝑞 ) + 𝐾𝑖 ∫(�̂�𝑟𝑞 𝑒𝑖𝑠𝑑 − �̂�𝑟𝑑 𝑒𝑖𝑠𝑞 )𝑑𝑡 (18) where 𝑒𝑖𝑠𝑑 = 𝑖𝑠𝑑 − �̂�𝑠𝑑 (19) 𝑒𝑖𝑠𝑞 = 𝑖𝑠𝑞 − �̂�𝑠𝑞 (20) 2.2. Decoupling and Current Sensor The direct-axis stator current 𝑖𝑠𝑑 (the rotor flux-producing component) and the quadrature-axis stator current 𝑖𝑠𝑞 (the torque-producing component) must be controlled separately for rotor flux-oriented vector control. The equations for the stator voltage components, on the other hand, are linked. 𝑢𝑠𝑑 , the direct axis component, and 𝑢𝑠𝑞 , the quadrature axis component, are both dependent on 𝑖𝑠𝑑 . For the rotor flux and electromagnetic torque, the stator voltage components 𝑢𝑠𝑑 and 𝑢𝑠𝑞 cannot be regarded as disconnected control variables. If the stator voltage equations are decoupled and the stator current components 𝑖𝑠𝑑 and 𝑖𝑠𝑞 are indirectly controlled by manipulating the induction motor's terminal voltages, the stator currents 𝑖𝑠𝑑 and 𝑖𝑠𝑞 can only be adjusted individually (decoupled control) [13]. The currents 𝑖𝑠𝑑 and 𝑖𝑠𝑞 are then controlled by Proportional Integral (PI) current sensor. The output current sensors are determined using equation [14] 𝑢𝑠𝑑 = (𝐾𝑖𝑑𝑝 + 𝐾𝑖𝑑𝑖 𝑠 ) (𝑖𝑠𝑑 ∗ − 𝑖𝑠𝑑 ) (21) 𝑢𝑠𝑞 = (𝐾𝑖𝑞𝑝 + 𝐾𝑖𝑞𝑖 𝑠 ) (𝑖𝑠𝑞 ∗ − 𝑖𝑠𝑞 ) (22) where 𝑖𝑠𝑑 = 1 𝑇𝑑𝑠+1 𝑖𝑠𝑑 ∗ (23) 𝑖𝑠𝑞 = 1 𝑇𝑑𝑠+1 𝑖𝑠𝑞 ∗ (24) International Journal of Applied Sciences and Smart Technologies Volume 4, Issue 1, pages 59-74 p-ISSN 2655-8564, e-ISSN 2685-9432 65 2.1.3. Speed Controller The reference current in q-reference frame ( * sq i ) in (24) is controlled by the Integral Proportional (IP) speed controller. The equation of IP speed controller [15] is 𝑖𝑠𝑞 ∗ = ∫ 𝐾𝑖 (𝜔𝑟 ∗ − 𝜔𝑟 )𝑑𝑡 − 𝐾𝑝𝜔𝑟 (25) where Kp and Ki are the speed controller gain. 2.2. Testing Method The system is tested using Matlab – Simulink – Cmex [16]. The simulation block diagram is shown in Figure 3. Figure 3. Simulation system The values of the various parameters are inputted to the current controller using the input port in the figure. With the control parameters Kp=0.5 and Ki=1, the reference speed is 100 rad/s. The stator and rotor resistance, stator and rotor inductance, and mutual inductance characteristics are all employed. In this test, the values of motor parameters in the controller vary as shown in Table 2, so they are different from the actual motor parameters. International Journal of Applied Sciences and Smart Technologies Volume 4, Issue 1, pages 59-74 p-ISSN 2655-8564, e-ISSN 2685-9432 66 Table 2. Variation Parameter Values of Induction Motor Parameter Percentage change (%) Values Unit s R -50 1.38 Ω -90 0.276 +50 4.14 +90 5.244 𝑅𝑟 -50 1.45 Ω -90 0.29 +50 4.35 +65 4.785 𝐿𝑠 -4 0.225504 H -5 0.223155 +5 0.246645 +10 0.25839 𝐿𝑟 -4 0.225504 H -5 0.223155 +5 0.246645 +10 0.25839 𝐿𝑚 -2 0.223342 H -3 0.221063 -5 0.216505 -10 0.20511 +2.5 0.233598 +3 0.234737 3 Results and Discussion The simulation result of the system using the right parameters is shown in Figure 4. It is shown that the actual speed (𝜔𝑟 ) can reach the reference speed (𝜔𝑟 ∗), i.e. 100 rad/s. Although the estimated speed at the transient is slightly different from the actual speed, the estimated speed has the same value as the actual and reference speed at a steady-state. This means that the sensorless control system is working well. The simulation result of the system using various parameters values are described in Figures 5 - 9. International Journal of Applied Sciences and Smart Technologies Volume 4, Issue 1, pages 59-74 p-ISSN 2655-8564, e-ISSN 2685-9432 67 Figure 4. Simulation result with normal parameter values Figure 5. Simulation result with the variation of Rs parameter values Figure 5 shows the value of Rs on the controller being varied. The figure shows that when the value of Rs in the controller is reduced to 90% (Figures 5.a and c), the actual speed can reach the reference speed, which is 100 rad/s, although there are differences in the transient conditions. When Rs is enlarged by 50% (Figure 5.b), the actual speed can International Journal of Applied Sciences and Smart Technologies Volume 4, Issue 1, pages 59-74 p-ISSN 2655-8564, e-ISSN 2685-9432 68 reach the reference speed, even though the estimated speed is oscillating. However, if the value of Rs is enlarged again, a steady state error occurs, where there is a difference between the actual and the reference speed, although only slightly (Figure 5.d). In this condition, the estimation speed oscillates with increasing amplitude. Thus, to get a good response, the difference in the value of Rs that can be applied is a maximum of +50%. Figure 6. Simulation result with the variation of Rr parameter values The condition for the change in the value of Rs is almost the same as the condition for the change in the value of Rr, as shown in Figure 6. The figure shows that when the value of Rr in the controller is reduced to 90% (Figures 6.a and c), the actual speed can reach the reference speed, namely 100 rad/s, although there is a difference in the transient conditions. In addition, when the Rr value is reduced, overshoot will occur (Figure 6.c), although the overshoot percentage is only slightly. When Rr is enlarged by 50% (Figure 5.b), the actual speed can reach the reference speed. However, if the value of Rr is enlarged again by 65%, the estimated speed and the actual speed oscillate (Figure 6.d). Thus, to get a good response, the difference in the value of Rr that can be applied is a maximum of 50%. International Journal of Applied Sciences and Smart Technologies Volume 4, Issue 1, pages 59-74 p-ISSN 2655-8564, e-ISSN 2685-9432 69 Figure 7. Simulation result with the variation of Ls parameter values The effect of differences in resistance values is different from differences in inductance values, as illustrated in Figures 7 - 9. In the three figures, to get a good response, the difference in inductance values between the inductance values in the controller and the actual is very small. The difference in the values of Ls (Figure 7) and Lr (Figure 8) is between -5% to +5% (Figure 7.a - c and Figure 8.a -c). When the difference gets bigger, i.e. 10%, the estimation speed oscillates (Figs 7.d and 8.d). In the two figures, it appears that the actual speed time to achieve stability (settling time) is longer than before. International Journal of Applied Sciences and Smart Technologies Volume 4, Issue 1, pages 59-74 p-ISSN 2655-8564, e-ISSN 2685-9432 70 Figure 8. Simulation result with the variation of Lr parameter values The small differences in the value of mutual inductance (Lm) between the controller and the actual value of the motor parameters have greatly affected the system response, as illustrated in Figure 9. It appears that to get a good response, the differences in the inductance value between the inductance value in the controller and the actual is smaller than Ls and Lr. The difference in Lm values that can be applied is between -3% to +3% (Figure 9.a, c, d, f). As the difference gets bigger, the estimation speed oscillates (Figs 9.b and d). In the two figures, it appears that the actual speed time to achieve stability (settling time) is longer than before. When the Lm value is enlarged (more than +3%) the system becomes an error. Therefore, the recommended Lm differences value is -3% to +3%. International Journal of Applied Sciences and Smart Technologies Volume 4, Issue 1, pages 59-74 p-ISSN 2655-8564, e-ISSN 2685-9432 71 Figure 9. Simulation result with the variation of Lm parameter values 4 Conclusion The differences in induction motor parameters between the controller and the actual value affect the system response. The value differences of Rr and Rs that can be applied are a maximum of 50%. However, the small differences in the inductance value greatly affect the system response. To get a good response, the value differences of Ls and Lr are between -5% to +5%, while the difference in the value of Lm is between -3% to +3%. International Journal of Applied Sciences and Smart Technologies Volume 4, Issue 1, pages 59-74 p-ISSN 2655-8564, e-ISSN 2685-9432 72 References [1] Z. Alpholicy X., S. S. Bhandari, P. P. Dsouza, and D. C. Raina, “Personal Assistant Robot,” Int. J. Appl. Sci. Smart Technol., 3(2), 145–152, 2021. [2] Y. E. Loho, D. Lestariningsih, and P. R. 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Control (E3-C3), INDICON 2015, 4–9, 2016. [13] F. Semiconductor, “3-Phase AC Induction Motor Vector Control Using a 56F8300 Device,” Memory, 2005. [14] R. Gunawan and F. Yusivar, “Reducing estimation error due to digitizing problem in a speed sensorless control of induction motor,” IECON Proc. (Industrial Electron. Conf., 2005(1), 1677–1682, 2005. [15] F. Yusivar, H. Haratsu, T. Kihara, S. Wakao, and T. Onuki, “Performance comparison of the controller configurations for the sensorless IM drive using the modified speed adaptive observer,” IEE Conf. Publ., 475, 194–200, 2000. [16] F. Yusivar and S. Wakao, “Minimum requirements of motor vector control modeling and simulation utilizing C MEX S-function in MATLAB/SIMULINK,” Proc. Int. Conf. Power Electron. Drive Syst., 1, 315–321, 2001. International Journal of Applied Sciences and Smart Technologies Volume 4, Issue 1, pages 59-74 p-ISSN 2655-8564, e-ISSN 2685-9432 74 This page intentionally left blank