HUNGARIAN JOURNAL OF INDUSTRY AND CHEMISTRY Vol. 53(1) pp. 27–37 (2025) hjic.mk.uni-pannon.hu DOI: 10.33927/hjic-2025-04 COMPARATIVE ANALYSIS OF MACHINE MODEL-BASED SENSORLESS DTC OF TWO PARALLEL CONNECTED FIVE-PHASE INDUCTION MACHINES KHALED MOHAMMED SAID BENZAOUI1*, SIFELISLAM GUEDIDA2, AHMED ZOUHIR KOUACHE1 AND ELAKHDAR BENYOUSSEF1 1 Laboratoire LAGE, Faculté des Sciences Appliquées, Université Kasdi Merbah Ouargla, Ouargla, 30000, ALGERIA 2 École Militaire Polytechnique, UER ELT, Algiers, 16111, ALGERIA In high-performance vector control methods, especially for critical applications, accurate and precise information about the state variables of a machine (speed, flux and torque) is necessary. Nevertheless, environmental conditions such as temperature, vibrations and EMI affect the performance of sensors. Sensorless control is a potential solution to address the aforementioned drawbacks as well as further enhance the reliability and performance of the system, in addition to reducing the cost and the size of the drive. Therefore, this work provides a comparative study of three different sensorless approaches of measuring the stator flux and speed estimation for the direct torque control (DTC) of two five-phase induction machines (FPIM) connected in parallel to a single two-level inverter based on a model reference adaptive system (MRAS), sliding mode observer (SMO) and high- gain observer (HGO). The independent control of such a drive is possible due to the additional degrees of freedom (DOF) provided by the five-phase system and the suitable rearrangement of phases. The open-loop estimator used in conventional DTC for estimating flux leads to drift and initial value problems as a result of using the pure integration method. Finally, an analytical study of the robustness, stability and effectiveness of the discussed sensorless control schemes is verified in terms of reference command tracking, low-speed operation and disturbance rejection. Keywords: sensorless direct torque control (DTC), sliding mode observer (SMO), model reference adaptive system (MRAS), high-gain observer (HGO), two parallel-connected FPIMs 1. Introduction Highly reliable and high-power critical applications such as in ship propulsion, electric vehicles and electric aircraft are the most suited applications for five-phase induction machines (FPIM) and permanent magnet synchronous machines (PMSM) [1]-[3]. This is due to several advantages that they boast, for instance, reduced current rating per phase, less flux and torque ripples as well as increased torque density on top of their fault- tolerant operation and additional degrees of freedom (DOF) compared to three-phase induction machines [4],[5]. These DOFs can be employed to achieve drive in two independent parallel-connected machines fed by a single inverter where the two machines operate independently under different conditions such as speed and load torque [6]. The direct torque control (DTC) of the FPIM is simple compared to field-oriented control and less sensitivity to the parameters of the machine with the Received: 19 Sept 2024; Revised: 6 Nov 2024; Accepted: 11 Nov 2024 *Correspondence: benzaoui.khaled@univ-ouargla.dz decoupled control of the flux and torque of the FPIM enabling high dynamic performance [7],[8]. The working principle of DTC is based on choosing the optimal voltage vector (VV) to be applied in each sampling period in order to obtain the desired level of performance regarding the stator flux and electromagnetic torque requirements [9]. High-quality speed and flux sensors, such as incremental encoders and the Hall effect, are a crucial part of vector control for drive systems like in automotive applications. Nevertheless, environmental conditions, costs and the sheer size of the system restrict their application in drive systems. Moreover, flux sensors require the machine to be redesigned. As a result, developing a speed and flux sensorless approach is greatly sought. Since mechanical sensors are heavy and susceptible to malfunction under demanding conditions, resulting in the degradation of the inherited robustness of the FPIM, sensorless technologies can improve the reliability and robustness of the drive system [10]. https://doi.org/10.33927/hjic-2025-04 mailto:benzaoui.khaled@univ-ouargla.dz BENZAOUI, GUEDIDA, KOUACHE AND BENYOUSSEF Hungarian Journal of Industry and Chemistry 28 As mentioned above, sensorless drive systems have attracted considerable attention over the last few years since implementing sensors increases hardware complexity and reduces noise immunity. Therefore, numerous methods for estimating the rotor speed and stator flux have been introduced to date classified as signal injections and machine model-based methods [11], as illustrated in Figure 1. The first method is the signal injection technique, which can be further classified into high-frequency and low-frequency methods. Meanwhile, good performance at low speeds and when stationary as well as reduced parameter sensitivity are achieved. Regardless, these methods require additional hardware, resulting in severe skin effects due to the injected signal and noticeable electromagnetic torque ripples. Furthermore, the machines need to be redesigned. The state-of-the-art nature of signal injection methods has been reported [11],[12]. The second method is a machine model-based (MM) technique, where the machine model and measured parameters (stator voltages and currents) are used to estimate state variables following a specific structure and direct physical phenomena interpretation. Such methods have gained much attention in the literature [10],[11]. Estimating the rotor speed using the model reference adaptive system (MRAS) is based on the error between the output of two models, that is, the reference and adjustable model, which is fed into an adaptive model to estimate the rotor speed. This approach is known for its accuracy, simplicity and ease of implementation [13]. In [14], a sliding mode observer (SMO) for the sensorless application of an induction machine was proposed. The SMO is comprised of a sliding surface and control law, ensuring the state variables are accurately estimated with reduced sensitivity to parameter variations and a faster dynamic response. The high-gain observer (HGO) can estimate state variables of the system based on the canonical form and reject model uncertainties. Furthermore, its simple structure and fast dynamic response have drawn interest concerning the sensorless control application of induction machines [15]. Therefore, this manuscript provides a comparative analysis of the three MM-based sensorless solutions, namely MRAS, SMO and HGO. Considering the merits and disadvantages of each technique, the structure of this paper is as follows: Section 2 describes the modeling of a dual parallel-connected FPIM drive. Section 3 deals with the DTC of the drive of the two machines. The MM- based sensorless schemes are described in detail in Section 4. In Section 5, the simulation results are interpreted. Finally, in the last section, conclusions are drawn. 2. Modelling drive The circuit diagram of the two FPIMs connected in parallel powered from a single two-level inverter is shown in Figure 2. Sensorless Techniques Signal Injection Methods Model-based Methods Low Frequency Methods High Frequency Methods Open-loop Estimator Model Reference Adaptive System Extended Kalman Filter Sliding Mode Observer High Gain Observer Full and Reduced Order Closed Loop Observer Figure 1: Classification of sensorless control techniques 2A S 1A S 2B S 1B S 2C S 1C S 2D S 1D S 2E S 1E S dc V A B C D E FPIM 1 FPIM 2 Figure 2: Schematic diagram of two FPIM drives connected in parallel COMPARATIVE ANALYSIS OF SENSORLESS METHODS 53(1) pp. 27–37 (2025) 29 The relationship between the inverter and the voltage of both machines is discussed as follows: 1 2 1 2 1 2 1 2 1 2 inv A SA SA inv B SB SC inv C SC SE inv SD SBD inv SE SDE v v v v v v v v v v vv v vv   =      =       ==     =       =   (1). The Clarke’s transformation matrix [C] describes the five-phase system in two orthogonal planes, 1 and 2 , as well as in terms of a zero sequence component as follows [16]: 1 cos (2 / 5) cos (4 / 5) cos (6 / 5) cos (8 / 5) 0 sin (2 / 5) sin (4 / 5) sin (6 / 5) sin (8 / 5) 2 1 cos (6 / 5) cos (2 / 5) cos (8 / 5) cos (4 / 5) [ ] 5 0 sin (6 / 5) sin (2 / 5) sin (8 / 5) sin (4 / 5) 1 1 1 1 1 2 2 2 2 2 C                         =           (2). The considered machines consist of five distributed star-connected windings spatially shifted by 72 electrical degrees. The two machines model in the stationary reference frame, described assuming the same simplifying assumption applied in three-phase machines, is given below [17]: 0 sj j sj j sj sj j rj j rj rj j s j sj s j mj rj j sj j sj sj j rj j rj rj j mj sj j rj j rj rj j d V R i dt d R i dt L i L i L i L i L i L i                       = +   = +   = +  =  = +  = (3). The electromagnetic torque is given as: 5 sin j mj emj sj rj j j sj rj p L T L L     = (4), where: j = 1 or 2, sj j V  stator voltages, sj j i  stator currents, s j stator flux linkages, rj j i  rotor currents, r j rotor flux linkages, sR stator resistance, sL stator leakage inductance, mL mutual inductance, rR rotor resistance, rL rotor leakage inductance, emT electromagnetic torque and p pair poles. 3. Direct torque control The DTC scheme is based on the direct application of the optimal VV to the drive, which is further explained by Equation 4. Over a sampling period, the stator flux is assumed to be constant due to the high time constant of the rotor circuit compared to the stator circuit [18]. From Equation 3, the stator flux is described as follows in the plane j : ^ ( )sj j sj j sj sj jv R i dt   = − (5). By neglecting the drop in the resistance voltage, Equation 5 can be written as: ^ sj j sj jv dt  =  (6). Therefore, by applying the optimal VV, the stator flux and electromagnetic torque angle δ are controlled (Equation 4) [18]. The implemented DTC for the proposed drive is shown in Figure 3. Two-level and seven-level HCs are utilized to compare the error between the reference and estimated values of both the flux and torque of the machines, thereby determining the required response, namely to increase or decrease the flux and torque. The obtained output of the HCs and the flux position are exploited to develop look-up table (Table 1) in order to select the optimal VV. The VVs of the two- level inverter are illustrated in Figure 4. Control of the two-machine drive requires two independent DTC controllers. Therefore, one controller is developed in the 1 plane and the other in the 2 plane taking into consideration the phase transposition given in Equation 1 and Figure 2, resulting in the current components generating torque/flux for one machine but not for the other and vice versa [19]. For further explanation if the stator flux vector of the first FPIM is in Sector II and the stator flux needs to be increased ( sj =1) and decreasing the electromagnetic torque ( Temj =3). Therefore, from Figure 4a, the selected VV is VM4 with a switching sequence of 11110. Furthermore, if the stator flux of the second FPIM is within Sector VI and the stator flux must be decreased while increasing the electromagnetic torque ( sj =0, Temj =3), the chosen VV is VL5 with a switching sequence of 01110 as shown in Figure 4b. Table 1: The look-up table Temj sj -1 1 3 VL(i+4) VL(i+1) 2 VM(i+4) VM(i+1) 1 VS(i+4) VS(i+1) 0 V0 V0 -1 VS(i+6) VS(i+9) -2 VM(i+6) VM(i+9) -3 VL(i+6) VL(i+9) BENZAOUI, GUEDIDA, KOUACHE AND BENYOUSSEF Hungarian Journal of Industry and Chemistry 30 The logic selection block alternates between the two selected VVs throughout the whole sampling period. For example, when VM4 is applied to the inverter over the first sampling period and VL5 over the second. 4. Machine model (MM)-based sensorless methods The basic common schematic structure of the MM-based sensorless methods which consist of the stator flux and/or rotor speed observer are depicted in Figure 5. This paper analyzes the principles of three MM- based stator flux and rotor speed observers. 4.1. Model reference adaptive system (MRAS) The MRAS discussed herein was proposed in [20]. The main objective of this observer is to estimate the rotor’s mechanical speed based on the measured stator voltages and currents. The MRAS constitutes three models as follows [21]: Stator flux and Torque estimation Logic selection PI PI First FPIM Second FPIM 5 Look-up table Look-up table 1Tem  2Tem 2 * em T 1 * em T 1m  2m  2m  * 1m  * dc V 2 2S V  2 2S I  1 1S I  1 1S V  1s 2s 1 * S  2 * S  ABCDE ACEBD 1  2 2emT ^ 2S ^ 2S ^ 1emT ^ 1S ^ 1S ^ Stator flux and Torque estimation 1emT ^ 1S ^ 2emT ^ 2S ^ 2S ^ 1S ^ S ACEBD S ABCDE I SABCDE I SACEBD S ABCDE S ABCDE 1 S ABCDE 2 2m  1m  Figure 3: DTC of two FPIMs connected in parallel VL1 11001 V L 9 V L 8 V L 7 VL5 VL6 VL2 VL10 VM 10 V M 9 V M 8 VM 7 VM6 VM 5 V M 3 V M 4 VM 2 VM1 V S 8 VS10 V S 9 VS7 VS6 VS5 V S 4 V S 3 VS2 VS1 01001 10000 11000 11101 11010 0 1 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 01110 00100 01010 101100111100110 00111 00010 00101 0 0 0 1 1 1 0 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 0 10001 11011 10101 [I] [VIII] [VII] [VI] [V] [IV] [III] [II] [X] [IX] V0 11111 00000 (a) 0 0 1 0 1 VL1 1011001001 V L 9 V L 8 V L 7 VL5 VL6 V L 4 V L 3 VL2 VL10 VM 10 V M 9 V M 8 VM 7 VM6 VM 5 V M 3 V M 4 VM 2 VM1 V S 8 VS10 V S 9 VS7 VS6 VS5 V S 4 V S 3 VS2 VS1 01111 1000011001 10100 10111 11100 1 0 1 0 1 0 0 1 0 0 1 0 0 0 1 1 1 1 0 1 0 0 1 1 1 01101 00001 01100 00110 01011 01000 00011 0 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1 1 0 1 0 0 0 0 2 0 1 1 0 0 0 10010 11110 10011 [I] [VIII] [VII] [VI] [V] [IV] [III] [II] [IX] [X] V0 11111 00000 (b) Figure 4: Mapping VVs: (a) in the 1 plane, (b) in the 2 plane OBSERVER GAIN FPIM MODEL Observer OBSERVED QUANTITIES MOTOR OUTPUT COMMAND VECTOR Figure 5: General presentation of the observers COMPARATIVE ANALYSIS OF SENSORLESS METHODS 53(1) pp. 27–37 (2025) 31 The reference model The voltage model of the machine is utilized given the independent nature of the mechanical speed of the rotor as follows: ( )       rj rj sj sj sj j sj sjj j j mj Ld V R I L I dt L        = − −        (7). The adaptive model The stator current model of the FPIM is implemented, where the estimated mechanical speed of the rotor is integrated into the model to adjust the rotor flux. ^ ^ ^ ^ ^ 1            sj j rj j rj jmj rj r sj jj rj rj rj j rj j ILd Idt T T     −      = − +                 (8) The adaptation mechanism The error between the estimated values of the rotor flux obtained by the two models is entered into the adaptation mechanism to estimate the mechanical speed of the rotor as described in the following equation: ^ ^ ^ estj rj j rj j rj j rj j rj pj estj ij estjK K dt             = −   = +   (9). The stator flux and electromagnetic torque can be calculated as follows: 2 ^ ^ 2 ^ ^ 2 2^ ^ ^ ^ 1 ^ ^ ^ ^ tan 5 ( ) 2 mj sj rj mj sj j rj sj j rj rj mj sj rj mj sj j rj sj j rj rj sj sj j sj j sj j sj sj j j emj sj j sj j sj j sj j L L L L i L L L L L L i L L p T i i                           −  − = +   − = +    = +      =          = −  (10). The schematic diagram of the MRAS is illustrated in Figure 6. 4.2. Sliding mode observer In this control method, the sliding mode control is used to construct the stator flux and speed observer to reconstruct the FPIM variables. Figure 7 shows the block diagram of the SMO. The observer design is based on two aspects, namely the sliding surface and the control law, which control the estimation process. With reference to the FPIM model in the j plane, the observer is expressed as follows [22]: ^ ^ ^ ^ ^ 1 6 2 3 1 1 1 2 2 ^ ^ ^ ^ ^ 6 1 3 2 1 3 1 4 2 ^ ^ 4 1 2 21 ^ ^ 4 3 1 4 2 s j s j s j s j s j s j i s i s s j s j s j s j s j s j i s i s s j s j s j s s s j s j s j s s d i a i a i a a bV A I A I dt d i a i a i a a bV A I A I dt d a i V A I A I dt d a i V A I A I dt                              = + + + + + +    = − + + + + + +  = + + + = + + +        (11), where ^ ^ ,s j s ji i  denote the estimated stator currents, ^ ^ ,s j s j   represent the estimated stator flux components, r stands for the estimated rotor speed, ( 1,2,3,4)ixA x = refer to the stator current gains and ( 1,2,3,4)xA x = are the stator flux gains. The sliding surface is defined as: 1 2 ( 1) ( 2) s s s I sign S I I sign S     = =       (12), where: ^ ^ 1 2 s js j s js j S i i S i i    = −   = − (13). Rs S jV  S ji  sL Rs S jV  S ji  1 rT m r L T 0 Kp Ki sL r m L L r m L L 1 rT m r L T r j r j ^ r j ^ r j ^ rj Figure 6: Schematic diagram of the MRAS Rs K sign SjV Sji Sj s V L 1 sL 1 s rL L Figure 7: Schematic diagram of the SMO BENZAOUI, GUEDIDA, KOUACHE AND BENYOUSSEF Hungarian Journal of Industry and Chemistry 32 The current and flux gains can be selected as follows: 1 2 1 1 2 2 0 0 i i i i A A D A A       =       (14), where r r s r s r r s s r R L L L D R L L L            =   −    (15) and ^ 1 1 1 2 ^ 1 2 1 2 rjrj rj j sj rj rj j sj rj R q L LA A A A R q L L               −         =      −  −       (16), where 1 , 2 , 1q and 2q are positive constants. The stability of the observer is related to its convergence onto the sliding surface [22]. A Lyapunov function is proposed to solve this problem as follows: 1 2 TV S S= (17). Since the stability of the observer is related to the estimated stator flux and their convergence onto the sliding surface, the observation error should be equal to zero. For this to be so, the derivative of the Lyapunov function must be strictly negative. . .1 0 2 TV S S=  (18) Therefore, the stability of the observer is ensured if the inequality below is verified: 1 2           (19). The stator flux vector and position, in addition to the rotor speed and electromagnetic torque, are computed as follows: 2 2^ ^ ^ ^ ^ 1 ^ ^ ^ ^ ^ ^ ^ ^ ^ tan 1/ ( ( ) ) 5 ( ) 2 sj sj j sj j sj j sj sj j mj j Ljsj j sj jj j sj j s j j mj j emj sj j sj jsj j sj j d J p i i T f dt p T i i                       −  = +      =       = − − −   = −  (20). 4.3. High-gain observer (HGO) The HGO technique has attracted much attention for its simple design and ease of tuning. The HGO is designed to directly reconstruct the stator variables of the machine from the measured quantities (stator voltages and currents) [23]. The block diagram of the HGO is illustrated in Figure 8. The expression of the observer, based on the FPIM model, can be given as: . ^ ^ ^ 1 1 1 ( ) ( ) ( ) m T m m x A x Bu v e v e x S C Ce − − −   = + +   =  V (21), ^ ^ 0 0 T s j s j s j s j T s j s jm s j s j x i i e i i i i           =       = − −    (22), where x denotes the state variables and me the stator current error. 1 1 2 2 1 2 2 1 1 2 2 1 ( , ) 0 ( ) ( , ) 1 ( , ( )) ( , ( )) diag I I with diag I I diag I kF with diag I F k      − −  =  =    =  =  V V (23) with 1 s k L = , ( ) r s r s r s s r R L L L kF R L L L             =   −    (24), where S represents the peculiar solution of the Lyapunov function.  1 1 2 2 2 2 2 2 22 T TS A S SA C C S C C I C I I I−  + + =   = =   (25) Replacing ( )mv e in Equation 21 yields: SjV Sji SjV Sji b1 a1/a6 a2/a3 2θ Sji ^ Sji ^ Sji a4 ^ Sj ^ Sj ^ Sj ji Sj ji  ^ Sj ji Sj ji  K K Figure 8: Schematic diagram of the HGO COMPARATIVE ANALYSIS OF SENSORLESS METHODS 53(1) pp. 27–37 (2025) 33 ^ ^ 1 6 2 3 1 ^ ^ 6 1 3 2 1 ^ ^ ^ 2 2 32 4 2 2 2 2 2 3 2 3 ^ 4 2 ( ) 2 ( ) ( ) ( ) s j s js j s j s j s j s j s j s j s js j s j s j s j s j s j s j s j s js j s j s j s j s j s j d i a i a i a a bV i i dt d i a i a i a a bV i i dt d aa a i V i i i i dt a a a a d a i dt                                 = + + + + + − = − + + + + + − = + + − − − + + = ^ ^ 2 2 2 2 2 2 2 2 3 2 3 3 ( ) ( )s j s js j s j s j aa V i i i i a a a a                   + + − − −  + + (26), where 1 s r s R R a L  + = −    , 2 r s r R a L L = , 3 s a L   = , 4 sa R= − , 6 ra = − , 1 1 s b L = and  is the observer gain. Estimation of the speed and torque are achieved based on the following formula: ^ ^ ^ ^ ^ ^ ^ 1/ ( ( ) ) ( ) j s j s jj s j s j L j em s j s jj s j s j d J p i i T f dt T p i i               = − − −   = − (27). 5. Simulation results This section uses simulation tests to evaluate the performance of the three MM-based observers in their transient/steady state and at low speeds as well as by taking into consideration implementation complexity and computational burden. The parameters of the two machines are depicted in Table 2. 5.1. First test The performance of the three MM-based sensorless methods is evaluated at different speeds, such as low and high, under their rated load torque. The test scenarios are summarized in Tables 3 and 4. 5.2. Second test This test examines the robustness and disturbance rejection of the three MM-based sensorless methods when the load torque is suddenly changed. The test scenario is described in Table 5. 5.3. Discussion The discussed MM-based sensorless methods to estimate accuracy using the FPIM over a wide range of reference speeds and by varying the load torque is evaluated (Figures 9-18). In Figures 9 and 14, the rotor speed responses of the two machines are shown. When the drive is operated in its steady and transient states, the speed of the machines Table 3: Test scenario of the first FPIM [ ]Time s [ / ]m rad s [ ]LT Nm 0 0.1→ 0 20→ 8 0.1 1→ 20 1 1.3→ 20 60→ 1.3 2→ 60 2 2.2→ 60 100→ 2.2 3→ 100 3 4→ 100 100→ − 4 4.5→ 100− Table 2: Parameters of the machine 1 Hp ; 200 V; 50 Hz; 1400 rpm 𝑅𝑠 [Ω] 10 𝑅𝑟 [Ω] 6.3 𝐿𝑠 [Ω] 0.4642 𝐿𝑟 [Ω] 0.4612 𝐿𝑚 [𝐻] 0.4212 𝐽 [𝐾𝑔. 𝑚2] 0.03 𝑓 [ 𝑁𝑚. 𝑠−1 𝑟𝑎𝑑 ] 0.0001 𝑇𝑒𝑚 [𝑁. 𝑚] 8 𝑝 2 Table 4: Test scenario of the second FPIM [ ]Time s [ / ]m rad s [ ]LT Nm 0 0.5→ 0 100→− 8 0.5 2→ 100− 2 2.6→ 100 20− → 2.6 3.4→ 20 3.4 3.7→ 20 60→ 3.7 4.2→ 60 4.2 4.4→ 60 100→ 4.4 5→ 100 Table 5: Second test scenario [ ]Time s [ / ]m rad s 1[ ]LT Nm 2[ ]LT Nm 0 0.5→ 0 100→ 8 -2 0.5 1→ 100 1 2→ 0 8 2 3→ -4 0 BENZAOUI, GUEDIDA, KOUACHE AND BENYOUSSEF Hungarian Journal of Industry and Chemistry 34 (a) (b) (c) (d) Figure 9: Rotor speed response: (a) DTC, (b) MRAS, (c) SMO and (d) HGO (a) (b) (c) (d) Figure 10: Electromagnetic torque response: (a) DTC, (b) MRAS, (c) SMO and (d) HGO (a) (b) (c) (d) Figure 11: Stator flux response: (a) DTC, (b) MRAS, (c) SMO and (d) HGO (a) (b) (c) Figure 12: Estimated rotor speed: (a) MRAS, (b) SMO and (c) HGO (a) (b) (c) Figure 13: Estimation error: (a) MRAS, (b) SMO and (c) HGO COMPARATIVE ANALYSIS OF SENSORLESS METHODS 53(1) pp. 27–37 (2025) 35 (a) (b) (c) (d) Figure 14: Rotor speed response: (a) DTC, (b) MRAS, (c) SMO and (d) HGO (a) (b) (c) (d) Figure 15: Electromagnetic torque response: (a) DTC, (b) MRAS, (c) SMO and (d) HGO (a) (b) (c) (d) Figure 16: Stator flux response: (a) DTC, (b) MRAS, (c) SMO and (d) HGO (a) (b) (c) Figure 17: Estimated rotor speed: (a) MRAS, (b) SMO and (c) HGO (a) (b) (c) Figure 18: Estimation error: (a) MRAS, (b) SMO and (c) HGO BENZAOUI, GUEDIDA, KOUACHE AND BENYOUSSEF Hungarian Journal of Industry and Chemistry 36 accurately follows reference commands regardless of the loading conditions of both machines with negligible over/undershoot. The estimated rotor speeds for the three observers are illustrated in Figures 12 and 17, respectively. These results show that the three methods accurately track the reference speed applied to the drive using different estimation errors, namely 4 rad/s for MRAS, ±3 rad/s for SMO and ±6 rad/s for HGO as presented in Figures 13 and 18, respectively. Given the electromagnetic torque response shown in Figures 10 and 15, it is quickly interpreted that in response to changes in speed and the load torque applied, the electromagnetic torque is precisely and quickly determined corresponding to the load torque in the steady state. The stator flux response exhibiting a circular trajectory following the reference command as well as a good degree of decoupling between the stator flux and electromagnetic torque of the two FPIMs is depicted in Figures 11 and 16. Therefore, neither the transient state nor sensorless operation affects the performance of the two-machine drive. The test results are summarized in Table 6 where a slight reduction in the torque and flux ripples is due to the accurate estimation of the three observers as opposed to the open-loop estimator used in the conventional method. An analytical comparison is given in Table 7 between the discussed sensorless control schemes based on a set of performance criteria. When compared to the SMO, the structures of the HGO and MRAS observers are less complex, only requiring one or two parameters to be tuned. However, the SMO performs better during the steady and transient states at high and low speeds with negligible estimation errors. 6. Conclusions An analytical study of three machine model-based sensorless schemes for a two-FPIM drive connected in parallel is presented where the sensorless operation of this two-machine drive is evaluated over a wide range of operations based on several criteria such as dynamic performance, low-speed operations as well as robustness to external disturbances such as variation in load torque, computational burden and implementation complexity. As illustrated in the simulation results, the three sensorless control schemes, namely the model reference adaptive system (MRAS), sliding mode observer (SMO) and high-gain observer (HGO), perform satisfactorily considering the reference speed and tracking of load torque when disturbances are ignored. 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