Design and Terminal Sliding Mode Control of Double Stator Bearingless Switched Reluctance Motor

This paper proposes an 18/15/8-pole bearingless switched reluctance motor (BSRM) with good decoupling performance, which optimizes the distance and number of teeth between the inner and outer stator and rotor, reduces the hysteresis force existing during phase change, improves the electromagnetic conversion efficiency, and reduces the interference of forces between torque and levitation. A three-layer rotor structure is used to design the magnetic separation frame, which eliminates the interference of magnetic lines between the torque system and the suspension system. The direct control idea is applied to control the torque and levitation forces. A new reaching law (RL) is proposed, and a torque sliding mode controller and a suspension force sliding mode controller are designed, which replace the traditional terminal sliding mode control and PID control, and improve the robustness of the control system and dynamic response. Ansoft Maxwell 2D software is used to perform electromagnetic analysis to verify the decoupling, and the control simulation model is established by MATLAB/Simulink simulation analysis, and the results are compared with the traditional controller method. The results show that the proposed control system effectively improves the dynamic response speed and robustness of the system, and verifies the effectiveness and superiority of the proposed control method.

the high-speed rotation of the traditional switched reluctance 23 motor are effectively solved [4]. Improve the critical speed 24 and service life of the motor. At the same time, it inherits the 25 high reliability and excellent speed regulation performance 26 of the SRM, which has been a research hotspot in the motor 27 The associate editor coordinating the review of this manuscript and approving it for publication was Amin Mahmoudi . field in the past two decades. It can be promoted in many 28 practical industrial fields such as flywheel energy storage 29 systems, electric vehicles, aerospace, etc [1], [5]. 30 However, the traditional BSRM has the problem of cou-31 pling torque control and suspension control. To solve these 32 problems, scholars in various countries have mainly con-33 ducted a lot of research on structure and control algorithms. 34 At present, there are four types of structural decoupling: 35 mixed stator tooth structure [6], wide rotor tooth structure [7], 36 double stator structure [8], and composite rotor structure [9]. 37 The basic principle of structural decoupling is to separate the 38 magnetic circuit and space of suspension control and torque 39 control, reduce the interference between each other, realize 40 decoupling with space, and be easier to control. For torque 41 control methods, there are mainly current square wave control 42 and direct torque control. For example, in literature [10], 43 [11], the current chopper control is used to directly adjust the 44 winding current through the current hysteresis loop controller 45 to generate and adjust the required torque. However, the in closed-loop radial displacement control using PID con-64 troller to produce a given radial suspension force value. In the sliding mode controllers are designed for torque and suspen-99 sion control respectively to replace the original PID control 100 to improve the tracking ability of the control signal and 101 improve the dynamic response-ability. A new RL combining 102 exponential RL and power RL is introduced in fast TSMC to 103 improve the arrival speed and reduce chattering. At the same 104 time, the torque control is divided into inherent torque and 105 disturbance torque in the torque sliding mode controller, and 106 the non-singular TSMC [21] is used to accurately control the 107 disturbance torque to suppress its chattering. In suspension 108 control, when the rotor is eccentric, permanent magnet and 109 excitation will occur permanent magnet and electromagnetic 110 conversion, and the instability of the system increases. Dou-111 ble hysteresis control is introduced based on TSMC control. 112 Finally, the control simulation model is established through 113 MATLAB/Simulink simulation analysis to verify the effec-114 tiveness and superiority of the proposed NRLTSMC torque 115 controller and NRLTSMC suspension controller.

118
As shown in Fig.1, the structure, winding mode, and phase 119 division of 18/15/8-pole DSBSRM are shown. The motor is 120 composed of the external stator, rotor, spacer ring, suspension 121 rotor ring, and internal stator. The winding of the coil adopts 122 a single winding mode. 18 pole outer stator is divided into 123 three phases, such as A1+ and A1-, A2+ and A2-, A3+ 124 and A3-constitute a pair of magnetic poles, and then three 125 pairs of magnetic poles constitute A phase, B phase, and C 126 phase constitute the same. The rotor is divided into a torque 127 rotor and a suspension rotor ring, which is connected by the 128 magnetic isolation ring made of magnetic isolation material. 129 The inner stator is 8 poles, and the inner stator teeth are 130 alternately distributed by permanent magnet and excitation, 131 forming a four-phase excitation phase of M1, M2, M3, M4 132 and a four-phase permanent magnet phase of N1, N2, N3, N4. 133 The operation principle of the motor is shown in Fig.2. 134 When a phase is connected, the force magnetic line generated 135 by the torque winding will enter the adjacent torque tooth, 136 forming a closed-loop of ' stator tooth-air gap-torque tooth-137 air gap-stator tooth'. According to the 'minimum reluctance 138 principle', the counterclockwise torque is generated. The 139  The motor rotor has good decoupling performance by using 153 a magnetic isolation ring to separate the torque system and  and stator, and there is no interference between the magnetic 163 densities. Therefore, the suspension system and the torque 164 system are not affected in terms of magnetic flux. The three 165 pairs of winding of one-phase torque winding are symmetri-166 cally distributed in the center, and the axial force generated 167 by them offset each other does not affect the suspension 168 force. Moreover, the magnetic flux generated by the levitation 169 system does not generate torque on the rotor ring, so the 170 levitation system does not affect the torque system when 171 operating. In conclusion, the torque system and suspension 172 system have good decoupling performance, which provides 173 the basis for independent control of the torque system and 174 suspension system. Sliding mode motion includes two processes: reaching 179 motion and sliding mode motion. The system tends to the 180 switching surface from any initial state until the motion reach-181 ing the switching surface is called approaching motion, which 182 is a process of s→0. According to the principle of sliding 183 mode variable structure, the reach ability condition of sliding 184 mode only ensures that any motion point in the state space 185 reaches the switching surface in a finite time, and there is no 186 restriction on the specific trajectory of reaching motion [19]. 187 The reaching law method can improve the dynamic quality of 188 reaching motion. The traditional exponential RL is designed 189 as follows [3]: where: s is the sliding mode plane λ 1 and λ 2 are the 192 switching gain and linear gain, respectively. λ 1 sgn(s) is the 193 VOLUME 10, 2022 constant-speed approximation term, and λ 2 s is the pure expo- The following figure is the function image of F(s) and sgn(s): 226 The NRL obtained is: Lyapunov stability theory, it should satisfy: Then: According to the dynamic principle, the motor torque balance 244 equation of DSSBSRM is: where J is the rotational inertia, B is the friction coefficient, B 247 is the electromagnetic torque, and r is the sum of disturbances. 248 where: T L is the load torque, and a and b are the internal 250 disturbance parameters.

251
Taking the reference torque T e = T e1 +T e2 , T e1 is the 252 inherent torque without considering the load and internal 253 interference, and T e2 is the interference torque caused by 254 internal and external interference. The solution of T e1 is as 255 follows:

256
Taking the speed tracking error e w as the difference 257 between the reference torque w * and the actual torque w: First, the sliding mode plane s w is designed: When the error enters the fast Terminal sliding mode, there 262 is: where: µ >0. Q and P are positive odd numbers.

283
On the above derivation: In conjunction (8), (19) and (20): Then: Then: So, the reference torque T e is: 296 Then, the torque NRL-TSMC simulation block diagram where: F is suspension force, m is rotor mass, m is the sum 306 of disturbances, and x is rotor displacement.

307
Taking the rotor displacement tracking error e x as the 308 difference between the reference displacement x * and the 309 actual displacement x: First design the sliding mode plane s x : When the error enters the fast Terminal sliding mode, there 314 is: According to (5), NRL in the suspension controller can be 317 expressed as: Then the combination (25), (28), and (29) can be obtained: 320 Then: Then combined with (31) in MATLAB/Simulink simula-326 tion of suspended NRL-TSMC simulation block diagram is 327 shown in Fig.6 below:

330
For the 18/15/8 pole DSBSRM control method, the torque 331 system and the suspension system are independently con-332 trolled by the analysis in Section 2.2. For torque control, 333 double closed-loop direct instantaneous torque control based 334 on a Terminal sliding mode controller is adopted. The outer 335 VOLUME 10, 2022 follows:  two is the input, ε min , and ε max are the limits of internal 373 and external hysteresis control respectively, and S(A) is the 374 switching signal of suspension winding. The suspension sys-375 tem uses the combination strategy of permanent magnet bias 376 and control winding, so the decoupling effect of the x-axis 377 and y-axis is also very good. The eccentricity control of the x-378 axis and y-axis can become two independent control systems. 379 However, when the x-axis is eccentric, the permanent magnet 380 and electromagnetic are transformed, and the instability of the 381 system is also increased. Therefore, the direct instantaneous 382 suspension force control using a double hysteresis controller 383 is adopted.

384
Combined with the above analysis, based on MATLAB/ 385 Simulink simulation analysis. Build 18/15/8 pole DSBSRM 386 control simulation model, as shown in Fig.9. 2N .m, 4N .m, and the reference speed is from 1500r/min 408 to 1800r/min. The mark red thickening part in the diagram 409 is the speed rising stage. We can see from the rough view 410 that the dynamic response-ability of SMC is significantly 411 better than that of PID under three load conditions, while 412 the dynamic response-ability of NRLTSMC is better than 413 that of SMC, whether in the start-up stage or the reference 414 speed change. In torque ripple suppression, NRLTSMC also 415 showed a smaller torque ripple. When the load is 4 N .m, 416 the PID control cannot meet the control requirements, and 417 the actual control speed cannot reach the reference speed. 418 To accurately analyze the data, the data in Fig.10, Fig.11, and 419 Fig.12 are quantitatively analyzed and arranged as follows: 420 To quantitatively analyze the torque, the total torque data 421 is measured by (33), and P is the torque ripple rate.  the torque ripple under NRLTSMC control is significantly 442 reduced, and the speed acceleration is significantly improved, 443 which proves the effectiveness of the NRLTSMC controller. 444 In this paper, NRLTSMC control has a better torque ripple 445 suppression effect and dynamic response ability than SMC 446 and PI control methods.

447
For radial displacement suspension control, according 448 to [8], the x-axis and y-axis directions of suspension struc-449 ture have good decoupling. Therefore, the y-axis direction 450 is selected for the simulation experiment. To highlight the 451 superiority of NRL design, the direct suspension force control 452 of the SMC controller with traditional exponential RL is 453 used as the control group. In the simulation process, the 454 parameters are set as λ 1 = 2, λ 2 = 48, q= 5, p= 9, ξ = 3, 455 α= 0.41,α2 = 0.71. Moreover, the disturbance is relatively 456 fixed in the suspension force control, so the external distur-457 bance F L is set to a constant value. In the control, the initial 458 reference displacement is set to 0, the deviation of −0.15 mm 459 is given at 0 s, the deviation of 0.5 mm is added at 0.2 s, 460 the simulation waveform of the NRLTSMC control strategy. 466 We can see from the control comparison simulation results in 467 Fig.13(a) and Fig.13(b)   working conditions, but also enhances the suspension force 491 output capacity. The structure is optimized for the cou-492 pling problem of torque and radial levitation force control 493 that exists in conventional BSRM. Due to good decou-494 pling performance, motor torque control and suspension con-495 trol are independently controlled. Subsequently, the TSMC 496 torque controller and suspension controller of the new RL is 497 designed. The torque of the controller is divided into inherent 498 torque and disturbance torque. The disturbance torque is pre-499 cisely controlled by nonsingular TSMC and NRL to suppress 500 torque disturbance. Combined with TSF direct instantaneous 501 torque control. The radial suspension control combines the 502 TSMC controller with double hysteresis control to reduce the 503 disturbance between the permanent magnet and the excitation 504 during eccentricity. Then simulation verification is carried 505 out. The direct instantaneous torque control with traditional 506 SMC and PID controller is used as the control group, and 507 the direct suspension force of SMC with CRL is used as the 508 control group. The results show that the motor structure can 509 decouple torque and levitation force, and the new RL TSMC 510 torque controller can effectively reduce the torque ripple and 511 speed jitter to improve the speed dynamic response-ability. 512 The TMC suspension force controller with the new RL has a 513 better dynamic response to ensure the radial stability of the 514 rotor.