Patent Publication Number: US-2023163710-A1

Title: Model predictive decomposition control method and device for open-winding five-phase permanent magnet synchronous motor

Description:
CROSS REFERENCE TO THE RELATED APPLICATIONS 
     This application is the national phase entry of International Application No. PCT/CN2020/133913, filed on Dec. 4, 2020, which is based upon and claims priority to Chinese Patent Application No. 202011347430.2, filed on Nov. 26, 2020, the entire contents of which are incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     The present invention belongs to the application field of a multi-phase motor open-winding topology system and relates to a model predictive decomposition control method and device for an open-winding five-phase permanent magnet synchronous motor. 
     BACKGROUND 
     The five-phase permanent magnet synchronous motor has the advantages of high efficiency, high power density, wide speed range, low torque ripple, and strong fault tolerance. The five-phase PMSM has been widely concerned and applied in aerospace, electric vehicles, and ship propulsion systems. At the same time, the open-winding motor topology system has excellent characteristics such as high output power, diverse power supply modes and voltage vector modulation methods, flexible control, redundancy, and fault tolerance. 
     Some researches in domestic and foreign have achieved specific results in the model predictive control method of the five-phase permanent magnet synchronous motor under the open-winding topology. The Chinese invention patent “A Model Predictive Current Control Method for Open Winding Permanent Magnet Synchronous Motors” (Patent No.: CN201910583770.6) discloses a control method for model prediction of open winding permanent magnet synchronous motors. This method uses a single power supply. Although the cost is saved, a zero-sequence current loop will inevitably be generated, and the flexibility of the open-winding system is not fully utilized at the same time. The Chinese invention patent “An Optimal Model Predictive Control Method for Dual Inverter Open Winding Induction Motors” (Patent No.: CN201910207446.4) discloses a control method optimized for the prediction of dual inverter open winding induction motors. Since this method only acts on one voltage vector for every control period, the characteristic is easy to implement but has poor steady-state performance. Therefore, the existing open-winding model prediction methods mostly use the same modulation method as the non-open-winding topology system but fails to take full advantage of the flexibility of the open-winding motor drive system and the advantages of numeric voltage vectors. At the same time, as the number of motor phases increases, especially the open-winding five-phase permanent magnet synchronous motor topology system, the switching loss of the drive system doubles. The control is flexible but complicated. Therefore, under the premise of maintaining the flexibility and freedom of the open-winding five-phase permanent magnet synchronous motor system, the main purpose of the present invention is reduce the switching loss of the inverter, and the calculation of the open-winding model prediction. 
     SUMMARY 
     The purpose of the present invention: Because of the problems in the prior technology, a model predictive decomposition control method and device for an open-winding five-phase permanent magnet synchronous motor is proposed, regarding the two inverters of the open-winding topology system as two independent units, takes the voltage vector provided by inverter 1 as the starting point, and selects different inverter switching sequences according to the needs of the open winding system under different working conditions. Under the premise of ensuring the port performance, the present invention can effectively reduce the switching times of the open winding five-phase permanent magnet synchronous motor control system to reduce the switching loss of the inverter. 
     Technical solution: To achieve the purpose as mentioned above of the present invention, the technical solution adopted by the present invention is as follows: 
     A model predictive decomposition control method for an open-winding five-phase permanent magnet synchronous motor includes the following steps: 
     Step 1) Obtaining q-axis and d-axis currents in a fundamental space of the open-winding five-phase permanent magnet synchronous motor in a rotating coordinate system as the given of a control system. 
     Step 2) Constructing a virtual voltage vector table of the open-winding five-phase permanent magnet synchronous motor in a static coordinate system. Applying a label of the virtual voltage vector in a previous control cycle and calculating voltage in the static coordinate system. Moreover, solving a back electromotive force of the motor according to a mathematical model of the motor; 
     Step 3) Using an Euler forward difference prediction equation to find a q-axis current and a d-axis current in this period; 
     Step 4) Substituting the voltage vector generated by inverter 1 as given in a current sampling period into the prediction model and cost function to find the currents of the q-axis and the d-axis of the rotating coordinate system in the next period. If the voltage vector provided by inverter 1 meets the requirements of the drive system, close inverter 2. Otherwise, clamping inverter 1 to the maximum available voltage vector, and open inverter 2 to provide the increment of the remaining part; 
     Step 5) Transmitting the selected voltage vector and its corresponding duty cycle signal the open-winding inverter to complete the closed-loop control of the drive system. 
     Further, the specific steps of Step 1) include: calculating a given speed n* and an actual speed n of the motor to obtain a required q-axis current reference value i q * by the proportional-integral controller. Moreover, the d-axis current reference value i d *=0. 
     Further, the specific steps of Step 2) include: 
     Step 2.1) Constructing an open-winding five-phase permanent magnet synchronous motor virtual voltage vector table; 
     Expressing the voltage vector generated by the five-phase voltage source inverter in the stationary coordinate system as: 
     
       
         
           
             
               
                 u 
                 s 
               
               = 
               
                 
                   2 
                   5 
                 
                 × 
                 
                   ( 
                   
                     
                       s 
                       a 
                     
                     + 
                     
                       
                         s 
                         b 
                       
                       ⁢ 
                       w 
                     
                     + 
                     
                       
                         s 
                         c 
                       
                       ⁢ 
                       
                         w 
                         2 
                       
                     
                     + 
                     
                       
                         s 
                         d 
                       
                       ⁢ 
                       
                         w 
                         3 
                       
                     
                     + 
                     
                       
                         s 
                         e 
                       
                       ⁢ 
                       
                         w 
                         4 
                       
                     
                   
                   ) 
                 
                 × 
                 
                   U 
                   dc 
                 
               
             
             ⁢ 
             
 
             
               ( 
               
                 w 
                 = 
                 
                   e 
                   
                     j 
                     ⁢ 
                     
                       2 
                       5 
                     
                     ⁢ 
                     π 
                   
                 
               
               ) 
             
           
         
       
     
     wherein, U dc  is the DC bus voltage, u s  is the voltage vector value in the static coordinate system, s i (i=a,b,c,d,e) is the switching state of each bridge arm, when the upper bridge arm is turned on, s i =1, when the lower bridge arm is turned on, s i =0; 
     Step 2.2) According to the size of the modulus length, divide the voltage vector obtained from Step 1) into large vector u B , medium vector u M , small vector u L , and zero vector u 0 . According to the principle that the harmonic subspace is equivalent to zero, constructing the virtual voltage vectors of the open winding five-phase permanent magnet synchronous: 
       VV i ( u   B   ,u   L )=0.618× u   M +(1−0.618)× u   B  
 
     wherein the scale factor 0.618 of the two voltage vectors makes the voltage vector of a five-phase permanent magnet synchronous motor equivalent to zero in the harmonic subspace; 
     Step 2.3) Expressing the voltage vector output value of the previous sampling period (k−1) as: 
     
       
         
           
             
               
                 u 
                 s 
               
               ( 
               
                 k 
                 - 
                 1 
               
               ) 
             
             = 
             
               
                 2 
                 5 
               
               × 
               
                 ( 
                 
                   
                     
                       s 
                       a 
                     
                     ( 
                     
                       k 
                       - 
                       1 
                     
                     ) 
                   
                   + 
                   
                     
                       
                         s 
                         b 
                       
                       ( 
                       
                         k 
                         - 
                         1 
                       
                       ) 
                     
                     ⁢ 
                     w 
                   
                   + 
                   
                     
                       
                         s 
                         c 
                       
                       ( 
                       
                         k 
                         - 
                         1 
                       
                       ) 
                     
                     ⁢ 
                     
                       w 
                       2 
                     
                   
                   + 
                   
                     
                       
                         s 
                         d 
                       
                       ( 
                       
                         k 
                         - 
                         1 
                       
                       ) 
                     
                     ⁢ 
                     
                       w 
                       3 
                     
                   
                   + 
                   
                     
                       
                         s 
                         e 
                       
                       ( 
                       
                         k 
                         - 
                         1 
                       
                       ) 
                     
                     ⁢ 
                     
                       w 
                       4 
                     
                   
                 
                 ) 
               
               × 
               
                 U 
                 dc 
               
             
           
         
       
       
         
           
             	 
             
               ( 
               
                 w 
                 = 
                 
                   e 
                   
                     j 
                     ⁢ 
                     
                       2 
                       5 
                     
                     ⁢ 
                     π 
                   
                 
               
               ) 
             
           
         
       
     
     Furthermore, the i q * and a actual speed to of the motor are calculated using Step 1). Hence, the instantaneous value of the q-axis and the d-axis back electromotive force of the open-winding five-phase permanent magnet synchronous motor in the rotating coordinate system at this time is expressed as: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       EMF 
                       d 
                     
                     = 
                     
                       ω 
                       × 
                       
                         L 
                         s 
                       
                       × 
                       
                         i 
                         q 
                         * 
                       
                     
                   
                 
               
               
                 
                   
                     
                       EMF 
                       q 
                     
                     = 
                     
                       
                         ω 
                         × 
                         
                           L 
                           s 
                         
                         × 
                         
                           i 
                           d 
                           * 
                         
                       
                       + 
                       
                         ω 
                         × 
                         
                           φ 
                           f 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     wherein Ls is the inductance of the motor, (f is the permanent magnet flux linkage of the motor, iq* is the q-axis current reference value in the rotating coordinate system, and id* is the d-axis current reference value in the rotating coordinate system. 
     Further, the specific steps of Step 3) include: 
     Step 3.1) Expressing the voltage equation in the rotating coordinate system of the five-phase permanent magnet synchronous motor as: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         di 
                         d 
                       
                       dt 
                     
                     = 
                     
                       
                         1 
                         
                           L 
                           s 
                         
                       
                       × 
                       
                         ( 
                         
                           
                             u 
                             d 
                           
                           - 
                           
                             
                               R 
                               s 
                             
                             × 
                             
                               i 
                               d 
                             
                           
                           - 
                           
                             
                               ω 
                               e 
                             
                             × 
                             
                               L 
                               s 
                             
                             × 
                             
                               i 
                               q 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         di 
                         q 
                       
                       dt 
                     
                     = 
                     
                       
                         1 
                         
                           L 
                           s 
                         
                       
                       × 
                       
                         ( 
                         
                           
                             u 
                             d 
                           
                           - 
                           
                             
                               R 
                               s 
                             
                             × 
                             
                               i 
                               d 
                             
                           
                           - 
                           
                             
                               ω 
                               e 
                             
                             × 
                             
                               L 
                               s 
                             
                             × 
                             
                               i 
                               d 
                             
                           
                           - 
                           
                             
                               ω 
                               e 
                             
                             × 
                             
                               φ 
                               f 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
     wherein, L s  is the inductance of the motor, R s  is the stator resistance of the motor, u d  is the voltage on the d-axis, u q  is the voltage on the q-axis, i d  is the d-axis current, and i q  is the q-axis current, ω e  is the electrical angle of the motor, φ f  is the permanent magnet flux linkage of the motor; 
     Step 3.2) Using an Euler forward difference method to find the currents of the q-axis and d-axis in the rotating coordinate system at time k+1: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         
                           
                             i 
                             d 
                           
                           ( 
                           
                             k 
                             + 
                             1 
                           
                           ) 
                         
                         - 
                         
                           
                             i 
                             d 
                           
                           ( 
                           k 
                           ) 
                         
                       
                       Ts 
                     
                     = 
                     
                       
                         1 
                         
                           L 
                           s 
                         
                       
                       × 
                       
                         ( 
                         
                           
                             
                               u 
                               d 
                             
                             ( 
                             k 
                             ) 
                           
                           - 
                           
                             
                               R 
                               s 
                             
                             × 
                             
                               
                                 i 
                                 d 
                               
                               ( 
                               k 
                               ) 
                             
                           
                           - 
                           
                             
                               EMF 
                               d 
                             
                             ( 
                             k 
                             ) 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         
                           
                             i 
                             q 
                           
                           ( 
                           
                             k 
                             + 
                             1 
                           
                           ) 
                         
                         - 
                         
                           
                             i 
                             q 
                           
                           ( 
                           k 
                           ) 
                         
                       
                       Ts 
                     
                     = 
                     
                       
                         1 
                         
                           L 
                           s 
                         
                       
                       × 
                       
                         ( 
                         
                           
                             
                               u 
                               q 
                             
                             ( 
                             k 
                             ) 
                           
                           - 
                           
                             
                               R 
                               s 
                             
                             × 
                             
                               
                                 i 
                                 q 
                               
                               ( 
                               k 
                               ) 
                             
                           
                           - 
                           
                             
                               EMF 
                               q 
                             
                             ( 
                             k 
                             ) 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
     wherein, u d (k) is the voltage on the d-axis at time k, u q (k) is the voltage on the q-axis at time k, i d (k) is the d-axis current, and i q (k) is the q-axis current, EMF d (k) is the d-axis back EMF of the motor at time k, and EMF q (k) is the q-axis back EMF of the motor at time k. 
     Further, the specific steps of Step 4) include: 
     Step 4.1) According to the mathematical model of open-winding five-phase permanent magnet synchronous motor, the currents of the q-axis and the d-axis in the rotating coordinate system at time k+1 is the sum of two inverters: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         u 
                         d 
                       
                       ( 
                       
                         k 
                         + 
                         1 
                       
                       ) 
                     
                     = 
                     
                       
                         
                           u 
                           d 
                           
                             INV 
                             ⁢ 
                             1 
                           
                         
                         ( 
                         
                           k 
                           + 
                           1 
                         
                         ) 
                       
                       - 
                       
                         
                           u 
                           d 
                           
                             INV 
                             ⁢ 
                             2 
                           
                         
                         ( 
                         
                           k 
                           + 
                           1 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         u 
                         q 
                       
                       ( 
                       
                         k 
                         + 
                         1 
                       
                       ) 
                     
                     = 
                     
                       
                         
                           u 
                           q 
                           
                             INV 
                             ⁢ 
                             1 
                           
                         
                         ( 
                         
                           k 
                           + 
                           1 
                         
                         ) 
                       
                       - 
                       
                         
                           u 
                           q 
                           
                             INV 
                             ⁢ 
                             2 
                           
                         
                         ( 
                         
                           k 
                           + 
                           1 
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
     wherein, u d   INV1 (k+1) is the d-axis voltage in the rotating coordinate system provided by inverter 1 at k+1 instance, u q   INV1 (k+1) is the q-axis voltage in the rotating coordinate system provided by inverter 1 at k+1 instance, u d   INV2 (k+1) is the d-axis voltage in the rotating coordinate system provided by inverter 2 at k+1 instance, u q   INV2 (k+1) is the q-axis voltage in the rotating coordinate system provided by inverter 2 at k+1 instance; 
     Step 4.2) Expressing the voltage vectors provided by inverter 1 and inverter 2 as: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         Δ 
                         ⁢ 
                         
                           i 
                           d 
                           
                             INV 
                             ⁢ 
                             1 
                           
                         
                       
                       = 
                       
                         
                           Ts 
                           
                             L 
                             s 
                           
                         
                         × 
                         
                           ( 
                           
                             
                               
                                 u 
                                 d 
                                 
                                   INV 
                                   ⁢ 
                                   1 
                                 
                               
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                             - 
                             
                               
                                 R 
                                 s 
                               
                               × 
                               
                                 
                                   i 
                                   d 
                                 
                                 ( 
                                 
                                   k 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 EMF 
                                 d 
                               
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         Δ 
                         ⁢ 
                         
                           i 
                           q 
                           
                             INV 
                             ⁢ 
                             1 
                           
                         
                       
                       = 
                       
                         
                           Ts 
                           
                             L 
                             s 
                           
                         
                         × 
                         
                           ( 
                           
                             
                               
                                 u 
                                 d 
                                 
                                   INV 
                                   ⁢ 
                                   1 
                                 
                               
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                             - 
                             
                               
                                 R 
                                 s 
                               
                               × 
                               
                                 
                                   i 
                                   q 
                                 
                                 ( 
                                 
                                   k 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 EMF 
                                 q 
                               
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               ⁢ 
               
 
               
                 { 
                 
                   
                     
                       
                         
                           Δ 
                           ⁢ 
                           
                             i 
                             d 
                             
                               INV 
                               ⁢ 
                               2 
                             
                           
                         
                         = 
                         
                           
                             - 
                             
                               Ts 
                               
                                 L 
                                 s 
                               
                             
                           
                           × 
                           
                             
                               u 
                               d 
                               
                                 INV 
                                 ⁢ 
                                 2 
                               
                             
                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           Δ 
                           ⁢ 
                           
                             i 
                             q 
                             
                               INV 
                               ⁢ 
                               2 
                             
                           
                         
                         = 
                         
                           
                             - 
                             
                               Ts 
                               
                                 L 
                                 s 
                               
                             
                           
                           × 
                           
                             
                               u 
                               d 
                               
                                 INV 
                                 ⁢ 
                                 2 
                               
                             
                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     wherein, Δi d   INV1  and Δi q   INV1  are d-axis current and q-axis currents in the rotating coordinate system generated by the inverter 1. Rs is the phase resistance of the motor windings. Ts is the control period of the controller. Ls is the stator inductance of the motor. EMF d (k+1) is the d-axis back-EMF of the motor at the k+1 instance. EMF q (k+1) is the q-axis back-EMF of the motor at the k+1 instance; 
     Step 4.3) If a Δi d   INV1  and a Δi q   INV1  is greater than or equal to the incremental requirement Δi d * and Δi q * at this moment, the voltage vector provided by inverter 1 is enough, and the inverter 2 is turn off. By applying multiple voltage vectors in one control period, express the voltage vector in the rotating coordinate system generated in the current sampling period as: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         
                           
                             u 
                             a 
                             
                               INV 
                               ⁢ 
                               1 
                             
                           
                           = 
                           
                             DR 
                             ⁢ 
                             
                               0 
                               
                                 INV 
                                 ⁢ 
                                 1 
                               
                             
                             × 
                             
                               ( 
                               
                                 
                                   
                                     DR 
                                     
                                       INV 
                                       ⁢ 
                                       1 
                                     
                                   
                                   × 
                                   
                                     real 
                                     ( 
                                     
                                       
                                         V 
                                         ⁢ 
                                         V 
                                       
                                       i 
                                       
                                         INV 
                                         ⁢ 
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   DR 
                                   
                                     INV 
                                     ⁢ 
                                     1 
                                   
                                 
                               
                               ) 
                             
                             × 
                             
                               real 
                               ( 
                               
                                 
                                   V 
                                   ⁢ 
                                   V 
                                 
                                 j 
                                 
                                   INV 
                                   ⁢ 
                                   1 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         
                           
                             u 
                             β 
                             
                               INV 
                               ⁢ 
                               1 
                             
                           
                           = 
                           
                             DR 
                             ⁢ 
                             
                               0 
                               
                                 INV 
                                 ⁢ 
                                 1 
                               
                             
                             × 
                             
                               ( 
                               
                                 
                                   
                                     DR 
                                     
                                       INV 
                                       ⁢ 
                                       1 
                                     
                                   
                                   × 
                                   
                                     imag 
                                     ⁡ 
                                     ( 
                                     
                                       
                                         V 
                                         ⁢ 
                                         V 
                                       
                                       i 
                                       
                                         INV 
                                         ⁢ 
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   DR 
                                   
                                     INV 
                                     ⁢ 
                                     1 
                                   
                                 
                               
                               ) 
                             
                             × 
                             
                               imag 
                               ⁡ 
                               ( 
                               
                                 
                                   V 
                                   ⁢ 
                                   V 
                                 
                                 j 
                                 
                                   INV 
                                   ⁢ 
                                   1 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             	 
             
               { 
               
                 
                   
                     
                       
                         u 
                         a 
                         
                           INV 
                           ⁢ 
                           2 
                         
                       
                       = 
                       0 
                     
                   
                 
                 
                   
                     
                       
                         u 
                         β 
                         
                           INV 
                           ⁢ 
                           2 
                         
                       
                       = 
                       0 
                     
                   
                 
               
             
           
         
       
     
     wherein, u α, u β, u α, and u β are the voltage vector in the stationary coordinate system selected by inverter 1 and inverter 2, respectively, distinguishing by superscripts; real(VV i   INV1 ) and imag(VV i   INV1 ) are the real and imaginary parts of the first virtual voltage vector to be selected by inverter 1 at this moment. real(VV j   INV1 ) and imag(VV j   INV1 ) are the real and imaginary parts of the second voltage vector to be selected by inverter 1. DR0 INV1  is the scale factors for the sum of the effective voltage vectors of inverter 1; DR INV1  is the scale factor between two effective voltage vectors inverter 1; 
     Expressing the scale factors of inverter 1 and inverter 2 when the increment provided by inverter 1 meets the current needs as: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         DR 
                         ⁢ 
                         
                           0 
                           
                             INV 
                             ⁢ 
                             1 
                           
                         
                       
                       = 
                       
                         m 
                         ⁡ 
                         ( 
                         
                           0 
                           ≤ 
                           m 
                           ≤ 
                           1 
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     
                       
                         DR 
                         
                           INV 
                           ⁢ 
                           1 
                         
                       
                       = 
                       
                         n 
                         ⁡ 
                         ( 
                         
                           0 
                           ≤ 
                           n 
                           ≤ 
                           1 
                         
                         ) 
                       
                     
                   
                 
               
               ⁢ 
               
 
               
                 { 
                 
                   
                     
                       
                         
                           DR 
                           ⁢ 
                           
                             0 
                             
                               INV 
                               ⁢ 
                               2 
                             
                           
                         
                         = 
                         0 
                       
                     
                   
                   
                     
                       
                         
                           DR 
                           
                             INV 
                             ⁢ 
                             2 
                           
                         
                         = 
                         0 
                       
                     
                   
                 
               
             
           
         
       
     
     Expressing a q-axis and a d-axis current in the rotating coordinate system of the motor at k+2 instance as: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         i 
                         d 
                       
                       ( 
                       
                         k 
                         + 
                         2 
                       
                       ) 
                     
                     = 
                     
                       
                         
                           i 
                           d 
                         
                         ( 
                         
                           k 
                           + 
                           1 
                         
                         ) 
                       
                       + 
                       
                         
                           Ts 
                           
                             L 
                             s 
                           
                         
                         × 
                         
                           ( 
                           
                             
                               
                                 u 
                                 d 
                                 
                                   INV 
                                   ⁢ 
                                   1 
                                 
                               
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                             - 
                             
                               
                                 R 
                                 s 
                               
                               × 
                               
                                 
                                   i 
                                   d 
                                 
                                 ( 
                                 
                                   k 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 EMF 
                                 d 
                               
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         i 
                         q 
                       
                       ( 
                       
                         k 
                         + 
                         2 
                       
                       ) 
                     
                     = 
                     
                       
                         
                           i 
                           q 
                         
                         ( 
                         
                           k 
                           + 
                           1 
                         
                         ) 
                       
                       + 
                       
                         
                           Ts 
                           
                             L 
                             s 
                           
                         
                         × 
                         
                           ( 
                           
                             
                               
                                 u 
                                 d 
                                 
                                   INV 
                                   ⁢ 
                                   1 
                                 
                               
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                             - 
                             
                               
                                 R 
                                 s 
                               
                               × 
                               
                                 
                                   i 
                                   q 
                                 
                                 ( 
                                 
                                   k 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 EMF 
                                 q 
                               
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     If a Δi d   INV1  and a Δi q   INV1  are less than the incremental required Δi d *, and Δi q * at the current moment, the required voltage vector is beyond the output capacity of the inverter 1. Furthermore, inverter 2 is required to output the remaining increments to meet the motor operating conditions. In this situation, inverter 1 outputs the maximum increments of Δi d   INV1  and Δi q   INV1 , inverter 2 acts on multiple voltage vectors in the current period. Express the voltage vector in the rotating coordinate system generated in the current sampling period as: 
     
       
         
           
             	 
             
               { 
               
                 
                   
                     
                       
                         u 
                         a 
                         
                           INV 
                           ⁢ 
                           1 
                         
                       
                       = 
                       
                         real 
                         ( 
                         
                           
                             V 
                             ⁢ 
                             V 
                           
                           i 
                           
                             INV 
                             ⁢ 
                             1 
                           
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     
                       
                         u 
                         β 
                         
                           INV 
                           ⁢ 
                           1 
                         
                       
                       = 
                       
                         imag 
                         ⁡ 
                         ( 
                         
                           
                             V 
                             ⁢ 
                             V 
                           
                           i 
                           
                             INV 
                             ⁢ 
                             1 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             { 
             
               
                 
                   
                     
                       
                         
                           
                             u 
                             a 
                             
                               INV 
                               ⁢ 
                               2 
                             
                           
                           = 
                           
                             DR 
                             ⁢ 
                             
                               0 
                               
                                 INV 
                                 ⁢ 
                                 2 
                               
                             
                             × 
                             
                               ( 
                               
                                 
                                   
                                     DR 
                                     
                                       INV 
                                       ⁢ 
                                       2 
                                     
                                   
                                   × 
                                   
                                     real 
                                     ( 
                                     
                                       
                                         V 
                                         ⁢ 
                                         V 
                                       
                                       i 
                                       
                                         INV 
                                         ⁢ 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   DR 
                                   
                                     INV 
                                     ⁢ 
                                     2 
                                   
                                 
                               
                               ) 
                             
                             × 
                             
                               real 
                               ( 
                               
                                 
                                   V 
                                   ⁢ 
                                   V 
                                 
                                 j 
                                 
                                   INV 
                                   ⁢ 
                                   2 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         
                           
                             u 
                             β 
                             
                               INV 
                               ⁢ 
                               2 
                             
                           
                           = 
                           
                             DR 
                             ⁢ 
                             
                               0 
                               
                                 INV 
                                 ⁢ 
                                 2 
                               
                             
                             × 
                             
                               ( 
                               
                                 
                                   
                                     DR 
                                     
                                       INV 
                                       ⁢ 
                                       2 
                                     
                                   
                                   × 
                                   
                                     imag 
                                     ⁡ 
                                     ( 
                                     
                                       
                                         V 
                                         ⁢ 
                                         V 
                                       
                                       i 
                                       
                                         INV 
                                         ⁢ 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   DR 
                                   
                                     INV 
                                     ⁢ 
                                     2 
                                   
                                 
                               
                               ) 
                             
                             × 
                             
                               imag 
                               ⁡ 
                               ( 
                               
                                 
                                   V 
                                   ⁢ 
                                   V 
                                 
                                 j 
                                 
                                   INV 
                                   ⁢ 
                                   2 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     wherein, u α, u β, u α, and u β are the voltage vector in the stationary coordinate system selected by inverter 1 and inverter 2, respectively, distinguishing by superscripts; real(VV i   INV1 ) and imag(VV i   INV1 ) are the real and imaginary parts of the first virtual voltage vector to be selected by inverter 1 at this moment; real(VV i   INV2 ) and imag(VV i   INV2 ) are the real and imaginary parts of the first candidate voltage vector of inverter 2; real(VV j   INV2 ) and imag(VV j   INV2 ) are the real and imaginary parts of the second candidate voltage vector of inverter 2 respectively; DR0 INV2  is the scale factor for the sum of the effective voltage vectors action; DR INV2  is the scale factor between the two effective voltage vector; 
     Expressing the scale factors of inverter 1 and inverter 2 when the increment provided by inverter 1 less than needs as: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         DR 
                         ⁢ 
                         
                           0 
                           
                             INV 
                             ⁢ 
                             1 
                           
                         
                       
                       = 
                         
                       1 
                     
                   
                 
                 
                   
                     
                       
                         DR 
                         
                           INV 
                           ⁢ 
                           1 
                         
                       
                       = 
                         
                       1 
                     
                   
                 
               
               ⁢ 
               
 
               
                 { 
                 
                   
                     
                       
                         
                           DR 
                           ⁢ 
                           
                             0 
                             
                               INV 
                               ⁢ 
                               2 
                             
                           
                         
                         = 
                           
                         
                           m 
                           ⁡ 
                           ( 
                           
                             0 
                             ≤ 
                             m 
                             ≤ 
                             1 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         
                           DR 
                           
                             INV 
                             ⁢ 
                             2 
                           
                         
                         = 
                           
                         
                           n 
                           ⁡ 
                           ( 
                           
                             0 
                             ≤ 
                             n 
                             ≤ 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Expressing the q-axis and d-axis current in the rotating coordinate system of the motor at k+2 instance as: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         i 
                         d 
                       
                       ( 
                       
                         k 
                         + 
                         2 
                       
                       ) 
                     
                     = 
                     
                       
                         
                           i 
                           d 
                         
                         ( 
                         
                           k 
                           + 
                           1 
                         
                         ) 
                       
                       + 
                       
                         
                           Ts 
                           
                             L 
                             s 
                           
                         
                         ⨯ 
                         
                           ( 
                           
                             
                               
                                 u 
                                 d 
                                 
                                   INV 
                                   ⁢ 
                                   1 
                                 
                               
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                             - 
                             
                               
                                 R 
                                 s 
                               
                               ⨯ 
                               
                                 
                                   i 
                                   d 
                                 
                                 ( 
                                 
                                   k 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             - 
                               
                             
                               
                                 EMF 
                                 d 
                               
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           Ts 
                           
                             L 
                             s 
                           
                         
                         ⨯ 
                         
                           
                             u 
                             d 
                             
                               INV 
                               ⁢ 
                               2 
                             
                           
                           ( 
                           
                             k 
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         i 
                         q 
                       
                       ( 
                       
                         k 
                         + 
                         2 
                       
                       ) 
                     
                     = 
                     
                       
                         
                           i 
                           q 
                         
                         ( 
                         
                           k 
                           + 
                           1 
                         
                         ) 
                       
                       + 
                       
                         
                           Ts 
                           
                             L 
                             s 
                           
                         
                         ⨯ 
                         
                           ( 
                           
                             
                               
                                 u 
                                 d 
                                 
                                   INV 
                                   ⁢ 
                                   1 
                                 
                               
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                             - 
                             
                               
                                 R 
                                 s 
                               
                               ⨯ 
                               
                                 
                                   i 
                                   q 
                                 
                                 ( 
                                 
                                   k 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             - 
                               
                             
                               
                                 EMF 
                                 q 
                               
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           Ts 
                           
                             L 
                             s 
                           
                         
                         ⨯ 
                         
                           
                             u 
                             d 
                             
                               INV 
                               ⁢ 
                               2 
                             
                           
                           ( 
                           
                             k 
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Step 4.4) According to the two different working conditions in Step 4.3), substituting the q-axis and d-axis current in the rotating coordinate system of the motor at k+2 instance into the cost function to obtain the expected voltage vector: 
       λ( i,j )=( i   d   *−i   d ( k+ 2)) 2 +( i   q   *−i   q ( k+ 2)) 2 .
 
     Further, the specific steps of Step 5) include: 
     Step 5.1) According to Step 4), the index number i of the optimal voltage vector has been found. The scale factor DR0 INV1  and DR INV1  for inverter 1, and the scale factor DR0 INV2  and DR INV2  for inverter 2. Outputting the duty cycle of each phase of the open-winding inverter by the virtual voltage vector table of the open winding five-phase permanent magnet synchronous motor, the details are proposed in Step 2; 
     If the Δi d   INV1  and the Δi q   INV1  are greater than the required increment Δi d * and Δi q * at the current moment, express the equation as: 
       DutyRatio x   INV1 =DR0 INV1 ×(DR INV1   ×s   x   INV1 ( i )+(1−DR INV1 )× s   x   INV1 ( j ))
 
       DutyRatio x   INV2 =0 
     wherein DutyRatio x   INV1  is the value calculated by the proposed algorithm, used for inverter 1 to generate the corresponding duty cycle waveform; DutyRatio x   INV2  is used for inverter 2 to generate the corresponding duty cycle waveform; s x   INV1 (i) is the first selected by inverter 1, is the switching function of a virtual voltage vector, wherein x=a,b,c,d,e; s x   INV1 (j) is the switching function of the second virtual voltage vector selected by inverter 1, wherein x-a,b,c,d,e; 
     If the Δi d   INV1  and the Δi q   INV1  are less than the required increments Δi d * and Δi q * at this moment, express the equation as: 
       DutyRatio x   INV1   =s   x   INV1 ( i ) 
       DutyRatio x   INV2 =DR0 INV2 ×(DR INV2   ×s   x   INV2 ( i )+(1−DR INV2 )× s   x   INV2 ( j ))
 
     wherein DutyRatio x   INV1  is the value calculated by the proposed algorithm for inverter 1 to generate the corresponding duty cycle waveform; DutyRatio x   INV2  is used for inverter 2 to generate the corresponding duty cycle waveform; s x   INV1 (i) is the first selected by inverter 1, is the switching function of a virtual voltage vector, wherein x=a,b,c,d,e; s x   INV1 (i) is the first selected by inverter 1, is the switching function of a virtual voltage vector, wherein x=a,b,c,d,e; s x   INV2 (j) is the switching function of the second virtual voltage vector selected by inverter 2, wherein x=a,b,c,d,e; 
     Step 5.2) Generating the pulse width modulation signal to the diver chip of the inverter by the obtained duty cycle of inverter 1 and the duty cycle of inverter 2 in Step 5.1). Finally, the corresponding voltage is output to the motor through the power semiconductor. 
     The device of a model predictive decomposition control for an open-winding five-phase permanent magnet synchronous motor mainly includes: 
     The data acquisition unit. Using a Hall-type current sensor ACS758 to sample the phase current of the motor; Using the relative position type encoder to obtain a electrical angle and a speed of the motor. Using the enhanced capture unit in the digital signal processor to calculate the electrical angular velocity ω and the electrical angle θ of the open-winding five-phase permanent magnet synchronous motor. Using the analog to digital converter module in the digital signal processor to sampling and transforming to obtain the current i d  and i q  of the q-axis and d-axis in the rotating coordinate system. 
     Command voltage input unit. Using a TMS320F28377S as the digital signal processor produced by Texas Instruments. Obtaining the reference values iq* and id* of the q-axis and d-axis currents in the rotating coordinate system by calculation through the program in the digital signal processor after the motor speed is given; 
     The increment judging unit. Judging whether the current voltage vector provided by the inverter 1 can meet the current demand of the motor drive system; 
     Inverter 1 acts solely unit. If the Δ d   INV1  and the Δ q   INV1  are greater than the current reference value through the internal program of the digital signal processor, inverter 1 output the duty cycle of DutyRatio x   INV1  solely, and inverter 2 output the duty cycle of 0; 
     Inverter 1 and inverter 2 work together unit. If the Δi d   INV1  and the Δi q   INV1  are less than the current reference value through the internal program of the digital signal processor, inverter 1 output the duty cycle of DutyRatio x   INV1  solely, and inverter 2 output the duty cycle of DutyRatio; 
     Pulse width modulation output unit. Using a voltage source inverter, modulating the duty cycle of inverter 1 and inverter 2 through a triangular carrier. Using a isolated 1ED020F12 driver chip to drives the power semiconductor to generate the corresponding phase voltage; 
     In conclusion, the data acquisition unit processes the sampled data through the digital signal processor as input of the system. According to the instructions input by the instruction voltage input unit, the digital signal processor controls and calculates the given value at this moment using the data acquired by the data acquisition unit. By judging the given value at the current moment selecting the mode suitable for the current state to output, the increment judgment unit judges between the inverter 1 acts solely unit and inverter 1 and inverter 2 work together unit. The pulse width modulation output unit transmits the duty cycle calculated by the above functions. The output acts on the motor winding to generate current. Then those current have been collected and calculated through the data acquisition unit, forming a complete hardware and software closed-loop control system. 
     The beneficial effects of the present invention: 
     1) The present invention is based on a five-phase permanent magnet synchronous motor model predictive control method. The advantages of this present invention is good dynamic performance, simple structure, and easy implementation. 
     2) By decomposing the open-winding five-phase permanent magnet synchronous motor system into the superposition of two independent inverters, the flexibility of the open-winding topology system can be improved. 
     3) The use of the dual inverter decomposition control method effectively reduce the switching times of the inverter, thereby reducing the overall switching loss of the five-phase open-winding permanent magnet synchronous motor system. 
     4) The proposed open-winding decomposition control method is universal and easy to implement, conducive to the engineering and practical application of the new theory. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    Open winding topology and peripheral device circuit. 
         FIG.  2    The internal implementation process of the device. 
         FIGS.  3 A- 3 B  The distribution of the space voltage vector generated by a single inverter; where  FIG.  3 A  is the voltage vector distribution of the fundamental subspace;  FIG.  3 B  is the voltage vector distribution of the third harmonic subspace. 
         FIG.  4    Single inverter virtual voltage vector synthesis. 
         FIG.  5    Virtual space voltage vector distribution of open-winding five-phase permanent magnet synchronous motor. 
         FIG.  6    decomposes the increments of inverter 1 and inverter 2 (take q-axis as an example). 
         FIG.  7    Inverter 1 acts alone. 
         FIG.  8    Inverter 1 and inverter 2 superimposed action. 
         FIG.  9    Switching sequence of dual inverter output when inverter 1 acts alone. 
         FIG.  10    Switching waveforms when inverter 1 and inverter 2 are superimposed. 
         FIG.  11    The A-phase current waveform when inverter 1 alone acts. 
         FIG.  12    The output torque when inverter 1 alone acts. 
         FIG.  13    The speed when inverter 1 alone acts. 
         FIG.  14    The A-phase current when inverter 1 and inverter 2 superimpose. 
         FIG.  15    The output torque when inverter 1 and inverter 2 superimpose. 
         FIG.  16    The speed when inverter 1 and inverter 2 superimpose. 
         FIGS.  17 A- 17 B  Phase A duty cycle inverters 1 and 2 individually act; where  FIG.  17 A  shows Phase A duty cycle when inverter 1 acts alone;  FIG.  17 B  shows Phase A duty cycle when inverter 2 acts on alone. 
         FIGS.  18 A- 18 B  A-phase duty cycle when inverter 1 and inverter 2 superimposed; where  FIG.  18 A  shows Phase A duty cycle when inverter 1 acts alone;  FIG.  18 B  shows Phase A duty cycle when inverter 2 acts on alone. 
         FIGS.  19 A- 19 B  Changes in the duty cycle of phase A under switching of operating conditions; where  FIG.  19 A  shows Phase A duty cycle when inverter 1 acts alone;  FIG.  19 B  shows Phase A duty cycle when inverter 2 acts on alone. 
     
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
     To make the objectives, technical solutions, and advantages of the present invention clearer, the following further describes the present invention in detail concerning the accompanying drawings and embodiments. The specific embodiments described here are only used to explain the present invention but not used to limit the present invention. 
     As shown in  FIG.  1    and  FIG.  2   , the present invention proposes a model predictive decomposition control method and device for an open-winding five-phase permanent magnet synchronous motor. The proposed method regards the two inverters of the open-winding topology system as two independent units, takes the voltage vector provided by inverter 1 as the starting point, and select different inverter switches sequence for the needs of the open-winding system under different working conditions. The present invention can effectively reduce the switching times of the open-winding five-phase permanent magnet synchronous motor control system and reduce the switching loss of the inverter under the premise of ensuring performance. 
     Among them, the adopted dual-power open-winding topology and the peripheral circuit of the device are shown in  FIG.  1   .  FIG.  2    is the flowchart of the proposed model predictive decomposition control device. 
     The specific implementation steps of the proposed model predictive decomposition control method for open-winding five-phase permanent magnet synchronous motors include: 
     Step 1) Obtaining the q-axis and d-axis currents in the fundamental space of the open-winding five-phase permanent magnet synchronous motor in the rotating coordinate system as the given of the control system. 
     Step 1.1) Calculating the speed error between the given speed n* and the actual speed n of the motor in real-time. The required q-axis current reference value i q * has been obtained through the PI regulator. The d-axis current reference value is set as i d *=0. 
     Step 1.2) Sampling the five-phase currents, from the five-phase natural coordinate system ABCDE to the two-phase rotating coordinate system. Obtaining the currents i d  and i q  of the q-axis and d-axis through the coordinate transformation from the five-phase stationary coordinate system to the two-phase rotating coordinate system. 
     Expressing the transformation matrix from the five-phase natural coordinate system ABCDE to the two-phase rotating coordinate system as: 
     
       
         
           
             
               T 
               
                 5 
                 ⁢ 
                 
                   s 
                   / 
                   2 
                 
                 ⁢ 
                 r 
               
             
             = 
             
               
                 2 
                 5 
               
               [ 
               
                 
                   
                     
                       cos 
                       ⁢ 
                          
                       
                         θ 
                         e 
                       
                     
                   
                   
                     
                       cos 
                       ⁢ 
                       
                         ( 
                         
                           
                             θ 
                             e 
                           
                           - 
                           α 
                         
                         ) 
                       
                     
                   
                   
                     
                       cos 
                       ⁢ 
                       
                         ( 
                         
                           
                             θ 
                             e 
                           
                           - 
                           
                             2 
                             ⁢ 
                             α 
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       cos 
                       ⁢ 
                       
                         ( 
                         
                           
                             θ 
                             e 
                           
                           - 
                           
                             3 
                             ⁢ 
                             α 
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       cos 
                       ⁢ 
                       
                         ( 
                         
                           
                             θ 
                             e 
                           
                           - 
                           
                             4 
                             ⁢ 
                             α 
                           
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     
                       
                         - 
                           
                         sin 
                       
                       ⁢ 
                          
                       
                         θ 
                         e 
                       
                     
                   
                   
                     
                       - 
                         
                       
                         sin 
                         ⁡ 
                         ( 
                         
                           
                             θ 
                             e 
                           
                           - 
                           α 
                         
                         ) 
                       
                     
                   
                   
                     
                       - 
                         
                       
                         sin 
                         ⁡ 
                         ( 
                         
                           
                             θ 
                             e 
                           
                           - 
                           
                             2 
                             ⁢ 
                             α 
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       - 
                         
                       
                         sin 
                         ⁡ 
                         ( 
                         
                           
                             θ 
                             e 
                           
                           - 
                           
                             3 
                             ⁢ 
                             α 
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       - 
                         
                       
                         sin 
                         ⁡ 
                         ( 
                         
                           
                             θ 
                             e 
                           
                           - 
                           
                             4 
                             ⁢ 
                             α 
                           
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     
                       cos 
                       ⁢ 
                          
                       3 
                       ⁢ 
                       
                         θ 
                         e 
                       
                     
                   
                   
                     
                       cos 
                       ⁢ 
                          
                       3 
                       ⁢ 
                       
                         ( 
                         
                           
                             θ 
                             e 
                           
                           - 
                           α 
                         
                         ) 
                       
                     
                   
                   
                     
                       cos 
                       ⁢ 
                          
                       3 
                       ⁢ 
                       
                         ( 
                         
                           
                             θ 
                             e 
                           
                           - 
                           
                             2 
                             ⁢ 
                             α 
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       cos 
                       ⁢ 
                          
                       3 
                       ⁢ 
                       
                         ( 
                         
                           
                             θ 
                             e 
                           
                           - 
                           
                             3 
                             ⁢ 
                             α 
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       cos 
                       ⁢ 
                          
                       3 
                       ⁢ 
                       
                         ( 
                         
                           
                             θ 
                             e 
                           
                           - 
                           
                             4 
                             ⁢ 
                             α 
                           
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     
                       
                         - 
                         sin 
                       
                       ⁢ 
                          
                       3 
                       ⁢ 
                       
                         θ 
                         e 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                       ⁢ 
                          
                       3 
                       ⁢ 
                       
                         ( 
                         
                           
                             θ 
                             e 
                           
                           - 
                           α 
                         
                         ) 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                       ⁢ 
                          
                       3 
                       ⁢ 
                       
                         ( 
                         
                           
                             θ 
                             e 
                           
                           - 
                           
                             2 
                             ⁢ 
                             α 
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                       ⁢ 
                          
                       3 
                       ⁢ 
                       
                         ( 
                         
                           
                             θ 
                             e 
                           
                           - 
                           
                             3 
                             ⁢ 
                             α 
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                       ⁢ 
                          
                       3 
                       ⁢ 
                       
                         ( 
                         
                           
                             θ 
                             e 
                           
                           - 
                           
                             4 
                             ⁢ 
                             α 
                           
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     
                       1 
                       2 
                     
                   
                   
                     
                       1 
                       2 
                     
                   
                   
                     
                       1 
                       2 
                     
                   
                   
                     
                       1 
                       2 
                     
                   
                   
                     
                       1 
                       2 
                     
                   
                 
               
               ] 
             
           
         
       
     
     wherein, α=0.4π, θ e  is the electrical angle of the motor. 
     Step 1.3) Obtaining the voltage components i α  and i β  in the stationary coordinate system through the q-axis, and d-axis voltage undergoes the coordinate transformation from the two-phase rotating coordinate system to the two-phase stationary coordinate system. 
     Expressing the transformation matrix from the two-phase rotating coordinate system to the two-phase stationary coordinate system as: 
     
       
         
           
             
               T 
               
                 2 
                 ⁢ 
                 
                   r 
                   / 
                   2 
                 
                 ⁢ 
                 s 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                       ⁢ 
                          
                       
                         θ 
                         e 
                       
                     
                   
                   
                     
                       
                         - 
                           
                         sin 
                       
                       ⁢ 
                          
                       
                         θ 
                         e 
                       
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     
                       sin 
                       ⁢ 
                          
                       
                         θ 
                         e 
                       
                     
                   
                   
                     
                       cos 
                       ⁢ 
                          
                       
                         θ 
                         e 
                       
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                       cos 
                       ⁢ 
                          
                       3 
                       ⁢ 
                       
                         θ 
                         e 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                       ⁢ 
                          
                       3 
                       ⁢ 
                       
                         θ 
                         e 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                       sin 
                       ⁢ 
                          
                       3 
                       ⁢ 
                       
                         θ 
                         e 
                       
                     
                   
                   
                     
                       cos 
                       ⁢ 
                          
                       3 
                       ⁢ 
                       
                         θ 
                         e 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Step 2) Constructing the virtual voltage vector table of the open-winding five-phase permanent magnet synchronous motor in the static coordinate system. Applying the label of the virtual voltage vector in the previous control cycle, and calculating the voltage in the static coordinate system. Moreover, solving the back electromotive force of the motor according to the mathematical model of the motor; 
     Step 2.1) Constructing an open-winding five-phase permanent magnet synchronous motor virtual voltage vector table: 
     Expressing the voltage vector generated by the five-phase voltage source inverter in the stationary coordinate system as: 
     
       
         
           
             
               u 
               s 
             
             = 
             
               
                 2 
                 5 
               
               ⨯ 
               
                 ( 
                 
                   
                     s 
                     a 
                   
                   + 
                   
                     
                       s 
                       b 
                     
                     ⁢ 
                     w 
                   
                   + 
                   
                     
                       s 
                       c 
                     
                     ⁢ 
                     
                       w 
                       2 
                     
                   
                   + 
                   
                     
                       s 
                       d 
                     
                     ⁢ 
                     
                       w 
                       3 
                     
                   
                   + 
                   
                     
                       s 
                       e 
                     
                     ⁢ 
                     
                       w 
                       4 
                     
                   
                 
                 ) 
               
               ⨯ 
               
                 U 
                 dc 
               
             
           
         
       
       
         
           
             ( 
             
               w 
               = 
               
                 e 
                 
                   j 
                   ⁢ 
                   
                     2 
                     5 
                   
                   ⁢ 
                   π 
                 
               
             
             ) 
           
         
       
     
     wherein, U dc  is the DC bus voltage; u s  is the voltage vector value in the static coordinate system; s i  (i=a,b,c,d,e) is the switching function of each bridge arm; when the upper bridge arm is turned on, s i =1, the lower bridge arm is turned on s i =0. 
     The space voltage vector distribution diagram of a single inverter in the five-phase open winding system is shown in  FIGS.  3 A- 3 B : 
     The space voltage vector distribution table of a single inverter is shown in Table 1: 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 The space voltage vector distribution diagram of a single inverter 
               
            
           
           
               
               
               
            
               
                 Voltage vector 
                 Length of vector 
                 Label of vector 
               
               
                   
               
            
           
           
               
               
               
               
            
               
                 Big vector (u B ) 
                 0.6472 
                 U dc   
                 V 24 V 25 V 28 V 12 V 14 V 6 V 7 V 3 V 19 V 17   
               
               
                 Meduim vector 
                 0.4 
                 U dc   
                 V 16 V 29 V 8 V 30 V 4 V 15 V 2 V 23 V 1 V 27   
               
               
                 (u M ) 
               
               
                 Little vector (u L ) 
                 0.2472 
                 U dc   
                 V 9 V 26 V 20 V 13 V 10 V 22 V 5 V 11 V 18 V 21   
               
            
           
           
               
               
               
            
               
                 Zero vector (u 0 ) 
                 0 
                 V 0 V 31   
               
               
                   
               
            
           
         
       
     
     Step 2.2) According to the principle that the third harmonic space voltage is equivalent to zero, synthesizing the voltage vector in the fundamental wave sub-plane. The synthesis principle is as follows: 
       VV i ( u   B   ,u   L )=0.618× u   M +(1−0.618)× u   B  
 
     wherein, the scale factor of 0.618 can make the two voltage vectors equivalent to zero in the harmonic subspace of the five-phase permanent magnet synchronous motor. 
     The synthesized virtual voltage vector is shown in  FIG.  4   : 
     According to the principle of the open-winding topology, the space voltage vector distribution of the open-winding five-phase permanent magnet synchronous motor is shown in  FIG.  5   . The voltage vector in the figure is the subtraction of the virtual voltage vector provided by inverter 1 and inverter 2. For example, 3-9′ means the virtual voltage vector in the open-winding topology obtained by subtracting the virtual voltage vector No. 3 of inverter 1 and the virtual voltage vector No. 9 of the inverter 2. 
     All the virtual space voltage vector distribution of the open-winding five-phase permanent magnet synchronous motor is shown in Table 2: 
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 virtual space voltage vector distribution of the open-winding 
               
               
                 five-phase PMSM 
               
            
           
           
               
               
               
               
               
            
               
                   
                 Label of 
                 Label of 
                   
                   
               
               
                   
                 virtual 
                 virtual 
                 Voltage vector in 
                 Voltage vector in 
               
               
                   
                 voltage 
                 voltage 
                 inverter 1 
                 inverter 2 
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                 vector in 
                 vector in 
                 Big 
                 Meduim 
                 Big 
                 Meduim 
               
               
                 Label 
                 inverter 1 
                 inverter 2 
                 vector 
                 vector 
                 vector 
                 vector 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 1 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 2 
                 0 
                 1 
                 0 
                 0 
                 25 
                 16 
               
               
                 3 
                 0 
                 2 
                 0 
                 0 
                 24 
                 29 
               
               
                 4 
                 0 
                 3 
                 0 
                 0 
                 28 
                 8 
               
               
                 5 
                 0 
                 4 
                 0 
                 0 
                 12 
                 30 
               
               
                 6 
                 0 
                 5 
                 0 
                 0 
                 14 
                 4 
               
               
                 7 
                 0 
                 6 
                 0 
                 0 
                 6 
                 15 
               
               
                 8 
                 0 
                 7 
                 0 
                 0 
                 7 
                 9 
               
               
                 9 
                 0 
                 8 
                 0 
                 0 
                 3 
                 23 
               
               
                 10 
                 0 
                 9 
                 0 
                 0 
                 19 
                 1 
               
               
                 11 
                 0 
                 10 
                 0 
                 0 
                 17 
                 27 
               
               
                 12 
                 0 
                 11 
                 0 
                 0 
                 31 
                 31 
               
               
                 13 
                 1 
                 0 
                 25 
                 16 
                 0 
                 0 
               
               
                 14 
                 1 
                 1 
                 25 
                 16 
                 25 
                 16 
               
               
                 15 
                 1 
                 2 
                 25 
                 16 
                 24 
                 29 
               
               
                 16 
                 1 
                 3 
                 25 
                 16 
                 28 
                 8 
               
               
                 17 
                 1 
                 4 
                 25 
                 16 
                 12 
                 30 
               
               
                 18 
                 1 
                 5 
                 25 
                 16 
                 14 
                 4 
               
               
                 19 
                 1 
                 6 
                 25 
                 16 
                 6 
                 15 
               
               
                 20 
                 1 
                 7 
                 25 
                 16 
                 7 
                 2 
               
               
                 21 
                 1 
                 8 
                 25 
                 16 
                 3 
                 23 
               
               
                 20 
                 1 
                 9 
                 25 
                 16 
                 19 
                 1 
               
               
                 23 
                 1 
                 10 
                 25 
                 16 
                 17 
                 27 
               
               
                 24 
                 1 
                 11 
                 25 
                 16 
                 31 
                 31 
               
               
                 25 
                 2 
                 0 
                 24 
                 29 
                 0 
                 0 
               
               
                 26 
                 2 
                 1 
                 24 
                 29 
                 25 
                 16 
               
               
                 27 
                 2 
                 2 
                 24 
                 29 
                 24 
                 29 
               
               
                 28 
                 2 
                 3 
                 24 
                 29 
                 28 
                 8 
               
               
                 29 
                 2 
                 4 
                 24 
                 29 
                 12 
                 30 
               
               
                 30 
                 2 
                 5 
                 24 
                 29 
                 14 
                 4 
               
               
                 31 
                 2 
                 6 
                 24 
                 29 
                 6 
                 15 
               
               
                 32 
                 2 
                 7 
                 24 
                 29 
                 7 
                 2 
               
               
                 33 
                 2 
                 8 
                 24 
                 29 
                 3 
                 23 
               
               
                 34 
                 2 
                 9 
                 24 
                 29 
                 19 
                 1 
               
               
                 35 
                 2 
                 10 
                 24 
                 29 
                 17 
                 27 
               
               
                 36 
                 2 
                 11 
                 24 
                 29 
                 31 
                 31 
               
               
                 37 
                 2 
                 0 
                 28 
                 8 
                 0 
                 0 
               
               
                 38 
                 2 
                 1 
                 28 
                 8 
                 25 
                 16 
               
               
                 39 
                 2 
                 2 
                 28 
                 8 
                 24 
                 29 
               
               
                 40 
                 3 
                 3 
                 28 
                 8 
                 28 
                 8 
               
               
                 41 
                 3 
                 4 
                 28 
                 8 
                 12 
                 30 
               
               
                 42 
                 3 
                 5 
                 28 
                 8 
                 14 
                 4 
               
               
                 43 
                 3 
                 6 
                 28 
                 8 
                 6 
                 15 
               
               
                 44 
                 3 
                 7 
                 28 
                 8 
                 7 
                 9 
               
               
                 45 
                 3 
                 8 
                 28 
                 8 
                 3 
                 23 
               
               
                 46 
                 3 
                 9 
                 28 
                 8 
                 19 
                 1 
               
               
                 47 
                 3 
                 10 
                 28 
                 8 
                 17 
                 27 
               
               
                 48 
                 3 
                 11 
                 28 
                 8 
                 31 
                 31 
               
               
                 49 
                 4 
                 0 
                 12 
                 30 
                 0 
                 0 
               
               
                 50 
                 4 
                 1 
                 12 
                 30 
                 25 
                 16 
               
               
                 51 
                 4 
                 2 
                 12 
                 30 
                 24 
                 29 
               
               
                 52 
                 4 
                 3 
                 12 
                 30 
                 28 
                 8 
               
               
                 53 
                 4 
                 4 
                 12 
                 30 
                 12 
                 30 
               
               
                 54 
                 4 
                 5 
                 12 
                 30 
                 14 
                 4 
               
               
                 55 
                 4 
                 6 
                 12 
                 30 
                 6 
                 15 
               
               
                 56 
                 4 
                 7 
                 12 
                 30 
                 7 
                 2 
               
               
                 57 
                 4 
                 8 
                 12 
                 30 
                 3 
                 23 
               
               
                 58 
                 4 
                 9 
                 12 
                 30 
                 19 
                 1 
               
               
                 59 
                 4 
                 10 
                 12 
                 30 
                 17 
                 27 
               
               
                 60 
                 4 
                 11 
                 12 
                 30 
                 31 
                 31 
               
               
                 61 
                 5 
                 0 
                 14 
                 4 
                 0 
                 0 
               
               
                 62 
                 5 
                 1 
                 14 
                 4 
                 25 
                 16 
               
               
                 63 
                 5 
                 2 
                 14 
                 4 
                 24 
                 29 
               
               
                 64 
                 5 
                 3 
                 14 
                 4 
                 28 
                 8 
               
               
                 65 
                 5 
                 4 
                 14 
                 4 
                 12 
                 30 
               
               
                 66 
                 5 
                 5 
                 14 
                 4 
                 14 
                 4 
               
               
                 67 
                 5 
                 6 
                 14 
                 4 
                 6 
                 15 
               
               
                 68 
                 5 
                 7 
                 14 
                 4 
                 7 
                 2 
               
               
                 69 
                 5 
                 8 
                 14 
                 4 
                 3 
                 23 
               
               
                 70 
                 5 
                 9 
                 14 
                 4 
                 19 
                 1 
               
               
                 71 
                 5 
                 10 
                 14 
                 4 
                 17 
                 27 
               
               
                 72 
                 5 
                 11 
                 14 
                 4 
                 31 
                 31 
               
               
                 73 
                 6 
                 0 
                 6 
                 15 
                 0 
                 0 
               
               
                 74 
                 6 
                 1 
                 6 
                 15 
                 25 
                 16 
               
               
                 75 
                 6 
                 2 
                 6 
                 15 
                 24 
                 29 
               
               
                 76 
                 6 
                 3 
                 6 
                 15 
                 28 
                 8 
               
               
                 77 
                 6 
                 4 
                 6 
                 15 
                 12 
                 30 
               
               
                 78 
                 6 
                 5 
                 6 
                 15 
                 14 
                 4 
               
               
                 79 
                 6 
                 6 
                 6 
                 15 
                 6 
                 15 
               
               
                 80 
                 6 
                 7 
                 6 
                 15 
                 7 
                 9 
               
               
                 81 
                 6 
                 8 
                 6 
                 15 
                 3 
                 23 
               
               
                 82 
                 6 
                 9 
                 6 
                 15 
                 19 
                 1 
               
               
                 83 
                 6 
                 10 
                 6 
                 15 
                 17 
                 27 
               
               
                 84 
                 6 
                 11 
                 6 
                 15 
                 31 
                 31 
               
               
                 85 
                 7 
                 0 
                 7 
                 2 
                 0 
                 0 
               
               
                 86 
                 7 
                 1 
                 7 
                 2 
                 25 
                 16 
               
               
                 87 
                 7 
                 2 
                 7 
                 2 
                 24 
                 29 
               
               
                 88 
                 7 
                 3 
                 7 
                 2 
                 28 
                 8 
               
               
                 89 
                 7 
                 4 
                 7 
                 2 
                 12 
                 30 
               
               
                 90 
                 7 
                 5 
                 7 
                 2 
                 14 
                 4 
               
               
                 91 
                 7 
                 6 
                 7 
                 2 
                 6 
                 15 
               
               
                 92 
                 7 
                 7 
                 7 
                 2 
                 7 
                 2 
               
               
                 93 
                 7 
                 8 
                 7 
                 2 
                 3 
                 23 
               
               
                 94 
                 7 
                 9 
                 7 
                 2 
                 19 
                 1 
               
               
                 95 
                 7 
                 10 
                 7 
                 2 
                 17 
                 27 
               
               
                 96 
                 7 
                 11 
                 7 
                 2 
                 31 
                 31 
               
               
                 97 
                 8 
                 0 
                 2 
                 23 
                 0 
                 0 
               
               
                 98 
                 8 
                 1 
                 3 
                 23 
                 25 
                 16 
               
               
                 99 
                 8 
                 2 
                 3 
                 23 
                 24 
                 29 
               
               
                 100 
                 8 
                 3 
                 3 
                 23 
                 28 
                 8 
               
               
                 101 
                 8 
                 4 
                 3 
                 23 
                 12 
                 30 
               
               
                 102 
                 8 
                 5 
                 3 
                 23 
                 14 
                 4 
               
               
                 103 
                 8 
                 6 
                 3 
                 23 
                 6 
                 15 
               
               
                 104 
                 8 
                 7 
                 3 
                 23 
                 7 
                 9 
               
               
                 105 
                 8 
                 8 
                 3 
                 23 
                 3 
                 23 
               
               
                 106 
                 8 
                 9 
                 3 
                 23 
                 19 
                 1 
               
               
                 107 
                 8 
                 10 
                 3 
                 23 
                 17 
                 27 
               
               
                 108 
                 8 
                 11 
                 3 
                 23 
                 31 
                 31 
               
               
                 109 
                 9 
                 0 
                 19 
                 1 
                 0 
                 0 
               
               
                 110 
                 9 
                 1 
                 19 
                 1 
                 25 
                 16 
               
               
                 111 
                 9 
                 2 
                 19 
                 1 
                 24 
                 29 
               
               
                 112 
                 9 
                 3 
                 19 
                 1 
                 28 
                 8 
               
               
                 113 
                 9 
                 4 
                 19 
                 1 
                 12 
                 30 
               
               
                 114 
                 9 
                 5 
                 19 
                 1 
                 14 
                 4 
               
               
                 115 
                 9 
                 6 
                 19 
                 1 
                 6 
                 15 
               
               
                 116 
                 9 
                 7 
                 19 
                 1 
                 7 
                 2 
               
               
                 117 
                 9 
                 8 
                 19 
                 1 
                 3 
                 23 
               
               
                 118 
                 9 
                 9 
                 19 
                 1 
                 19 
                 1 
               
               
                 119 
                 9 
                 10 
                 19 
                 1 
                 17 
                 27 
               
               
                 120 
                 9 
                 11 
                 19 
                 1 
                 31 
                 31 
               
               
                 121 
                 10 
                 0 
                 17 
                 27 
                 0 
                 0 
               
               
                 122 
                 10 
                 1 
                 17 
                 27 
                 25 
                 16 
               
               
                 123 
                 10 
                 2 
                 17 
                 27 
                 24 
                 29 
               
               
                 124 
                 10 
                 3 
                 17 
                 27 
                 28 
                 8 
               
               
                 125 
                 10 
                 4 
                 17 
                 27 
                 12 
                 30 
               
               
                 126 
                 10 
                 5 
                 17 
                 27 
                 14 
                 4 
               
               
                 127 
                 10 
                 6 
                 17 
                 27 
                 6 
                 15 
               
               
                 128 
                 10 
                 7 
                 17 
                 27 
                 7 
                 9 
               
               
                 129 
                 10 
                 8 
                 17 
                 27 
                 3 
                 23 
               
               
                 130 
                 10 
                 9 
                 17 
                 27 
                 19 
                 1 
               
               
                 131 
                 10 
                 10 
                 17 
                 27 
                 17 
                 27 
               
               
                 132 
                 10 
                 11 
                 17 
                 27 
                 31 
                 31 
               
               
                 133 
                 11 
                 0 
                 31 
                 31 
                 0 
                 0 
               
               
                 134 
                 11 
                 1 
                 31 
                 31 
                 25 
                 16 
               
               
                 135 
                 11 
                 2 
                 31 
                 31 
                 24 
                 29 
               
               
                 136 
                 11 
                 3 
                 31 
                 31 
                 28 
                 8 
               
               
                 137 
                 11 
                 4 
                 31 
                 31 
                 12 
                 30 
               
               
                 138 
                 11 
                 5 
                 31 
                 31 
                 14 
                 4 
               
               
                 139 
                 11 
                 6 
                 31 
                 31 
                 6 
                 15 
               
               
                 140 
                 11 
                 7 
                 31 
                 31 
                 7 
                 2 
               
               
                 141 
                 11 
                 8 
                 31 
                 31 
                 3 
                 23 
               
               
                 142 
                 11 
                 9 
                 31 
                 31 
                 19 
                 1 
               
               
                 143 
                 11 
                 10 
                 31 
                 31 
                 17 
                 27 
               
               
                 144 
                 11 
                 11 
                 31 
                 31 
                 31 
                 31 
               
               
                   
               
            
           
         
       
     
     Step 2.3) Calculating the voltage vector output value of the previous sampling period: 
     
       
         
           
             
               
                 u 
                 s 
               
               ( 
               
                 k 
                 - 
                 1 
               
               ) 
             
             = 
             
               
                 2 
                 5 
               
               ⨯ 
               
                 ( 
                 
                   
                     
                       s 
                       a 
                     
                     ( 
                     
                       k 
                       - 
                       1 
                     
                     ) 
                   
                   + 
                   
                     
                       
                         s 
                         b 
                       
                       ( 
                       
                         k 
                         - 
                         1 
                       
                       ) 
                     
                     ⁢ 
                     w 
                   
                   + 
                   
                     
                       
                         s 
                         c 
                       
                       ( 
                       
                         k 
                         - 
                         1 
                       
                       ) 
                     
                     ⁢ 
                     
                       w 
                       2 
                     
                   
                   + 
                   
                     
                       
                         s 
                         d 
                       
                       ( 
                       
                         k 
                         - 
                         1 
                       
                       ) 
                     
                     ⁢ 
                     
                       w 
                       3 
                     
                   
                   + 
                   
                     
                       
                         s 
                         e 
                       
                       ( 
                       
                         k 
                         - 
                         1 
                       
                       ) 
                     
                     ⁢ 
                     
                       w 
                       4 
                     
                   
                 
                 ) 
               
               ⨯ 
               
                 U 
                 dc 
               
             
           
         
       
       
         
           
             	 
             
               ( 
               
                 w 
                 = 
                 
                   e 
                   
                     j 
                     ⁢ 
                     
                       2 
                       5 
                     
                     ⁢ 
                     π 
                   
                 
               
               ) 
             
           
         
       
     
     Furthermore, the instantaneous value of the q-axis and d-axis back electromotive force of the open-winding five-phase permanent magnet synchronous motor in the rotating coordinate system at this time can be calculated by the i q * and the actual speed ω of the motor: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       EMF 
                       d 
                     
                     = 
                     
                       ω 
                       ⨯ 
                       
                         L 
                         s 
                       
                       ⨯ 
                       
                         i 
                         q 
                         * 
                       
                     
                   
                 
               
               
                 
                   
                     
                       EMF 
                       q 
                     
                     = 
                     
                       
                         ω 
                         ⨯ 
                         
                           L 
                           s 
                         
                         ⨯ 
                         
                           i 
                           d 
                           * 
                         
                       
                       + 
                       
                         ω 
                         ⨯ 
                         
                           φ 
                           f 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     wherein, Ls is the inductance of the motor, φ f  is the permanent magnet flux linkage of the motor, iq* is the q-axis current reference value in the rotating coordinate system, and id* is the d-axis current reference value in the rotating coordinate system. 
     Step 3) The Euler forward difference prediction equation has been used to find the q-axis current and d-axis current in this period; 
     Step 3.1) Expressing the voltage equation in the rotating coordinate system of the five-phase permanent magnet synchronous motor as: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         di 
                         d 
                       
                       dt 
                     
                     = 
                     
                       
                         1 
                         
                           L 
                           s 
                         
                       
                       ⨯ 
                       
                         ( 
                         
                           
                             u 
                             d 
                           
                           - 
                           
                             
                               R 
                               s 
                             
                             ⨯ 
                             
                               i 
                               d 
                             
                           
                           - 
                           
                             
                               ω 
                               e 
                             
                             ⨯ 
                             
                               L 
                               s 
                             
                             ⨯ 
                             
                               i 
                               q 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         di 
                         q 
                       
                       dt 
                     
                     = 
                     
                       
                         1 
                         
                           L 
                           s 
                         
                       
                       ⨯ 
                       
                         ( 
                         
                           
                             u 
                             d 
                           
                           - 
                           
                             
                               R 
                               s 
                             
                             ⨯ 
                             
                               i 
                               d 
                             
                           
                           - 
                           
                             
                               ω 
                               e 
                             
                             ⨯ 
                             
                               L 
                               s 
                             
                             ⨯ 
                             
                               i 
                               d 
                             
                           
                           - 
                           
                             
                               ω 
                               e 
                             
                             ⨯ 
                             
                               φ 
                               f 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
     wherein, L s  is the inductance of the motor, R s  is the stator resistance of the motor, u d  is the voltage on the d-axis, u q  is the voltage on the q-axis, i d  is the d-axis current, and i q  is the q-axis current, ω e  is the electrical angle of the motor, ϕ f  is the permanent magnet flux linkage of the motor; 
     Step 3.2) Euler forward difference method has been used to find the currents of the q-axis and d-axis in the rotating coordinate system at time k+1: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         
                           
                             i 
                             d 
                           
                           ( 
                           
                             k 
                             + 
                             1 
                           
                           ) 
                         
                         - 
                         
                           
                             i 
                             d 
                           
                           ( 
                           k 
                           ) 
                         
                       
                       Ts 
                     
                     = 
                     
                       
                         1 
                         
                           L 
                           s 
                         
                       
                       ⨯ 
                       
                         ( 
                         
                           
                             
                               u 
                               d 
                             
                             ( 
                             k 
                             ) 
                           
                           - 
                           
                             
                               R 
                               s 
                             
                             ⨯ 
                             
                               
                                 i 
                                 d 
                               
                               ( 
                               k 
                               ) 
                             
                           
                           - 
                           
                             
                               EMF 
                               d 
                             
                             ( 
                             k 
                             ) 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         
                           
                             i 
                             q 
                           
                           ( 
                           
                             k 
                             + 
                             1 
                           
                           ) 
                         
                         - 
                         
                           
                             i 
                             q 
                           
                           ( 
                           k 
                           ) 
                         
                       
                       Ts 
                     
                     = 
                     
                       
                         1 
                         
                           L 
                           s 
                         
                       
                       ⨯ 
                       
                         ( 
                         
                           
                             
                               u 
                               q 
                             
                             ⁢ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                           - 
                           
                             
                               
                                 R 
                                 s 
                               
                               ⨯ 
                               
                                 i 
                                 q 
                               
                             
                             ⁢ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                           - 
                           
                             
                               EMF 
                               q 
                             
                             ⁢ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
     wherein, u d (k) is the voltage on the d-axis at time k, u q (k) is the voltage on the q-axis at time k, i d (k) is the d-axis current, and i q (k) is the q-axis current, EMF d (k) is the d-axis back EMF of the motor at time k, and EMF q (k) is the q-axis back EMF of the motor at time k. 
     Step 4) The voltage vector generated by inverter 1 as given in the current sampling period has been substituted into the prediction model and cost function to find the current of the q-axis current and d-axis current of the rotating coordinate system in the next period, meeting with the given by the control system. If the voltage vector provided by inverter 1 meets the requirements of the drive system, inverter 2 will be closed. Otherwise, inverter 1 will be clamped to the maximum voltage vector, and inverter 2 will provide the increment of the remaining part; 
     Step 4.1) According to the mathematical model of open-winding five-phase permanent magnet synchronous motor, expressing the currents of the q-axis and d-axis in the rotating coordinate system at time k+1: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         u 
                         d 
                       
                       ( 
                       
                         k 
                         + 
                         1 
                       
                       ) 
                     
                     = 
                     
                       
                         
                           u 
                           d 
                           
                             INV 
                             ⁢ 
                             1 
                           
                         
                         ( 
                         
                           k 
                           + 
                           1 
                         
                         ) 
                       
                       - 
                       
                         
                           u 
                           d 
                           
                             INV 
                             ⁢ 
                             2 
                           
                         
                         ( 
                         
                           k 
                           + 
                           1 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         u 
                         q 
                       
                       ⁢ 
                       
                         ( 
                         
                           k 
                           + 
                           1 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           u 
                           q 
                           
                             INV 
                             ⁢ 
                             1 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             k 
                             + 
                             1 
                           
                           ) 
                         
                       
                       - 
                       
                         
                           u 
                           q 
                           
                             INV 
                             ⁢ 
                             2 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             k 
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Step 4.2) Expressing the voltage vectors that inverter 1 and inverter 2 provided as: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       Δ 
                       ⁢ 
                       
                         i 
                         d 
                         
                           INV 
                           ⁢ 
                           1 
                         
                       
                     
                     = 
                     
                       
                         Ts 
                         
                           L 
                           s 
                         
                       
                       ⨯ 
                       
                         ( 
                         
                           
                             
                               u 
                               d 
                               
                                 INV 
                                 ⁢ 
                                 1 
                               
                             
                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                           - 
                           
                             
                               R 
                               s 
                             
                             ⨯ 
                             
                               
                                 i 
                                 d 
                               
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               EMF 
                               d 
                             
                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     
                       Δ 
                       ⁢ 
                       
                         i 
                         q 
                         
                           INV 
                           ⁢ 
                           1 
                         
                       
                     
                     = 
                     
                       
                         Ts 
                         
                           L 
                           s 
                         
                       
                       ⨯ 
                       
                         ( 
                         
                           
                             
                               u 
                               d 
                               
                                 INV 
                                 ⁢ 
                                 1 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               
                                 R 
                                 s 
                               
                               ⨯ 
                               
                                 i 
                                 q 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               EMF 
                               q 
                             
                             ⁢ 
                             
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             { 
             
               
                 
                   
                     
                       Δ 
                       ⁢ 
                       
                         i 
                         d 
                         
                           INV 
                           ⁢ 
                           2 
                         
                       
                     
                     = 
                     
                       - 
                       
                         
                           Ts 
                           
                             L 
                             s 
                           
                         
                         ⨯ 
                         
                           
                             u 
                             d 
                             
                               INV 
                               ⁢ 
                               2 
                             
                           
                           ( 
                           
                             k 
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       Δ 
                       ⁢ 
                       
                         i 
                         q 
                         
                           INV 
                           ⁢ 
                           2 
                         
                       
                     
                     = 
                     
                       
                         - 
                         
                           
                             Ts 
                             
                               L 
                               s 
                             
                           
                           ⨯ 
                           
                             u 
                             d 
                             
                               INV 
                               ⁢ 
                               2 
                             
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           k 
                           + 
                           1 
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
     wherein, Δi d   INV1  and Δi q   INV1  are the d-axis current and q-axis currents in the rotating coordinate system generated by the inverter 1. Rs is the phase resistance of the motor windings. Ts is the control period of the controller. Ls is the stator inductance of the motor. EMF d (k+1) is the d-axis back-EMF of the motor at the k+1 instance. EMF q (k+1) is the q-axis back-EMF of the motor at the k+1 instance; 
     As shown in  FIG.  6   , the difference between the q-axis current and the q-axis reference current at time k+1 has been calculated by using the principle of dead-beat control in the q-axis current under the rotating coordinate system: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       Δ 
                       ⁢ 
                       
                         i 
                         d 
                       
                     
                     = 
                     
                       
                         
                           
                             i 
                             d 
                           
                           ( 
                           
                             k 
                             + 
                             2 
                           
                           ) 
                         
                         - 
                         
                           
                             i 
                             d 
                           
                           ( 
                           
                             k 
                             + 
                             1 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           Δ 
                           ⁢ 
                           
                             i 
                             d 
                             
                               INV 
                               ⁢ 
                               1 
                             
                           
                         
                         + 
                         
                           Δ 
                           ⁢ 
                           
                             i 
                             d 
                             
                               INV 
                               ⁢ 
                               2 
                             
                           
                              
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       Δ 
                       ⁢ 
                       
                         i 
                         q 
                       
                     
                     = 
                     
                       
                         
                           
                             i 
                             q 
                           
                           ⁢ 
                           
                             ( 
                             
                               k 
                               + 
                               2 
                             
                             ) 
                           
                         
                         - 
                         
                           
                             i 
                             q 
                           
                           ⁢ 
                           
                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                       = 
                       
                         
                           Δ 
                           ⁢ 
                           
                             i 
                             q 
                             
                               INV 
                               ⁢ 
                               1 
                             
                           
                         
                         + 
                         
                           Δ 
                           ⁢ 
                           
                             i 
                             q 
                             
                               INV 
                               ⁢ 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Step 4.3) Traversing the virtual voltage vector generated by inverter 1, and predict the q-axis and d-axis currents in the rotating coordinate system of the five-phase permanent magnet synchronous motor at time k+2: 
     If a Δi d   INV1  and a Δi q   INV1  is greater than or equal to the requirement Δi d * and Δi q * at this moment, the voltage vector provided by inverter 1 is enough, and the inverter 2 is turn off. By applying multiple voltage vectors in one cycle, express the voltage vector in the rotating coordinate system generated in the current sampling period as: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       u 
                       a 
                       
                         INV 
                         ⁢ 
                         1 
                       
                     
                     = 
                     
                       DR 
                       ⁢ 
                       
                         
                           0 
                           
                             INV 
                             ⁢ 
                             1 
                           
                         
                         ⨯ 
                         
                           ( 
                           
                             
                               
                                 DR 
                                 
                                   INV 
                                   ⁢ 
                                   1 
                                 
                               
                               ⨯ 
                               
                                 real 
                                 ( 
                                 
                                   VV 
                                   i 
                                   
                                     INV 
                                     ⁢ 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     DR 
                                     
                                       INV 
                                       ⁢ 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                               ⨯ 
                                 
                               
                                 real 
                                 ( 
                                 
                                   VV 
                                   j 
                                   
                                     INV 
                                     ⁢ 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       u 
                       β 
                       
                         INV 
                         ⁢ 
                         1 
                       
                     
                     = 
                     
                       DR 
                       ⁢ 
                       
                         
                           0 
                           
                             INV 
                             ⁢ 
                             1 
                           
                         
                         ⨯ 
                         
                           ( 
                           
                             
                               
                                 DR 
                                 
                                   INV 
                                   ⁢ 
                                   1 
                                 
                               
                               ⨯ 
                               
                                 imag 
                                 ( 
                                 
                                   VV 
                                   i 
                                   
                                     INV 
                                     ⁢ 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     DR 
                                     
                                       INV 
                                       ⁢ 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                               ⨯ 
                                 
                               
                                 imag 
                                 ( 
                                 
                                   VV 
                                   j 
                                   
                                     INV 
                                     ⁢ 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             { 
             
               
                 
                   
                     
                       u 
                       a 
                       
                         INV 
                         ⁢ 
                         2 
                       
                     
                     = 
                     0 
                   
                 
               
               
                 
                   
                     
                       u 
                       β 
                       
                         INV 
                         ⁢ 
                         2 
                       
                     
                     = 
                     0 
                   
                 
               
             
           
         
       
     
     wherein, u α, u β, u α, and u β are the voltage vector in the stationary coordinate system selected by inverter 1 and inverter 2, respectively, distinguishing by superscripts; real(VV i   INV1 ) and imag(VV i   INV1 ) are the real and imaginary parts of the first virtual voltage vector to be selected by inverter 1 at this moment. real(VV j   INV1 ) and imag(VV j   INV1 ) are the real and imaginary parts of the second voltage vector to be selected by inverter 1. DR0 INV1  is the scale factors for the sum of the effective voltage vectors of inverter 1; DR INV1  is the scale factor between two effective voltage vectors inverter 1; 
     Expressing the scale factors of inverter 1 and inverter 2 when the increment provided by inverter 1 meets the current needs as: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       DR 
                       ⁢ 
                       
                         0 
                         
                           INV 
                           ⁢ 
                           1 
                         
                       
                     
                     = 
                     
                       m 
                       ⁡ 
                       ( 
                       
                         0 
                         ≤ 
                         m 
                         ≤ 
                         1 
                       
                       ) 
                     
                   
                 
               
               
                 
                   
                     
                       DR 
                       
                         INV 
                         ⁢ 
                         1 
                       
                     
                     = 
                     
                       n 
                       ⁡ 
                       ( 
                       
                         0 
                         ≤ 
                         n 
                         ≤ 
                         1 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             { 
             
               
                 
                   
                     
                       DR 
                       ⁢ 
                       
                         0 
                         
                           INV 
                           ⁢ 
                           2 
                         
                       
                     
                     = 
                     0 
                   
                 
               
               
                 
                   
                     
                       DR 
                       
                         INV 
                         ⁢ 
                         2 
                       
                     
                     = 
                     0 
                   
                 
               
             
           
         
       
     
     Expressing the q-axis and d-axis current in the rotating coordinate system of the motor at k+2 instance as: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         i 
                         d 
                       
                       ( 
                       
                         k 
                         + 
                         2 
                       
                       ) 
                     
                     = 
                     
                       
                         
                           i 
                           d 
                         
                         ( 
                         
                           k 
                           + 
                           1 
                         
                         ) 
                       
                       + 
                       
                         
                           Ts 
                           
                             L 
                             s 
                           
                         
                         ⨯ 
                         
                           ( 
                           
                             
                               
                                 u 
                                 d 
                                 
                                   INV 
                                   ⁢ 
                                   1 
                                 
                               
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                             - 
                             
                               
                                 R 
                                 s 
                               
                               ⨯ 
                               
                                 
                                   i 
                                   d 
                                 
                                 ( 
                                 
                                   k 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 EMF 
                                 d 
                               
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         i 
                         q 
                       
                       ⁢ 
                       
                         ( 
                         
                           k 
                           + 
                           2 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           i 
                           q 
                         
                         ⁢ 
                         
                           ( 
                           
                             k 
                             + 
                             1 
                           
                           ) 
                         
                       
                       + 
                       
                         
                           Ts 
                           
                             L 
                             s 
                           
                         
                         ⨯ 
                         
                           ( 
                           
                             
                               
                                 u 
                                 d 
                                 
                                   INV 
                                   ⁢ 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   k 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 
                                   R 
                                   s 
                                 
                                 ⨯ 
                                 
                                   i 
                                   q 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   k 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 EMF 
                                 q 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   k 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     wherein u d   INV1 (k+1) and u q   INV1 (k+1) are the voltages of the q-axis and d-axis in the rotating coordinate system provided by the inverter 1. 
     As shown in  FIG.  7   , the virtual voltage vector provided by the inverter has been able to meet the state of the motor at the current moment, and the inverter 2 will be shut down in time, thereby reducing the loss of the system. 
     If a Δi d   INV1  and a Δi q   INV1  are less than the incremental required Δi d * and Δi q * at the current moment, the required voltage vector is beyond the output capacity of the inverter 1. Furthermore, inverter 2 is required to output the remaining increments to meet the motor operating conditions. In this situation, inverter 1 outputs the maximum increments of Δi d   INV1  and Δi q   INV1 , inverter 2 acts on multiple voltage vectors in the current period. Express the voltage vector in the rotating coordinate system generated in the current sampling period as: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       u 
                       a 
                       
                         INV 
                         ⁢ 
                         1 
                       
                     
                     = 
                     
                       real 
                       ( 
                       
                         VV 
                         i 
                         
                           INV 
                           ⁢ 
                           1 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   
                     
                       u 
                       β 
                       
                         INV 
                         ⁢ 
                         1 
                       
                     
                     = 
                     
                       imag 
                       ( 
                       
                         VV 
                         i 
                         
                           INV 
                           ⁢ 
                           1 
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             { 
             
               
                 
                   
                     
                       u 
                       a 
                       
                         INV 
                         ⁢ 
                         2 
                       
                     
                     = 
                     
                       DR 
                       ⁢ 
                       
                         
                           0 
                           
                             INV 
                             ⁢ 
                             2 
                           
                         
                         ⨯ 
                         
                           ( 
                           
                             
                               
                                 DR 
                                 
                                   INV 
                                   ⁢ 
                                   2 
                                 
                               
                               ⨯ 
                               
                                 real 
                                 ( 
                                 
                                   VV 
                                   i 
                                   
                                     INV 
                                     ⁢ 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     DR 
                                     
                                       INV 
                                       ⁢ 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                               ⨯ 
                                 
                               
                                 real 
                                 ( 
                                 
                                   VV 
                                   j 
                                   
                                     INV 
                                     ⁢ 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       u 
                       β 
                       
                         INV 
                         ⁢ 
                         2 
                       
                     
                     = 
                     
                       DR 
                       ⁢ 
                       
                         
                           0 
                           
                             INV 
                             ⁢ 
                             2 
                           
                         
                         ⨯ 
                         
                           ( 
                           
                             
                               
                                 DR 
                                 
                                   INV 
                                   ⁢ 
                                   2 
                                 
                               
                               ⨯ 
                               
                                 imag 
                                 ( 
                                 
                                   VV 
                                   i 
                                   
                                     INV 
                                     ⁢ 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     DR 
                                     
                                       INV 
                                       ⁢ 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                               ⨯ 
                                 
                               
                                 imag 
                                 ( 
                                 
                                   VV 
                                   j 
                                   
                                     INV 
                                     ⁢ 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     wherein, u α, u β, u α, and u β are the voltage vector in the stationary coordinate system selected by inverter 1 and inverter 2, respectively, distinguishing by superscripts; real(VV i   INV1 ) and imag(VV i   INV1 ) are the real and imaginary parts of the first virtual voltage vector to be selected by inverter 1 at this moment; real(VV i   INV2 ) and imag(VV i   INV2 ) are the real and imaginary parts of the first candidate voltage vector of inverter 2; real(VV j   INV2 ) and imag(VV j   INV2 ) are the real and imaginary parts of the second candidate voltage vector of inverter 2 respectively; DR0 INV2  is the scale factor for the sum of the effective voltage vectors action; DR INV2  is the scale factor between the two effective voltage vector; 
     Expressing the scale factors of inverter 1 and inverter 2 when the increment provided by inverter 1 less than needs as: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         DR 
                         ⁢ 
                         
                           0 
                           
                             INV 
                             ⁢ 
                             1 
                           
                         
                       
                       = 
                       1 
                     
                   
                 
                 
                   
                     
                       
                         DR 
                         
                           INV 
                           ⁢ 
                           1 
                         
                       
                       = 
                       1 
                     
                   
                 
               
               ⁢ 
               
 
               
                 { 
                 
                   
                     
                       
                         
                           DR 
                           ⁢ 
                           
                             0 
                             
                               INV 
                               ⁢ 
                               2 
                             
                           
                         
                         = 
                         
                           m 
                           ⁡ 
                           ( 
                           
                             0 
                             ≤ 
                             m 
                             ≤ 
                             1 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         
                           DR 
                           
                             INV 
                             ⁢ 
                             2 
                           
                         
                         = 
                         
                           n 
                           ⁡ 
                           ( 
                           
                             0 
                             ≤ 
                             n 
                             ≤ 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Expressing The currents of the q-axis and d-axis of the motor in the rotating coordinate system at time k+2 as: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         i 
                         d 
                       
                       ( 
                       
                         k 
                         + 
                         2 
                       
                       ) 
                     
                     = 
                     
                       
                         
                           i 
                           d 
                         
                         ( 
                         
                           k 
                           + 
                           1 
                         
                         ) 
                       
                       + 
                       
                         
                           Ts 
                           
                             L 
                             s 
                           
                         
                         ⨯ 
                         
                           ( 
                           
                             
                               
                                 u 
                                 d 
                                 
                                   INV 
                                   ⁢ 
                                   1 
                                 
                               
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                             - 
                             
                               
                                 R 
                                 s 
                               
                               ⨯ 
                               
                                 
                                   i 
                                   d 
                                 
                                 ( 
                                 
                                   k 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             - 
                             
 
                             
                               
                                 EMF 
                                 d 
                               
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           Ts 
                           
                             L 
                             s 
                           
                         
                         ⨯ 
                         
                           
                             u 
                             d 
                             
                               INV 
                               ⁢ 
                               2 
                             
                           
                           ( 
                           
                             k 
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         i 
                         q 
                       
                       ⁢ 
                       
                         ( 
                         
                           k 
                           + 
                           2 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           i 
                           q 
                         
                         ⁢ 
                         
                           ( 
                           
                             k 
                             + 
                             1 
                           
                           ) 
                         
                       
                       + 
                       
                         
                           Ts 
                           
                             L 
                             s 
                           
                         
                         ⨯ 
                         
                           ( 
                           
                             
                               
                                 u 
                                 d 
                                 
                                   INV 
                                   ⁢ 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   k 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 
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     As shown in  FIG.  8   , the virtual voltage vector provided by inverter 1 can no longer meet the current state of the motor. Inverter 1 outputs the maximum voltage vector, and inverter 2 provide the remaining current increment, to achieve the expected given value. 
     Step 4.4) According to the two different working conditions in Step 4.3), the currents of the q-axis and d-axis in the motor rotating coordinate system at time k+2 is substituted into the cost function to obtain the expected voltage vector: 
       λ( i,j )=( i   d   *−i   d ( k+ 2)) 2 +( i   q   *−i   q ( k+ 2)) 2  
 
     Step 5.1) According to Step 4), the index number i of the optimal voltage vector has been found. The scale factor DR0 INV1  and DR INV1  for inverter 1, and the scale factor DR0 INV2  and DR INV2  for inverter 2. Outputting the duty cycle of each phase of the open-winding inverter by the virtual voltage vector table of the open winding five-phase permanent magnet synchronous motor, the detail is proposed in Step 2; 
     If the Δi d   INV1  and the Δi q   INV1  are greater than the required increment Δi d * and Δi q * at the current moment, express the equation as: 
       DutyRatio x   INV1 =DR0 INV1 ×(DR INV1   ×s   x   INV1 ( i )+(1−DR INV1 )× s   x   INV1 ( j ))
 
       DutyRatio x   INV2 =0 
     wherein DutyRatio x   INV1  is the value calculated by the proposed algorithm, used for inverter 1 to generate the corresponding duty cycle waveform; DutyRatio x   INV2  is used for inverter 2 to generate the corresponding duty cycle waveform; s x   INV1 (i) is the first selected by inverter 1, is the switching function of a virtual voltage vector, wherein x=a,b,c,d,e; s x   INV1 (j) is the switching function of the second virtual voltage vector selected by inverter 1, wherein x=a,b,c,d,e; 
     If the Δi d   INV1  and the Δi q   INV1  are less than the required increments Δi d * and Δi q * at this moment, express the equation as: 
       DutyRatio x   INV1   =s   x   INV1 ( i ) 
       DutyRatio x   INV2 =DR0 INV2 ×(DR INV2   ×s   x   INV2 ( i )+(1−DR INV2 )× s   x   INV2 ( j ))
 
     wherein DutyRatio x   INV1  is the value calculated by the proposed algorithm for inverter 1 to generate the corresponding duty cycle waveform; DutyRatio x   INV2  is used for inverter 2 to generate the corresponding duty cycle waveform; s x   INV1 (i) is the first selected by inverter 1, is the switching function of a virtual voltage vector, wherein x=a,b,c,d,e; s x   INV1 (i) is the first selected by inverter 1, is the switching function of a virtual voltage vector, wherein x-a,b,c,d,e; s x   INV2 (j) is the switching function of the second virtual voltage vector selected by inverter 2, wherein x=a,b,c,d,e; 
     As shown in  FIG.  10   , the switching sequence on the left is the switching sequence of inverter 1, and the switching sequence on the right is the switching sequence of inverter 2. In the current working condition, inverter 1 outputs the maximum voltage vector and only switches twice in each control cycle, and inverter 2 normally outputs the switching waveform so that the overall switching loss of the drive system is reduced. 
     Step 5.2) Generating the pulse width modulation signal to the diver chip of the inverter by the obtained duty cycle of inverter 1 and the duty cycle of inverter 2 in Step 5.1). Finally, the corresponding voltage is output to the motor through the power semiconductor. 
       FIG.  11   - FIG.  13    show the simulated waveforms of phase A current, output torque, and speed when inverter 1 acts alone. 
       FIG.  14   - FIG.  16    show the simulated waveforms of phase A current, output torque, and speed when inverters 1 and 2 are superimposed. 
       FIGS.  17 A- 17 B  and  FIGS.  18 A- 18 B  respectively show the change of the A-phase duty cycle of inverter 1 when inverter 1 acts alone and when inverter 1 and inverter 2 are superimposed on each other. 
       FIGS.  19 A- 19 B  show the change of the A-phase duty ratio of inverter 1 when the open-winding five-phase permanent magnet synchronous motor is switched from inverter 1 alone to inverter 1 and inverter 2 acts superimposed. 
     The present invention for open-winding five-phase permanent magnet synchronous motors proposed by the present invention can reduce system losses, maximize the flexibility of the open-winding drive system from the above simulation waveforms. The vector traversal of 144 times is reduced to 24 times, reducing the amount of calculation. 
     The above embodiments are only used to illustrate the design ideas and features of the present invention. The purpose is to enable those skilled in the art to understand the content of the present invention and implement them accordingly. The protection scope of the present invention is not limited to the above embodiments. Therefore, all equivalent changes or modifications made according to the principles and design ideas disclosed in the present invention fall within the protection scope. 
     Although the embodiments of the present invention have been shown and described, those of ordinary skill in the art can understand that various changes, modifications, substitutions, and modifications can be made to these embodiments without departing from the principle and purpose the present invention. The claims and their equivalents define the scope of the present invention.