Abstract:
A motor control system comprising a motor configured to operate at a rotational velocity and a control module in communication with the motor is provided. The control module is configured to receive a torque command indicating a desired amount of torque to be generated by the motor, obtain a rotational velocity of the motor, receive a desired phase advance angle for driving the motor; and generate a voltage command indicating a voltage magnitude to be applied to the motor based on the rotational velocity of the motor, the motor torque command, and the desired phase advance angle by using a plurality of dynamic inverse motor model equations that (i) allow the desired phase advance angle to exceed an impedance angle of the motor and (ii) specify that the voltage magnitude is a function of a voltage magnitude of a previous voltage command.

Description:
CROSS-REFERENCES TO RELATED APPLICATIONS 
       [0001]    This patent application claims priority to U.S. Provisional Patent Application Ser. No. 61/769,262, filed Feb. 26, 2013, which is incorporated herein by reference in its entirety. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    Electrical power steering (EPS) systems in vehicles use an electric motor connected to the steering gear or steering column that is electronically controlled to provide a torque to assist a driver in steering the vehicle. EPS systems typically include an electric motor and a controller. The controller receives steering torque information from a torque sensor and controls the motor to transmit assist torque to the wheels, e.g., by applying the torque to the steering column. One type of an electric motor is a Permanent Magnet (PM) brushless motor. 
         [0003]    Sinusoidal Brushless Motor Control is a technique used to control brushless motors in EPS systems. Some such techniques utilize a feedforward motor voltage command/control utilizing a steady state representation of the motor characteristics. 
       SUMMARY OF THE INVENTION 
       [0004]    In an embodiment of the invention, a motor control system comprises a motor configured to operate at a rotational velocity and a control module in communication with the motor is provided. The control module is configured to receive a torque command indicating a desired amount of torque to be generated by the motor, obtain a rotational velocity of the motor, receive a desired phase advance angle for driving the motor; and generate a voltage command indicating a voltage magnitude to be applied to the motor based on the rotational velocity of the motor, the motor torque command, and the desired phase advance angle by using a plurality of dynamic inverse motor model equations that (i) allow the desired phase advance angle to exceed an impedance angle of the motor and (ii) specify that the voltage magnitude is a function of a voltage magnitude of a previous voltage command. 
         [0005]    In another embodiment of the invention, a method for controlling a motor of an electronic power steering (EPS) system comprises receiving a torque command indicating a desired amount of torque to be generated by the motor. The method obtains a rotational velocity of the motor. The method receives a desired phase advance angle for driving the motor. The method generates a voltage command indicating a voltage magnitude to be applied to the motor based on the rotational velocity of the motor, the motor torque command, and the desired phase advance angle by using a plurality of dynamic inverse motor model equations. The inverse motor model equations allow the desired phase advance angle to exceed an impedance angle of the motor and specify that the voltage magnitude is a function of a voltage magnitude of a previous voltage command. 
         [0006]    These and other advantages and features will become more apparent from the following description taken in conjunction with the drawings. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0007]    The subject matter which is regarded as the invention is particularly pointed out and distinctly claimed in the claims at the conclusion of the specification. The foregoing and other features, and advantages of the invention are apparent from the following detailed description taken in conjunction with the accompanying drawings in which: 
           [0008]      FIG. 1  is an exemplary schematic illustration of a motor control system in accordance with exemplary embodiments; 
           [0009]      FIG. 2  is a phasor diagram of a motor in accordance with exemplary embodiments; 
           [0010]      FIG. 3  is a diagram illustrating the four quadrants of operation for a motor in accordance with exemplary embodiments; 
           [0011]      FIG. 4   a  is a block diagram that illustrates calculation of desired d-axis current in accordance with exemplary embodiments; 
           [0012]      FIG. 4   b  illustrates lookup tables for finding desired d-axis current in accordance with exemplary embodiments; 
           [0013]      FIG. 5   a  is a block diagram that illustrates calculation of a voltage command in accordance with exemplary embodiments; 
           [0014]      FIG. 5   b  is a block diagram that illustrates calculation of a desired amount of direct axis current in accordance with exemplary embodiments; 
           [0015]      FIG. 6   a  illustrates an implementation of a derivative filter in accordance with exemplary embodiments; 
           [0016]      FIG. 6   b  illustrates an implementation of a derivative filter in accordance with exemplary embodiments; 
           [0017]      FIG. 7  illustrates an implementation of a low pass filter in accordance with exemplary embodiments; 
           [0018]      FIG. 8  is a block diagram that illustrates calculation of a voltage command in accordance with exemplary embodiments; 
           [0019]      FIG. 9  is a block diagram that illustrates calculation of a final voltage command in accordance with exemplary embodiments; and 
           [0020]      FIG. 10  is flow diagram illustrating a motor control method in accordance with exemplary embodiments. 
       
    
    
     DETAILED DESCRIPTION 
       [0021]    Embodiments of the invention provide a controller for controlling a motor of an electric power steering (EPS) system by supplying a voltage command at a phase advance angle up to 90 degrees and beyond 90 degrees (i.e., the phase advance angle above the impedance angle of the motor). The controller uses a motor model that includes equations for calculating a voltage command based on inputs that include a motor velocity, a torque command, and a phase advance angle. The controller receives the inputs, calculates voltage commands specifying required voltages according to the motor model, and sends the voltage commands to the electric motor to control the torque generated by the motor. In one embodiment, the motor model allows for calculating the voltage commands even when the torque generated from the motor voltage is opposite to the rotational direction of the motor (i.e., when the motor operates in quadrant II and IV) and the phase advance angle is up to or greater than 90 degrees. In one embodiment, the motor model allows for such calculation by limiting motor regenerative current in quadrants II and IV. 
         [0022]    Referring now to the Figures, where the invention will be described with reference to specific embodiments, without limiting same,  FIG. 1  illustrates a motor control system  10  in accordance with one aspect of the invention. In the exemplary embodiment as shown, the motor control system  10  includes a motor  20 , an inverter  22 , a supply voltage  24 , and a control module  30  (also referred to as a controller). The voltage supply  24  supplies a supply voltage V B  to the motor  20 . In one embodiment, the voltage supply  24  is a 12 volt battery. However, it is to be understood that other types of voltage supplies may be used as well. The inverter  22  is connected to the motor  20  by a set of three connections  32  that are labeled as ‘A’, ‘B’ and ‘C’. In one embodiment, the motor  20  is a polyphase, permanent magnet (PM) brushless motor. In this example, the motor  20  is a three-phase PM motor. The control module  30  is connected to the motor  20  through the inverter  22 . The control module  30  receives a motor torque command T CMD  from a source  34  such as, for example, a steering control system. The control module  30  includes control logic for sending a motor voltage command V LL  to the motor  20  through the inverter  22 . 
         [0023]    Referring now to  FIGS. 1 and 2 , the motor  20  is operated such that a phase of the motor voltage command V LL  shifts with respect to a phase of a developed back electromotive force (BEMF) voltage E g  of the motor  20 . A phasor diagram of the motor  20  is shown in  FIG. 2  and illustrates a voltage vector V having a magnitude that is the motor voltage command V LL . A BEMF voltage vector E has a magnitude that is the BEMF voltage E g . An angle between voltage vector V and the BEMF voltage vector E is defined and is referred to as a phase advance angle  8 . A stator phase current is referred to as I, a stator phase current in the quadrature axis (q-axis) is referred to as I q , a stator phase current in the direct axis (d-axis) is referred to as I d , a stator phase reactance in the respective d-axis is referred to as X d , the stator phase reactance in the q-axis is referred to as X q , and a stator phase resistance at phase A is referred to as R a . 
         [0024]    In one embodiment, an encoder  36  (shown in  FIG. 1 ) is used to measure an angular position θ of a rotor (not shown in  FIG. 1 ) of the motor  20 . The angular position θ of the motor  20  is used to determine the input phase voltages V a , V b  and V c , where input phase voltage V a  corresponds with connection A, input phase voltage V b  corresponds with connection B, and input phase voltage V c  corresponds with connection C. The control module  30  includes control logic for calculating input phase voltages V a , V b , and V c  by the following equations: 
         [0000]        V   a   =V  sin(δ+θ)  Equation 1
 
         [0000]        V   b   =V  sin(δ+θ+120°)  Equation 2
 
         [0000]        V   c   =V  sin(δ+θ+240°)  Equation 3
 
         [0025]    The motor  20  rotates in a clockwise as well as a counterclockwise direction, and may also produce torque in both the clockwise and counterclockwise direction during operation. Therefore, the motor  20  is capable of operating in all four quadrants of operation, which is illustrated in  FIG. 3 .  FIG. 3  is an exemplary diagram illustrating the four quadrants of operation for the motor  20 , where quadrant I includes positive velocity and positive torque, quadrant II includes negative velocity and positive torque, quadrant III includes negative velocity and negative torque, and quadrant IV includes positive velocity and negative torque. In the event that the motor  20  is operating in either quadrant II or quadrant IV, the motor  20  may create a regenerative current that is sent back into the DC power supply  24  (shown in  FIG. 1 ). 
         [0026]    The control module  30  includes control logic for monitoring the motor  20  for a rotational velocity. Specifically, the control module  30  may be in communication with a speed measuring device (not shown in  FIG. 1 ) that provides an output indicating an angular velocity ω m  of the motor  20 . Alternatively, the angular velocity ω m  of the motor  20  may be calculated by differentiating the angular position θ, where dθ/dt=ω m . The angular velocity ω m  may also be referred to as the mechanical velocity of the motor  20 , and is measured in radians/second. The control module  30  also includes control logic for also calculating an electrical velocity ω e  of the motor  20 , where the electrical velocity is calculated by multiplying the mechanical velocity ω m  by a number of poles N p  of the motor  20 , and dividing the product of the mechanical velocity ω m  and the number of poles N p  by two. 
         [0027]    In one embodiment, a memory (not shown) of the control module  30  stores several motor circuit parameters. Specifically, in one embodiment, the motor circuit parameters include a motor voltage constant K e  that is measured in volts/radian/second, a motor and control module output circuit resistance R that is measured in Ohms, and motor inductances L q  and L d  that are measured in Henries. In another embodiment, the control module  30  may include control logic for calculating the motor circuit parameters including motor voltage constant K e , the motor and control module output circuit resistance R, and the motor inductances L q  and L d . In such an embodiment, the control logic may adjust the calculated motor output circuit resistance R and the calculated motor voltage constant K e  based on the temperature of the motor. The control logic may also adjust the calculated motor voltage constant K e  and the calculated motor inductances L q  and L d  with respect to the motor current in order to comprehend the saturation effects. The control module  30  also includes control logic for monitoring the supply voltage V B  to the motor  20 . 
         [0028]    In an embodiment, the control module  30  is configured to generate a voltage command using a motor control model. An example of a motor control model for a sinusoidal permanent magnet (PM) motor includes the following equations: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           V 
                            
                           
                               
                           
                            
                           
                             cos 
                              
                             
                               ( 
                               δ 
                               ) 
                             
                           
                         
                         = 
                           
                          
                         Vq 
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       L 
                                       q 
                                     
                                     R 
                                   
                                    
                                   s 
                                 
                                 + 
                                 1 
                               
                               ) 
                             
                              
                             
                               RI 
                               q 
                             
                           
                           + 
                           
                             
                               K 
                               e 
                             
                              
                             
                               ω 
                               m 
                             
                           
                           + 
                           
                             
                               L 
                               d 
                             
                              
                             
                               ω 
                               e 
                             
                              
                             
                               I 
                               d 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   4 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             - 
                             V 
                           
                            
                           
                               
                           
                            
                           
                             sin 
                              
                             
                               ( 
                               δ 
                               ) 
                             
                           
                         
                         = 
                           
                          
                         Vd 
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       L 
                                       q 
                                     
                                     R 
                                   
                                    
                                   s 
                                 
                                 + 
                                 1 
                               
                               ) 
                             
                              
                             
                               RI 
                               q 
                             
                           
                           - 
                           
                             
                               L 
                               q 
                             
                              
                             
                               ω 
                               e 
                             
                              
                             
                               I 
                               q 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   5 
                 
               
             
             
               
                 
                   
                     T 
                     e 
                   
                   = 
                   
                     
                       K 
                       e 
                     
                      
                     
                       I 
                       q 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   6 
                 
               
             
           
         
       
     
         [0000]    where:
 
V is the magnitude of the voltage applied to the motor, i.e., the motor voltage V LL ;
 
V q  is the q-axis vector component of motor voltage in phase with the motor BEMF;
 
V d  is the d-axis vector component of motor voltage 90 degrees out of phase with the motor BEMF;
 
δ is the angle of the applied voltage relative to the BEMF in radians (the phase advance angle);
 
L q  and L d  are the stator q-axis and d-axis inductances, respectively (Henries);
 
R is the motor circuit resistance, including the motor stator and controller hardware (Ohms);
 
K e  is the motor voltage constant (Voltage/Radian/second);
 
ω m  is the rotor mechanical velocity (Radian/second);
 
ω e  is the rotor electrical velocity (Radian/second);
 
I d  is the direct (d) axis current (Amperes);
 
I q  is the quadrature (q) axis current (Amperes);
 
T e  is the electromagnetic torque (Newton meter); and
 
s is the Laplace operator.
 
         [0029]    Setting the desired motor torque command T CMD  equal to T e  in the above equations 4-6 and solving for the voltage and phase advance angle required to deliver the desired torque yields the following: 
         [0000]    
       
         
           
             
               
                 
                   
                     V 
                     q 
                   
                   = 
                   
                     
                       
                         T 
                         CMD 
                       
                        
                       
                         R 
                         
                           K 
                           e 
                         
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 L 
                                 q 
                               
                               R 
                             
                              
                             s 
                           
                           + 
                           1 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         K 
                         e 
                       
                        
                       
                         ω 
                         m 
                       
                     
                     + 
                     
                       
                         
                           
                             ω 
                             e 
                           
                            
                           
                             L 
                             d 
                           
                         
                         R 
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 
                                   T 
                                   CMD 
                                 
                                 
                                   K 
                                   e 
                                 
                               
                                
                               
                                 ω 
                                 e 
                               
                                
                               
                                 L 
                                 q 
                               
                             
                             - 
                             
                               V 
                                
                               
                                   
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   δ 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   L 
                                   d 
                                 
                                 R 
                               
                                
                               s 
                             
                             + 
                             1 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   7 
                 
               
             
             
               
                 
                   
                     V 
                     d 
                   
                   = 
                   
                     
                       
                         R 
                          
                         
                           ( 
                           
                             
                               
                                 
                                   L 
                                   d 
                                 
                                 R 
                               
                                
                               s 
                             
                             + 
                             1 
                           
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           
                             
                               V 
                                
                               
                                   
                               
                                
                               
                                 cos 
                                  
                                 
                                   ( 
                                   δ 
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 T 
                                 CMD 
                               
                                
                               
                                 R 
                                 
                                   K 
                                   e 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         L 
                                         q 
                                       
                                       R 
                                     
                                      
                                     s 
                                   
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 K 
                                 e 
                               
                                
                               
                                 ω 
                                 m 
                               
                             
                           
                           
                             
                               ω 
                               e 
                             
                              
                             
                               L 
                               d 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         
                           T 
                           CMD 
                         
                         
                           K 
                           e 
                         
                       
                        
                       
                         ω 
                         e 
                       
                        
                       
                         L 
                         q 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   8 
                 
               
             
           
         
       
     
         [0030]    In one embodiment, the control module  30  is configured to use the equations 7 and 8 to solve for the final motor voltage magnitude V (i.e., V LL  in  FIG. 1 ) for all four quadrants. For the quadrants I and III, the control module  30  uses the equation 7 to solve for V q  and to divide V q  by cosine of the phase advance angle as shown in the following equation 9: 
         [0000]    
       
         
           
             
               
                 
                   V 
                   = 
                   
                     
                       V 
                       q 
                     
                     
                       cos 
                        
                       
                         ( 
                         δ 
                         ) 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   9 
                 
               
             
           
         
       
     
         [0000]    For the quadrants II and IV (e.g., when the phase advance angle is over 90 degrees), the control module  30  uses the equations 7 and 8 to solve for V q  and V d  and uses the V q  and V d  in the following equation 10: 
         [0000]        V= Sign( V   q )√{square root over ( V   q   2   +V   d   2 )}  Equation 10
 
         [0000]    where Sign( ) is a function that outputs the sign (e.g., positive or negative) of a value. It is to be noted that the control module  30  may use the equation 10 to compute the final motor voltage magnitude V for all four quadrants I-IV. 
         [0031]    In an embodiment, the control module  30  uses a simplified equation to calculate the final voltage magnitude. Specifically, the equation 8 for calculating V d  may be simplified by using a directly-commanded d-axis current I d . That is, using a directly commanded value as I d  instead of using I d  as a variable in the equation 8 allows for avoiding the implementation of the double derivative operation (i.e., 
         [0000]    
       
         
           
             
               ( 
               
                 
                   
                     
                       L 
                       q 
                     
                     R 
                   
                    
                   s 
                 
                 + 
                 1 
               
               ) 
             
             × 
             
               ( 
               
                 
                   
                     
                       L 
                       q 
                     
                     R 
                   
                    
                   s 
                 
                 + 
                 1 
               
               ) 
             
           
         
       
     
         [0000]    for 
         [0000]    
       
         
           
             
               
                 T 
                 CMD 
               
                
               
                 R 
                 
                   K 
                   e 
                 
               
             
             ) 
           
         
       
     
         [0000]    of the equation 8. Calculating I d  in order to directly command I d  is described further below by reference to  FIGS. 4   a  and  4   b.    
         [0032]    In an embodiment, the control module  30  implements the following equation 11 to avoid a double derivative operation: 
         [0000]    
       
         
           
             
               
                 
                   
                     V 
                     d 
                   
                   = 
                   
                     
                       
                         I 
                         d_des 
                       
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         R 
                          
                         
                           ( 
                           
                             
                               
                                 
                                   L 
                                   d 
                                 
                                 R 
                               
                                
                               s 
                             
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                     - 
                     
                       
                         
                           T 
                           CMD 
                         
                         
                           K 
                           e 
                         
                       
                        
                       
                         ω 
                         e 
                       
                        
                       
                         L 
                         q 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   11 
                 
               
             
           
         
       
     
         [0000]    The control module  30  may be configured to use the equations 7 and 11 to solve for the final motor voltage magnitude V for all four quadrants. For quadrants I and III, the control module  30  uses the equation 7 to solve for V q  and uses the V q  value in the equation 9 to solve for V. For quadrants II and IV, the control module  30  uses the equations 7 and 11 to solve for V, and V d , and use the V, and V d  values in the equation 10 to solve for V. 
         [0033]    The equation 11 includes an I d     —     des  value that represents a desired amount of I d  current. I d     —     des  may be a signal from a phase control sub-function representing the desired d-axis current I d . This desired amount of I d  current, in one embodiment, is calculated as an input to a regenerative current limiting function. An example of such a function is described in U.S. Patent Application Publication No. 2013/0154524, entitled “Motor Control System for Limiting Regenerative Current,” filed on Dec. 15, 2011, the entire contents of which are incorporated herein by reference. A regenerative current limiting function is also described further below after the description of  FIG. 11 . 
         [0034]    An exemplary motor control system as described in the above-incorporated U.S. Patent Application Publication No. 2013/0154524 provides techniques for limiting negative supply current, or regenerative current, that is produced by an electric motor when operating in either quadrant II or quadrant IV. The system calculates a value of a target field weakening current I dTARGET , which is used as an input to calculate a motor voltage. The value I d     —     des  may be used by the system as the I dTARGET  current. 
         [0035]    In one embodiment, the equations 7, 8 and 11 for V q  and V d  above include derivative terms (e.g., 
         [0000]    
       
         
           
             
               
                 
                   
                     T 
                     CMD 
                   
                    
                   
                     R 
                     
                       K 
                       e 
                     
                   
                    
                   
                     ( 
                     
                       
                         
                           
                             L 
                             q 
                           
                           R 
                         
                          
                         s 
                       
                       + 
                       1 
                     
                     ) 
                   
                 
                  
                 
                   
 
                 
                  
                 and 
                  
                 
                   
 
                 
                  
                 
                   
                     I 
                     d_des 
                   
                    
                   
                     
                       3 
                     
                     2 
                   
                    
                   
                     R 
                      
                     
                       ( 
                       
                         
                           
                             
                               L 
                               d 
                             
                             R 
                           
                            
                           s 
                         
                         + 
                         1 
                       
                       ) 
                     
                   
                 
               
               ) 
             
             . 
           
         
       
     
         [0000]    Discretizing a derivative can produce noise due to sampling and resolution effects at high frequencies. To address the noise, the controller  30  of one embodiment includes a digital filter (not shown in  FIG. 1 ) to implement the derivative function utilizing a Fourier Series representation of a derivative with a Hamming window applied. 
         [0036]      FIG. 4   a  is a block diagram that illustrates an implementation of desired d-axis current I d     —     des  calculation module  400 . This d-axis current is calculated for directly commanding the I d . This directly commanded I d     —     des  is used for V d  calculations (e.g., in the equation 11) for quadrants II and IV only. In one embodiment, I d     —     des    405  is a function of the motor velocity ω m    410  and the absolute value of the torque command T CMD    415  as shown. In one embodiment, a sign block  425  identifies the sign (e.g., positive or negative) of the torque command T CMD    415 , and the multiplier  430  multiplies the motor velocity ω m    410  by the sign value (e.g., −1 or +1) of the torque command T CMD    415 . In one embodiment, the resulting product is used to look up values from a look up table. The sign of this resulting product indicates the quadrant in which the motor is operating. For instance, when the sign of the resulting product is negative because the sign of the torque command is opposite to the sign of the motor velocity, the motor is operating in quadrants II or IV. When the sign of the resulting product is positive because the sign of the torque command is the same as the sign of the motor velocity, the motor is operating in quadrants I or III. The absolute value block  435  takes the amplitude of the torque command T CMD    415  and the amplitude is used to select a look up table. 
         [0037]    In one embodiment, I d     —     des  can be determined from a set of calibratable, interpolated, fixed x, variable y lookup tables depicted as curves in a graph  420 , each defined at a specific torque command. In one embodiment, the x-axis of the graph  420  represents the motor velocity multiplied by the sign of the torque command and the y-axis of the graph represents the desired current I d     —     des .  FIG. 4   b  illustrates exemplary lookup tables depicted as curves  440 - 470  in a graph  475 . The x-axis of the graph  475  represents the motor velocity multiplied by the sign of the torque command in revolutions per minute (RPM). The y-axis represents the desired current I d     —     des  in amperes. In one embodiment, when the amplitude of the torque command does not exactly match any of the torque command amplitudes for the different curves, an interpolation technique is employed to find the desired current I d     —     des  value. In one embodiment, when the motor velocity multiplied by the sign of the torque command is positive (i.e., when the motor is operating in quadrants I or III), I d     —     des  is set to zero. The corresponding portion of the graph is not depicted in  FIG. 4   b.    
         [0038]      FIGS. 5-10  illustrate exemplary implementations of the equations 7, 8, and 11 for calculating voltage commands. Specifically,  FIGS. 5   a  and  5   b  illustrate an exemplary approach for calculating V q  (i.e., using the equation 7) and other terms of the equations 8 and 11. Implementations of static versions of the equations 7, 8, and 11 (i.e., without the derivative term s) are described in the above-incorporated U.S. Patent Application Publication No. 2013/0154524. An exemplary implementation of the equation 7 is also described in U.S. Pat. No. 7,157,878, the entire content of which is incorporated herein by reference. 
         [0039]      FIG. 5   a  is a block diagram that illustrates an implementation of a voltage command calculation module  500 . Specifically, this module  500  calculates V q    595  according to the equation 8 in one embodiment. As shown, the module  500  takes as inputs the torque command  415 , the motor velocity ω m    410 , and a phase advance angle δ  599 , and outputs V q    595 . 
         [0040]    An R/K e  block  515 , an L q /R block  520 , a derivative filter  525 , and an adder  527  together implement the first term 
         [0000]    
       
         
           
             
               T 
               CMD 
             
              
             
               R 
               
                 K 
                 e 
               
             
              
             
               ( 
               
                 
                   
                     
                       L 
                       q 
                     
                     R 
                   
                    
                   s 
                 
                 + 
                 1 
               
               ) 
             
           
         
       
     
         [0000]    of the equation 7. Exemplary implementations of the derivative filter  525  are described further below by reference to  FIGS. 6   a  and  6   b . A K e  block  530  implements the second term K e ω m  of the equation 7. A L q /R block  535 , a pole number block  540 , an L q /R block  550 , a low pass filter (LPF)  555 , a sine block  560 , multipliers  565 - 580 , and an adder  585  together implement the third term 
         [0000]    
       
         
           
             
               
                 
                   ω 
                   e 
                 
                  
                 
                   L 
                   q 
                 
               
               R 
             
              
             
               ( 
               
                 
                   
                     
                       
                         T 
                         CMD 
                       
                       
                         K 
                         e 
                       
                     
                      
                     
                       ω 
                       e 
                     
                      
                     
                       L 
                       q 
                     
                   
                   - 
                   
                     V 
                      
                     
                         
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         δ 
                         ) 
                       
                     
                   
                 
                 
                   
                     
                       
                         L 
                         d 
                       
                       R 
                     
                      
                     s 
                   
                   + 
                   1 
                 
               
               ) 
             
           
         
       
     
         [0000]    of the equation 7. An exemplary implementation of the LPF  555  is described further below by reference to  FIG. 7 . In one embodiment, the LPF  555  may be bypassed. 
         [0041]    The phase advance angle δ  599  that the sine block  560  takes as an input may be calculated by an I d     —     des  calculation block, which will be described further below by reference to  FIG. 5   b . The motor electrical velocity ω e    542  is related to the motor mechanical velocity ω m    410  and the number of motor poles by the equation ω e =(number of motor poles N p /2)×ω m . The first, second, and third terms of the equation 7 is summed by an adder  590  to output V q    595  of the equation 7. 
         [0042]      FIG. 5   b  is a block diagram that illustrates calculation of I d     —     des  Specifically,  FIG. 5   b  illustrates an I d     —     des  calculation block  513 . The I d     —     des  calculation block  513  is similar to the I d     —     des  calculation module  400  for quadrants II and IV described above by reference to  FIG. 4   a . The I d     —     des  calculation block  513  takes as inputs the torque command  415 , the motor velocity ω m    410 , a supply voltage  503 , and a set of motor circuit parameters  508 . The I d     —     des  calculation block  513  calculates the I d     —     des    405  and the phase advance angle δ  599 . In one embodiment, the set of motor circuit parameters  508  includes an estimation of the motor circuit resistance R, an estimation of the q-axis stator inductance L q , an estimation of the d-axis stator inductance L d , an estimation of the motor voltage constant K e , and a number of poles of the motor N p . 
         [0043]      FIG. 6   a  illustrates an exemplary non-recursive implementation of the derivative filter  525  of  FIG. 5  in various embodiments. The derivative filter  525  may be used for calculating derivative terms of the equations 7, 8, and 11. For different derivative terms from different equations, the values of the derivative input  605  would be different. Also, the gain values that the derivative filter gain block  610  uses would be different for calculating the different derivative terms. In one embodiment, the gain values are in a range of float 0-200. In one embodiment, the V q  filter coefficients include six constants, which are depicted in  FIG. 6  as C0, C1, C2, C4, C5, and C6. Different sets of these constants are used for calculating the different derivative terms in one embodiment. It is to be noted that, in this example, the derivative filter  525  is non-recursive—i.e., not reusing the output as an input. In one embodiment, the derivative filter  525  is a finite impulse response (FIR) filter. 
         [0044]      FIG. 6   b  illustrates another exemplary implementation of the derivative filter  525  of  FIG. 5  in various embodiments. Compared to the implementation illustrated in  FIG. 6   a , the implementation illustrated in  FIG. 6   b  is simplified by having a symmetric structure for the filter coefficients. For example, C6 is set to −C0, C5 is set to −C1, and C4 is set to −C2. Having such symmetric coefficients limits the number of multiplication operations (i.e., from six to three as indicated by the six multipliers  625 - 640  in  FIG. 6   a  and the three multipliers  655 - 665  in  FIG. 6   b ) and enforces the symmetry of the coefficients (i.e., prevents a set of six coefficients that are not symmetric from being loaded). 
         [0045]      FIG. 7  illustrates an exemplary implementation of the LPF  555  of  FIG. 5 . 
         [0046]    As shown, an output of the LFP  555  is calculated based on inputs  745  and  750 . In one embodiment, the input  745  is L d /R and the input  750  is 
         [0000]    
       
         
           
             
               
                 
                   T 
                   CMD 
                 
                 
                   K 
                   e 
                 
               
                
               
                 ω 
                 e 
               
                
               
                 L 
                 q 
               
             
             - 
             
               V 
                
               
                   
               
                
               
                 
                   sin 
                    
                   
                     ( 
                     δ 
                     ) 
                   
                 
                 . 
               
             
           
         
       
     
         [0000]    In one embodiment, the LPF  555  also includes a look up table  705  to find a cut off value for the input  750  based on the input  745 . In one embodiment, a ratio of one millisecond and the input  745  is used to find the cut off value from the look up table  705 . Alternatively, in one embodiment, the look up table  705  may be replaced with an equation 
         [0000]    
       
         
           
             
               1 
               - 
               
                 e 
                 
                   - 
                   
                     T 
                     τ 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where T is a sampling time period (e.g., one millisecond) and τ is the input  745 . The rest of this exemplary implementation of the LPF includes a multiplier  720 , adders  725  and  730 , and 1/Z blocks  735  and  740 . In one embodiment, the initial conditions for the 1/Z blocks  735  and  740  are set to zero. 
         [0047]      FIG. 8  is a block diagram that illustrates an implementation of a voltage command calculation module  800 . Specifically, this module  800  calculates V d    895  according to the equation 11 in one embodiment. As shown, the module  800  takes as inputs the torque command  415 , the motor velocity ω m    410 , and the directly commanded I d     —     des    405 . 
         [0048]    A square root(3)/2 block  805 , the L q /R block  535 , the derivative filter  525 , a resistance block  810 , multipliers  815 - 825 , and an adder  830  together implement the first term 
         [0000]    
       
         
           
             
               I 
               d_des 
             
              
             
               
                 3 
               
               2 
             
              
             
               R 
                
               
                 ( 
                 
                   
                     
                       
                         L 
                         d 
                       
                       R 
                     
                      
                     s 
                   
                   + 
                   1 
                 
                 ) 
               
             
           
         
       
     
         [0000]    of the equation 11. Exemplary implementations of the derivative filter  525  are described above by reference to  FIGS. 6   a  and  6   b . The pole number block  540 , the L q /R block  550 , the R/K e  block  515 , and an adder  840  implement the second term 
         [0000]    
       
         
           
             
               
                 T 
                 CMD 
               
               
                 K 
                 e 
               
             
              
             
               ω 
               e 
             
              
             
               L 
               q 
             
           
         
       
     
         [0000]    of the equation 11. An adder  835  adds the calculated first term of the equation 11 and the negative of the second term of the equation 11 to output V d    895 . 
         [0049]      FIG. 9  is a block diagram that illustrates an implementation of a final voltage command calculation module  900  that computes the final voltage command V mag    945 . The final voltage command calculation module  900  takes as inputs the phase advance angle  599 , V q    595 , and V d    895 . A cosine block  905  and the divider  910  compute the voltage magnitude according to the equation 9 based on the phase advance angle  599  and V q    595 . The hypotenuse block  915  and the sign block  920  compute the voltage magnitude according to the equation 10 based on V q    595  and V d    895 . The selector  935  selects one of the voltage magnitudes computed according to the equations 9 and 10 based on the quadrant in which the motor is operating. That is, the selector  935  selects the voltage magnitude computed according to the equation 9 when the motor is operating in quadrants I or III. Otherwise, the selector  935  selects the voltage magnitude computed according to the equation 10. In one embodiment, the selected voltage magnitude signal may be limited to a range of values by a saturator  940 . In one embodiment, the final voltage magnitude  945  is fed back into the equations 7 and 8 to generate the next final voltage magnitude. 
         [0050]    Referring now to  FIG. 10 , a flow diagram illustrates a motor control method that can be performed by the control module  30  of  FIG. 1  in one embodiment. As can be appreciated in light of the disclosure, the order of operation within the method is not limited to the sequential execution as illustrated in  FIG. 10 , but may be performed in one or more varying orders as applicable and in accordance with the present disclosure. 
         [0051]    In one example, the method at  1010  receives a torque command indicating a desired amount of torque to be generated by the motor. In one embodiment, the torque command originates from another module (not shown in  FIG. 1 ) that monitors the hand wheel movement (e.g., for the hand wheel angle and the hand wheel torque) and computes a desired amount of torque based on the hand wheel movement. 
         [0052]    At  1020 , the method obtains a rotational velocity of the motor. In one embodiment, the control module  30  receives an angular position θ of a rotor of the motor  20  of  FIG. 1  periodically and computes the rotational velocity based on the received angular positions. At  1030 , the method receives a desired phase advance angle for driving the motor. 
         [0053]    At  1040 , the method generates a voltage command indicating a voltage magnitude to be applied to the motor based on the rotational velocity of the motor, the motor torque command, and the desired phase advance angle by using the equations 7-11 that allow the desired phase advance angle to exceed ninety degrees. 
         [0054]    A regenerative current limiting function used for deriving the equation 11 will now be described. In various embodiments, when controlling a sinusoidally excited motor, the phase advance angle may be selected based on various design goals. For example, optimal phase advance equations are derived to minimize the peak motor current. To derive the optimal phase advance equations, the steady state motor equations are written, for example, in motor q-axis and d-axis coordinates as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     V 
                     q 
                   
                   = 
                   
                     
                       RI 
                       q 
                     
                     + 
                     
                       
                         ω 
                         e 
                       
                        
                       
                         L 
                         d 
                       
                        
                       
                         I 
                         d 
                       
                        
                       
                         
                           3 
                         
                         2 
                       
                     
                     + 
                     
                       
                         K 
                         e 
                       
                        
                       
                         ω 
                         m 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   12 
                 
               
             
             
               
                 
                   
                     V 
                     d 
                   
                   = 
                   
                     
                       
                         - 
                         
                           ω 
                           e 
                         
                       
                        
                       
                         L 
                         q 
                       
                        
                       
                         I 
                         q 
                       
                     
                     + 
                     
                       
                         RI 
                         d 
                       
                        
                       
                         
                           3 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   13 
                 
               
             
             
               
                 
                   
                     T 
                     e 
                   
                   = 
                   
                     
                       K 
                       e 
                     
                      
                     
                       I 
                       q 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   14 
                 
               
             
             
               
                 
                   
                     ω 
                     m 
                   
                   = 
                   
                     
                       2 
                       
                         N 
                         p 
                       
                     
                      
                     
                       ω 
                       e 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   15 
                 
               
             
           
         
       
     
         [0055]    The phase advance angle of the motor voltage command with respect to the motor BEMF waveform is represented by δ and may be calculated with the following equation 16: 
         [0000]    
       
         
           
             
               
                 
                   δ 
                   = 
                   
                     
                       Tan 
                       
                         - 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           - 
                           
                             V 
                             d 
                           
                         
                         
                           V 
                           q 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   16 
                 
               
             
           
         
       
     
         [0000]    It is to be noted that the d-axis vector is considered positive when the vector is pointing to the right side as shown in  FIG. 2 . 
         [0056]    In order to minimize the peak current, the d-axis current should be zero whenever possible. For motor voltages below the available supply voltage, the optimal phase advance may be computed by the following equation, which is derived by setting I d =0 in the above equations 12-15 and solving for δ using the equation 16. The result is referred to as optimal phase advance angle δ 2 . 
         [0000]    
       
         
           
             
               
                 
                   
                     δ 
                     2 
                   
                   = 
                   
                     
                       Tan 
                       
                         - 
                         1 
                       
                     
                     ( 
                     
                       
                         
                           ω 
                           e 
                         
                          
                         
                           L 
                           q 
                         
                          
                         
                           
                             T 
                             CMD_SCL 
                           
                           
                             K 
                             e 
                           
                         
                       
                       
                         
                           
                             K 
                             e 
                           
                            
                           
                             ω 
                             e 
                           
                            
                           
                             2 
                             
                               N 
                               p 
                             
                           
                         
                         + 
                         
                           TR 
                           
                             K 
                             e 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   17 
                 
               
             
           
         
       
     
         [0057]    In one embodiment, the magnitude of the value computed for δ 2  is limited by the maximum δ 2  phase advance (δ 2 MAX) equation 18 given below (save the sign of the computed δ 2  and reapply after limiting). This magnitude limiting should be performed because the noise on motor velocity near zero motor velocity could potentially cause the sign of the limit value to be opposite of the sign of δ 2 . 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       δ 
                       2 
                     
                      
                     MAX 
                   
                   = 
                   
                     
                       Tan 
                       
                         - 
                         1 
                       
                     
                      
                     
                       
                         
                            
                           
                             ω 
                             e 
                           
                            
                         
                          
                         
                           L 
                           q 
                         
                       
                       R 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   18 
                 
               
             
           
         
       
     
         [0058]    When the supply voltage limit is reached, the I d  current is allowed to be nonzero to continue to get the desired torque out of the motor (this is referred to as field weakening). Using phase advance with field weakening allows the torque vs. speed performance of a given motor/control module to be expanded. In order to derive the equation for the optimal phase advance δ 1  at the supply voltage limit, the equations 12-15 above may be solved again with the voltage set constant at the supply voltage (e.g., modulator input voltage, or DC Link Voltage). 
         [0000]    
       
         
           
             
               
                 
                   X 
                   = 
                   
                     
                       
                         
                           
                             T 
                             cmd_SCL 
                           
                           
                             K 
                             e 
                           
                         
                          
                         
                           ( 
                           
                             
                               R 
                               2 
                             
                             + 
                             
                               
                                 ω 
                                 e 
                                 2 
                               
                                
                               
                                 L 
                                 q 
                               
                                
                               
                                 L 
                                 d 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           2 
                            
                           
                             ω 
                             e 
                           
                            
                           
                             K 
                             e 
                           
                            
                           R 
                         
                         
                           N 
                           p 
                         
                       
                     
                     DCLinkLimit 
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   18 
                 
               
             
             
               
                 
                   
                     δ 
                     1 
                   
                   = 
                   
                     
                       Tan 
                       
                         - 
                         1 
                       
                     
                     ( 
                     
                       
                         
                           X 
                           2 
                         
                         - 
                         
                           R 
                           2 
                         
                       
                       
                         
                           R 
                            
                           
                               
                           
                            
                           
                             ω 
                             e 
                           
                            
                           
                             L 
                             d 
                           
                         
                         + 
                         
                           X 
                            
                           
                             
                               
                                 R 
                                 2 
                               
                               + 
                               
                                 
                                   ω 
                                   e 
                                   2 
                                 
                                  
                                 
                                   L 
                                   q 
                                 
                                  
                                 
                                   L 
                                   d 
                                 
                               
                               - 
                               
                                 X 
                                 2 
                               
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   19 
                 
               
             
           
         
       
     
         [0059]    The value computed for δ 1  may be limited by the maximum δ 1  phase advance (δ 1 MAX) equation given below: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       δ 
                       1 
                     
                      
                     MAX 
                   
                   = 
                   
                     
                       Tan 
                       
                         - 
                         1 
                       
                     
                      
                     
                       
                         
                           ω 
                           e 
                         
                          
                         
                           L 
                           q 
                         
                       
                       R 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   20 
                 
               
             
           
         
       
     
         [0060]    One or more of the following exceptions to the above calculations of phase advance angle may apply. The first exception is when operating in quadrant III, the minimum, or most negative value, instead of the maximum should be used for the equation 11. When the torque command is zero, the maximum should be used if the motor speed is positive and the minimum should be used if the motor speed is negative. The second exception is that after calculating δ, a limit is applied to the calculated value to ensure δ is within a legal range. 
         [0061]    Another embodiment of phase advance calculation is described when used for control of the supply regeneration current. Equation 22 described below for phase advance may be used for quadrants II and IV when the option to limit the amount of supply current regenerated to the vehicle supply is required. This equation allows the amount of supply current regenerated to be calibratable by setting a non-zero desired value of I d  current in quadrants II and IV, targeted to provide just enough supply current limiting to meet motor design requirements. An embodiment of this equation for phase advance to be used in quadrants II and IV is as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     δ 
                     3 
                   
                   = 
                   
                     
                       Tan 
                       
                         - 
                         1 
                       
                     
                     ( 
                     
                       
                         
                           
                             ω 
                             e 
                           
                            
                           
                             L 
                             q 
                           
                            
                           
                             
                               T 
                               CMD_SCL 
                             
                             
                               K 
                               e 
                             
                           
                         
                         - 
                         
                           
                             RI 
                             d_des 
                           
                            
                           
                             
                               3 
                             
                             2 
                           
                         
                       
                       
                         
                           
                             K 
                             e 
                           
                            
                           
                             ω 
                             e 
                           
                            
                           
                             2 
                             
                               N 
                               p 
                             
                           
                         
                         + 
                         
                           TR 
                           
                             K 
                             e 
                           
                         
                         + 
                         
                           
                             ω 
                             e 
                           
                            
                           
                             L 
                             d 
                           
                            
                           
                             I 
                             d_des 
                           
                            
                           
                             
                               3 
                             
                             2 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   22 
                 
               
             
           
         
       
     
         [0000]    The numerator of the input to the arc tangent in the equation 22 is a steady state version of the equation 11. 
         [0062]    In an embodiment, the pre-calculated terms from a voltage control sub-function may be used instead of the equation 22, as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     Term_D 
                     = 
                     
                       
                         ω 
                         e 
                       
                        
                       
                         L 
                         d 
                       
                        
                       
                         I 
                         
                           d 
                           des 
                         
                       
                        
                       
                         
                           3 
                         
                         2 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     Term_E 
                     = 
                     
                       
                         RI 
                         d_des 
                       
                        
                       
                         
                           3 
                         
                         2 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       V 
                       q 
                     
                     = 
                     
                       Term_A 
                       + 
                       
                         Term_B 
                         ss 
                       
                       + 
                       Term_D 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   23 
                 
               
             
             
               
                 
                   
                     V 
                     d 
                   
                   = 
                   
                     
                       
                         - 
                         
                           Term_B 
                           ss 
                         
                       
                       * 
                       
                         Term_X 
                         q 
                       
                     
                     + 
                     Term_E 
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   24 
                 
               
             
             
               
                 
                   
                     δ 
                     3 
                   
                   = 
                   
                     
                       Tan 
                       
                         - 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           - 
                           
                             V 
                             d 
                           
                         
                         
                           V 
                           q 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   25 
                 
               
             
           
         
       
     
         [0063]    While the invention has been described in detail in connection with only a limited number of embodiments, it should be readily understood that the invention is not limited to such disclosed embodiments. Rather, the invention can be modified to incorporate any number of variations, alterations, substitutions or equivalent arrangements not heretofore described, but which are commensurate with the spirit and scope of the invention. Additionally, while various embodiments of the invention have been described, it is to be understood that aspects of the invention may include only some of the described embodiments. Accordingly, the invention is not to be seen as limited by the foregoing description.