Abstract:
A closed-loop system may include a plant (an electric machine requiring control) and a fractional-order proportional-resonant controller. The fractional-order proportional-resonant controller may have an order greater than zero and less than or equal to one. The order for the fractional-order proportional-resonant controller may be selected to yield a target amplitude and target slope for frequency response. The frequency response may be such that a steady-state error associated with a speed of the electric machine is inversely proportional to the target amplitude and less than a predetermined threshold. The order of the controller may be 0.9.

Description:
TECHNICAL FIELD 
       [0001]    This disclosure relates to fractional-order proportional-resonant controllers for sinusoidal references. 
       BACKGROUND 
       [0002]    Various types of electric vehicles or power systems draw power from a battery bank or direct current source (e.g., photovoltaic cells, fuel cells, capacitors). Direct current from the battery is fed into an inverter to generate alternating current, which is received by an electric machine or power grid. An error may develop between the demand (reference) and output, commonly referred to as steady-state error. A controller may be used to ensure the output tracks the reference such that the steady-state error is as close to zero as possible. Numerous strategies have been implemented to reduce steady-state error, but generally have drawbacks. The drawbacks may be related to the amount of processing required to control the signal or limited degrees of freedom to properly tune the control system transfer functions. 
       SUMMARY 
       [0003]    A closed-loop system may include a plant (e.g., an electric machine requiring control) and a fractional-order proportional-resonant controller. The fractional-order proportional-resonant controller may have an order greater than zero and less than or equal to one. The order for the fractional-order proportional-resonant controller may be selected to yield a target amplitude and target slope for frequency response. The frequency response may be such that a steady-state error associated with a speed of the electric machine is inversely proportional to the target amplitude and less than a predetermined threshold. The order of the controller may be 0.9. The bandwidth about the natural frequency or resonant frequency of the controller may be one radian per second. The natural frequency of the controller may be 120π Hz. The proportional gain of the controller may be four. The integral gain of the controller may be 300. The controller may have a gain of at least 50 dB at 60 Hz. 
         [0004]    A vehicle may include an inverter for an electric machine and a fractional-order proportional-resonant controller. The fractional-order proportional-resonant controller may have at least three degrees of freedom and an order, greater than zero and less than or equal to one, selected to yield a target amplitude and target slope for frequency response such that a steady-state error associated with an output signal of the electric machine is inversely proportional to the target amplitude and less than a predetermined threshold. The order of the controller may be 0.9. The bandwidth about the natural frequency may be one. The natural frequency of the system may be 120n Hz. The proportional gain may be four. The integral gain may be 300. The controller may have a gain of at least 50 dB at 60 Hz. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0005]      FIG. 1  depicts an electric vehicle or plant. 
           [0006]      FIG. 2  depicts a FO-PR controller for an electric vehicle or three-phase inverter. 
           [0007]      FIGS. 3A and 3B  depict the system response at a 120π natural frequency. 
       
    
    
     DETAILED DESCRIPTION 
       [0008]    Embodiments of the present disclosure are described herein. It is to be understood, however, that the disclosed embodiments are merely examples and other embodiments may take various and alternative forms. The figures are not necessarily to scale; some features could be exaggerated or minimized to show details of particular components. Therefore, specific structural and functional details disclosed herein are not to be interpreted as limiting, but merely as a representative basis for teaching one skilled in the art to variously employ the present invention. As those of ordinary skill in the art will understand, various features illustrated and described with reference to any one of the figures may be combined with features illustrated in one or more other figures to produce embodiments that are not explicitly illustrated or described. The combinations of features illustrated provide representative embodiments for typical applications. Various combinations and modifications of the features consistent with the teachings of this disclosure, however, could be desired for particular applications or implementations. 
         [0009]    Controllers are used to achieve an objective output from a particular plant or system. The controller may be implemented as an interconnected system of parts having assigned functions. Some control systems are used to minimize steady state error and ensure proper system response. Typically, these systems employ feedback loops to create a closed-loop system, which reduces steady state error. Resonant systems provide particular challenges to controllers because of their continuously oscillating output. 
         [0010]    Closed-loop proportional integral (PI) controllers may be implemented to reduce steady state error of resonant systems. A closed-loop PI controller includes a proportional gain and integral gain. Typically, proportional integral controllers have a transfer function similar to Equation 1 below. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       G 
                       PI 
                     
                      
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       K 
                       P 
                     
                     + 
                     
                       
                         K 
                         I 
                       
                       s 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where K P  is the proportional gain and K I  is the integral gain. The PI controller can be used to controller resonant systems by transforming the resonant components of the input signal using Park&#39;s transformation or direct-quadrature-zero transformation. PI controllers also require additional feedfoward loops to control resonant systems. 
         [0011]    Closed-loop proportional-resonant (PR) controllers may be implemented to reduce steady state error of resonant systems. PR controllers also include proportional gain and integral gain. Typically, second-order PR controllers have a transfer function similar to Equation 2 below. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       G 
                       PR 
                     
                      
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       K 
                       P 
                     
                     + 
                     
                       
                         
                           K 
                           I 
                         
                          
                         s 
                       
                       
                         
                           s 
                           2 
                         
                         + 
                         
                           ω 
                           n 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where K P  is the proportional gain, K I  is the integral gain, and ω n  is the natural frequency. The PR controller above is an ideal PR controller. The controller is said to have “second-order” because the Laplace operator, s, is raised to the second power. The ideal PR controller has infinite gain but may cause stability problems due to practical limitations for implementing the signal processing systems. 
         [0012]    For these reasons, a non-ideal PR transfer function is used instead. The transfer function for a practical, non-ideal, second-order PR controller is shown in Equation 3 below. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       G 
                       PR 
                     
                      
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       K 
                       P 
                     
                     + 
                     
                       
                         
                           K 
                           I 
                         
                          
                         
                           ω 
                           c 
                         
                          
                         s 
                       
                       
                         
                           s 
                           2 
                         
                         + 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             ω 
                             c 
                           
                            
                           s 
                         
                         + 
                         
                           ω 
                           n 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where K P  is the proportional gain, K I  is the integral gain, and ω n  is the natural frequency. The resonant frequency may be adjusted to appropriately widen the bandwidth of the controller. Widening of the controller bandwidth may decrease the maximum gain of the controller. The second-order PR controller is said to have three degrees of freedom because the controller may be tuned using K P , K I , and ω C . Proportional-resonant controllers can operate on a stationary frame signal derived from a Clarke transformation. Proportional-resonant controllers do not require feedforward 
         [0013]    A fractional-order or arbitrary order controllers may be implemented using fractional-orders integrators and differentiators to enhance an engineer&#39;s ability to tune the controller. A fractional-order PID controller is said to increase the degrees of freedom of a PID controller, which has three degrees of freedom, to five degrees of freedom. The fractional-order PID controller, as depicted in Equation 4, includes five degrees of freedom (i.e., K P , K I , α, K D , β). 
         [0000]    
       
         
           
             
               
                 
                   
                     G 
                      
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       K 
                       P 
                     
                     + 
                     
                       
                         K 
                         I 
                       
                       
                         s 
                         α 
                       
                     
                     + 
                     
                       
                         K 
                         D 
                       
                        
                       
                         s 
                         β 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where K P  is the proportional gain, K I  is the integral gain, α is the fractional-order of the integral component, K D  is the derivative gain, and β is the fractional-order of the derivative component. 
         [0014]    As discussed above, PI controllers are poorly situated to control resonant systems and PR controllers lack sufficient degrees of freedom to provide fine tuning of the controller. A fractional-order, proportional-resonant (FO-PR) controller can provide additional degrees of freedom and avoid drawbacks related to the implementation of a PI controller on a resonant system. One such FO-PR controller is described in Equation 5. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       G 
                       
                         FO 
                         - 
                         PR 
                       
                     
                      
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       K 
                       P 
                     
                     + 
                     
                       
                         
                           K 
                           I 
                         
                          
                         
                           s 
                           α 
                         
                       
                       
                         
                           s 
                           
                             2 
                              
                             α 
                           
                         
                         - 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             cos 
                              
                             
                               ( 
                               
                                 απ 
                                 2 
                               
                               ) 
                             
                           
                            
                           
                             ω 
                             n 
                             α 
                           
                            
                           
                             s 
                             α 
                           
                         
                         + 
                         
                           ω 
                           n 
                           
                             2 
                              
                             α 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where K P  is the proportional gain, K I  is the integral gain, α is the fractional-order of the integral component, and ω n  is the natural frequency at frequency (2πf). The controller may be adjusted for different frequencies and added in parallel using a similar reference and feedback system as disclosed in  FIG. 2 , as shown in Equation 6. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       G 
                       
                         FO 
                         - 
                         PR 
                       
                     
                      
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         ∞ 
                       
                        
                       
                           
                       
                        
                       
                         K 
                         Pj 
                       
                     
                     + 
                     
                       
                         
                           K 
                           Ij 
                         
                          
                         
                           s 
                           α 
                         
                       
                       
                         
                           s 
                           
                             2 
                              
                             α 
                           
                         
                         - 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             cos 
                              
                             
                               ( 
                               
                                 απ 
                                 2 
                               
                               ) 
                             
                           
                            
                           
                             ω 
                             nj 
                             α 
                           
                            
                           
                             s 
                             α 
                           
                         
                         + 
                         
                           ω 
                           nj 
                           
                             2 
                              
                             α 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0015]    Adding all of the frequencies in parallel allows for an FO-PR controller to control all of the frequencies anticipated when designing the controller for a particular plant. For example, a vehicle may demand various frequencies for given conditions. The FO-PR controller can provide accurate control of the resonant signal at each of those frequency demands with a smaller steady-state error and higher gain than other controllers. The coefficients, K Pj  is the proportional gain, K Ij  is the integral gain, α j  is the fractional-order of the integral component, and ω nj , of the FO-PR controller of Equation 6 may be adjusted independently for each frequency. This tuning provides separate controllers with new sets of parameters, which are then paralleled, to address each frequency required. 
         [0016]    PI controllers are generally limited to operate on a rotating dq reference frame. The three-phase reference signal, as required by most electric machines and utility grids, is converted into the rotating dq reference frame by a Park transformation. PR controllers are capable of controlling a stationary reference frame on an α-β or x-y plane. A stationary reference frame is generally derived using a Clarke transformation of a three-phase reference signal. 
         [0017]    An example vehicle having an electric machine is depicted in  FIG. 1  and referred to generally as a vehicle  16 . It is noted that a vehicle&#39;s electric machine and inverter is only one example of a plant that may be controlled by a fractional-order proportional-resonant controller. For example, a universal power supply could also be a plant controlled using a similar method. Continuing, the vehicle  16  includes a transmission  12  and is propelled by at least one electric machine  18  and may have selective assistance from an internal combustion engine (not shown). The electric machine  18  may be an alternating current (AC) electric motor depicted as “motor”  18  in  FIG. 1 . The electric machine  18  receives electrical power and provides torque for vehicle propulsion. The electric machine  18  also functions as a generator for converting mechanical power into electrical power through regenerative braking. 
         [0018]    The vehicle  16  includes an energy storage device, such as a traction battery  52  for storing electrical energy. The battery  52  is a high-voltage battery that is capable of outputting electrical power to operate the electric machine  18 . The battery  52  also receives electrical power from the electric machine  18  and the second electric machine  24  when they are operating as generators. The battery  52  is a battery pack made up of several battery modules (not shown), where each battery module contains a plurality of battery cells (not shown). Other embodiments of the vehicle  16  contemplate different types of energy storage devices, such as capacitors and fuel cells (not shown) that supplement or replace the battery  52 . A high-voltage bus electrically connects the battery  52  to the electric machine  18  and to the second electric machine  24 . 
         [0019]    The vehicle includes a battery energy control module (BECM)  54  for controlling the battery  52 . The BECM  54  receives input that is indicative of vehicle conditions and battery conditions, such as battery temperature, voltage and current. The BECM  54  calculates and estimates battery parameters, such as battery state of charge and the battery power capability. The BECM  54  provides output (BSOC, P cap ) that is indicative of a battery state of charge (BSOC) and a battery power capability (P cap ) to other vehicle systems and controllers. 
         [0020]    The vehicle  16  includes a DC-DC converter or variable voltage converter (VVC)  10  and an inverter  56 . The VVC  10  and the inverter  56  are electrically connected between the traction battery  52  and the electric machine  18 . The VVC  10  “boosts” or increases the voltage potential of the electrical power provided by the battery  52 . The VVC  10  also “bucks” or decreases the voltage potential of the electrical power provided to the battery  52 , according to one or more embodiments. The inverter  56  inverts the DC power supplied by the main battery  52  (through the VVC  10 ) to AC power for operating the electric machine  18 . The inverter  56  also rectifies AC power provided by the electric machine  18  to DC for charging the traction battery  52 . Other embodiments of the transmission  12  include multiple inverters (not shown), such as one invertor associated with each electric machine  18 . The VVC  10  includes an inductor assembly  14 . 
         [0021]    The transmission  12  includes a transmission control module (TCM)  58  for controlling the electric machine  18 , the VVC  10 , and the inverter  56 . The TCM  58  is configured to monitor, among other things, the position, speed, and power consumption of the electric machine  18 . The TCM  58  also monitors electrical parameters (e.g., voltage and current) at various locations within the VVC  10  and the inverter  56 . The TCM  58  provides output signals corresponding to this information to other vehicle systems. 
         [0022]    The vehicle  16  includes a vehicle system controller (VSC)  60  that communicates with other vehicle systems and controllers for coordinating their function. Although it is shown as a single controller, the VSC  60  may include multiple controllers that may be used to control multiple vehicle systems according to an overall vehicle control logic, or software. 
         [0023]    The vehicle controllers, including the VSC  60  and the TCM  58  generally includes any number of microprocessors, ASICs, ICs, memory (e.g., FLASH, ROM, RAM, EPROM and/or EEPROM) and software code to co-act with one another to perform a series of operations. The controllers also include predetermined data, or “look up tables” that are based on calculations and test data and stored within the memory. The VSC  60  communicates with other vehicle systems and controllers (e.g., the BECM  54  and the TCM  58 ) over one or more wired or wireless vehicle connections using common bus protocols (e.g., CAN and LIN). The VSC  60  receives input (PRND) that represents a current position of the transmission  12  (e.g., park, reverse, neutral or drive). The VSC  60  also receives input (APP) that represents an accelerator pedal position. The VSC  60  provides output that represents a desired wheel torque, desired engine speed, and generator brake command to the TCM  58 ; and contactor control to the BECM  54 . 
         [0024]    If the vehicle  16  is a plug-in electric vehicle, the battery  52  may periodically receive AC energy from an external power supply or grid, via a charge port  66 . The vehicle  16  also includes an on-board charger  68 , which receives the AC energy from the charge port  66 . The charger  68  is an AC/DC converter which converts the received AC energy into DC energy suitable for charging the battery  52 . In turn, the charger  68  supplies the DC energy to the battery  52  during recharging. It is understood that the electric machine  18  may be implemented on other types of electric vehicles, such as a hybrid-electric vehicle or a fully electric vehicle. 
         [0025]    Now referring to  FIG. 2 , a fractional-order, proportional-resonant controller  100  is shown. The controller  100  includes a stationary frame x-component current input derived from the summation  120  of i xd *, i x    110 , and i x *  136 . Input i xd * is derived from i dd *  102  having a rotating reference frame d and space vector block  104 , which incorporates the feedback phase shift angle  118 . The feedback phase shift angle  118  is derived from the phase-locked loop (PLL)  116  controller and voltage signal  114 . Input i x    110  is the stationary frame x-component of the three-phase feedback current signal i  106  from the output of the inverter (not shown). The i x *  136  is a stationary frame reference signal from a grid current or other reference current, i*  135 . Similarly, the controller  100  includes an y-component current input derived from the summation  122  of i y    112  and i y *  138 . i y    112  is the y-component of the feedback current signal from the output of the inverter (not shown). The i y *  138  is a reference signal from a grid current or other reference current, i*  135 . 
         [0026]    The summation blocks  120 ,  122  are individually fed into respective FO-PR transfer functions. For example, signal i x  is adjusted by a proportional gain K P    124  and a fractional-order, resonant transfer function  126  having adjustable constants K I , α, and ω. The proportional and integral components are joined at summation block  128  having an out of voltage U x *  140 . Similarly, signal i x  is adjusted by a proportional gain K P    130  and a fractional-order, resonant transfer function  132  having adjustable constants K I , α, and ω. The proportional and integral components are joined at summation block  134  having an output of voltage, U y *  142 . The outputs U x *  140  and U y *  142  are fed to the space vector modulation block (SVM)  144 . The SVM  144  allows the generation of pulse width modulation signals without converting the space vector outputs from the controller, U x *  140  and U y *  142 , into three-phase values first. 
         [0027]    Referring to  FIGS. 3A-B , the numerical results of the FO-PR controller are depicted.  FIG. 3A  discloses the gain of the FO-PR controller having wherein the natural frequency is 120π Hz, the proportional gain is four, the integral gain is 300, and alpha (fractional order) is 0.9. As shown, the reduced bandwidth PR controller  202  having a reduced bandwidth of 0.4, ω B , where ω B =(−1.196ζ+1.85)ω n . Where is the damping ratio and ω n  is the natural frequency. As shown, the 1.0 bandwidth PR controller  204  has a wider bandwidth than the reduced bandwidth controller, but a smaller gain. The FO-PR controller  206  has substantially higher gain than either PR controller  202 ,  204  at over 70 dB and provides a wider bandwidth.  FIG. 3B  depicts the phase-shift response of each controller. The reduced bandwidth PR controller  202  has a larger phase-shift near the resonant frequency of 60 Hz than the 1.0 bandwidth PR controller  204 . The FO-PR controller  206  has the best phase-shift response, as shown. 
         [0028]    The words used in the specification are words of description rather than limitation, and it is understood that various changes may be made without departing from the spirit and scope of the disclosure. As previously described, the features of various embodiments may be combined to form further embodiments of the invention that may not be explicitly described or illustrated. While various embodiments could have been described as providing advantages or being preferred over other embodiments or prior art implementations with respect to one or more desired characteristics, those of ordinary skill in the art recognize that one or more features or characteristics may be compromised to achieve desired overall system attributes, which depend on the specific application and implementation. These attributes may include, but are not limited to cost, strength, durability, life cycle cost, marketability, appearance, packaging, size, serviceability, weight, manufacturability, ease of assembly, etc. As such, embodiments described as less desirable than other embodiments or prior art implementations with respect to one or more characteristics are not outside the scope of the disclosure and may be desirable for particular applications.