Abstract:
A method for compensating for torque ripple in pulse width modulated machines including providing damping for transient disturbances utilizing a fixed feedback controller and rejecting steady disturbances utilizing an adaptive controller.

Description:
BACKGROUND OF THE INVENTION 
     This invention relates generally to control systems, and more specifically, to control systems where fluctuations in the torque produced by a motor are undesirable. 
     Permanent Magnet Synchronous Machines (PMSM), when driven by a pulse width modulation scheme, generate unwanted fluctuations, e.g. ripples, in the torque produced by the motor. This torque ripple is undesirable. Torque ripple is a major concern in many general motion applications. For example, one application where torque ripple is a major concern, and where removal of adverse torque ripple is beneficial, is semiconductor wafer handling machines. During manufacture, a manufacturer does not want to disturb a wafer in any fashion while moving the wafer from station to station. Currently, at least some known expensive motors are used to overcome torque ripple through a design incorporated into the motor. 
     The most widely used torque ripple compensation technique is the feed forward approach. A requirement of the feed forward approach is either prior knowledge of the motor construction and/or prior measurement of a torque ripple signal. The measured signal, referenced to the motor rotor, is then fed forward through the control into the motor. The signal application results in attenuation of torque ripple. Feed forward compensation is successful in a broad class of problems and is arguably a preferred approach when complete knowledge of the torque ripple signal is available. However, in certain situations either due to environmental, physical constraints or usability issues, complete knowledge of the torque ripple signal is unavailable. 
     BRIEF SUMMARY OF THE INVENTION 
     In one aspect of the invention, a method is provided for compensating for torque ripple in pulse width modulated machines. The method includes providing damping for transient disturbances utilizing a fixed feedback controller, and rejecting steady disturbances utilizing an adaptive controller. 
     In another aspect, a control system configured to compensate for torque ripple is provided. The control system includes a plant to be controlled, a fixed feedback controller configured to provide damping for transient disturbances, and an adaptive controller configured to reject steady disturbances. 
     In further aspect, a control system is provided including a fixed feedback controller configured to provide damping for transient disturbances and an adaptive controller configured to reject steady disturbances. The control system is configured to determine Q P  to minimize system output where system output is defined as output=P 11 d+Q p P 12 e. The adaptive controller is configured to adjust Q P  utilizing a least means square (LMS) algorithm according to: 
     
       
           e ( n )= d ( n )−w T   u ( n ) 
       
     
     
       
           ŵ ( n +1)= ŵ ( n )+μ u ( n ) e *( n ) 
       
     
     for each time step, where: 
     M=number of taps 
     μ=step−size 
     u(n)=M by 1 input vector 
     d(n)=desired response 
     ŵ(n+1)=estimate of weighting factors. 
     In yet another aspect, a control system is provided that includes a fixed feedback controller configured to provide damping for transient disturbances, and an adaptive controller configured to reject steady disturbances. The control system is configured to determine Q P  to minimize system output where system output is defined as output=P 11 d+Q p P 12 e. The adaptive controller is configured to adjust Q p  utilizing a recursive least squares (RLS) algorithm according to:          k        (   n   )       =         λ     -   1            P        (     n   -   1     )            u        (   n   )           1   +       λ     -   1              u   T          (   n   )            P        (     n   -   1     )            u        (   n   )                                   α( n )= d ( n )− ŵ   1 ( n −1) u ( n ) 
     
       
           ŵ ( n )= ŵ ( n −1)+ k ( n )α*( n ) 
       
     
     
       
           P ( n )=λ −1   P ( n −1)−λ −1   k ( n ) u   T ( n ) P ( n −1) 
       
     
     for each time step, where: 
     ŵ(n)=tap weight factor 
     k(n)=gain factor 
     α(n)=priori estimation error 
     P(n)=correlation matrix inverse 
     and includes initialization values of: 
     
       
         P(0)=δ −1   I   
       
     
     
       
         ŵ(0)=0 
       
     
     where δ is a positive number less than one. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a block diagram of an adaptive Q controller. 
     FIG. 2 is a detailed block diagram of the controller shown in FIG. 1 illustrating the adaptive Q algorithm. 
     FIG. 3 is a block diagram of a plant and disturbance model. 
     FIG. 4 is a block diagram of an augmented system. 
     FIG. 5 is a block diagram of a motor control system incorporating an optimal disturbance rejection controller. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     In one embodiment of the present invention, a hybrid control method combines traditional fixed feedback control with adaptive feedback techniques. The hybrid control provides damping for transient disturbances via a fixed controller and rejection of steady disturbances, for example torque ripple, via an adaptive controller. 
     The adaptive controller technique differs from other control techniques in that a measurement of an external signal coherent with the disturbance is not needed, nor is a knowledge of how the disturbance enters the system. As stability is an issue, the adaptive controller described herein is implemented to minimize stability problems. Further, an adaptive control technique for torque ripple compensation is based upon an adaptive feedback control technique called adaptive-Q disturbance estimate feedback (Adaptive-Q). 
     The adaptive controller technique differs from traditional feedback techniques since information concerning how the torque ripple signal enters the system is not utilized. The objective of a disturbance rejection control is to make the transfer function from disturbance input to system output have a desirable frequency response. In most cases, the controller attenuates or eliminates the disturbance. 
     FIG. 1 is a block diagram of an Adaptive-Q control system  20  including a fixed feedback controller, commonly referred to as a linear quadratic gaussian (LQG) controller, comprising a state estimator (Kalman filter)  22  and a state feedback gain  24 . The LQG is part of an adaptive feedback structure  26 , which further includes an adaptive filter (Q p )  28 . Feedback structure  26  is used to control a plant  30 , for example, a motor. The fixed feedback controller provides a pre-determined amount of system damping for transient disturbances that adaptive filter  28  may be unable to adapt to and suppress quickly enough. Additionally, it has been shown that a fixed feedback controller increases the adaptation speed of the resulting control. 
     In control system  20  a control input  32  affects state estimator  22 , assuming perfect system identification, in the same manner as plant  30  causing an estimation error to remain constant. Therefore, a transfer function from s to e is zero. Since the transfer function is zero any stable adaptive filter (Q p )  28  placed in the loop will not drive the resulting system unstable. Additionally, by varying a Q p  transfer function all controllers for stabilizing the plant are swept. 
     Q p    28  is adjusted to provide a desired system output. System output is defined as output=P 11 d+P 12 Q p e. If it is assumed that y, u, e and s are scalars, then the equation is rewritten in standard output error format as:output=P 11 d+Q p P 12 e. Restating, it is desired to determine a Q p    28  that minimizes the system output. Since Q p    28  sweeps all stabilizing controllers, a system that is guaranteed stable is obtained, assuming a perfect quality system identification. 
     FIG. 2 is a block diagram of a system  40  implementing an adaptive Q algorithm shown in more detail than the system of FIG.  1 . Referring specifically to FIG. 2, a fixed feedback controller  42  is a standard linear quadratic gaussian (LQG) control. Controller  42  contains no knowledge of how a disturbance enters the system. Rather, controller  42  is configured to use an identity matrix as a model to estimate how disturbances enter the system. Q p  filter structure  44  includes a finite impulse response (FIR) filter  46  and ensures a stable system transfer function. FIR filter  46  with a limited number of coefficients allows filter structure  44  to sweep a subset of the stabilizing controllers. Adaptation algorithms  48  used are gradient dissent algorithms. For example, a least mean squares (LMS) algorithm and a recursive least squares (RLS) algorithm. 
     Referencing the block diagram of system  40 , shown in FIG. 2, a set of simulation equations implemented in system  40  is shown below. Discrete time LQG (D-T) state equations are: 
     
       
           x ( k +1)= Ax ( k )+ Bu ( k )+ E??d ( k ) 
       
     
     
       
           y ( k )= Cx ( k ) 
       
     
     
       
           u ( k )=− Kx ′( k )+ r ( k )+ s ( k ) 
       
     
     
       
           x′ ( k +1)= Ax ′( k )+ Bu ( k )+ F ( y ( k )− y ′( k )) 
       
     
     
       
           y ′( k )= Cx ′( k ) 
       
     
     where F and K are calculated using an appropriate Ricatti equation. An identity matrix serves as a model for the E?? matrix for the linear quadratic estimator (LQE) design. Combination of the above equations results in a single set of D-T state equations as shown below.                [           x        (     k   +   1     )                   x   ′          (     k   +   1     )             ]     =         [         A         -   BK             FC         A   -   BK   -   FC           ]          [           x        (   k   )                   x   ′          (   k   )             ]       +       [         B           B         ]          r        (   k   )         +       [           E      ??             0         ]          d        (   k   )                         [           y        (   k   )                   y   ′          (   k   )             ]     =       [         C       0           0       C         ]          [           x        (   k   )                   x   ′          (   k   )             ]                                    
     Prior to using an estimation error signal, which is defined as e(k)=y(k)−y′(k) in the adaptation algorithm, the error signal is filtered through the transfer function P 12 , shown as error filter  50  in FIG.  2 . The state equations are X p (k+1)=Ax p (k)+Be(k) and y p (k)=Cx(k), where the signals y p (k) and y(k), are desired inputs to adaptive algorithm  48  which adjusts the filter weights for FIR filter  46 . 
     As described above, adaptive algorithm  48  is responsible for adjusting Q p  filter weights (FIR filter coefficients). Two exemplary algorithms used for adjusting Q p  filter weights are a least mean squares (LMS) algorithm and a recursive least squares (RLS) algorithm. 
     The RLS algorithm is summarized below, where 
     ŵ(n)=tap weight factor 
     k(n)=gain factor 
     α(n)=priori estimation error 
     P(n)=correlation matrix inverse 
     with initialization values of 
     
       
           P (0)=δ −1   I   
       
     
     
       
           ŵ (0)=0 
       
     
     providing, for each step of time,          k        (   n   )       =         λ     -   1            P        (     n   -   1     )            u        (   n   )           1   +       λ     -   1              u   T          (   n   )            P        (     n   -   1     )            u        (   n   )                                   α( n )= d ( n )− ŵ   1 ( n −1) u ( n ) 
     
       
           ŵ ( n )= ŵ ( n −1)+ k ( n )α*( n ) 
       
     
     
       
           P ( n )=λ −1   P ( n −1)−λ −1   k ( n ) u   T ( n ) P ( n −1). 
       
     
     The LMS algorithm is summarized below, where 
     M=number of taps 
     μ=step−size 
     u(n)=M by 1 input vector 
     d(n)=desired response 
     ŵ(n+1)=estimate of weighting factors 
     providing a computation for each step of time as 
     
       
           e ( n )= d ( n )− w   T   u ( n ) 
       
     
     
       
           ŵ ( n +1)= ŵ ( n )+μ u ( n ) e *( n ). 
       
     
     The LMS algorithm has the advantages of being relatively simple and numerically efficient to compute. However, a disadvantage of the LMS algorithm is a fixed step size. The RLS algorithm uses a variable step size, but is numerically intensive to compute. 
     LQG Disturbance Rejection Simulation 
     A LQG disturbance rejection (LQGDR) controller differs in two ways from the LQG control structure implemented for the Adaptive-Q controller. A first difference is in a Kalman filter (LQE) estimator design. The LQGDR is an ideal control model that permits comparisons to be made to determine an effectiveness rating. The estimator design contains full knowledge of disturbance frequency content and further includes a model from disturbance input to system output. The LQE portion uses the same information concerning state and output noise as LQG design described above for Adaptive-Q control. Derivation of the LQGDR begins with augmenting the existing plant state equations to include a model of the disturbance frequency spectrum. The disturbance is modeled according to the state equations x d (k+1)=A d x(k)+B d d(k) and y d (k)=C d x d (k), where the feed through term D d  is assumed to be zero. 
     Disturbance state equations augment plant disturbance input to create colored noise for the Kalman filter as shown in plant and disturbance model  70 , shown in FIG.  3 . The disturbance model in the above equation is a discrete time (D-T) model, while the disturbance is a continuous time (C-T) signal, a possible source of error. However, at a D-T system sampling rate of 1000 Hz errors are negligible. 
     In one embodiment, augmented plant model state equations include          [           x        (     k   +   1     )                   x   d          (     k   +   1     )             ]     =         [         A         EC   d             0         A   d           ]          [           x        (   k   )                   x   d          (   k   )             ]       +       [         B       0           0         B   d           ]          [         u           d         ]                   y        (   k   )       =       [     C                 0     ]          [           x        (   k   )                   x   d          (   k   )             ]                              
     where system matrices are defined as          A   aug     =         [         A         EC   d             0         A   d           ]                     B   aug       =     [         B           0         ]                 C   aug     =         [     C                 0     ]                     E   aug       =       [         0             B   d           ]     .                              
     An augmented system model  80 , shown in FIG. 4, is used to determine LQE and LQG sections. A resulting control structure including the LQGDR controller  100  is shown in FIG. 5, where a plant model  102  replaces augmented model  80 , shown in FIG.  4 . 
     The state equations for the LQGDR controller  104 , therefore, are          [           x        (     k   +   1     )                   x   ′          (     k   +   1     )             ]     =       [         A         -   BK             FC           A   aug     -       B   aug        K     -     FC   aug             ]                   [           x        (   k   )                   x   ′          (   k   )             ]     +       [         B             B   aug           ]          r        (   k   )         +       [         E           0         ]          d        (   k   )            
          y        (   k   )           =         [     C                 0     ]          [           x        (   k   )                   x   ′          (   k   )             ]       .                                  
     While the invention has been described in terms of various specific embodiments, those skilled in the art will recognize that the invention can be practiced with modification within the spirit and scope of the claims.