Abstract:
A method and apparatus for use with a motor controller that receives a command velocity and that applies voltages to drive a motor at the command velocity, the apparatus comprising a dual inertia lost motion assembly including a motor and a load couplable to the motor, the lost motion assembly characterized by at least some lost motion between the motor and the load, the motor and load together characterized by a total assembly inertia, an acceleration error determiner for generating an acceleration error that is the difference between a derivative of the command velocity and a motor acceleration value and a low pass acceleration error filter filtering the acceleration error and having a gain set as a percentage of the total assembly inertia, the acceleration error filter providing the filtered signal to the controller, the controller using the filtered signal to adjust the applied voltages.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS  
       [0001]     Not applicable.  
       STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT  
       [0002]     Not applicable.  
       BACKGROUND OF THE INVENTION  
       [0003]     The field of the invention is control systems for dual inertia lost motion mechanical systems and more specifically optimal acceleration feedback for use by a controller in controlling a system characterized by lost motion.  
         [0004]     This section of this document is intended to introduce various aspects of art that may be related to the present invention described and/or claimed below and provides background information to facilitate a better understanding of the various aspects of the present invention. It should be understood that the statements in this section of this document are to be read in this light, and not as admissions of prior art.  
         [0005]     High performance servo drives are used in many different applications and in many different industries. For example, in the media printing industry servo drives are routinely employed to accelerate and decelerate large spools of paper web to wind and unwind and position sections of the web precisely with respect to printer heads and other system components to facilitate application of information on the paper material surfaces. In printing applications and in many other applications precise motor-load control is extremely important.  
         [0006]     Typically a velocity command signal is provided to a drive indicating a desired motor load velocity and the drive is configured to apply AC voltages to the motor causing motor velocity to converge toward the command velocity. The relationship between applied voltages and motor velocity is effected by motor-load assembly inertia. For instance, assume first and second motor-load assemblies where a first assembly inertia is half as large as a second assembly inertia. When motor velocity is to be reduced, a much smaller force is required to increase the first assembly than to slow the second assembly at the same rate. Similarly, when motor velocity is to be increased, a much larger force is required to increase the first assembly velocity than to increase the second assembly velocity at the same rate.  
         [0007]     Many systems include one or more feedback loops that provide motor-load operating characteristics to the drive for comparison to the command velocity and derivatives thereof so that the drive can adjust motor velocity in a suitable fashion and increase performance. Thus, for instance, some drives include a position feedback loop and/or a velocity feedback loop. In these cases, when the motor position or velocity do not match the commanded position (e.g., the integral of the commanded velocity) and/or the commanded velocity, the drive alters the applied voltages to correct the error.  
         [0008]     In theory, in an ideal directly linked motor-load assembly, the load is linked to the motor via an infinitely stiff coupling and the motor and load inertias appear as a single mass or total assembly inertia. A control system for a motor-load assembly including an infinitely stiff coupling is easily constructed with the application of velocity and/or position feedback loops.  
         [0009]     In reality, most direct linked motor-load assemblies do not include an infinitely stiff coupling and instead are characterized by some degree of flexibility. In these cases the flexible motor-load coupling can be thought of as a spring between the motor and load masses such that angular load position typically is slightly different than the angular position of the motor.  
         [0010]     When a drive includes one or more feedback loops and a motor-load assembly includes a flexible coupling, the combination often causes resonance within the drive-motor-load system. To this end, when a velocity command is received and a drive accelerates a motor toward a command velocity, in the case of a flexible coupling, the motor achieves the desired velocity prior to the lagging load. Because of the spring effect of the flexible coupling, the load often shoots past the command velocity thereby loading the coupling with another spring force that places a torque on the motor which causes the motor velocity to exceed the commanded velocity so that a velocity error is identified by the control system in the drive (e.g., the feedback velocity instantaneously exceeds the commanded velocity). The drive in turn alters the applied voltages to slow the motor-load assembly.  
         [0011]     Eventually the spring force within the coupling causes the load velocity to slow such that the load velocity converges toward the motor velocity. However, here again, as the force within the coupling is released, the load velocity may shoot through the motor velocity thereby placing a slowing force on the motor. In response to the slowing motor velocity the drive again alters the applied voltages, this time to increase the motor speed. This oscillating process continues at what is typically referred to as a system resonant frequency.  
         [0012]     While many motor-load assemblies include a direct linkage between the motor and the load, other assemblies include some type of indirect coupling such as a gear box, gear train, etc., that introduces lost motion between the motor and the load. Here, the phrase “lost motion” is used to refer to an actual disconnect between the motor and the load that occurs under certain operating circumstances because there is some mechanical “play” or “slack” within the coupling components. For instance, in a simple example, a particularly sloppy gear train may result in a coupling where five degrees of motor rotation separate the relative positions in which the motor engages the load through the gear train in opposite directions. Thus, when rotating in a forward direction the motor-gear train may be engaged with the load at a first angle and the motor may have to be rotated five degrees in the reverse direction relative to the load to engage the load in the other direction. In this case, assume that the motor is initially rotating in the forward direction to drive a coupled load and that the command velocity is reduced to slow motor speed. Unless load friction slows the load at a rate at least as fast as the rate at which the motor speed is reduced, at the instant that the motor speed is reduced, the motor and load become decoupled.  
         [0013]     Motor-load assemblies that include lost motion couplings add complex dynamics to drive systems that complicate control tasks appreciably. In this regard, the instantaneous effective inertia associated with the motor changes whenever a lost motion coupling causes the motor-load to couple or decouple. Here, the phrase “effective inertia” is used to refer to the motor inertia alone when the load is decoupled from the motor and to the sum of the motor inertia and the load inertia when the load is coupled to the motor. In many cases the load inertia will be much larger than the motor inertia so that the difference between the effective coupled inertia and effective decoupled inertia is appreciable.  
         [0014]     Because inertia effects the relationship between applied voltages and velocity and the effective inertias are different during coupled and decoupled motor-load conditions, drive velocity adjustments that occur during coupled and decoupled motor-load conditions end up causing different velocity modifications. Thus, for instance, given a specific velocity error, independent of whether or not the error occurs when the motor is coupled to the load and when the motor is decoupled from the load, the error will result in the same change in applied voltages to eliminate the error. Here, however, because the coupled and decoupled inertias are different, the effect on velocity is different. Thus each of spring related resonance and lost motion coupling dynamics hamper motor-load assembly control.  
         [0015]     One way to eliminate system resonance is to provide filters within the feedback loops that filter out disturbances at the resonant frequencies. Thus, for instance, where a system resonant frequency is known, a notch filter may be provided that specifically eliminates the resonant frequency signal from the feedback loop to the drive.  
         [0016]     While filters are routinely used to reduce system resonance, such filters typically have not been employed to minimize the effects of lost motion couplings because they generally do not provide the bandwidth necessary to meet application requirements.  
       BRIEF SUMMARY OF THE INVENTION  
       [0017]     Certain aspects commensurate in scope with the originally claimed invention are set forth here. It should be understood that these aspects are presented merely to provide the reader with a brief summary of certain forms the invention might take and that these aspects are not intended to limit the scope of the invention. Indeed, the invention may encompass a variety of aspects that may not be set forth below.  
         [0018]     It has been recognized that an acceleration feedback loop can be provided in addition to a velocity feedback loop in a system characterized by at least some degree of lost motion where the acceleration feedback loop is optimally tuned as a function of the total motor-load inertia. In this regard, it has further been recognized that a low pass filter having a gain of between 0.4 and 0.6 times the total motor-load inertia minimizes system dynamics in a lost motion system. Moreover, it has been recognized that the filter should also be selected to have a time constant of approximately three times the time constant of a velocity loop filter to provide optimal results.  
         [0019]     Consistent with the above, in at least some embodiments the invention includes an apparatus for use with a motor controller that receives a command velocity and that applies voltages to drive a motor at the command velocity, the apparatus comprising a dual inertia lost motion assembly including a motor and a load couplable to the motor, the lost motion assembly characterized by at least some lost motion between the motor and the load, the motor and load together characterized by a total assembly inertia, an acceleration error determiner for generating an acceleration error that is the difference between a derivative of the command velocity and a motor acceleration value and a low pass acceleration error filter filtering the acceleration error and having a gain set as a percentage of the total assembly inertia, the acceleration error filter providing the filtered signal to the controller, the controller using the filtered signal to adjust the applied voltages. In some cases the gain is between 40% and 60% of the total assembly inertia. In some cases the gain is approximately 50 percent of the total assembly inertia.  
         [0020]     According to another aspect, in some embodiments the controller includes a velocity error filter having a velocity error time constant and wherein the acceleration error filter has an acceleration error time constant that is between two and four times the velocity error time constant. In some cases the acceleration error time constant is approximately three times the velocity error time constant.  
         [0021]     Some embodiments of the invention include an apparatus for use with a motor controller that receives a command velocity and that applies voltages to drive a motor at the command velocity, the apparatus comprising a dual inertia lost motion assembly including a motor and a load couplable to the motor, the lost motion assembly characterized by at least some lost motion between the motor and the load, the motor and load together characterized by a total assembly inertia and an acceleration feedback loop including a low pass acceleration error filter where the loop gain is between 40% and 60% of the total inertia, the low pass filter providing a filtered signal to the controller.  
         [0022]     Still other embodiments include an apparatus for use with a motor controller that receives a command velocity and that applies voltages to drive a lost motion motor-load assembly including a motor and a load couplable to the motor, the lost motion assembly characterized by at least some lost motion between the motor and the load, the motor and load together characterized by a total assembly inertia, the apparatus comprising an acceleration feedback loop including a low pass acceleration error filter where the loop gain is between 40% and 60% of the total inertia, the low pass filter providing a filtered signal to the controller.  
         [0023]     The invention also includes a method for use with a motor controller that receives a command velocity and that applies voltages to drive a lost motion motor-load assembly including a motor and a load couplable to the motor, the lost motion assembly characterized by at least some lost motion between the motor and the load, the motor and load together characterized by a total assembly inertia, the method comprising the steps of providing an acceleration feedback loop from the motor to the controller, providing a low pass acceleration error filter within the feedback loop where the filter gain is between 40% and 60% of the total inertia.  
         [0024]     These and other objects, advantages and aspects of the invention will become apparent from the following description. In the description, reference is made to the accompanying drawings which form a part hereof, and in which there is shown a preferred embodiment of the invention. Such embodiment does not necessarily represent the full scope of the invention and reference is made therefore, to the claims herein for interpreting the scope of the invention. 
     
    
     BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS  
       [0025]     The invention will hereafter be described with reference to the accompanying drawings, wherein like reference numerals denote like elements, and:  
         [0026]      FIG. 1  is a diagram illustrating the effect of a motor inertia on how a disturbance affects a motor velocity;  
         [0027]      FIG. 2  is similar to  FIG. 1 , albeit illustrating the effect of a motor inertia and coupled load inertia on how a disturbance affects motor velocity;  
         [0028]      FIG. 3  is a diagram illustrating a prior art motor control system;  
         [0029]      FIG. 4  is a diagram equivalent to the diagram of  FIG. 3 ;  
         [0030]      FIG. 5  is a diagram equivalent to the diagram of  FIG. 4  and including an inner acceleration feedback loop;  
         [0031]      FIG. 6  is a diagram similar to  FIG. 1  albeit including an acceleration feedback loop;  
         [0032]      FIG. 7  is a diagram illustrating a control system according to the present invention where filter delays are represented; and  
         [0033]      FIG. 8  is a diagram illustrating a control system according to at least one aspect of the present invention. 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0034]     Referring now to the drawings wherein like reference numerals correspond to similar elements throughout the several views and, more specifically, referring to  FIGS. 1 and 2 , simple diagrams  60  and  61  are provided to illustrate the effect that different inertias associated with a motor that is decoupled from a load and a motor that is coupled to a load have on how a torque disturbance T d  effects motor velocity ω m  are illustrated. In this regard, diagram  60  includes a divider block  64  and an integrator block  66 . The torque disturbance T d  is provided to divider block  64  which divides disturbance T d  by a motor inertia m 1  thereby providing a motor acceleration value a m . Acceleration value a m  is provided to integrator  66  which integrates the acceleration value yielding motor velocity ω m .  
         [0035]     In  FIG. 2 , diagram  61  is similar to diagram  60  of  FIG. 1  except that the inertia m 1 +m 2  associated with block  68  replaces motor inertia m 1 . Here, torque disturbance T d  is provided to divider block  68  which divides disturbance T d  by combined inertias m 1 +m 2  providing acceleration value a m . Integrator  66  integrates value a m  to yield motor velocity ω m .  
         [0036]     In the case of a motor-load system characterized by at least some degree of lost motion, it should be clear from  FIGS. 1 and 2  that the effects of a torque disturbance T d  on motor velocity ω m  are different when a load is coupled to a motor and when the same load is decoupled from the motor. When the load is decoupled from the motor as in  FIG. 1 , the torque disturbance Td has a greater effect on motor velocity ω m  than when the load is decoupled from the motor as in  FIG. 2 . In effect, the larger coupled inertia dampens the effect of disturbance T d . Hereinafter the ratio of gains (m 1 +m 2 )/m 1  will be referred to as ratio R.  
         [0037]     In the description that follows, first, a diagram corresponding to a classic proportional/integral/derivative (PID) velocity regulator is described. Thereafter, various diagrams that are equivalent to the classic regulator initially described are described to develop a velocity regulator diagram including direct acceleration feedback where the acceleration feedback gain is a function of motor/load inertia. After the desired regulator form is derived, transfer functions corresponding to the inner acceleration loop under different operating conditions are derived and used to identify an optimal acceleration error gain value. More specifically, separate transfer functions where a load is coupled to and decoupled from the load for the inner loop are developed to identify an optimal gain equation and then common R values (i.e., (m 1 +m 2 )/m 1 ) are used to solve the gain equation and find optimal gain values. After an optimal gain value for most common R values is determined, transfer functions including filter time constant factors are developed for the coupled and decoupled conditions which are then analyzed in light of the optimal gain value to identify an optimal acceleration error time constant for common R values.  
         [0038]     Referring now to  FIG. 3 , an exemplary classic PID velocity regulator diagram  10  receives a velocity command signal ω* and causes a motor and load linked thereto to operate at a motor velocity ω m  equal to the commanded velocity ω*. Regulator  10  includes three summers,  14 ,  26  and  30 , two derivative blocks  16  and  22 , two integrators  18  and  35 , four amplifiers  20 ,  28 ,  24  and  32  and one divider  34 . Velocity ω* is provided to derivative block  16  which generates a command acceleration value a*. Acceleration value a* is provided to amplifier  20  which multiplies acceleration value a* by the combined or total motor/load inertia m t  (i.e., m t =m 1 +m 2 ) thereby generating a scaled command acceleration value which is provided to summer  26 .  
         [0039]     Referring still to  FIG. 3 , a feedback loop provides a velocity feedback signal ω fb  equal to motor velocity ω m  to summer  14 . Summer  14  subtracts feedback signal ω fb  from command velocity ω* thereby generating a velocity error signal. The velocity error signal is provided each of integrator block  18 , amplifier  24  and derivative block  22 . Integrator block  18  integrates the velocity error signal and provides its output to amplifier  28 . Amplifier  28  multiplies the integrated velocity error value by an integral gain k i  and provides the scaled up value to summer  26 .  
         [0040]     Derivative block  22  takes the derivative of the velocity error signal and provides the derivative value to amplifier  32 . Amplifier  32  multiplies the received value by a derivative gain k d  and provides the scaled value to summer  26 . Amplifier  24  multiples the velocity error value by a proportional gain k p  and provides the scaled value to summer  26 .  
         [0041]     Summer  26  adds all of the received values and provides a motor torque T m  to summer  30 . Summer  30  subtracts a load torque value T l  from motor torque T m  and provides a combined torque value to divider  34 . The combined torque value is akin to the disturbance torque T d  in  FIGS. 1 and 2 . Divider  34  divides the combined torque value by the total motor/load inertia m t  thereby generating a motor acceleration value a m . Acceleration value a m  is provided to integrator  35  which yields the motor velocity value ωm.  
         [0042]     Referring now to  FIG. 4 , a diagram  11  similar to the diagram  10  in  FIG. 3  is provided. In  FIG. 4 , many of the components are identical to the components of  FIG. 3  and therefore, in the interest of simplifying this explanation, components in  FIG. 4  that are identical to components in  FIG. 3  are identified by the same numbers and are not again described here in detail. Thus, for instance, derivative block  16  in  FIG. 4  is identical to derivative block  16  in  FIG. 3 , summer  30  in  FIG. 4  is identical to summer  30  in  FIG. 3  and so on.  
         [0043]     There are three main differences between  FIGS. 3 and 4 . First, instead of having a single feedback loop as in  FIG. 3 ,  FIG. 4  includes both an outer loop and an inner loop. In  FIG. 4 , the outer loop is identical to the feedback loop in  FIG. 3  and thus provides the velocity feedback signal ω fb  to summer  14 . The inner feedback loop in  FIG. 4  replaces the feedforward loop in  FIG. 3  that included block  22  and amplifier  32 . More specifically, the inner loop in  FIG. 4  provides the velocity feedback value ω fb  directly to derivative module  22  (i.e., the velocity error value from summer  14  is no longer provided to derivative block  22 ).  
         [0044]     Second, to compensate for the fact that gain k d  is now multiplied by the derivative of the velocity feedback signal ω fb  instead of the derivative of the velocity error value, amplifier  20  in  FIG. 3  had been replaced by amplifier  38  in  FIG. 4  which has a larger gain equal to k d +m t .  
         [0045]     Third, summer  26  in  FIG. 3  has been replaced by summer  27  in  FIG. 4  that subtracts the output of amplifier  32  from the sum of the other three signals received by summer  27 .  
         [0046]      FIG. 5  is an equivalent circuit to  FIG. 4 . The primary difference between  FIGS. 4 and 5  is that derivative gain value k d  has been replaced by a gain expression Gm t  that includes the total motor/load inertia value m t . This change is represented in  FIG. 5  by amplifiers  50  and  48  that replace amplifier  38  and  32  and that include inertia associated gain values of (G+1)m t  and Gm t , respectively.  
         [0047]     Referring now to  FIG. 6 , a diagram  70  similar to the diagrams illustrated in  FIGS. 1 and 2  is provided where the inner feedback loop of  FIG. 5  has been added. In  FIG. 6 , value m e  is an “effective inertia” value which is equal to either motor inertia m 1  or the motor/load inertia m 1 +m 2 , depending on whether or not the load is coupled or decoupled to the motor. Thus, where the load is decoupled from the motor, effective inertia m e  is equal to motor inertia m 1  and, where the load is coupled to the motor, inertia m e  is equal to the total or combined inertia (m 1 +m 2 ). In  FIG. 6 , diagram  70  includes a summer  72 , a divider  74 , an integrator  75 , a derivative block  77  and feedback gain block  76 .  
         [0048]     Referring still to  FIG. 6 , when the load inertia m 2  is engaged or coupled to the motor inertia m 1 , the transfer function for diagram  70  can be expressed as:  
                 ω   m       T   d       =       1     (     1   +   G     )       ·     1       (       m   1     +     m   2       )     ⁢   s                 Eq   .           ⁢   1             
 
 Similarly, when load inertia m 2  is decoupled from motor inertia m 1 , the transfer function for diagram  70  can be expressed as:  
                 ω   m       T   d       =       R     1   +   GR       ·     1       (       m   1     +     m   2       )     ⁢   s                 Eq   .           ⁢   2             
 
 Thus, the minimum gain with load inertia m 2  coupled to motor inertia m 1  is 1/(1+G) and the maximum gain with load inertia m 2  coupled to motor inertia m 1  is R/(1+RG). 
 
         [0051]     In order to provide an optimally stable system, it is necessary to optimize the gain deviation around a nominal value. Thus, to minimize gain deviation between the coupled and decoupled conditions, ideally, the following equation should be satisfied:  
                 1     (     1   +   G     )       ·     R     (     1   +   GR     )         =   1           Eq   .           ⁢   3             
 
 Solving for gain value G, we obtain the following equation:  
             G   =         -     (     1   +   R     )         2   ⁢   R       +           5   ⁢     R   2       -     2   ⁢   R     +   1         2   ⁢   R                 Eq   .           ⁢   4             
 
         [0053]     Solutions to equation 4 for common values of ratio R are summarized in the following table.  
                                                           Solutions to Equation 5                R   G                            4   .443           6   .500           8   .529           10   .547           12   .558           14   .567                      
 
         [0054]     The average G value in the table above is 0.524 and, the range of values is generally between 0.4 and 0.6. Thus, an optimal value for gain G is between 0.4 and 0.6, and is approximately 0.500 independent of which common ratio R exists in a system.  
         [0055]     Referring now to  FIG. 7 , a diagram  80  similar to diagram  70  of  FIG. 6  is provided. In addition to the components of  FIG. 6 ,  FIG. 7  includes additional or modified components that represent filter delays. In this regard, an acceleration error filter delay has been added to the inner loop gain block  100  via the factor  1 /(T f s+1) including filter time constant T f . In addition, a velocity error filter delay has been added via block  86  and is represented as two first order filter by factor  
         1       (         T   v     ⁢   s     +   1     )     2       .       
 
         [0056]     In addition to the modifications to represent filter delays,  FIG. 7  has been modified so that the torque disturbance T d  is represented by the difference between a motor torque value T m  and the load torque value T l .  
         [0057]     Referring still to  FIG. 7 , diagram  80  includes three summers  84 ,  90  and  96 , a velocity error filter block  86 , a PI regulator  88 , a divider block  98 , an integrator block  99 , a derivative block  101  and the acceleration feedback block  100  including a delay term in the denominator. Velocity command ω* is provided to summer  84  which subtracts a velocity feedback signal ω fb  from the command value ω* and generates a velocity error signal.  
         [0058]     The velocity error signal is provided to velocity error filter  86  and the output thereof is provided to PI regulator  88 . Regulator  88  steps up the received signal and provides its output to summer  90 . Summer  90  subtracts the output of filter/delay block  100  from the output of regulator  88  and provides its output as a motor torque value to summer  96 . Summer  96  subtracts load torque value T l  from the motor torque value providing the combined torque value to divider block  98 . Divider block  98  divides the combined torque value by the effective inertia m e  providing an acceleration value to integrator block  99 . Integrator block  99  integrates the acceleration value thereby generating the motor velocity value which is also the velocity feedback ω fb . The velocity feedback ω fb  is provided to derivative block  101  which provides its output to block  100 .  
         [0059]     Referring still to  FIG. 7 , the transfer function for the inner loop corresponding to components  90 ,  96 ,  98 ,  99 ,  101  and  100  is different when the load is coupled to and decoupled from the motor. When the load is coupled to the motor, the transfer function for the inner loop can be expressed as:  
               TF   ⁡     (   s   )       =       1       (     1   +   G     )     ⁢       (       m   1     +     m   2       )     ·   s         ·       (     1   +       T   f     ·   s       )       [     1   +         T   f     ·   s       (     1   +   G     )         ]                 Eq   .           ⁢   5             
 
 When the load is decoupled from the motor, the transfer function can be expressed as:  
               TF   ⁡     (   s   )       =       R       (     1   +   GR     )     ⁢       (       m   1     +     m   2       )     ·   s         ·       (     1   +       T   f     ·   s       )       [     1   +         T   f     ·   s       1   +   GR         ]                 Eq   .           ⁢   6             
 
         [0061]     Referring to equation 5 above, when the load is coupled to the motor, the impedance term is:  
               1   +       T   f     ·   s         [     1   +         T   f     ·   s       (     1   +   G     )         ]             Eq   .           ⁢   7             
 
 A first order approximation of the impedance term can be represented as:  
             1   +       T   f     ·       G   ·   s       1   +   G                 Eq   .           ⁢   8             
 
         [0063]     Referring once again to  FIG. 7 , adding an impedance term corresponding to velocity error filter block  86 , the impedance term for diagram  80  when the motor and load are coupled can be expressed as:  
               Y   ⁡     (   s   )       =       1   +       T   f     ·     Gs     (     1   +   G     )               (     1   +       T   v     ⁢   s       )     2               Eq   .           ⁢   9             
 
 Next, let:  
               T   v     =       T   f     ·     G     (     1   +   G     )                 Eq   .           ⁢   10             
 
 Combining equations 9 and 10 yields equation 11:  
               Y   ⁡     (   s   )       =     1     1   +       T   v     ·   s                 Eq   .           ⁢   11             
 
 and, pole-zero cancellation occurs. 
 
         [0067]     Referring to equation 10 and the values provided in the table above, it should be appreciated that with any of the gain G values in the table, equation 10 yields an inner loop time constant T f  which is approximately 3 times the velocity error filter time constant T v . More specifically, when nominal gain G value of 0.5, the inner loop or acceleration error filter time constant T f  is exactly 3T v .  
         [0068]     Referring once again to equation 6, the impedance term in equation 6 is:  
               1   +       T   f     ·   s         1   +         T   f       (     1   +   GR     )       ·   s               Eq   .           ⁢   12             
 
 which can be approximated as:  
             1   +       T   f     ·       G   ·   R       1   +   GR       ·   s             Eq   .           ⁢   13             
 
 Referring again to  FIG. 7 , adding an impedance term corresponding to velocity error filter block  86  to equation 13, the impedance term for diagram  80  when the load is disconnected from the motor can be expressed as:  
               Y   ⁡     (   s   )       =       [     1   +       T   f     ·     GRs     1   +   GR           ]         (     1   +       T   v     ⁢   s       )     2               Eq   .           ⁢   14             
 
 Approximating the denominator in equation 14 as first order term yields the following equations:  
               Y   ⁡     (   s   )       =       [     1   +       T   f     ·     GR     1   +     G   ·   R         ·   s       ]       (     1   +     2   ⁢     T   v     ⁢   s       )               Eq   .           ⁢   15             
 
         [0072]     Next, assuming an inner loop filter time constant T f  of 3T v  and exemplary gain G and ratio R values of 0.5 and 6, respectively (see again table above), equation 15 can be simplified as:  
               Y   ⁡     (   s   )       =       (     1   +       9   4     ⁢       T   v     ·   s         )       (     1   +     2   ⁢       T   v     ·   s         )               Eq   .           ⁢   16             
 
 As seen in equation 16, when the load is decoupled from the motor, there is near perfect pole-zero cancellation. 
 
         [0074]     In summary, where the acceleration feedback loop gain is set equal to approximately 0.5 and the acceleration error filter time constant T f  is approximately 3 times the velocity error filter time constant T v , for typical ratio R values, the resulting system is very stable and there is virtually no high frequency attenuation of quantization noise.  
         [0075]     Referring now to  FIG. 8 , a exemplary relatively more complex system  110  which reflects a real motor/load system more accurately is illustrated. While system  110  is more complex than the systems described above, the analysis above is still sound and the manner in which system  10  values should be tuned or selected is the same as above. Generally, in  FIG. 8 , a lost motion block  144  and a spring block  146  are provided which represent disturbances that occur in a typical motor/load configuration characterized by at least some lost motion. Here, the lost motion and spring characteristics along with the load torque T l  and the load inertia m 2  impact overall system operation via a summer  130  which provides its output to a divider block  136 . The output of block  136  is an acceleration value which is provided a summer  132 . In addition to receiving the acceleration value from block  136 , summer  132  also receives command acceleration value a* derived from command velocity ω*. Summer  132  subtracts the feedback acceleration value from the command acceleration value a* and provided its output filter block  134 . Filter block  134  is tuned according to the teachings above to have a gain G of approximately 0.5 and has a filter time constant T f  which is 3 times the velocity loop time constant T v  (not illustrated in  FIG. 8  but nevertheless present in PI regulator  124 ).  
         [0076]     Referring still to  FIG. 8 , system  110  includes six summer  114 ,  126 ,  130 ,  132 ,  142  and  148 , a derivative module  116 , five amplifier blocks  120 ,  134 ,  136 ,  146  and  150 , a PI regulator  124 , a current regulator  128 , four integrator blocks  138 ,  140 ,  151  and  154 , lost motion block  144  and spring block  146 . Velocity command ω* is provided to derivative block  116  which generates command acceleration value a*. Command acceleration value a* is provided to amplifier  120  which multiplies that value by the total motor/load inertia m t  providing its output to summer  126 . Velocity command ω* is also provided to summer  114  which subtracts a velocity feedback signal ω fb  from command velocity ω* providing a velocity error value to PI regulator  124 . Regulator  124  steps up the error signal and provides is output to summer  126 .  
         [0077]     Referring still to  FIG. 8 , summer  132  subtracts an acceleration feedback value a fb  from command acceleration value a* providing an acceleration error to filter  134 . Filter  134  is tuned as described above and provides its output to summer  126 . Summer  126  adds the output of filter  134  to the sum of the other values received and provides is output to current regulator  128 . Regulator  128  provides a motor torque value to summer  130 .  
         [0078]     Summer  130  subtracts the output of spring block  146  from the motor torque value and provides its output to amplifier  136 . Amplifier  136  divides the received signal by motor inertia m 1  thereby generating a motor acceleration value a m  which is provided to integrator block  138 . Block  138  integrates the motor acceleration value a m  thereby generating a motor velocity value ω m  which is provided to integrator block  140 . Block  140  integrates the motor velocity value ω m  thereby generating motor position value P m . Position P m  is provided to summer  142 . Summer  142  subtracts a load position value P l  from the motor position value P m  and provides its output to lost motion block  144 . Lost motion block  144  introduces lost motion dynamics and provides its output to spring block  146 . Spring block  146  provides spring type dynamics as a function of a spring constant k and provides its output to each of summers  130  and  148 . Summer  148  subtracts the load torque T l  from the output of spring block  146  and provides its output to amplifier  150 . Amplifier  150  divides the output of summer  148  by load inertia m 2  and provides its output to integrator  151 . Integrator  151  integrates the received value thereby generating a load velocity signal which is provided to integrator block  154 . Block  154  integrates the received value to generate load position value P l  which is fed back to summer  142 .  
         [0079]     It should be appreciated that the total inertia (m 1 +m 2 ) has to be determined for the coupled motor-load assembly during a commissioning procedure. Such procedures are well known in the art and therefore will not be described here in detail. After total inertia m t  is determined, that value along with the optimal gain G value are used to set the acceleration error filter gain.  
         [0080]     While the invention may be susceptible to various modifications and alternative forms, specific embodiments have been shown by way of example in the drawings and have been described in detail herein. However, it should be understood that the invention is not intended to be limited to the particular forms disclosed.  
         [0081]     To apprise the public of the scope of this invention, the following claims are made: