Abstract:
A method and apparatus for use with a plant drive system that receives a reference command signal and generates a torque command to drive a plant, the method for estimating plant inertia and comprising the steps of providing a reference model that models the plant, the model receiving the reference command signal and generating a model output signal as a function thereof, identifying a plant output signal, mathematically combining the reference command signal and the plant output signal to generate a first error value, mathematically combining the plant output signal and the model output signal to generate a second error value, mathematically combining the first and second error values to generate an inertia estimate value and using the inertia estimate value to modify the torque command thereby providing a modified torque command signal to drive the plant.

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 present invention relates to motor controllers and more specifically to a method and apparatus for adaptively adjusting a motor controller as a function of a real time inertia estimate.  
         [0004]     General-purpose industrial motor drive manufacturers supply standard drives for a, large variety of applications such as fans, pumps, conveyors, and web lines. A typical drive includes, among other components, a comparator, a load velocity sensor, a proportional-integral (PI) regulator and a current regulator. The comparator receives a feedback signal from the sensor and a reference command signal (e.g., a command velocity signal) and generates a difference or error signal. The PI regulator receives the error signal and steps up that value, as the label implies, proportionally and integrally, to generate a regulated value. The regulated value is provided to the current regulator that generates a motor torque command signal for driving an associated motor/load.  
         [0005]     Because loads and performance requirements are different for each application, typically, a standard drive has to be “tuned” for a specific application to achieve desired results (i.e., proportional and integral gain factors have to be set as a function of a specific motor (i.e., the “plant”) and load driven by the drive). To properly tune a drive, ideally, system parameters such as plant inertia, friction, damping, and load must be known. During an off-line commissioning procedure (e.g., a procedure typically performed prior to normal operation of a drive), system parameters can be determined and used to tune the drive.  
         [0006]     As known in the industry, at least some system parameters can vary over time and with different operating conditions and consequently it is difficult to keep a drive running optimally even if it is initially tuned off-line. In at least some applications inertia may vary over time.  
         [0007]     One solution for dealing with changing operating parameters has been to develop model reference adaptive controllers (MRACs) that automatically tune a drive to follow a desired or model behavior. A block diagram of an exemplary MRAC  10  is shown in  FIG. 1  and includes a controller  12 , a reference model module  14 , a plant  16 , a summer  18  and an adaptive module  20 . In  FIG. 1 , a reference command r is provided to reference model  14 , controller  12  and adaptive module  20 . A controller output u drives plant  16  to produce a plant output. A plant output signal x is sent back to controller  12  for closed loop control. A reference model output signal x m  is subtracted from the plant output x to form an error signal ε. The reference model output signal x m  and the error signal ε are also received by adaptive module  20  which calculates new controller gain(s) K in an attempt to force the closed loop system to behave like model  14 . Some known systems have used the reference command signal r to calculate gain K changes. Other known schemes have used the model output signal x m  in place of reference command signal r to calculate gain K changes.  
         [0008]     Unfortunately, prior art techniques that use the reference command signal r or the model output signal x m  to calculate gain K changes drastically change the structure of the simple PI regulator control loop that has been implemented in many industrial drives.  
       BRIEF SUMMARY OF THE INVENTION  
       [0009]     It has been recognized that a drive can be configured that includes a conventional PI regulator structure that adapts to changing inertia. To this end, an exemplary drive includes an adaptive module that uses the difference between a command signal and a plant output signal as well as the different between the plant output signal and an ideal model output signal to generate an inertia estimate which is in turn used to alter the output of a PI regulator to adjust for real time changes in system inertia.  
         [0010]     Consistent with the above, at least one inventive embodiment Includes a method for use with a plant drive system that receives a reference command signal and generates a torque command to drive a plant, the method for estimating plant inertia and comprising the steps of providing a reference model that models the plant, the model receiving the reference command signal and generating a model output signal as a function thereof, identifying a plant output signal, mathematically combining the reference command signal and the plant output signal to generate a first error value, mathematically combining the plant output signal and the model output signal to generate a second error value, mathematically combining the first and second error values to generate an inertia estimate value and using the inertia estimate value to modify the torque command thereby providing a modified torque command signal to drive the plant.  
         [0011]     In some embodiments the step of mathematically combining to generate the first error value includes subtracting the plant output signal from the reference command signal. In some cases the step of mathematically combining to generate the second error value includes subtracting the model output signal from the plant output signal. In some cases the step of mathematically combining the first and second error values includes multiplying the first error value and the second error value to generate a combined error value. In some embodiments the step of mathematically combining the first and second error values further includes the step of multiplying the combined error value by a gain function to generate the inertia estimate.  
         [0012]     In some embodiments the step of multiplying the combined error value by a gain function includes multiplying the combined error value by −γ (K p s+K i )/s)(e com ) where e com  is the combined error value, K p  is a proportional gain value, K i  is an integral gain value and g is a rate adaptation gain value. In some cases the method further includes the step of setting the adaptation gain value g to a value between zero and one.  
         [0013]     In some cases the plant drive system also includes a current regulator that receives the modified torque command signal and uses the modified torque command signal to generate a motor torque value used to drive the plant, the method further including the step of mathematically combining the plant output signal, the motor torque value and the inertia estimate to generate a load torque estimate, the step using the inertia estimate value to modify the torque command including using both the inertia estimate value and the load torque estimate to provide a modified torque command signal to drive the plant. In some cases the step of mathematically combining the plant output signal, the motor torque value and the inertia estimate to generate a load torque estimate includes taking the derivative of the plant output signal to generate an acceleration value, multiplying the acceleration value by the inertia estimate value to generate an intermediate value and subtracting the intermediate value from the motor torque value to generate the load torque estimate.  
         [0014]     Some embodiments include a method for use with a plant drive system that receives a reference command signal and generates a torque command to drive a plant, the method for estimating plant inertia and comprising the steps of setting an adjustment gain value, providing a reference model that models the plant, the model receiving the reference command signal and generating a model output signal as a function thereof, identifying a plant output signal, subtracting the plant output signal from the reference command signal to generate a first error value, subtracting the model output signal from the plant output signal to generate a second error value, multiplying the first and second error values to generate a combined error value, multiplying the combined error value by the adjustment gain value to generate the inertia estimate value and using the inertia estimate value to modify the torque command thereby providing a modified torque command signal to drive the plant. In some cases the adjustment gain value is −γ((K p s+K i )/s), where K p  is a proportional gain value, K i  is an integral gain value and g is a rate adaptation gain value and wherein the step of setting the adjustment gain value includes setting values K p , K i  and γ where value γ is set to a value between zero and one.  
         [0015]     Some embodiments include an adaptive apparatus for use with a plant drive system that receives a reference command signal and generates a torque command to drive a plant, the apparatus for estimating plant inertia and comprising a reference model that models the plant, the model receiving the reference command signal and generating a model output signal as a function thereof, a observer for identifying a plant output signal and a processor running a program to perform the steps of mathematically combining the reference command signal and the plant output signal to generate a first error value, mathematically combining the plant output signal and the model output signal to generate a second error value, mathematically combining the first and second error values to generate an inertia estimate value and using the inertia estimate value to modify the torque command thereby providing a modified torque command signal to drive the plant.  
         [0016]     In some cases the processor mathematically combines to generate the first error value by subtracting the plant output signal from the reference command signal. In some cases the processor mathematically combines to generate the second error value by subtracting the model output signal from the plant output signal. In some cases the processor mathematically combines the first and second error values by multiplying the first error value and the second error value to generate a combined error value. In some cases the processor mathematically combines the first and second error values by further multiplying the combined error value by a gain function to generate the inertia estimate. In some cases the gain function is γ((K p s+K i )/s) where K p  is a proportional gain value, K i  is an integral gain value and γ is a rate adaptation gain value.  
         [0017]     In some embodiments the plant drive system also includes a current regulator that receives the modified torque command signal and uses the modified torque command signal to generate a motor torque value used to drive the plant, the processor further performing the program to mathematically combine the plant output signal, the motor torque value and the inertia estimate to generate a load torque estimate, the processor using the inertia estimate value to modify the torque command by using both the inertia estimate value and the load torque estimate to provide a modified torque command signal to drive the plant. In some cases the processor mathematically combines the plant output signal, the motor torque value and the inertia estimate to generate a load torque estimate by taking the derivative of the plant output signal to generate an acceleration value, multiplying the acceleration value by the inertia estimate value to generate an intermediate value and subtracting the intermediate value from the motor torque value to generate the load torque estimate.  
         [0018]     Some embodiments include an apparatus that receives a reference command signal and generates a torque command signal to drive a plant, the apparatus comprising a plant model receiving the reference command signal and generating a model output signal, an observer for identifying a plant output signal, an inertia estimator receiving the reference command signal, the plant output signal and the model output signal and generating an inertia estimate value, a proportional-integral regulator receiving the reference command signal and stepping up the reference command signal to produce a regulated value and a multiplier for multiplying the regulated value by the inertia estimate value to generate the torque command signal to drive the plant.  
         [0019]     In some cases the inertia estimator generates the inertia estimate value by subtracting the plant output signal from the reference command signal to generate a first error value, subtracting the model output signal from the plant output signal to generate a second error value, multiplying the first and second error values to generate a combined error value and multiplying the combined error value by the adjustment gain value to generate the inertia estimate value.  
         [0020]     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  
       [0021]      FIG. 1  is a schematic diagram illustrating an exemplary model reference adaptive controller;  
         [0022]      FIG. 2  is a schematic diagram illustrating a controller according to at least one embodiment of the present invention;  
         [0023]      FIG. 3  is similar to  FIG. 2 , albeit including a model module and an adaptive module according to at least some embodiments of the present invention;  
         [0024]      FIG. 4  is a graph illustrating motor torque and load torque related to a first simulation using the controller of  FIG. 3 ;  
         [0025]      FIG. 5  is a graph illustrating a command velocity signal, a model output signal and a plant output signal that result during a first simulation using the controller of  FIG. 3 ;  
         [0026]      FIG. 6  is a graph illustrating a changing inertia estimate curve that results from the first simulation using the controller of  FIG. 3 ;  
         [0027]      FIG. 7  is a graph illustrating command, model and plant velocity signals as well as an inertia curve generated during a second simulation using the circuit of  FIG. 2 ;  
         [0028]      FIG. 8  is a graph illustrating an inertia estimate curve similar to the curve illustrated in  FIG. 6 , albeit derived using a controller that included a low pass filter in series with the adaptive module of  FIG. 3 ;  
         [0029]      FIG. 9  is a graph illustrating inertia estimate curves generated with different adaptation rate gain values where the rate gain values were 0.1, 0.2, 0.5 and 1.0;  
         [0030]      FIG. 10  is a graph illustrating different inertia estimate curves given a single adaptation rate gain value γ where system inertias were 0.04, 04 and 4.0;  
         [0031]      FIG. 11  is a schematic diagram illustrating a controller similar to the controller of  FIG. 3 , albeit including a load observer module;  
         [0032]      FIG. 12  is a graph illustrating inertia estimate curves generated with and without the load observer module of  FIG. 11 ;  
         [0033]      13  is a graph illustrating inertia estimate curves generated using the controller of  FIG. 11  where the proportional gain was set to zero and the low pass filter was removed;  
         [0034]      FIG. 14  is a graph illustrating motor torque and load torque curves similar to the curves shown in  FIG. 4 ;  
         [0035]      FIG. 15  is a graph illustrating command, plant and model velocity signals during one additional simulation using the controller of  FIG. 11 ; and  
         [0036]      FIG. 16  is a graph similar to  FIG. 6 , albeit illustrating an inertia estimate curve resulting from one additional simulation. 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0000]     A. Development of Adaptive Equation  
         [0037]     An exemplary controller/plant model system  26  is shown in  FIG. 2  where it can be seen that a controller receives a command velocity signal ω c  and causes a plant  42  to operate at a plant velocity cap. In addition to plant  42  that is modeled as a simpler inertia J, model  26  includes a conventional PI regulator that includes an integrator  30 , an integral gain  34 , a proportional gain  32  and a summer  44 , an adaptive inertia estimate multiplier block  36 , a current regulator modeled in block  38  and first and second summers  28  and  40 , respectively. Summer  28  receives each of the command velocity signal ω c  and the plant velocity signal ω p  and subtracts the plant velocity signal ω p  from the command velocity signal to generate an error value. The error value is provided to the PI regulator which in turn generates an intermediate command signal α c . The proportional and integral gains K p  and K i , are a function of the desired damping ζ and bandwidth ω n  for the system as indicated by the following equations: 
 
K p =2ζω n   Eq. 1 
 
K i =ω n   2   Eq. 2 
 
 Some optional tuning criteria may be utilized to set the values of gains K i  and K p . Friction, damping, and similar terms can be set to zero without appreciably effecting accuracy of the system for two reasons. First, inertia is the dominant parameter in many drive systems. Knowing the inertia is the key to tuning the system to have good base-band performance. Second, additional parameters complicate development of a workable adaptation law derivation. The intermediate command signal □c is provided to adaptive inertia estimate multiplier block  36  which, as the label implies, multiplies the intermediate command signal α c  by an essentially real time inertia estimate J est  to generate a command torque value τ c . 
 
         [0038]     Referring still to  FIG. 2 , block  38  models operation of a current regulator and related dynamics. To simplify development of the inventive adaptive control law, it can be assumed that the current regulator  38  has unity gain. Unity gain is a realistic assumption because the current loop dynamics are typically an order of magnitude or more greater than the velocity loop dynamics. Current regulator  38  receives the command torque value τ c  and uses that value to generate a motor torque value τ m .  
         [0039]     Summer  40  receives a load torque value τ l  as well as the motor torque value τ m . The load torque value τ l  models load torque and system disturbances. Summer  40  adds the τ m  and τ l  values to generate an output that is provided to plant model block  42 . Thus, the present invention contemplates a model that is consistent with a control scheme that includes a conventional PI regulator structure and an adaptive gain parameter J est  that is equal to a real time system inertia.  
         [0040]     Equation 3 represents the command response transfer function of the velocity output ω p  with respect to the velocity input ω c .  
                 ω   p       ω   c       =           K   p     ⁢   s     +     K   i             J     J   est       ⁢     s   2       +       K   p     ⁢   s     +     K   i                 Eq   .           ⁢   3             
 
 If the estimate J est  of the inertia is equal to the plant inertia J, then the characteristic equation of the transfer function is a well-behaved second order model as defined by the parameters in Equations 1 and 2. 
 
         [0041]     The disturbance rejection transfer function of the velocity output ω p  with respect to the load torque input τ l  is shown in Equation 4:  
                 ω   p       τ   l       ⁢         1   J     ⁢   s           J     J   est       ⁢     s   2       +       K   p     ⁢   s     +     K   i                 Eq   .           ⁢   4             
 
 Equation 4 has the same dynamics as the transfer function shown in Equation 3 and both transfer functions demonstrate optimal performance with large gain values K p  and K i . The magnitudes of gains K p  and K i  are limited by the physical properties of the torque producing components in the system. 
 
         [0042]     Referring again to  FIG. 2 , the reference command signal ω c , plant output signal ω p  and output of block  36  may be represented by labels r, x and u, respectively, to be consistent with the labels in  FIG. 1 . Here, utilizing a state space notation where the current regulator gain is unity, the plant  42  output x  can be represented as: 
 
 {dot over (x)}=ax+bu   Eq. 5 
 
 where a=0 and b=1/J. 
 
         [0043]     In the case of the model, it is assumed that the inertia estimate J est  is equal to the actual plant inertia J (i.e., J est  and J in Equation 3 above cancel) and therefore the model output can be represented by the following equation:  
               x   m     =             K   p     ⁢   s     +     K   i           s   2     +       K   p     ⁢   s     +     K   i         ⁢   r             Eq   .           ⁢   6             
 
 or equivalently:  
               x   m     =           2   ⁢           ⁢   ξ   ⁢           ⁢     ω   n     ⁢   s     +     ω   n   2           s   2     +     2   ⁢   ξ   ⁢           ⁢     ω   n     ⁢   s     +     ω   n   2         ⁢   r             Eq   .           ⁢   7             
 
 where K p  and K i  are defined in Equations 1 and 2 above. Here, because the model as shown in  FIG. 2  includes a conventional PI regulator, a control law representing the output signal of block  36  can be represented by the following equation:  
             u   =     κ   *     (           K   p     ⁢   s     +     K   i       s     )     ⁢     (     r   -   x     )               Eq   .           ⁢   8             
 
 where κ* is an unknown optimal gain. Gains K p  and K i  are the PI regulator gains and (r−x) is the error between the reference command signal r and the actual plant output signal x. 
 
         [0044]     Equations 5 and 8 can be combined and the terms rearranged to yield the following equation:  
               x   r     =         b   ⁢           ⁢   κ   *     K   p     ⁢   s     +     b   ⁢           ⁢   κ   *     K   i                 s   2     ⁡     (       b   ⁢           ⁢   κ   *     K   p       -   α     )       ⁢   s     +     b   ⁢           ⁢   κ   *     K   i                   Eq   .           ⁢   9             
 
 Equations 6 and 9 can be combined to yield the following relationships:  
               x   r     =           b   ⁢           ⁢   κ   *     K   p     ⁢   s     +     b   ⁢           ⁢   κ   *     K   i             s   2     +       (       b   ⁢           ⁢   κ   *     K   p       -   α     )     ⁢   s     +     b   ⁢           ⁢   κ   *     K   i           =             K   p     ⁢   s     +     K   i           s   2     +       K   p     ⁢   s     +     k   i         =       x   m     r                 Eq   .           ⁢   10             
 
 Thus, the optimal gain κ* is equal to the instantaneous plant inertia J (i.e., where b=1/J, κ* cancels b if κ*=J). In Equation 10, note that factor a has an assumed zero value. 
 
         [0045]     Thus, an adaptive law can be established by starting with the desired result that: 
 
{dot over (x)}={dot over (x)} m   Eq. 11 
 
 and the error between the actual plant output x and model output x m  can be expressed by the following equation:  
             e   =       x   -     x   m       =     ɛ   =       b     s   +     K   p     +       K   i     ⁢     1   s           ⁢       k   ~     ⁡     (           K   p     ⁢   s   ⁢           ⁢   K     +     K   i       s     )       ⁢     (     r   -   x     )                   Eq   .           ⁢   12             
 
 where {tilde over (k)}=k−k*, k is the active gain in the controller and k* is the desired gain or parameter value. 
 
         [0046]     Assuming a strictly positive real (SPR) Lyapunov function, then:  
             v   =         ɛ   2     2     +           k   ~     2       2   ⁢           ⁢   γ       ⁢        b                    Eq   .           ⁢   13             
 
 And the time derivative can be expressed as:  
               V   .     =       ɛ   ⁢           ⁢     ɛ   .       +         κ   ~     γ     ⁢        b        ⁢     κ     ~   .                   Eq   .             
 
 After finding {dot over (ε)} and substituting into Equation 14, the following adaptation law results:  
               κ     ~   .       =       -   γ     ⁢           ⁢     sgn   ⁡     (   b   )       ⁢     (           K   p     ⁢   s     +     K   i       s     )     ⁢     (     r   -   x     )     ⁢   ɛ             Eq   .           ⁢   15             
 
 where γ is a positive adaptation rate gain and (r−x)ε is a combined error value e com . 
 
         [0047]     The adaptation law represented by Equation 15 differs from previous laws because the gain is a function of two error signals. The first error signal ε represents the difference between the reference model output x m  and the plant output x. The second error (r−x) represents the difference between the reference command signal r and the plant output x. The proportional and integral gains from the controller are also present in the adaptive law. The sign function of b (i.e., sgnb), can be ignored because parameter b is equal to the inertia of the system which is always positive.  
         [0000]     B. Simulation Results  
         [0048]     The adaptive law and inertia estimate represented by Equation 15 can be used to modify  FIG. 2  resulting in the controller  50  of  FIG. 3  linked to plant model  42 . Controller  50  includes summers  28 ,  44  and  40 , a PI regulator including components  30 ,  34  and  32 , an adaptive inertia estimate multiplier block  36  and a current regulator  38  in common with the controller of  FIG. 2 . In addition, controller  50  includes a reference model module  76  and an adaptive module  56 . Model module  76  receives the reference command velocity signal ω c  and outputs a model signal com consistent with Equation 7 above.  
         [0049]     Adaptive module  56  implements the Equation 15 adaptive law. To this end, module  56  includes a summer  78 , a multiplier block  68 , a PI gain block  66  and an adaptation rate gain block  69 . Summer  78  receives the model output signal ω m  and the plant output signal ω p  and subtracts the model signal □m from the plant output signal ω p  to generate error value ε. Multiplier block  68  multiplies the error from summer  28  by error ε and its output is provided to block  66 . Block  66  proportionally and integrally steps up the output of block  68  and provides its output to adaptation rate gain block  69  which in turn generates inertia estimate J est  which is provided to block  36  for stepping up intermediate command signal α c .  
         [0050]     For the purposes of the simulation, the reference model and closed loop system were chosen to be critically damped with ζ=1 and to have a bandwidth ωn=10 rad/sec and the adaptive rate gain γ was set to 0.5. In a per unit system, when rated torque=1 is applied to the system, the time it takes to reach rated velocity=1 is the equivalent system inertia with units of seconds. In the exemplary simulation here, inertia J=0.6 sec.  
         [0051]     Referring to  FIG. 4 , during the simulation, a sinusoidal velocity command  81  was applied to the system with an amplitude of 1 and a frequency of 1 Hz. A step load  83  disturbance=1 was applied at 5 seconds. Resulting velocity profiles including a command velocity  21 , a model velocity  23  and a plant velocity  25  are shown in  FIG. 5 .  
         [0052]     As seen in  FIG. 5 , model velocity  23  follows the command velocity  21  with a slight lag and 15% overshoot because of the chosen design parameters and command frequency. Plant velocity  25  identically tracks (and hence is difficult to discern) the command velocity  21  (i.e., the reference signal) except during start-up (i.e., just after time 0 in  FIG. 4 —see  31 ) and when the load is applied (i.e., just after the 5 second in  FIG. 4 —see  33 ).  
         [0053]     Referring to  FIG. 6 , a graph illustrating the adaptive inertia estimate signal  27  is shown for the above described simulation. In  FIG. 6  it can be seen that the system inertia is quickly and adaptively identified after startup with some overshoot. When the disturbance is applied at the 5 second time, the inertia estimate is refigured within a short time.  
         [0054]     Referring again to  FIG. 3 , if current regulator  38  becomes saturated, the adaptive module must be controlled so that the inertia estimate does not continue to integrate up to an incorrect value. Thus, if current regulator  38  becomes saturated, the adaptation mechanism is held constant or disabled in at least some applications.  
         [0055]     As illustrated in  FIG. 5 , in the previous simulation, the velocity command was sinusoidal. A more typical velocity command may have a trapezoidal profile. To this end, referring to  FIG. 7 , during a second simulation a velocity command signal  89  was ramped up and down over a time period of 0.3 sec with a dwell time of 0.2 sec both at unity and 0 velocities. Also shown in  FIG. 7  are a model output signal  91 , a plant output signal  93  and an inertia estimate curve  95 . Inertia curve  95  has the same axis as the velocity signals with units of seconds. It should be appreciated that the inertia estimate with the trapezoidal excitation has a smoother profile than the estimate with the sinusoidal excitation as illustrated in  FIG. 6 . Once again the model output signal  91  slightly lags the reference or command signal  89  and overshoots the command signal by approximately 15%. The plant output signal  93  tracks the mode output signal  91  well and only deviates during start up (see  97 ) and when the load torque is applied at the five second time (see  99 ).  
         [0056]     Plant inertia in industrial applications usually changes slowly over time. For this reason, in at least some applications, a filter can be added in series with the inertia estimate loop to eliminate or reduce oscillations that occur in the inertia estimate. In addition, a filter helps eliminate disturbances that occur when a load torque is applied to the system. The sinusoidal excitation that produced the curves in  FIGS. 5 and 6  was repeated with a first order low pass filter (LPF) in series with module  69  (see phantom  169  in  FIG. 3 ) with a time constant of τ=0.12 seconds. The resulting inertia estimate signal  51  is shown in  FIG. 8 . Comparing the inertia curves in  FIGS. 6 and 8 , the inertia estimate determination in  FIG. 8  is relatively smoother when a filter is added to the inertia estimate path. In addition to the filter, a rate limiter and/or saturation filter could be placed in series with the inertia estimate branch of the controller configured in  FIG. 3 .  
         [0057]     To show the effects of gain γ on the rate at which the adaptive module  56  calculates the inertia estimate, simulations were run with different values of γ.  FIG. 9  shows inertia estimate curves  101 ,  103 ,  105  and  107  that correspond to γ values of 0.1; 0.2; 0.5 and 1.0, respectively, where the plant inertia J was equal to 0.6 seconds for all four simulations. During these simulations no filters where placed in series with the inertia estimate and the control parameters were set to ζ=1 and ωn=10 rad/sec. It can be seen that as value γ is increased toward one, the inertia estimate is calculated more quickly but that the overshoot becomes greater.  
         [0058]     For a given rate gain γ, the adaptive module can identify a wide range of inertia values depending on the plant included in a system.  FIG. 10  includes curves  112 ,  114  and  116  corresponding to inertia values of 0.04 seconds, 0.4 seconds and 4.0 seconds, respectively, that were identified with a fixed gain γ=0.75. For comparison, the inertia estimates were normalized for the plot by dividing through by the actual inertia. To generate the  FIG. 10  curves, a LPF was placed in series with the inertia estimate where the filter had a time constant τ=0.75 sec and parameters ζ and ωn were set to 1 and 4 rad/second, respectively. It can be seen that as the system inertia value decreases, at startup the adaptive module requires less time to accurately estimate the inertia but overshoot is greater. It can also be seen that when the system inertia is very small, a load torque disturbance has a large affect on the transient inertia estimate and that after a disturbance it takes a relatively long time for the inertia estimate to be accurately redetermined.  
         [0059]     In at least some cases where system inertia is small, an additional estimator to identify the load torque is helpful to reduce the error in the transient inertia estimate. To this end,  FIG. 11  illustrates another controller configuration  50 ′ that is linked to a plant  42 ′ that, in addition to the components described above, includes a load estimator  158 . In  FIG. 11 , components that are common with the components of  FIG. 3  are identified by the same numeric labels followed by a prime. For instance, the current regulators in  FIGS. 3 and 11  are referenced via numbers  38  and  38 ′, respectively, the motor models are represented by numerals  72  and  72 ′, respectively, etc. Components that are common among controllers  50  and  50 ′ are not described again here in detail in order to simplify this explanation.  
         [0060]     Referring still to  FIG. 11 , load estimator  158  includes a derivative module  200 , a multiplier block  202 , first and second summers  204  and  208  and a low pass filter block  206 . Derivative module  200  receives the plant output signal ω p  and, as the label implies, calculates the derivative thereof to generate an acceleration signal that is provided to block  202 . Block  202  receives the acceleration signal and multiplies that signal by the inertia estimate J est  providing its output to summer  204 . Summer  204  subtracts the value received from block  202  from the motor torque value τ m  to generate a load torque estimate τ lest  that is filtered by filter  206 . Summer  208  receives the filtered torque estimate τ lest  and the intermediate value α c  and adds those two values to generate a command torque value τ c  which is provided to current regulator  38 . Here, the LPF  206  is included to filter out the high frequency noise generated by taking the derivative via block  200 .  
         [0061]     Referring to  FIG. 12 , a graph illustrates inertia estimate curves  130  and  132  with and without the load torque estimator in place, respectively, where the plant inertia was 0.04 seconds, the design parameters were ζ=1 and ωn=4 rad/second, the rate gain was γ=0.75 and the adaptation filter time constant and load observer filter time constant were 0.75 and 0.1 seconds, respectively. As illustrated, the load observer reduced the inertia estimate overshoot by approximately 30%. The proportional gain K p  in the adaptive module  56  contributes to an aggressive inertia estimate. The LPF helps to reduce the noise in the estimate.  
         [0062]     Referring again to  FIG. 11 , it has been recognized that similar results can be achieved by eliminating the proportional gain in the adaptive module and eliminating the LPF  206 . The simulation conditions used to generate the curves in  FIG. 10  were reproduced with K p =0 and K i =ω n   2  for the adaptive module and the resulting normalized inertia curves  131 ,  133  and  135  are shown in  FIG. 13  for inertia values of 0.04, 0.4 and 4.0, respectively. Comparing  FIGS. 10 and 13 , it can be seen that similar inertia estimates are produced with only integral gain versus proportional-integral gain. In addition, LPF  206  (see again  FIG. 11 ) does not have to be implemented where the proportional gain is set to a zero value.  
         [0063]     One final simulation applied a clipped sinusoidal type load disturbance where a velocity command signal was sinusoidal.  FIG. 14  is a graph including curves of the motor torque  140  required to drive the motor to follow the velocity command and the load torque disturbance  142 . The clipped sinusoidal load torque was applied after 4 seconds. This disturbance signal was programmed to have the same magnitude and frequency as the motor command torque and to be 180 degrees out of phase. Consequently, the load disturbance was always opposing the motion of the motor and drive. The peak magnitude of the load disturbance sine wave was clipped to hold a constant value of 2 pu torque.  
         [0064]     The resulting velocity signal curves are shown in  FIG. 15  and include a command velocity  150 , a model output velocity  152  and a plant output velocity  154 . As illustrated, the plant output velocity signal  154  quickly tracks the model signal  152  until there is a slight drop in magnitude when the load signal is applied at four seconds. After the four second time, the adaptation algorithm quickly recovers and corrects the velocity feedback signal.  
         [0065]     The inertia estimate curve  170  corresponding to the curves in  FIG. 15  is shown in  FIG. 16 . The inertia estimate integrates up to the programmed value of 1 pu. When the load torque is applied after 4 seconds, the inertia estimate continues to integrate higher. The new steady state value has a mean of 1.4 seconds. The motor controller observes an opposing torque in the lower velocity feedback signal. The controller increases motor torque command to overcome the disturbance. The adaptation mechanism integrates to a higher value because it interprets the higher torque signal as an increase in inertia. Thus, it should be appreciated that the adaptive module  56  cannot distinguish between an increase in inertia and a load torque signal that always opposes motion of the regulator.  
         [0066]     Although not presented here, experimental results substantially confirmed the simulation results described above.  
         [0067]     One or more specific embodiments of the present invention have been described above. It should be appreciated that in the development of any such actual implementation, as in any engineering or design project, numerous implementation-specific decisions must be made to achieve the developers&#39; specific goals, such as compliance with system-related and business related constraints, which may vary from one implementation to another. Moreover, it should be appreciated that such a development effort might be complex and time consuming, but would nevertheless be a routine undertaking of design, fabrication, and manufacture for those of ordinary skill having the benefit of this disclosure.  
         [0068]     Thus, the invention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the following appended claims.  
         [0069]     To apprise the public of the scope of this invention, the following claims are made: