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Digital Control of Repetitive Errors in Disk Drive Systems (PowerPoint)

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					  Digital Control of Repetitive Errors
         in Disk Drive Systems
  Kok Kia Chew and Masayoshi Tomizuka

                Presented by Corey Drechsler
as a part of the DSSC Servos and Channels Seminar Series




                    10 August 2011
                       The System

 The plant is an 8 in. IBM hard drive, but is still
  mechanically similar to modern hard drives.
  Spindle Speed: 3125 RPM
  Track Density: 1200 TPI
                               Spindle
                                Motor                          VCM




                               Disk                                  Head



                                  ~450mm
                                         8-in. IBM 62PC Hard Drive
                          The Problem :
                          Periodic Disturbances
 Simple frequency response
  techniques can be used for track
  following, given constant or
  slowly varying disturbances.
 Periodic disturbances can be
  large and more difficult to
  eliminate.
    Additionally, there are often many
     harmonics of the signal present
 Changing existing (analog)
  controllers is undesirable.
                     Controller Layout

 An add-on repetitive controller can reduce periodic
  disturbances.
                           Computational Limits

 A 16 MHz Intel 80386 CPU is used for all digital
  processing (approximately 250 kFLOPS)
 Spindle speed of 3125 RPM (52 Hz)
 Up to 54 samples per track (356 µsec sample time)
             cpu _ speed _ in _ flops
                                       operations
               N  spindle _ speed
                250000 operations
                        sec ond
                                              89 operations
            54 revolution  52 revolutions
                samples                            sample
                                 sec ond


 Limited to 89 operations at the highest sampling rate!
 Higher sample rates allow more harmonics to be
  removed, but allows less time for computation.
                      Repetitive Controller Design:
                      Overview
 The block diagram is redrawn from the perspective
  of the repetitive controller.




 The “system” is not just the analog controller and
  the plant in series, but is the closed-loop response
  from ur(k) to er(k).
 Reducing the magnitude of the output/error signal
  er(k) is the goal of the repetitive controller design.
                        Repetitive Controller Design:
                        System Identification
 The system transfer function can be represented in
  the following form:
                          
                    GS z 1 
                                         
                              z  d B z 1
                                 A z 1 
  where
                Az 1   1  a1 z 1    a n z  n
                Bz 1   b0  b1 z 1    bm z m

 The numerator B(z-1) is then factored into two parts,
  B+(z-1) and B-(z-1), which are the cancelable (stable)
  and uncancelable parts of B
                        Repetitive Controller Design:
                        System Identification Example
 For the test system, a least-squares identification
  algorithm was used. At N=54 (sample time of 356
  µsec), the function was estimated as:
                            z 3  0.0972  0.0848 z 1 
              G S z 1  
                            1  1.6054 z 1  0.7791 z  2
                                                                      Pole-Zero Map
                                                     1




 In this case, there are no
                                                   0.8

                                                   0.6


  unstable zeros, so B+ = B                        0.4



  and B- = 1
                                  Imaginary Axis
                                                   0.2


                                                     0


                                                   -0.2

                                                   -0.4


                                                   -0.6

                                                   -0.8

                                                    -1
                                                          -1   -0.5        0          0.5   1
                                                                        Real Axis
                                 Repetitive Controller Design:
                                 Ideal Controller
 The ideal repetitive controller is then:

        
   GC z 1   
                              
               k r z  N  d A z 1 B   z 
                                                        0  kr  2
                               
                  1  z  N bB  z 1                   
                                                b  max B e     j
                                                                          2
                                                                               ,0 

 A and B+ cancel the stable portion of the system.
 B- cancels only the phase of the unstable zeros,
  while b cancels their magnitude.
 kr sets the controller gain, but is limited to ensure
  robust stability.
 The z-N/(1-z-N) is the internal model of the disturbance
  and cancels the fundamental frequency and all of the
  un-aliased harmonics.
                                                           Repetitive Controller Design:
                                                           Ideal Controller Example
                       The controller for the test system becomes:
                                   1
                              GC z                        
                                           z 54 1 - 1.605 z -1 + 0.7791z -2                                     
                                                             
                                         1  z 54 - 0.0972 z -3 - 0.0848 z -4                                   
                        where kr=b=1
                                     Bode Diagram                                                    Pole-Zero Map
                    100                                                               1

                     80                                                             0.8
   Magnitude (dB)




                     60
                                                                                    0.6
                     40
                                                                                    0.4
                     20




                                                                   Imaginary Axis
                                                                                    0.2
                      0

                    -20                                                               0
                    720
                                                                                    -0.2
                    540
Phase (deg)




                                                                                    -0.4
                    360

                    180                                                             -0.6

                      0                                                             -0.8

                    -180                                                             -1
                         0       1                     2       3
                       10       10                    10      10                      -1.5   -1   -0.5               0   0.5   1
                                     Frequency (Hz)                                                      Real Axis
                            Repetitive Controller Design:
                            Robust Controller
 However, this system is only stable if the system
  model is sufficiently accurate.
 To ensure robust stability, a (typically low pass) filter
  q(z-1) is added to the controller, sacrificing high-
  frequency regulation for stability:

                   
                     1
                          
                                          
                            k r z  N  d q z 1 A z 1 B   z 
              GC z
                                          
                               1  q z 1 z  N bB  z 1

 This typically does not hurt the tracking
  performance, as higher harmonics of the
  disturbance tend to be lower in magnitude.
                         Repetitive Controller Design:
                         Stability
 The system is provably stable if:
               Ae  j B  e j Ba e  j       1
        1  kr                                            ,0 
                 bB e Aa e 
                        j          j
                                                   qe 
                                                       j


  where Aa and Ba represent the actual system transfer
  function

 Note that if A = Aa, B = Ba (i.e. the model is perfect),
  and 0 < kr < 2, the left-hand side is always less than
  unity, and is therefore stable for any of the choices
  of q investigated in the paper.
                     Repetitive Controller Design:
                     Filter Selection
 The figures below show the frequency response and
  stability bounds for three different filters.
 For the test system, q2 is used, as it provides the
  largest range of robustness, while not significantly
  affecting the results.     0
                                         Bode Diagram



                                           -2



                         Magnitude (dB)
                                           -4



                                           -6     q1 = 1

                                                  q2 = (z + 2 + z-1)/4
                                           -8
                                                  q3 = (z + 6 + z-1)/8

                                          -10 0        1                 2    3
                                            10      10                 10    10
                                                           Frequency (Hz)
                          Repetitive Controller Design:
                          Robust Controller Example
 The robust controller becomes:
                z  2  z 1  54
                             z
        
   GC z 1  
                     4       
                                           
                                      1 - 1.605 z -1 + 0.7791z -2                                  
                                z 
                                            
              z  2  z 1  54  - 0.0972 z -3 - 0.0848 z -4
            1  
                                                                                                   
                                 
                      4                                                  Pole-Zero Map

                                                          1

                                                        0.8



 The high frequency poles                              0.6

                                                        0.4

  move away from the unit

                                       Imaginary Axis
                                                        0.2

                                                          0
  circle, decreasing the gain                           -0.2


  (and hence rejection) at                              -0.4

                                                        -0.6

  higher harmonics                                      -0.8

                                                         -1

                                                          -1.5   -1   -0.5        0          0.5       1   1.5
                                                                               Real Axis
Experimental Results:
Original vs. Repetitive Control
Experimental Results:
Varying Sample Rates
                     Conclusions

 Discrete time repetitive control can significantly
  reduce the effect of a periodic disturbance and its
  harmonics.
 This controller can be added to an existing control
  system without the need to modify the original
  controller.
 Computational limits restrict how many harmonics
  can be removed by the controller.
 Modeling errors also restrict the amount of reduction
  for high frequency harmonics.
                       References

 T. Yamaguchi, Modeling and control of a disk file head-
  position system. Proc. Instn. Mech. Engrs., Vol 215, Part
  I, 2001, pp. 549-568
 K. Chew and M. Tomizuka, Digital control of repetitive
  errors in disk drive systems. IEEE Control Systems
  Mag., Vol 10, Issue 1, 1990, pp. 16-19
 K. Chew and M. Tomizuka, Digital control of repetitive
  errors in disk drive systems. Proc. Amer. Cont. Conf.,
  Vol 1, 1989, pp. 540-548

				
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