Physikalische ModellIdentifikation einer Achse

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Physikalische Modell-Identifikation einer Achse

• Semesterarbeit 2003/2004
• Bryn Lloyd
• Betreuer
• Dr. Kraus, Herr Gretler, Herr Eger, Prof. Morari

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Introduction

• Project objective
• Identification of a physical model of the axis of
  the gear grinding machine RZ 400 for Reishauer
  AG, Wallisellen.
• Important Identification of a physical model.
• Behavior of axis corresponds to a (stiff)
  two-mass-oscillator.

Reishauer
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Introduction

• Voltage u and angle f2 are measured.
• Input for identification is M K u
• Output is angular velocity d/dt f2
• Objective Identification of physical parameters
  J1, J2, c1, d0, d1 from Input-Output data.

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transfer function D(s)
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Frequency Response (typical)

• Real Pole 1.1 Hz
• complex Poles 451 exp(j91) Hz
• Zero 7957 Hz

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Identifiability

• Assumptions
• The experiment are informative enough
• The true system corresponds to a two-mass
  oscillator
• Then the coefficients of the transfer function
  can be identified using the input-output data.
• Can the physical parameters be identified?
• Yes. There is an explicit mapping of the
  coefficient-space into the physical parameter
  space.
• ? The physical parameters can be identified from
  the input-output behavior, for any identification
  method, if above assumptions are valid.

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Identifiability Resistance d00
Only 3 independent equations but 4 phys.
Variables ? No explicit mapping physical
parameters can not be identified
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Simulation

• The model of the two-mass was implemented in
  Matlab Simulink.
• Input M Linear Chirp (1 2200 Hz)
• No noise
• Sampling frequency 5000 Hz
• ? Mk
• ? f2k ? d/dt f2k (f2k - f2k-1)/Ts

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Method 1

• Matlab System Identification Toolbox
• ? idgrey model structure.
• User-defined mfile, which returns a
  continuous-time model in state space. The model
  is calculated from the physical parameters
• The model is converted into a discrete-time
  model, so it can be evaluated using the data ?
  loss function.
• The parameters are optimized such that the loss
  function is minimized.

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Method 1
idgrey-Methode
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Results Method 1

• To find difficulties using this method, only 1
  parameter was estimated. The other parameters
  were assumed to be known.
• Method does not work sufficiently!
• Spring c1 is hard to estimate.
• Why?

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Results Method 1Stiff System (spring c1 too
large)
loss function (c1)
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Results Fundamental problemImportant
information at high frequencies (zero of
transfer function at 8000 Hz, sampling frequency
is 5000Hz)
Zero very uncertain ? Coefficient b2 very
uncertain ?physical parameters not identifiable
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Problem zero at 8000 Hz
Sampling frequency (5000Hz)
Zero of true system
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Results Method 1 Summary

• Loss function multimodal (several minima).
• and/or loss function is very flat ? not sensitive
  to change in physical parameters.
• System is very stiff (spring c1 large).
• Zero of transfer function at 8000Hz.

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Conclusion Method 1

• Remember These results were obtained for the
  SIMPLE problem of estimating ONLY ONE phys.
  Parameter!
• ? We need to find a different method
• The fundamental problem will remain
• We have no information at very high frequencies
• Zero of transfer function is at 8000 Hz
• Sampling frequency is 5000 Hz

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Method 2

• First Non-physical model identification
• The non-physical model will be estimated using
  n4sid-method from the System Identification
  Toolbox.
• This method returns a state-space model.
• The frequency response is calculated at N
  frequency points from the non-physical model.
• These points are used for a non-linear least
  square optimization of the physical model.

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Method 2
Data from experiment
Initial values J1,0, J2,0 , c1,0 etc
n4sid-model
physical model
Discrete points of frequency response
adapt the parameters J1,i, J2,i etc
J1,N, J2,N etc
criterion
Loss function
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Method 2

• Without any extra information, this method cannot
  work either.
• ? We need some pre-knowledge to estimate the
  physical parameters.
• Candidates
• Zero of transfer function
• J1
• Two experiments. A defined change of one
  parameter,
• e.g.
• Experiment 1 with J2
• Experiment 2 with J2 ? (? defined)

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Method 2zero is assumed to be known

• Example
• Assumption Zero is at 5.1e4 rad/s
• But True System has a zero at 5.0e4 rad/s
• 5.1 / 5 1.02 ? 2
• Can the phys. Parameters still be estimated
  correctly?
• J1 -28
• J2 154.9
• c1 83.9
• d0 0
• d1 80.3
• VERY SENSITIVE

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Method 2zero is assumed to be known
Zero
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Method 2J1 is assumed to be known

• Example
• Assumption J11.2241.021.2485
• But True System J11.224
• 1.2485 / 1.224 1.02 ? 2
• Can the phys. Parameters still be estimated
  correctly?
• J1 2
• J2 -11
• c1 9.3
• d0 0
• d1 9.3
• VERY SENSITIVE

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Method 2 ?J2 is assumed to be known

• We need two experiments.
• Experiment 1 J1, J2, c1, d0, d1
• Experiment 2 J1, (J2?), c1, d0, d1

Experiment 2
Experiment 1
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Method 2 ?J2 is assumed to be known
2 experiments Input PRBS Sampling 5000
Hz ?J20.147 n4sid model order 14 Frequency
response calculated at 512 points. Estimate of
phys. Parameters 0.5-2.5
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Method 2 Summary

• If we have good pre-knowledge about the system,
  i.e. J1, zero of transfer function or ?J2 , we
  can estimate the (unknown) physical parameters.
• The physical parameters are very sensitive to J1
  and the zero.
• Using two experiments with different J2, the
  difference ?J2 being known, we can estimate all
  five physical parameters quite well.

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Project Conclusions

• The identification of the physical parameters is
  not a simple problem.
• Experiments are not informative enough to
  estimate all physical parameters, using sampling
  frequency 5000 Hz.
• Most promising method is method 2, using 2
  experiments. All other attempts were not
  successful.
• Method 1 works on simulated data, if f1 is
  measured, instead of f2 . Resulting transfer
  function has zero at 415 Hz.
• The methods have not yet been tested with real
  data.

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Method 2J1 is assumed to be known
J2
c1
d1
d0
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Gear grinding machine RZ 400

• Work piece Outside diameter 400 mm
• Root diameter 10 mm
• Number of teeth 5-999
• Module 0.5-8 mm
• Helix angle 45
• Slide travel (Z-axis) 300 mm
• Weight 300 kg