Position Control using Lead Compensators

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Position Control using Lead Compensators

• Bill Barraclough

Sheffield Hallam University
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Technology considered

• A small d.c. motor actually to drive the system
• Torque (and therefore acceleration) depends on
  applied voltage
• Back e.m.f. of the motor means the T.F. is of the
  form K/s(1 Ts)
• So it inherently contains integration !

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Possible Controllers

• Proportional Derivative (Stability problems
  will arise if we include integration)
• Velocity feedback using a tachogenerator
• Lead Compensator
• We will concentrate on the lead compensator but
  we will also mention the other possibilities

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The lead compensator

• These controllers often provide good performance
  without some of the drawbacks of the p.i.d.
• We will obtain the transfer function of a
  suitable lead compensator for a small d.c. motor
  used to control position ...
• ... and produce a digital version.

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The Motor

• We will base the work on a motor type which we
  have in the laboratory ...
• ... and on which you will have the opportunity to
  try out the resulting controllers !

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The Lead Compensator

• Its transfer function (and that of the lag
  compensator) is of the form

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The Motor

• The laboratory motors have a transfer function
  approximately

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The Procedure

• Obtain the TF in s of the lead compensator
• Digitise it
• Implement it !

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Two Approaches

• Decide to replace the motors pole by a faster
  one. This determines a ...
• ... and use trial and error to find K and b.
• Or decide the closed-loop T.F. we require and
  deduce the controller T.F. needed to achieve it.

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Method 1 Trial and Error

• Controller transfer function

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MATLAB/SIMULINK to the rescue!

• Use of MATLAB and SIMULINK suggested that good
  performance would result from the following
  controller

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We have two methods of digitising this T.F.

• The simple method
• The Tustin method

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Which is better ?

• The simple method is easier algebraically
• but ...
• The Tustin method leads to a controller which
  performs more nearly like the analogue version.

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The Simple Method

• We will do the conversion by the simple method
  using an interval Ts of 0.1 s.
• 1.33(s2.5)/(s7) becomes ...
• 1.33(1-z-1)/0.1 2.5/(1-z-1)/0.1 7
• which by algebra gives
• (0.9782 - 0.7824z-1)/(1 - 0.5882z-1)

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The Tustin Method

• Now the sum becomes (since 2/Ts 20)
• 1.3320(1-z-1)/(1z-1)2.5/20(1-z-1)/(1z-1)
  7
• giving by unreliable Barraclough mathematics
• (1.1085 - 0.8619z-1)/(1 - 0.4815z-1)

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How do the controllers perform ?

• Both digital versions have slightly more
  overshoot than the analogue version.
• The Tustin one is nearer to the analogue version
  than is the simple one.
• Both digital versions give a reasonably good
  performance.

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Designing for a particular closed-loop performance

• Suppose we decide we require an undamped natural
  frequency of 5 rad/s ...
• ... and a damping ratio of 0.8.
• This means that the closed-loop transfer function
  needs to be
• 25/(s2 8s 25)

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The required controller T.F. ?

• We have
• So forward path D(s) x G(s) ..
• and the CLTF is D(s)G(s)/1 D(s)G(s)

D(s)
G(s)

_
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The sum continues ...

• This means that
• D(s)G(s)/1 D(s)G(s) 25/(s2 8s 25)
• and as G(s) 12/s(s 2.5)
• we will show that
• D(s) must be 2.08(s 2.5)/(s 8)
• to produce the required performance.

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Your turn !

• If we use a sampling interval of 0.1 s again
• What will the digitised transfer functions be
• using the simple method ...
• ... and the Tustin method ?
• We can check the Tustin one by MATLAB
• using the c2dm command.

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Your Turn continued

• The syntax is
• nd,ddc2dm(num,den,ts,tustin)
• num and den represent the T.F. in s
• ts is the sampling interval
• nd and dd represent the T.F. in z.

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Summary

• Lead compensators are often useful in position
  control systems using a d.c. motor
• with a Type 1 transfer function.
• We have examined two methods of doing the
  digitisation.
• The Tustin method gives the best approximation to
  the analogue performance for a given sampling
  interval.