Lead-Lag Compensators

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Lead-Lag Compensators
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1 Introduction to Lead-Lag Compensation

Generally the purpose of the Lead-Lag compensator is to create a controller which has has an overall magnitude of approximately 1. The lead-lag compensator is largely used for phase compensation rather than magnitude. A pole is an integrator above the frequency of the pole. A zero is a derivative above the frequency of the zero.

Adding a pole to the system changes the phase by -90 deg and adding a zero changes the phase by +90 deg. So if the system needs +90 deg added to the phase in a particular frequency band then you can add a zero at a low frequency and follow that zero with a pole at a higher frequency.

2 Lead or Lag Control

Lead and lag control are used to add or reduce phase between 2 frequencies. Typically these frequencies are centered around the open loop crossover frequency. A lead filter typically has unity gain (0 dB) are low frequencies while the lag provides a nonunity gain at low frequencies.[1]


LaTeX: C\left(s\right)=\frac{\tau_{1}s+1}{\tau_{2}s+1} Lead[2]


where

LaTeX: \tau_{1}>\tau_{2} makes this a lead filter and
LaTeX: \tau_{1}<\tau_{2} makes this a lag filter.

2.1 Lead-Lag Control

This is an extension of the Lead and Lag network described above. We can always stick a gain in front of something so reformulate LaTeX: C\left(s\right) above to look like this

LaTeX: C\left(s\right)=k\frac{\tau_{a}s+1}{\tau_{b}s+1}



Then a lead-lag controller would have the form

LaTeX: C\left(s\right)=k\left(\frac{\tau_{a}s+1}{\tau_{b}s+1}\right)\left(\frac{\tau_{c}s+1}{\tau_{d}s+1}\right) [3]


where

LaTeX: k is a gain,
LaTeX: \tau_{a}>\tau_{b} makes the first part a lead filter and
LaTeX: \tau_{c}<\tau_{d} makes the last part a lag filter.

3 References

  • Goodwin, G. C., Graebe, S. F., and Salgado, M. E. 2000 Control System Design. 1st. Prentice Hall PTR. ISBN 0139586539
  • Franklin, G. F., Emami-Naeini, A., and Powell, J. D. 1993 Feedback Control of Dynamic Systems. 3rd. Addison-Wesley Longman Publishing Co., Inc. ISBN 0201527472

3.1 Notes

  1. Franklin et all, pg. 412
  2. Goodwin et all, pg. 170, Eqn. 6.6.1
  3. Goodwin et all, pg. 170, Eqn. 6.6.1