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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.

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