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 Lead-Lag Compensators  In order to prevent spam, users must register before they can edit or create articles.

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.

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

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

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.