12.7: The Recipe for Loopshaping
- Page ID
- 50957
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In the above analysis, we have extensively described what the open loop transfer function \(PC\) should look like, to meet robustness and performance specifications. We have said very little about how to get the compensator \(C\), the critical component. For clarity, let the designed loop transfer function be renamed, \(L = PC\). It suffices to just pick
\[ C \, = \, L/P. \]
This simple step involves a plant inversion: the idea is to first shape \(L\) as a stable transfer function meeting the requirements of stability and robustness, and then divide through by the plant transfer function.
- When the plant is stable and has stable zeros (minimum-phase), the division can be made directly.
- One caveat for the stable-plant procedure is that lightly-damped poles or zeros should not be canceled verbatim by the compensator, because the closed-loop response will be sensitive to any slight change in the resonant frequency. The usual procedure is to widen the notch or the peak in the compensator, through a higher damping ratio.
- Non-minimum phase or unstable behavior in the plant can usually be handled by performing the loopshaping for the closest stable model, and then explicitly considering the effects of adding the unstable parts.
- In the case of unstable zeros, we find that they impose an unavoidable frequency limit for the crossover. In general, the troublesome zeros must be faster than the closed-loop frequency response.
- In the case of unstable poles, the converse is true: The feedback system must be faster than the corresponding frequency of the unstable mode.