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I have a system that has the following transfer function:

$G(s) = \frac{1}{s^2(s^2+4+1)}$

As can be seen it is a 4th order system. This is the bode plot of the system: bode plot of plant

I need to use a lead compensator in order to generate sufficient phase margin. If I use a tame PDD controller I am limiting my crossover frequency around the position of the green arrow, as the double derivative action can theoretically only provide 180 phase lead. However if I use a tame PDDD action I can place my crossover frequency in the vicinity of the orange arrow. This also means that I can make my system infinitely fast, if there is no limit on control action...

My question is: is it realistic to use a triple tame derivative action? I can't find a similar controller anywhere on the internet!

Thanks in advance!

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In the synthetic example you provide there is no real disadvantage to the triple derivative. In a practical application the story is different. Generally, the triple derivative will amplify (high-frequent) measurement noise and additional low-pass filtering is needed to guarantee performance. The phase lag associated with the low-pass filter will limit the achievable bandwidth.

It is generally (in any controller) a good idea to include a low-pass filter as this will limit control action at higher frequencies. This is desirable as generally measurement noise will increase at higher frequencies (which limits the effectiveness of the feedback loop) and it is generally necessary to limit forces generated by the actuators (e.g. to limit heat dissipation, to limit the maximum generated force or because the available actuator bandwidth is limited).

Alternatives to including a triple derivative with additional low-pass filters in the controller include lead-lag filters or band-limited differentiators.

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