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I have the following transfer function $$10000 \cdot\frac{(s-0.012)(s+1.05)(s-18)}{(s+0.22)^2(s+45)(s+1000)}$$ For which I am trying to tune a PID controller for. I'm using the pidTuner in MATLAB to do so. I wish to have the settle time be less than 3 seconds if possible, however, the best I have been able to do is 77 seconds with the settings $K_p = 0.049955$, $K_i = 0.0087525$ and $K_d=0$. At these values, going in any direction seems to increase the settling time more. Any help would be really great!

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You have 2 right-half plane zeroes : 0.012 and 18. The zero at +18 is "fast" and will not affect the performance much, but your slow zero at 0.012 will severely limit your performance. You can't cancel this right-half plane zero with a right-half plane pole in your controller, your controller output will be unbounded.

Is this homework or a real life problem?

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  • $\begingroup$ Thanks for the response. This is for school. If that's the case then is there anything I can do? This transfer function seems to model the system quite well. $\endgroup$
    – cicey60307
    Apr 10 at 18:21
  • $\begingroup$ Would a controller other than a PID controller work better? $\endgroup$
    – cicey60307
    Apr 10 at 18:46
  • $\begingroup$ I doubt it. The slow RHP zero is probably the limiting factor. Maybe you made a mistake in your transfer function ? How did you obtain it? $\endgroup$
    – Ben
    Apr 10 at 18:50
  • $\begingroup$ Estimated poles and zeros off of a Bode plot, then used the bode function in MATLAB to guess and check the signs of the zeros until the bode plots matched each other. $\endgroup$
    – cicey60307
    Apr 10 at 18:56
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    $\begingroup$ Ohh, yes. That's nasty. That sharp initial negative response, followed by a slow increase to a positive response is the time-domain manifestation of your slow unstable zero. You are not going to have an easy time controlling that for a fast, precise response. It's not within the bounds of your question but -- can you change the plant? Part of the real job of a control systems engineer is to inform the mechanical team -- diplomatically or with the use of clue-by-fours -- that they need to change things. $\endgroup$
    – TimWescott
    Apr 10 at 20:57

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