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For a plant G(s), the transfer function is given by: $$G(s) = \frac{(10s+50)}{s(s^2+20s+2500)}$$

Using Ziegler-Nicholas method to get the parameter: $k_p = \dfrac{-1000}{3}$ and $w =-28.868j$.

Is it correct? what should I do in case of negative $k_p$ and how do I calculate the PID parameters?

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The Ziegler-Nichols cycling method is not appropriate for this system. Because of the zero located in -5, the closed loop system is always stable and will not oscillate for proportional compensator (poles will never be conjugated on top of the $j\omega$ axis), as can be evidenced by the root locus plot:

enter image description here

If you still wish to use the Ziegler-Nichols table for this system you should use a different approach, like the step response method.

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  • $\begingroup$ I got all the parameters for a PID controller using the PID tunner in MATLAB, but I need to show the calculations for the parameters. So which approach should I use? Our course content only has Z-N oscillation method in detail.. $\endgroup$ – Hassan Riaz May 19 '18 at 2:16
  • $\begingroup$ Since you already have a "desired" closed-loop transfer function (from the tuning you've done), you could use the diophantine equation approach. Link: en.wikibooks.org/wiki/Control_Systems/Polynomial_Design $\endgroup$ – Vicente Cunha May 19 '18 at 15:53

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