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Consider the following transfer function : $$ T(s)=\frac{4}{s^{2}+2s+4} $$ I wish to design a lead-lag compensator satisfying a set of requirements. What matters for now is the process done to meet these requirements. The poles for this transfer function are $s=-1\pm j\sqrt{3}$ i.e., we have a pair of complex poles.

The damping ratio for this system can be extracted from the desired or targeted percentage overshoot and $\omega_{n}$ can also be extracted via the desired settling time. From both $\omega_{n}$ and the damping ratio $\zeta$ I was able to determine the desired poles and they turned out to be of the form $s=a\pm bj$.

The open-loop transfer function of the lead compensator cascaded with the plant is : $$ G'=\frac{4G}{s^{2}+2s+4} $$ where : $$ G=\frac{K(s+z')}{s+p'} $$ where $z'$ is the lead-zero and $p'$ is the lead-pole.

Here's the issue I am facing :

The procedure for designing a lead-compensator requires that $z'$ be placed in such a way that minimizes the effect of the dominant closed-loop poles.

But this is troubling since my closed-loop poles are both complex. Therefore, is it allowable to place a "zero" on one of the complex-pair poles to cancel it out? Because in all examples I had encountered, this was usually done when you have a closed-loop pole lying on the negative real axis but in my case I don't have any real poles.

I would hope for some advice regarding this issue and how to overcome it to design a lead-lag compensator for a control system. Note that the lag compensator part is mainly for the steady-state error for a ramp input which shouldn't affect the cause of this question at the moment.

Edit: The results that I have obtained are a satisfactory settling time but a huge overshoot percent, I wish to make sure of the following: When applying the angle equations (sum of angles of zeros - sum of angles poles = -180) do we calculate the angle of both $s=-1\pm j\sqrt{3}$ with respect to the desired pole, or do we take just take one complex pole only?

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  • $\begingroup$ Neither closed loop pole is dominant, or they are equally dominant, so why not place the zero at $z'=-1+0j$ (or maybe a little to the right of that), to equally reduce the influence of both poles as best it can? $\endgroup$
    – Andy Walls
    Nov 24, 2021 at 20:33
  • $\begingroup$ Ah I see, so I must include the zero in the negative-real axis only as close as possible to the poles. Thank you for your suggestion I will retry again and check back @AndyWalls $\endgroup$
    – CynthiaZ1998
    Nov 24, 2021 at 20:35
  • $\begingroup$ You can place complex zeros to cancel complex poles; but in that case both poles are equivalent contributors to the system so such a cancellation would require two zeros. It is likely the real zero Andy suggests would provide a suitable result with a single zero as well as the additional single pole for stability. $\endgroup$
    – Dan Boschen
    Nov 24, 2021 at 21:07
  • $\begingroup$ Think of the $|T(s)|$ as a tent canvas. Poles push the tent canvas up at that point and the surrounding canvas slopes up to it. Zeros anchor the tent canvas to the ground at that point and the surrounding canvas slopes down to it. The farther you get from either type of point, the less its affect on the canvas. $\endgroup$
    – Andy Walls
    Nov 24, 2021 at 21:10
  • $\begingroup$ Thank you very much @AndyWalls and Dan for your answers, I have edited my question and would hope for your assitance $\endgroup$
    – CynthiaZ1998
    Nov 24, 2021 at 22:54

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