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I am currently solving a question on pole placement method and in it we develop a required characteristic equation for the system using % overshoot and settling time.

I know that in a 2nd-order system, the characteristic equation is \$s^2+2\zeta ws+w^2\$, and then we replace the values of \$\zeta\$ and \$w\$ with the values we get from the values which we get from % overshoot and settling time.

How would we do this in a 3rd-order system? My guess is that we take a general equation like \$(s+p)(s^2+2\zeta w s+w^2)\$ and then find \$p\$, \$\zeta\$ and \$w\$, but I don't know if this is correct or how would I even do that.

The question I am currently solving is an example from "Control Systems Engineering" by Norman Nise. This is the first example of chapter 12 "Design via State Space":

Given the plant \$G(s)\$ = some transfer function design the phase-variable feedback gains to yield 9.5% overshoot and a settling time of 0.74 second.

In the solutions, it simply says:

This equation must match the desired characteristic equation, \$s^3 + 15.9s^2 + 136.08s + 413.1 = 0\$

What is the method for finding this equation?

TimWescott
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    Can you _edit your question_ to cite the book, or tell us what class you're taking? The "right" answer depends on the context. From the roots of that characteristic equation my guess is that's from an introductory course in dynamic systems control, or an introduction to optimal control that hasn't gotten around to worrying about robustness or disturbance rejection. – TimWescott Apr 14 '23 at 22:41
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    Their desired characteristic equation has \$p=5.1\$, \$\omega=9\$ and \$\zeta=0.6\$. In this case, the assumption of 2nd order dominance (\$p\gt 5\,\omega\,\zeta\,\$) is ***false*** and therefore the overshoot and settling time were configured without relying upon that assumption. Which means applying the final value theorem of Laplace with the inverse Laplace, I think. I've +1 the question. You will have two equations and two unknowns here, I believe. – periblepsis Apr 14 '23 at 22:55
  • I have edited the question, so that people don't need to dig through the comments for the book citation. – TimWescott Apr 15 '23 at 14:46
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    I started to answer this question and then I backed out, because -- there's just too many ways to do this, each acceptable in its own venue. I'd pretty much have to read that book chapter, and possibly bits of prior ones, to know what approach the author might be taking. The fact that he's so very specific in his numbers could mean that he's being overly precise, or it could mean that he's presented you with some optimal control technique that you're missing in the text (maybe freshly presented in that section, or perhaps in some prior chapter). – TimWescott Apr 15 '23 at 14:54
  • @TimWescott The example has a very specific plant block which the questioner didn't disclose. In it, it contains a zero at -5. The book author specifically chose to cancel this with -5.1 to help better demonstrate the effect of the 3rd pole as well as the design process steps and simulation needs. There really isn't enough information in the question. – periblepsis Apr 15 '23 at 20:45
  • @TimWescott [Here's the example from the book.](https://i.stack.imgur.com/FWn3c.png) Zoom up as needed. It's not a grainy picture. (The snapshot meets 'fair use' at least so far as my own use here is concerned, as courts have consistently found non-profit educational purposes qualify. Whether it's use still qualifies on this website may be an issue for EESE lawyers to worry over. But that's not my problem.) – periblepsis Apr 15 '23 at 20:57

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