# Control Systems - How to find dominant poles *without* MATLAB?

1. Feb 14, 2017

### ctrlhelp10

1. The problem statement, all variables and given/known data
Design a lag-lead compensator for the system of Figure 9.37 so that the system will operate with 20% overshoot and a twofold reduction in settling time. Further, the compensated system will exhibit a tenfold improvement in steady-state error for a ramp input.

2. Relevant equations
$\zeta=\frac{-ln(\frac{percentOS}{100})}{\sqrt(\pi^2+ln^2(\frac{percentOS}{100}))}$

3. The attempt at a solution
Using the equation and 20% overshoot, $\zeta = 0.456$. How do I find the dominant poles by hand, WITHOUT matlab? Every single example in my book and the ones I've tried looking for online ALL use Matlab...

2. Feb 14, 2017

### LvW

Can you solve a quadratic equation?
You have nothing to do than to find the closed-loop function and set the denominator equal to zero. This gives you the pole distribution.

3. Feb 14, 2017

### ctrlhelp10

Is it simply:
Gol = $\frac{K}{(s)(s+6)(s+10)}$
Gcl = $\frac{Gol}{1+Gol}=\frac{K}{s^3+16s^2+60s+k}$
s^3+16s^2+60s+k = 0

What do I do from here?

4. Feb 14, 2017

### magoo

Leave the denominator in factored form and find the 3 solutions for s(s+6)(s+10) = 0

It should be pretty straightforward.

5. Feb 21, 2017

### ctrlhelp10

The problem asks for a 20% overshoot, and the damping ratio corresponding to that is 0.456. The book says we have to drag the poles in matlab until we get our desired damping ratio shown at the bottom, and then the poles are there (-1.79+-3.5j)

Is there a way to do this completely by hand without the use of the Root locus plot or matlab?