# Design a controller using Matlab

1. Nov 5, 2016

### PhysicoRaj

1. The problem statement, all variables and given/known data
For the plant:
$G(s)=\frac{1}{s(s+2)(0.4s+1)}$
Design a controller in Matlab such that $K_v=4$ , phase margin = $60^o$ and zero steady state error for step input.

2. Relevant equations
$e_{ss} = Lim_{s->0} \frac{s R(s)}{1+D(s)G(s)}$
Lead/Lag compensator structure: $D(s) = K\frac{(\alpha s + 1)}{(\beta s + 1)}$

3. The attempt at a solution
From the data that $e_{ss}=0$ for step input and finite non zero velocity error for ramp reference, the system desired is type-1. The plant already has a pole at $s=0$, hence the controller need not add poles at $s=0$ .
I chose the lead/lag compensator to tune the phase margin.
Using matlab I found the plant has a phase margin of $65.7499^o$ at phase crossover of $0.4777 rad s^{-1}$.
To find the controller gain K:
using $R(s) = \frac{1}{s^2}$ (for ramp input) and $e_{ss} = \frac{1}{4}$ in the steady state error equation, I got $K = 8$ .
So my compensator becomes:
$D(s) = 8\frac{(\alpha s + 1)}{(\beta s + 1)}$

I loaded the plant in the Matlab sisotool and tried adjusting the phase margin to 60 degrees but its not going beyond 46 degrees:

While with controller gain K around 2, the pm can be adjusted to 60 degrees:

Is my choice of controller or controller gain wrong? Or have i missed anything in this route? Are other assumptions and calculations right? How do I proceed further?
Also, how would I verify the steady state error of the closed loop system after design (from the ramp response? [attachment3]) .

Thank you.

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• ###### attachment3.JPG
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2. Nov 10, 2016

### Greg Bernhardt

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

3. Nov 24, 2016

### PhysicoRaj

I finally solved it. I assumed the controller gain $K = 8$ to be the inherent gain of the plant and for that plant designed a restructured controller of the form:
$D(s) = \frac{(\alpha_1 s+1)(\alpha_2 s+1)}{(\beta_1 s+1)(\beta_2 s+1)}$
leaving with two poles and two zeros.

Closed loop characteristics were satisfactory.

Thanks.

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