- #1
PhysicoRaj
Gold Member
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Homework Statement
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.
Homework 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)}##
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.