Right... this should be a simple question, but I just can't seem to solve it. I have a simple unity gain control system with a constant gain controller in the forward channel, before the plant.
basically like the figure below;
with H(s)=1 as it's unity gain feedback, and the controller having a constant gain Kc.
the transfer function for the plant is
with K=10, Km=0.5, R=1, J=0.003, b=0.1
I've modelled it in simulink and when simulated it shows that the settling time of the system increases as the gain Kc of the controller is increased. This makes sense as a higher gain causes a larger overshoot and so the signal is obviously going to take longer to settle.
The problem is that whenever I do the maths and calculate everything, I find that according to the equations, the gain Kc of the controller always gets cancelled out and no matter what it equals I get a settling time of 0.069s.
I know the settling time (to within 2% of final value) is defined as 4/ζωn. I've found the closed loop transfer of the system to be
which gives
ωn = √(KcKKm/J)
and
ζ = [(b+Km^2)/J] / 2√(KcKKm/J)
just looking at these you can see that when multiplied together, the parts containing Kc are simply cancelled out.
What am I doing wrong??
(sorry if there are any mistakes in there i'm writing all the transfer functions from memory, should be correct though)
basically like the figure below;
with H(s)=1 as it's unity gain feedback, and the controller having a constant gain Kc.
the transfer function for the plant is
Rich (BB code):
(K*Km)/(s[R(Js+b)+Km^2])
I've modelled it in simulink and when simulated it shows that the settling time of the system increases as the gain Kc of the controller is increased. This makes sense as a higher gain causes a larger overshoot and so the signal is obviously going to take longer to settle.
The problem is that whenever I do the maths and calculate everything, I find that according to the equations, the gain Kc of the controller always gets cancelled out and no matter what it equals I get a settling time of 0.069s.
I know the settling time (to within 2% of final value) is defined as 4/ζωn. I've found the closed loop transfer of the system to be
Rich (BB code):
[(Kc*K*Km)/J] / s^2 + [(b+Km^2)/J]s + [(Kc*K*Km)/J]
ωn = √(KcKKm/J)
and
ζ = [(b+Km^2)/J] / 2√(KcKKm/J)
just looking at these you can see that when multiplied together, the parts containing Kc are simply cancelled out.
What am I doing wrong??
(sorry if there are any mistakes in there i'm writing all the transfer functions from memory, should be correct though)
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