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)

Last edited: