Overshoot Control: Determining k to Avoid Overshoot

[peripatein]

Hello,
Suppose I have the following step response:
y_step(t)=(kP/1+kP)(1-e^(-t/τ))
where k is constant and P is the plant.
How may I determine the values of k for which there would be no overshoot?

 

[jim hardy]

No takers yet ?

Seems to me you'd have to plug in whatever f(t) describes P and find the damping.

 

[peripatein]

Here are the relevant functions:
P(s)=s^2+a1*s+a0
Y_step(s)=[kP(s)]/[1+kP(s)]
Now, how exactly do I find the values of k which would prevent an overshoot?

 

[jim hardy]

you're on the right track...

Laplace Transfer function of plant then is P(s) = s² + a₁s +a₀

and when connected to feedback to close the loop takes that form KP/(1+KPh), h being feedback gain

so you'll have to expand that by plugging in P(s) to get Laplace transform of the closed loop,

and multiply that by a step, which in Laplace is 1/s,

which will result in a pretty long fraction
that'll have to be resolved by algebra

but since this looks like a homework problem it'll be quite do-able, for textbooks are that way.

Once the closed loop response is boiled down to a quadratic form it should be straightforward to extract damping.

Now - it was 1965 when i took modern control theory course
and my algebra has grown rusty, if you'll pardon a cheap excuse for not solving this and typing it out in latex.

Above is the approach i'd have used in 1965. I remember struggling with the algebra of this type problems, and still dread them.
Hopefully someone who's fresh will chime in now, i wanted to prime the pump for you. Could be they're teaching an easier method nowadays.

 

[milesyoung]

... Suppose I have the following step response:
y_step(t)=(kP/1+kP)(1-e^(-t/τ))
where k is constant and P is the plant ...

This looks an awful lot like you're mixing up frequency- and time-domain expressions. I assume P is a (complex-valued) transfer function. How did you arrive at this expression?

Here are the relevant functions:
P(s)=s^2+a1*s+a0
Y_step(s)=[kP(s)]/[1+kP(s)] ...

Are you sure you have the right P(s)? I ask because it's not a proper transfer function, i.e. it cannot represent any physically realizable system, which is a tad unusual in introductory control theory.