# Designing a PID controller - control systems engineering

1. Oct 12, 2012

### RSDXzec

1. The problem statement, all variables and given/known data

I am given the transfer function
Gp (s) = 1/(s+16)
and I have to design a controller of the form
Gc (s) = K(1 + 1/T*s)
(which is a PID controller)

To meet the following requirements:

Maximum steady state error to a steadily changing desired position = 5%

Maximum percentage overshoot to a step change in desired position = 4.4%

Maximum settling time allowed following a step change in desired position = 0.5sec

2. Relevant equations

%OS = %overshoot
ζ = damping ratio
ω = natural frequency

%OS = 100e^(-pi(ζ/sqrt(1-ζ^2)))

Settling time = 4/ζ

3. The attempt at a solution

First I did the calculations knowing the settling time and overshoot and found
ζ = 0.705
ω = 11.3464
although I don't know how useful that working is.

Then to make progress on the controller I modified the original equation:
Gc (s) = K(1 + 1/T*s)
to
Gc (s) = K((s + Z)/s)
where Z = 1/T

I derived the open loop function by doing
G(s) = Gc(s)*Gp(s)
= (K(s + Z))/(s(s+16))

And with the steady state error limit I tried to get some progress done on that but couldn't get very far there either.

Ess = lim(s->0) (1/(1+G(s)))
5% = 1/(1 + K*Z)

Any help here would really be appreciated, I feel like I'm missing something really simple here and it's starting to get frustrating.

Cheers.

2. Oct 13, 2012

### rude man

Your controller is a PI, not a PID.

What is the overall closed-loop transfer function of your system, including the controller?

Also, settling time = 4/ζω.

Last edited: Oct 13, 2012