# 2nd Order Control system PD controller

#### DC2R

Hi,

This question on PD control is from a practice quiz.

1. Homework Statement

If you can't see it- the question asks to find values for Kp and Kd such that the system achieves 5% OS and has a settling time Ts of 3s.
Cs = 3
Cd = 2
m = 5

2. Homework Equations
ω_n^2/(s^2 + 2ζw_n + ω_n^2) - 2nd order form

Overall Transfer Function I found:
Y/R = CG/1+CG = [ (Kp + Kds)/m ] / [ s^2 + ((Cd + Kd)/m) + ((Cs + Kp)/m)]

Ts = 4/σ (2% of final value); where σ = ζω_n

3. The Attempt at a Solution
From the transfer function I found above i noticed that it is not exactly in the 2nd order form. However I was told that we still set

ω_n = SQRT[ ((Cs + Kp)/m) ] {1}
because when we add a derivative controller it only adds a zero which effects the numerator, not the poles on the denominator.

firstly we know for 5%OS → ζ≈0.7
from Ts = 3 = 4/ζω_n
→ ω_n = 4/3ζ = 40/21

from eqn {1} - solving for Kp
Kp = (ω_n^2)m - Cs = (5)(40/21)^2 - 3 = 15.14

At this point I stopped because their answers were Kp = 17.93 and Kd = 12.12.
Could you explain if the process i used is incorrect because I cant understand what I am doing wrong.

Thank you
(btw I just joined Physics Forums like an hour ago)

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#### donpacino

Gold Member
There are often many ways to solve control problems. Plot your solution. Does it meet the design criteria?

#### DC2R

hi, can someone please have a look at my working to see if im on the right track, I spent many hours on it and im not even sure if this is the correct way.

donpacino, Im a beginner in Matlab, Im not sure how a plot would verify my results.

#### donpacino

Gold Member
hi, can someone please have a look at my working to see if im on the right track, I spent many hours on it and im not even sure if this is the correct way.

donpacino, Im a beginner in Matlab, Im not sure how a plot would verify my results.
run a simulation to see if your KP and KI values meet the criteria for rise time and settling time.

There are often multiple solutions that will work for problems like this

"2nd Order Control system PD controller"

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