Transient response of a system (control)

[silentwf]

Homework Statement

Find T,K value such that the system's percent overshoot is .254 and peak time is 3
[PLAIN]http://img560.imageshack.us/img560/6139/problemz.png

Homework Equations

[PLAIN]http://img46.imageshack.us/img46/51/releq.png

The Attempt at a Solution

[PLAIN]http://img12.imageshack.us/img12/4818/solz.png
(I had problems using the LATEX built into this forum, so i used another tool to generate the equations. sorry about the loading times).

The main question i have up to now is i have no idea how to solve the two equations. Another thing i want to confirm is whether there is anything wrong with my equations. I've worked on this problem a few times, and i always get stuck on the two equations. any idea on how to solve this problem?

 

[CEL]

Homework Statement

Find T,K value such that the system's percent overshoot is .254 and peak time is 3
[PLAIN]http://img560.imageshack.us/img560/6139/problemz.png

Homework Equations

[PLAIN]http://img46.imageshack.us/img46/51/releq.png

The Attempt at a Solution

[PLAIN]http://img12.imageshack.us/img12/4818/solz.png
(I had problems using the LATEX built into this forum, so i used another tool to generate the equations. sorry about the loading times).

The main question i have up to now is i have no idea how to solve the two equations. Another thing i want to confirm is whether there is anything wrong with my equations. I've worked on this problem a few times, and i always get stuck on the two equations. any idea on how to solve this problem?

First of all, you must divide both the numerator and the denominator of your TF by T, in order to make the calculations.

 

[silentwf]

Hey,
Okay, that was something I missed. After reworking it, there's still something wrong with it:
[PLAIN]http://img35.imageshack.us/img35/6645/solk.png
as you can see, although i have solved for the values for %OS and Tpeak in the 3rd and 4th line of the picture, the solutions do not match the last part of the second line, which is that damping = 1 / (2*w).

any idea?

forgot to add, the answer provides that T=1.09 and K=1.424, which gives w=1.143 and damping=0.437. but with this answer (damping), %OS becomes .217...so not sure where went wrong

 

[CEL]

Hey,
Okay, that was something I missed. After reworking it, there's still something wrong with it:
[PLAIN]http://img35.imageshack.us/img35/6645/solk.png
as you can see, although i have solved for the values for %OS and Tpeak in the 3rd and 4th line of the picture, the solutions do not match the last part of the second line, which is that damping = 1 / (2*w).

any idea?

forgot to add, the answer provides that T=1.09 and K=1.424, which gives w=1.143 and damping=0.437. but with this answer (damping), %OS becomes .217...so not sure where went wrong

You have three equations for two unknowns ($$\zeta$$ and $$\omega_n$$). The system is overdetermined.
It is interesting that using $$\zeta = 0.3998$$ in the third equation, gives the wanted solution for $$\omega_n$$ and substituting this value in the first equation gives the wanted solution for $$\zeta$$.
The problem is ill formulated.

 

[silentwf]

I agree. Thanks for your help though, really saved me from wasting hours on this problem.