What is the Correct Interpretation of Damping Ratio in Control Systems?

[engnrshyckh]

Homework Statement

In order to produce a damping ratio of 0.5, the fraction of derivative of error needed will be

(A) 1
(B) 0.8
(C) 0.08
(D) 0.008

Relevant Equations

The general characteristic eq of 2nd order system in s plane is
S^2+2(zeta) wn(s) +wn^2=0

I don't know how to calclute error in derivative for daming ratio of 0.5

 

[Jody]

Is the equation $s^2+2 \zeta \omega_n s + w_n^2$?

$\zeta$ is your damping ratio; so you'll want to fix that to 0.5. I've never heard of "the fraction of derivative of error." I'm going to guess that's your steady-state error. What's your input? Is it a step? May you provide the equation for the steady-state error or for this "fraction of derivative of error"? I get the feeling once you get that and plug in $\zeta = 0.5$ you're going to be very close to what you want if not already done with the problem.

 

[engnrshyckh]

Is the equation $s^2+2 \zeta \omega_n s + w_n^2$?

$\zeta$ is your damping ratio; so you'll want to fix that to 0.5. I've never heard of "the fraction of derivative of error." I'm going to guess that's your steady-state error. What's your input? Is it a step? May you provide the equation for the steady-state error or for this "fraction of derivative of error"? I get the feeling once you get that and plug in $\zeta = 0.5$ you're going to be very close to what you want if not already done with the problem.

I know we can calculte steady state error by using final value theorem. I don't know about the input. Its the complete statement.

 

[Jody]

Does it have any graphs or other specifications?

 

[engnrshyckh]

N

Does it have any graphs or other specifications?

Does it have any graphs or other specifications?

No graphs are there

 

[Jody]

Does your textbook have a definition of "the fraction of derivative of error" or is it the steady-state error? Which book are you using? I'm looking through Modern Control Systems by Dorf.

Interestingly he refers to $\zeta$ as the damping ratio, but also $\zeta \omega_n$ as the closed-loop damping constant. I never noticed that before (maybe I just liked its reciprocal $\tau$ too much); I bring this up because I'm wondering if 0.5 is the damping ratio or the damping constant. We might have an extra variable hidden in subtle vocabulary I was unaware of.

I'm struggling to really understand the question not knowing what this "fraction of derivative of error" is. I would have done the same approach final value theorem for the steady-state error, but once you set $s$ to zero it makes that damping irrelevant; it's more relevant for the overshoot and the time it takes to settle. Overshoot doesn't seem to match the context, but I went for it anyways and got about 16%. Looks nothing like the options above.

The only other thing I stumbled upon is after you do the Laplace transform of the second order system with response to a step it's 1 minus some damped sinusoidal. The amplitude of that sinusoidal is $1/\sqrt{1-\zeta^2}$. Still not exactly what I was hoping for, but the $\sqrt{1-\zeta^2}$ is about 0.86; this is just out of desperation for an answer and exploring- I don't think it's right.

 

[engnrshyckh]

Does your textbook have a definition of "the fraction of derivative of error" or is it the steady-state error? Which book are you using? I'm looking through Modern Control Systems by Dorf.

Interestingly he refers to $\zeta$ as the damping ratio, but also $\zeta \omega_n$ as the closed-loop damping constant. I never noticed that before (maybe I just liked its reciprocal $\tau$ too much); I bring this up because I'm wondering if 0.5 is the damping ratio or the damping constant. We might have an extra variable hidden in subtle vocabulary I was unaware of.

I'm struggling to really understand the question not knowing what this "fraction of derivative of error" is. I would have done the same approach final value theorem for the steady-state error, but once you set $s$ to zero it makes that damping irrelevant; it's more relevant for the overshoot and the time it takes to settle. Overshoot doesn't seem to match the context, but I went for it anyways and got about 16%. Looks nothing like the options above.

The only other thing I stumbled upon is after you do the Laplace transform of the second order system with response to a step it's 1 minus some damped sinusoidal. The amplitude of that sinusoidal is $1/\sqrt{1-\zeta^2}$. Still not exactly what I was hoping for, but the $\sqrt{1-\zeta^2}$ is about 0.86; this is just out of desperation for an answer and exploring- I don't think it's right.

the question is from ELECRTRONICS AND COMMUNICATION (ch name is control system) objective book written by HANDA. In this book only multiple choice questions are present with no/little literature. I am following this book to prepare for junior engineer test conducted in my country. They mostly take question from this book if you want i can post complete ch in pdf