Homework 11, due April 27, 2005.

Due Wednesday, April 27, 2005.
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goodwine
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Homework 11, due April 27, 2005.

Post by goodwine »

Unless otherwise indicated, these problems were written to be completed sequentially, YMMV, however.
  1. Poles and zeros.
    Consider
    • Image
    1. What are the poles and zeros of G(s)?
    2. Plot them on the complex plane.
  2. One real pole.
    Consider
    • Image
    1. What are the pole(s) and zero(s) of G(s)?
    2. For p = -5, -1, 1, 5
      • plot them on the complex plane; and,
      • use the matlab step() function to plot the step response of G(s).
    3. Describe the effect of changing the location of a single real pole on the step response.
  3. Purely imaginary poles.
    Consider
    • Image
    1. What are the pole(s) and zero(s) of G(s)?
    2. For a = -5, -1, 1, 5
      • plot them on the complex plane; and,
      • use the matlab step() function to plot the step response of G(s).
    3. Describe the effect of changing the location of a complex conjugate purely imaginary pair of poles on the step response.
  4. Complex conjugate poles.
    If a second order transfer function has a unit step response that has a peak time of 1 second, a 1% settling time of 10 seconds and a maximum percentage overshoot of 15%, where are the two poles of the transfer function located? Write the transfer function and plot the location of the poles.
  5. Complex conjugate poles.
    Consider
    • Image
    Assume that a = 2 and b = 9.
    1. If a is increased, what will be the effect on
      • the peak time;
      • the rise time;
      • the settling time; and,
      • the maximum percent overshoot for the transfer function's step response?
      • If a is decreased, will the effect be the the opposite for each specification?
    2. If b is increased, what will be the effect on
      • the peak time;
      • the rise time;
      • the settling time; and,
      • the maximum percent overshoot for the transfer function's step response?
      • If b is decreased, will the effect be the the opposite for each specification?
  6. Time domain specifications.
    On the complex plane, indicate the region where a complex conjugate pair of poles may be located such that the peak time is less than 2 second, the maximum percentage overshoot is less than 30% and the 1% settling time is less than 20 seconds.
  7. Consider the following block diagram.
    • Image
    1. Plot the location of the poles of Y(s)/R(s) for k = 0.1, .25, .5, .75, 1, 2, 3, 5, 10, 20, 40, 100.
    2. For what approximate range of values for k does the step response of Y(s)/R(s) have a peak time less than 1/2 second? Verify your answer using the step() command in matlab for a value of k typical of the range you specified.
    3. For what approximate range of k values is the maximum overshoot less than 10%. Verify your answer using the step() command in matlab for a value of k that is typical for the range you specified.
    4. Is it possible to have a peak time less than 6 seconds and a maximum overshoot less than 10% for any k value?
  8. Effect of additional poles far to the left.
    1. Plot the pole and zero locations and step response of
      • Image
    2. Plot the pole and zero locations and step response of
      • Image
    3. Plot the pole and zero locations and step response of
      • Image
    4. Plot the pole and zero locations and step response of
      • Image
    5. Plot the pole and zero locations and step response of
      • Image
    Note that compared to part (a), each of the subsequent transfer function has an additional pole, and when the pole is far to the left the step response is nearly the same as the step response for part (a). Explain why this happens.
Last edited by goodwine on Thu Jun 15, 2006 4:17 pm, edited 1 time in total.
NDChevy07

Problem 8

Post by NDChevy07 »

Is that an "s" above the fractions in parts b through e?
goodwine
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Re: Problem 8

Post by goodwine »

NDChevy07 wrote:Is that an "s" above the fractions in parts b through e?
Yes, s/20, s/15, s/10 and s/5.
Bill Goodwine, 376 Fitzpatrick
NDChevy07

Max. Percentage overshoot

Post by NDChevy07 »

For problems 4 and 6, I am finding values for a and b from the peak time and settling time relationships that you gave us in class:
tp=pi/wd and ts=-1/(zeta*wn)*ln(s/100).
And also noticing that wd=b and zeta*wn=a .
The values that I get for these, however, do not correspond to the max. percent overshoot value we need to find. Am I right to assume that this is ok as long as the value for Mp is less than what you give us in the problem using the a and b found above?
goodwine
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Re: Max. Percentage overshoot

Post by goodwine »

NDChevy07 wrote:For problems 4 and 6, I am finding values for a and b from the peak time and settling time relationships that you gave us in class:
tp=pi/wd and ts=-1/(zeta*wn)*ln(s/100).
And also noticing that wd=b and zeta*wn=a .
The values that I get for these, however, do not correspond to the max. percent overshoot value we need to find. Am I right to assume that this is ok as long as the value for Mp is less than what you give us in the problem using the a and b found above?
Yes, in problem 4 I provided three specifications, but you only have two parameters to vary (essentially the real and imaginary parts of the eigenvalues). Thus it is unlikely that you will be able to exactly satisfy the three conditions. So, the correct thing to do is think of them as specifications and that doing better than specified is o.k. Just try to meet two of them and beat the other one, if possible. If that's not possible, then try to simply do better than as many as possible.
Bill Goodwine, 376 Fitzpatrick
kdormuth

Post by kdormuth »

For M_p: it's defined as the percent overshoot, but the equation (Ox-xss)/xss looks like you would need to multiply by 100 in order to get a whole number percent. So, for example, if we need to keep the percent overshoot less than 10%, is M_p=10, or is M_p=0.10 ?
Thanks,
Kristin
goodwine
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Post by goodwine »

kdormuth wrote:For M_p: it's defined as the percent overshoot, but the equation (Ox-xss)/xss looks like you would need to multiply by 100 in order to get a whole number percent. So, for example, if we need to keep the percent overshoot less than 10%, is M_p=10, or is M_p=0.10 ?
Thanks,
Kristin
It's the latter.
Bill Goodwine, 376 Fitzpatrick
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