Homework 8, due March 24, 2010.

[goodwine]

Reading: sections 9.7 and 9.8 from the course text. (Skip 9.6 for now).
Exercises: 9.6, 9.7, 9.10, 9.11, 9.12.

[Josh]

Professor,
Problem 9.10 part 3 asks us to determine the maximum value for k so that the percentage overshoot remains under 20 %. Equation 9.19 gives the percentage overshoot, but there is no k in that equation; it only depends on the damping ratio. Wouldn't the k's cancel if we are looking for the percentage overshoot?

[goodwine]

Professor,
Problem 9.10 part 3 asks us to determine the maximum value for k so that the percentage overshoot remains under 20 %. Equation 9.19 gives the percentage overshoot, but there is no k in that equation; it only depends on the damping ratio. Wouldn't the k's cancel if we are looking for the percentage overshoot?

The damping ratio depends on where in the complex plane the poles are. From the root locus plot you should be able to determine those regions. It may be worth waiting until after tomorrow's class to try that one.

[AL089]

Professor, I keep getting this response when I try to use rlocus to check my answer for 9.10. Is the syntax right?

>> rlocus([4],[1 4 3])
??? No appropriate method or public field FreqUnits for class plotopts.TimePlotOptions.

Error in ==> rlocus at 125
rlocus(sys,k)

[AL089]

Never mind. I had another figure open, and that's what was screwing it up.

[AL089]

Professor, for root locus plots where there are break-in and break-out points (from Rule 9.8.17), is there any way for us to know what the curve will look like, other than that it will be smooth and not so complicated. The text said that it was computed numerically, but is there any indicator that might hint at the height of the curve (since it is apparently only a "near circle")?

[goodwine]

Professor, for root locus plots where there are break-in and break-out points (from Rule 9.8.17), is there any way for us to know what the curve will look like, other than that it will be smooth and not so complicated. The text said that it was computed numerically, but is there any indicator that might hint at the height of the curve (since it is apparently only a "near circle")?

The only thing we really know is that the curve is not too crazy since it's related to the roots of a relatively low order polynomial.

It is possible to get a more accurate curve than the rules indicate. For example, in the semi-circle types, you could pick a real value that is inside the circle and fix the real part at that value, and then compute the imaginary values that would give an angle of 180 degrees. Such detail usually isn't necessary and, frankly, is sort of pointless in the age of matlab.

Being able to plot them isn't pointless, because knowing how different things contribute to different features of the plots will allow us to be intelligent when we design controllers.

[Josh]

Professor,
Problem 12 asks about the stability of the response of the system under unity control feedback. Stability appears to be covered in section 9.6, the section you told us to skip. Should we actually read this section (or at least enough of it to do the problem), or should we not worry about stability for this assignment?

[goodwine]

Professor,
Problem 12 asks about the stability of the response of the system under unity control feedback. Stability appears to be covered in section 9.6, the section you told us to skip. Should we actually read this section (or at least enough of it to do the problem), or should we not worry about stability for this assignment?

You don't need section 9.6. Stability in that case relates to recognizing that if the poles move into the right half of the complex plane, that corresponds to unstable solutions.

[alibardi]

For 9.10 part 3, I was able to calculate the damping ratio and natural frequency to keep the overshoot below 20%, but I can't figure out how to get a value of k from that. Where should I go from there?

[goodwine]

For 9.10 part 3, I was able to calculate the damping ratio and natural frequency to keep the overshoot below 20%, but I can't figure out how to get a value of k from that. Where should I go from there?

That's the subsection called "Determining the Gain" 9.8.5.

[alibardi]

I'm still missing something I think. I understand how to find gain at a point on asymptotes or circle thing, but I don't know what this has to do with overshoot because I don't know what point to use to measure the gain. I sort of think it might have something to do with what we did in last week's homework, like in Figure 9.21 where the pole is at -zeta*omega_n, and I just solved for zeta and omega_n. But there's not a pole there, the value I'm getting is -2, which is directly between the poles and is where the asymptote is. This seemed like it was probably a good thing that the point I got was on the asymptote, but maybe not, because I don't know what to do now. So...any recommendations?

Another question I'm still having is sort of the same idea. For part 2 of 9.10 and 9.11, once I get the transfer function, I'm not sure how to relate the overshoot to k again. I got the transfer function and put in G(s), but then I don't know what to do. I'm not sure what I'm missing.

[goodwine]

I'm still missing something I think. I understand how to find gain at a point on asymptotes or circle thing, but I don't know what this has to do with overshoot because I don't know what point to use to measure the gain. I sort of think it might have something to do with what we did in last week's homework, like in Figure 9.21 where the pole is at -zeta*omega_n, and I just solved for zeta and omega_n. But there's not a pole there, the value I'm getting is -2, which is directly between the poles and is where the asymptote is. This seemed like it was probably a good thing that the point I got was on the asymptote, but maybe not, because I don't know what to do now. So...any recommendations?

If it's a second order system, the overshoot is related to the angle of the poles from the imaginary axis, because the damping ratio is related to that and the overshoot depends only on the damping. Look for the example problem that has two angled lines from the origin with a shaded region in between. (Fig 9.25 on p. 272).

Another question I'm still having is sort of the same idea. For part 2 of 9.10 and 9.11, once I get the transfer function, I'm not sure how to relate the overshoot to k again. I got the transfer function and put in G(s), but then I don't know what to do. I'm not sure what I'm missing.

It's the same thing, if I understand your question. Each possible k value corresponds to some pole locations on the root locus. You want them to be in the regions that have the specified desired properties. Again, the problems with the shaded regions do that.