Bill Goodwine's Courses

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Professor I'm having problems with the first problem of hw. 9 where G(s)=2/((s+1)(s+2)), plotting the root locus I obtain two poles on the real axis at -1 and -2 and the vertical asymptotes at -1.5. When asked for the k needed for overshoot less than 10%, the distance of the poles to the point where the overshoot lines intercepts the root locus is ~2.55. Now as I understand it to obtain the magnitude i need to multiply the distances from BOTH poles together and divide by the distances from the zeros. In this case there are no zeros. The problem now is if just multiply the distances together I won't get the right answer. I would have to do it through the method outlined in the solutions. Is the distance magnitude method only for situations with one or more zeros?

I meant to say hw 10. sorry

frodrig3 wrote:Professor I'm having problems with the first problem of hw. 9 where G(s)=2/((s+1)(s+2)), plotting the root locus I obtain two poles on the real axis at -1 and -2 and the vertical asymptotes at -1.5. When asked for the k needed for overshoot less than 10%, the distance of the poles to the point where the overshoot lines intercepts the root locus is ~2.55. Now as I understand it to obtain the magnitude i need to multiply the distances from BOTH poles together and divide by the distances from the zeros. In this case there are no zeros. The problem now is if just multiply the distances together I won't get the right answer. I would have to do it through the method outlined in the solutions. Is the distance magnitude method only for situations with one or more zeros?

If you solve 1 + k G(s) = 0 for k you get

|k| = |1/G(s)|

If you are off by a factor of 2, it's because you probably forgetting the 2 that's in the numerator of G.

1) I'm confused when it comes to finding gains in homework ten. Let's use problem 10.1 as an example. For calculating K in part b I multiplied the distances between the poles and the point where the overshoot is 10 %. Then, I divided by 2 because there is a 2 in the numerator of G(s). This gave me the correct answer. I used this same process in part g after I found the lead compensator. However, in this case my answer was exactly half of your solution because you didn't divide by two. Why did you divide the product of the distances by the coefficient in the numerator in part b but not in part g? I know you used a different method for calculating K but shouldn't the product of the distances method give the same answer? Lastly, I checked my answer with MATLAB for part g and it gave me exactly the answer I calculated.

2) Did the HW 10 grader know that there were numerous different answers for the lag compensators? I got all of them wrong but I thought I was doing them correctly.

thenisey wrote:1) I'm confused when it comes to finding gains in homework ten. Let's use problem 10.1 as an example. For calculating K in part b I multiplied the distances between the poles and the point where the overshoot is 10 %. Then, I divided by 2 because there is a 2 in the numerator of G(s). This gave me the correct answer. I used this same process in part g after I found the lead compensator. However, in this case my answer was exactly half of your solution because you didn't divide by two. Why did you divide the product of the distances by the coefficient in the numerator in part b but not in part g? I know you used a different method for calculating K but shouldn't the product of the distances method give the same answer? Lastly, I checked my answer with MATLAB for part g and it gave me exactly the answer I calculated.

You are right, it looks like the solution is wrong because of the 3 in the denominator of k. The 3 came from the distance from the zero added by the compensator to the desired pole location. So, k should be half of that value due to the 2 in the numerator of G(s).

2) Did the HW 10 grader know that there were numerous different answers for the lag compensators? I got all of them wrong but I thought I was doing them correctly.

I do not know. If you want it regraded you can submit it for that, but they were probably graded consistently so it's probably not worth it.

Professor,

I was just wondering what the statistics were for the last test. Thanks!

Whoops, never mind, I found it.

I and several other students had the same situation as the above person said about the lead/lag compensators on the homework, so I thought I would ask just to make sure I am doing it correctly:

for lead compensators: pole is to left of zero, p>z
Pick a point in the region that you want the R-L to pass through, let the zero be on the real axis underneath this point, then use the angle formula -180= sum(angles from zeros to s) - sum(angles from poles to s), solve for the angle from the unknown pole on the real axis), then use this angle to find the distance to the pole, to get lead compensator (s+z)/(s+p).

for lag compensators: zero is to left of pole, z>p
To reduce steady state error by a certain factor (I'll use 2 as an ex.) either find the exact ratio of z/p needed by setting
(error_without_lag)*(reducingfactor,like 1/2)=(error_with_lag, z/p) and solving for z/p, or just approximate z/p=2.
Let p= a small # like .01, solve for z.

That is the correct process, right? If so, I think the homework was graded incorrectly.

cplagema wrote:I and several other students had the same situation as the above person said about the lead/lag compensators on the homework, so I thought I would ask just to make sure I am doing it correctly:

for lead compensators: pole is to left of zero, p>z
Pick a point in the region that you want the R-L to pass through, let the zero be on the real axis underneath this point, then use the angle formula -180= sum(angles from zeros to s) - sum(angles from poles to s), solve for the angle from the unknown pole on the real axis), then use this angle to find the distance to the pole, to get lead compensator (s+z)/(s+p).

for lag compensators: zero is to left of pole, z>p
To reduce steady state error by a certain factor (I'll use 2 as an ex.) either find the exact ratio of z/p needed by setting
(error_without_lag)*(reducingfactor,like 1/2)=(error_with_lag, z/p) and solving for z/p, or just approximate z/p=2.
Let p= a small # like .01, solve for z.

That is the correct process, right? If so, I think the homework was graded incorrectly.

That is all exactly correct and a good summary of how to do it.

Professor,

In studying for the final, is it important that we are familiar with each of the methods you presented on how to solve for NONhomogeneous systems of first order ODEs, such as diagonalization, variation of parameters, and method of undetermined coefficients? Or would it suffice to study whichever one we are most comfortable with? I guess, would you ever request that we solve one of these problems with a certain method, or would we be free to choose? Thanks.

ngeraci wrote:In studying for the final, is it important that we are familiar with each of the methods you presented on how to solve for NONhomogeneous systems of first order ODEs, such as diagonalization, variation of parameters, and method of undetermined coefficients? Or would it suffice to study whichever one we are most comfortable with? I guess, would you ever request that we solve one of these problems with a certain method, or would we be free to choose? Thanks.

In the past I've asked questions both ways: either allowing you to pick whatever method you want or by specifying that you must use a particular method. Since I personally don't like being told what to do, more often than not I've not specified a method, but I can't guarantee that I won't. One reason, for example, I might specify the method to use is that there was some theoretical connection with another problem that some of the methods had, so to fully test things I would require another method. When I write a test I categorize problems in my mind on an "easy" to "difficult" scale and try to have a balance. If I were to specify a method, that would definitely push the problem farther up the scale toward the difficult end.

My recommendation would be in preparing for the exam make sure you know one method pretty well and to revisit the other methods if you think you're pretty well prepared otherwise. If you want to be 100% prepared, however, then you must know all the methods.

Professor Goodwine,

Is pole placement fair game for the final? It seems that such problems would take a lot of algebra to solve by hand.

lawnoy wrote:Professor Goodwine,

Is pole placement fair game for the final? It seems that such problems would take a lot of algebra to solve by hand.

Yes. For a small enough problem you can do them by hand. In fact, Exam 2 had a problem where you found the ks to place the poles for a second order system.

Professor, in Homework 5 we proved that for diagonalization C(t)=exp(At)*C(0), where exp(At)=T*exp(lambda*t)*T^-1

Does this method still hold if there is a g(t) added to the A-matrix? If so, where does g(t) appear in the solution? Would we otherwise have to do the process outlined in the course notes?

Josh wrote:Professor, in Homework 5 we proved that for diagonalization C(t)=exp(At)*C(0), where exp(At)=T*exp(lambda*t)*T^-1

Does this method still hold if there is a g(t) added to the A-matrix? If so, where does g(t) appear in the solution? Would we otherwise have to do the process outlined in the course notes?

Check out Equation 6.32. It's easy to diagonalize A, but the g(t) get's more complicated because it gets multiplied by T^{-1}.

Professor could you post the solutions to Exam 1 and Exam 2.
Thank You

elazaroa wrote:Professor could you post the solutions to Exam 1 and Exam 2.
Thank You

I don't have those available.

Professor, will you be posting solutions or our grades from the final?