Homework 9, due April 3, 2009.

[goodwine]

Reading: all of Chapter 9 except sections 9.10 and 9.11.

Exercises: For any problem that requires you to use a root locus plot, you must sketch the plot by hand and include all the computations necessary to accurately sketch the plot. In addition, you must verify your answer using matlab and submit a printout of the matlab root locus. If a problem requires you to use simulink, you must submit a printout of your system for that as well as any of the responses you determine or verify from simulink.

1. 9.7
2. 9.9
3. 9.17
4. 9.29 In addition for this problem:
   1. For the uncompensated system (part 1), is there a k that gives the closed loop system a damping ration of 0.5?
   2. For the uncompensated system (part 1), what is the k value where the natural frequency of the closed loop response is 20? What is the damping ratio for that k value?
   3. For the compensated system (part 2), what is the k value where the natural frequency of the closed loop response is 20? What is the damping ratio for that k value?
   4. Use simulink to construct the feedback system (Figure 104) and using the k values from the previous two parts, plot the step response. Does the response have the characteristics you would expect from the location of the closed loop poles for those two k values (overshoot, rise time, settling time)? Explain any discrepancies.
5. Consider
   • 
   1. From a root locus plot for this system, determine the k value where the root locus plot crosses the imaginary axis. Verify this using the Routh array.
   2. Determine the k value where the closed loop system will have a damping ration of .707. Use simulink to construct the closed-loop system and verify that the step response has the properties you would expect from the location of the poles for that k value.
6. Consider
   • 
   1. From the root locus plot, does there exist a k such that the closed loop system has a setting time less than 10?
   2. Design a lead compensator with the zero at s=-1 and the pole to the left of that (which you need to determine) such that the closed loop system is stable and has a damping ratio of approximately 0.5. It is permissible to use rlocus() and trial and error in matlab to determine the pole location.
   3. Use simulink to verify the properties of the step response of the compensated system.

[mnguye10]

For Exercise 4: Problem 9.29 additional part, when it says find the corresponding zeta for the k value that gives omega-n to be 20:

Would the omega-n value be the same for both the (s + 20) and the (s^2 + s + 10) parts at the same k? Or is it like, for an arbitrary k value, the omega-n of each pole location isn't necessarily the same? (So in this case, that would mean the omega-n of the (s^2 + s + 10) part is 20 and the (s + 20) might have a different value, but we don't care what it is because it's relatively far to the left of the imaginary axis compared to the (s^2 + s + 10) part)?

[goodwine]

For Exercise 4: Problem 9.29 additional part, when it says find the corresponding zeta for the k value that gives omega-n to be 20:

Would the omega-n value be the same for both the (s + 20) and the (s^2 + s + 10) parts at the same k? Or is it like, for an arbitrary k value, the omega-n of each pole location isn't necessarily the same? (So in this case, that would mean the omega-n of the (s^2 + s + 10) part is 20 and the (s + 20) might have a different value, but we don't care what it is because it's relatively far to the left of the imaginary axis compared to the (s^2 + s + 10) part)?

Two things: when you add the lead compensator, the k value that gives a particular natural frequency in general will be different.

Also, it is not correct to think that the s+20 parts has its own k. For any given gain value there are two poles (for the uncompensated system) or three poles (for the compensated system). In matlab when you click on the lines, you will find that each branch goes from 0 to +infty, so for the closed loop system, if you computed the poles you will get two or three, respectively. Because of the structure of the closed loop system, there is only one k value.

Related to the added pole being far to the left, yes, the step response should be basically what you expect because it is far to the left.

[cpascual]

For problem 9.9 part 4, what is the ohm or omega symbol supposed to represent because I don't know what that means or how it can be related to w (angular velocity)?

[goodwine]

For problem 9.9 part 4, what is the ohm or omega symbol supposed to represent because I don't know what that means or how it can be related to w (angular velocity)?

That's an upper case omega, so it is the Laplace transform of the angular velocity.