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.

- 9.7
- 9.9
- 9.17
- 9.29 In addition for this problem:
- For the uncompensated system (part 1), is there a k that gives the closed loop system a damping ration of 0.5?
- 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?
- 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?
- 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.

- Consider
- 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.
- 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.

- Consider
- From the root locus plot, does there exist a k such that the closed loop system has a setting time less than 10?
- 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.
- Use simulink to verify the properties of the step response of the compensated system.