Homework 10, due April 23, 2008.

[goodwine]

Simulink models from class: pendulumand lead/lag example.

1. (50 points) Conventional wisdom among runners is that it is better to run on an asphalt surface than concrete because the former is softer, and hence is less damaging to the runner's body. Develop a model (equations) describing the essential mechanics of a foot inside a shoe impacting the ground and determine whether or not this conventional wisdom is true.
2. (10 points for each part) Consider the system illustrated in Figure 4.1 of the course text. Let m=1, b=3 and k=1.
   1. Determine the transfer function from the input, f(t) to the output x(t).
   2. What is the transfer function from a desired position, x_d(t) to the actual position, x(t) if the applied force is proportional to the difference between the desired and actual positions of the mass? Construct the block diagram representation for this transfer function in simulink.
   3. Use a root locus plot to predict what the rise time for the system will be if the input is a unit step function when the damping ratio is 0.7071. What must the proportional gain be in that case? What will the settling time and percent overshoot be?
   4. Use your simulink model to verify that the rise time, settling time and percent overshoot is what you predict.
   5. Sketch the region in the complex plan where the poles of the closed loop system must be in order for the percent overshoot to stay the same as above and the rise time to be reduced by at least 25%.
   6. Determine the zero and pole locations for a lead compensator so that this specification (at least 25% reduced rise time) is met. You do not have to compute exactly where the pole and zero must be to exactly reduce the rise time by 25%. Determining them by trail and error so that the response is better than that is fine. Verify your compensator works as you want by simulating it in simulink.
   7. Repeat the previous two parts, but rather than reducing the rise time by 25%, reduce the percent overshoot by 25%.
   8. Is it possible to both reduce the rise time and precent overshoot by 25% with one lead compensator?
   9. Using one of your lead compensators, compute that the steady state value of x(t) will be if the desired x_d(t) is a unit step input. Verify your computation in your simulink model.
   10. Design a lag compensator so that the stead state error is reduced by a factor of 10. Verify it with your simulink model.