Homework 3, due February 26, 2014.

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Reading: Ogata Chapter 7, sections 1 - 6.

Exercises (all B-7-): 2-6.

Also, consider the four following "sensitivity functions"

1. S = 1/(1 + PC), the sensitivity function
2. T = PC/(1 + PC), the complementary sensitivity function
3. PS = P/(1 + PC), the load sensitivity function, and
4. CP = C/(1 + PC), the noise sensitivity function.

For the following block diagram

where R(s) is the reference signal, E(s) is the error signal, D(s) is a disturbance, N(s) is measurement noise, U(s) is the input signal and X(s) is the output, show that

1. the sensitivity function is the transfer function from the measurement noise to the output and from the input to the error,
2. the complementary sensitivity function is the transfer function from input to output,
3. the load sensitivity function is the transfer function from disturbance to the output and,
4. the noise sensitivity function is the transfer function from the noise to U(s).

We will see in the next few lectures that many performance specifications for a feedback system may be characterized by the sensitivity and complementary sensitivity functions. Also, important advanced topics in robust stability and performance depend on these two sensitivity functions as well.

Consider the plant G(s) = k/(s+1). When this is the only element in the feedforward path of a unity feedback system, what is the sensitivity function? Plot the magnitude of S versus frequency (you may use bode() for this in matlab, but make sure you put the right transfer function in there!) for k = 1, 10, 20. You may put them on the same plot if you want.

Next construct a simulink model for this. Have the input be a sinusoid and determine the error for several frequencies and for the different gain values you used when plotting the sensitivity function. How does this error relate to your plot of the sensitivity function?

Finally, as a goal but not a formal part of this assignment, you should have the system identification portion of the project substantially finished and perhaps start writing that part of the report.