Homework 10, due April 20, 2005.

[goodwine]

1. Find the transfer function Y(s)/R(s) for the following block diagram.
   • 
2. Find the transfer function Y(s)/R(s) for the following block diagram.
   • 
3. Consider the block diagram illustrated in the following figure.
   • 
   Compute the step response of the system (i.e., find y(t) assuming R(s)=1/s) for the following values of a and b and verify the answer using the matlab step() command. Hint: it is easiest to do the partial fraction expansion using the variables a and b and then substituting the values instead of computing a partial fraction expansion multiple times.
   1. a = -1, b = 1;
   2. a = -2, b = 1;
   3. a = -1, b = 2; and,
   4. a = 1, b = 1.
   What general conclusions can you draw regarding the effect of changing a and b?
4. Consider the block diagram in the following figure.
   • 
   Compute the step response of the system for the following values of a and b and verify your answer using the matlab step() command:
   1. a = -1, b = 1;
   2. a = 1, b = 1;
   3. a = -1, b = -1; and,
   4. a = 1, b = -1.
   What general conclusions can you draw regarding the effect of changing a and b?
5. For the transfer function in problem 4 let a=-1 and b=-3. Compute the partial fraction expansion by hand and plot the three terms in the response (the term due to A/s, the term due to B/(s-a) and the term due to C/(s-b)) and verify that the response from the step() command is the sum of these three terms.

[mightyduck]

For number 4, letters a) and d) contain the same values for a and b. I'm assuming d) should be a = 1 and b = -1??

[goodwine]

For number 4, letters a) and d) contain the same values for a and b. I'm assuming d) should be a = 1 and b = -1??

Yes. Thanks.

[kp316]

For problem 4, parts b & c, I am finding that the denominators of several of the y(t) terms are zero, which makes little sense to me. If a and b are the same then isn't 1/(a^2 - ab) undefined? I don't know if I'm missing the point to this problem or what, but if someone could point me in the right direction it would be much appreciated.

[goodwine]

For problem 4, parts b & c, I am finding that the denominators of several of the y(t) terms are zero, which makes little sense to me. If a and b are the same then isn't 1/(a^2 - ab) undefined? I don't know if I'm missing the point to this problem or what, but if someone could point me in the right direction it would be much appreciated.

If a=b you can't use the partial fractions that I did in class since there was a division by (a-b) or (b-a) at some point in it.

If a=b you need to partial fraction

A/s + (Bs+C)/(s+a)^2 = 1/(s(s+a)^2).

You can either do it twice with the numbers substituted, or try it with just an a or b so you only have to do it once, but I'm not sure how bad the algebra gets in the latter case.

[NDChevy07]

I went to do the step function analysis on Matlab and it said that there is not function called step.m???

[goodwine]

I went to do the step function analysis on Matlab and it said that there is not function called step.m???

If you are using your own student version, perhaps the controls package isn't part of it.