E12 Homework Assignment #1

You may use Matlab wherever you choose, unless otherwise directed.

Problem 1.   (From Feedback Control Systems, 4th Edition, Phillips and Harbor).

 

Problem 2.  

Problem 3. 

 

Problem 4.  Repeat problem 1 (parts a, b and e)with

Problem 5.   Consider x.

  1. Show that the peak of |G(jω)| occurs at x. 
    Hint, if X is positive, then the peak of X is the same as the peak of X2, and the peak of X is the minimum of 1/X.
  2. Show that the peak has amplitude r

Problem 6.  A transfer function can have a time delay associated with it.  For example, a first order system with a 1 second time constant and a 0.5 second delay is given by:

Figure out how to calculate the contribution of the time delay term (the exponential) to the magnitude and the phase of the transfer function and then plot (by hand) the Bode plot of G(s).

Check with MatLab.  You can type "G=tf(1,[1 1],'InputDelay',0.5)" at the MatLab prompt to define the transfer function.