# HW8 Problem 1 Hints

Problem 1a) The "Gang of Four" refers to the transfer functions at the bottom of page 315. Note that some of these rational expressions may not have zeros after substituting the plant and controller, and consequently, there may not be an analytical formula for the zeros. If this is the case, say so in your answer. Don't forget to also give analytical formulas for the poles for each member of the "Gang of Four."

Problem 1b) The relationship between gain crossover frequency, phase margin, resonance peak, and percentage overshoot vs. k can be determined by analytical methods or by using the suggested commands. Note that the command 'stepinfo' will be useful to plot percentage overshoot vs k. You will need to extract the value from the corresponding structure by using the appropriate syntax, i.e. S(i,j).Overshoot. Note that the resonance peak you are asked to plot is from the closed-loop bode plot and not from the step response. Finally, you can also calculate explicit formulas such as for the gain crossover frequency as a function of k, if you prefer. Don't forget to use log scale for the values of k.

Problem 1c) should be straight forward once you do part b)

Problem 1d) Use the relationship between phase margin and k from part 1b) to determine what k values correspond to 30deg, 45deg, and 60deg phase margin and plot the step response for those particular k values. --Soto 21:36, 4 December 2007 (PST)