HW 7 Prob 3 Hint
This question's purpose is to give you some intuition about how controller gains affect the closed loop system, particularly the poles.
Part a is a simple manipulation. You are given the form of the denominator of the closed loop system (T(s)= L(s)/(1+L(s))), the plant, and the controller. You must then find expressions for k_p and T_i in terms of the other parameters. Part b follows from this part, solving for a specific system.
Part c asks you to manipulate the specific system in part b by making k_p a parameter once again and plotting how the closed loop poles vary as k_p is varied. The first set of plots asked for should have 4 subplots: Re(pole 1) vs k_p, Im(pole 1) vs k_p, Re(pole 2) vs k_p, and Im(pole 2) vs k_p. The second plot is a root locus plot, with k_p acting as the loop gain (as discussed in class on Wednesday)- don't forget to label gains at interesting features!
--Braman 10:14, 25 November 2007 (PST)