1. (Direct pole placement): The closed loop structure in Figure 1 and a controller C(s) are used to control an unstable plant with transfer function P(s) = a. What is the minimum order of a biproper controller capable of arbitrarily placing all poles of the resultant closed loop system? s+1 8-1 b. In an attempt to design a controller that will yield satisfactory 2% settling time (~4 s) and overshoot (~20%) in response to a step input, find the coefficients of a closed loop controller of the form C(s) = bis + bo s + ao that places the dominant (i.e., rightmost) poles at $1,2 = -12j and any remaining poles at s3 = 84 = ... = -5. (Here we assume the right most underdamped poles, with appropriate settling time and overshoot, will dominant the closed loop step response.) c. Use computer software to find the 2% settling time and maximum percent overshoot of the step response for the system you obtained in part b. (You may use the stepinfo(sys) command in Matlab.) Explain why the step response of the system obtained in part b differs from that of the following system with the same poles R(s) C(s) P(s) Y(s) Figure 1: Use this figure for Problem