R + E C(s) P(s) Figure 1: Unity-Feedback System Use correct labeling of axes for each plot. The sketches need not be to scale. Consider the plant transfer function 10 P(s) = (s+ 1)(s+10) Design Objectives: Find a controller C(s) such that the following are satisfied: i) The closed-loop system is stable. = r(t) Yss due to a unit-step reference input r(t) = 1(t) is equal to zero. ii) The steady-state error ess iii) The percent overshoot of the unit-step response of y(t) does not exceed PO = 16.3%. iv) The peak time tp of the step-response is not more than 2 seconds. Method: The root locus design method focusses on choosing controller parameters so that the closed-loop poles lie in some desirable' region of the complex plane. The design objectives apart from stability are specified in the time domain. Your first task, therefore, is to find a desirable' region called D such that the design objectives are at least approximately satisfied if the closed-loop poles lie in D. Your second task will be to find a controller transfer function C'(s) such that the closed-loop poles lie in D. Your third task will be to check that the time domain design objectives are actually met. If not, D can be changed and the process repeated. Step 0: 15 points: Determine the desirable region D in the complex plane such that the design requirements (i), (iii), (iv) are satisfied approximately if the closed-loop poles lie in this region D. You would do this based on the second-order prototype system. The equations for maximum overshoot Mp and the peak time tp are: where 1- = ev = etan T == = Wd 2 cos. Provide a sketch of the s-plane with the desirable region D clearly marked. Step I: First try a proportional controller C(s) = Kp 1) 5 points: What is the closed-loop transfer function Hyr(s)? 2) 5 points: What is the range of Kp for closed-loop stability? 3) 10 points: 4) 10 points: 5) 10 points: Can design objective (ii) be satisfied? Explain why or why not. What is the range of Kp to satisfy design objective (iii)? What is the range of Kp to satisfy design objective (iv)? 6) 5 points: Can objectives (iii) and (iv) be satisfied at the same time? If so, what is the range of K? 7) 20 points: Sketch the root locus and draw the desirable region D on the root locus as well. Mark all the intersections on the plot and write the values of Kp and s at each point where the root locus intersects the boundaries of the region D. The root locus should have the Kp values at all important points, as well as jw-axis crossings, breakaways, asymptotic centers, angles of asymptotes, etc. You can use MATLAB for the calculations and to verify your root-locus but submit hand-sketched root-locus.