Problem-4 (25p). Consider again the same unity feedback system of Problem-3, with 20(s + 2) G(S) = s(s+5)(s + 7) Design a controller to obtain a critically damped system with a settling time of 2 seconds. Place the third pole 5 times as far from the imaginary axis as the dominant pole pair. Required Steps: 1. Use controllable canonical form for full state-variable feedback. 2. Obtain the gain matrix of K by means of coefficient matching method of Ackermann's formula by hand. You may validate your results with the "acker" or "place" function in MATLAB. Problem-3 (25p). Design a lag compensator so that the system given in the figure below, where K(s+4) G(s) = (s + 2)(8 + 6) (s +8) R(3) C(s) operates with a steady-state error of ess = 0.02 and a 15% G(S) overshoot (corresponds to a damping ratio of ? = 0.517 and a phase margin of 53). Notes: Please draw Bode plots by hand neatly and clearly on the blank semi-logarithmic planes provided below (you may change the limit values of the axes if needed). Please write the transfer function of the lag compensator separately after the design process for clarity. Bode Diagram 40 30 20 10 Magnitude (dB) -10 -20 -40 -50 90 45 0 -45 Phase (deg) -90 -135 - 180 -225 -270 10 10 102 To 100 Frequency (rad/s) Problem-4 (25p). Consider again the same unity feedback system of Problem-3, with 20(s + 2) G(S) = s(s+5)(s + 7) Design a controller to obtain a critically damped system with a settling time of 2 seconds. Place the third pole 5 times as far from the imaginary axis as the dominant pole pair. Required Steps: 1. Use controllable canonical form for full state-variable feedback. 2. Obtain the gain matrix of K by means of coefficient matching method of Ackermann's formula by hand. You may validate your results with the "acker" or "place" function in MATLAB. Problem-3 (25p). Design a lag compensator so that the system given in the figure below, where K(s+4) G(s) = (s + 2)(8 + 6) (s +8) R(3) C(s) operates with a steady-state error of ess = 0.02 and a 15% G(S) overshoot (corresponds to a damping ratio of ? = 0.517 and a phase margin of 53). Notes: Please draw Bode plots by hand neatly and clearly on the blank semi-logarithmic planes provided below (you may change the limit values of the axes if needed). Please write the transfer function of the lag compensator separately after the design process for clarity. Bode Diagram 40 30 20 10 Magnitude (dB) -10 -20 -40 -50 90 45 0 -45 Phase (deg) -90 -135 - 180 -225 -270 10 10 102 To 100 Frequency (rad/s)