4. The proportional control system of Figure 3 was designed according to the Ziegler-Nichols rules. Due to customer demands for more stringent performance specifications, the proportional controller is to be replaced with a PID controller. R(s+ C(s) 15 6.4- Figure 3 (a) Determine the critical gain for this system, Kc, and apply it to determine the frequency of unstable oscillations, coc, when the gain is set at Kc-(Hint: with K-K. two of the closed-loop poles will be at s = ja). 7 marks] (b) Using the results of part (a), design a replacement PID controller for the proportional controller of Figure 3. Simplify the expression for the resulting controller transfer function as much as possible. 7 marks] (c) Determine the steady-state error for the system of part (b) for r(t) = t; ie, a unit ramp input. [6 marks] 4. The proportional control system of Figure 3 was designed according to the Ziegler-Nichols rules. Due to customer demands for more stringent performance specifications, the proportional controller is to be replaced with a PID controller. R(s+ C(s) 15 6.4- Figure 3 (a) Determine the critical gain for this system, Kc, and apply it to determine the frequency of unstable oscillations, coc, when the gain is set at Kc-(Hint: with K-K. two of the closed-loop poles will be at s = ja). 7 marks] (b) Using the results of part (a), design a replacement PID controller for the proportional controller of Figure 3. Simplify the expression for the resulting controller transfer function as much as possible. 7 marks] (c) Determine the steady-state error for the system of part (b) for r(t) = t; ie, a unit ramp input. [6 marks]