Consider the standard unity-feedback system shown below: R(s) E(s) Y(s) input: r(t), output: y(t) F(s) C(s) P(s) error: e(t) = r(t) y(t) Y(s) closed-loop transfer-function: Hyr (s) = R(s) 10 where P(s) = s(1 + 0, 1s)(1 + 0.01s)* The open-loop transfer function is G(s) = P(s)C(s). The design specifications are given as follows: Speci For unit-ramp input at r(t), the steady-state error e. < 0.005. Spec2 Phase Margin PM > 55. Spec3 For unit-step input at r(t), the response y(t) has Percent Overshoot PO < 2.5%, with ty as small as possible. Designl: Proportional Control Let F(s) = 1 and C(s) = K. Determine the range of K values that stabilize P. Determine the minimum value Kmin that satisfies Spec1. Is the closed-loop system stable for Kmin? Draw the Nyquist plot and the Bode plot for G(s) = P(s). Determine the phase and gain margins for G(s) = Kmin P(s). Design2: Lead Control Let F(s) = 1 and Ca(s) = K. with a > 1. Design C, to meet Specl and satisfy PM > 30. Draw the Bode plot. Comment on whether Spec2 can be satisfied with a single lead controller. Draw the step-response of the closed-loop with C2. What is the steady-state error es for a unit-step reference input? Determine the Percent Overshoot PO, peak-time t, and the settling time (2%) t.. Draw the root-locus and compare the actual PO with the approximate PO based on the dominant closed-loop poles. Design3: Lead-Lead Control Let F(s) = 1 and Ca(s) = KC(s), where a > I and Ca(s) is the lead control solution in Design2 above. Determine the second lead stage of this lead-lead controller so that Specl and Spec2 are both satisfied. What is the steady-state error c, for a unit-step reference input? Determine the Percent Overshoot PO, peak-time t, and the settling time (2%) t, . Is Spec3 satisfied? Design4: Lead-Lag Control Let F(s) = 1 and Ca(s) = KC(s), where a < 1 and C,(s) is the lead control solution in Design2 above. Determine the second lag stage of this lead-lag controller so that Specl and Spec2 are both satisfied. What is the steady-state error e,, for a unit-step reference input? Determine the Percent Overshoot PO, peak-time t, and the settling time (2%) t,. Is Spec3 satisfied? Design5: Lead-Lead Control with Prefilter For the lead-lead controller solution Ca(s) in Design3, determine a suitable prefilter F(s) such that all three design specs Specl and Spec2 and Spec3 are satisfied. Draw the step-response for this final design. What is the steady-state error c for a unit-step reference input? Determine the Percent Overshoot PO, peak-time t, and the settling time (2%) t.. Draw the step responses in Design2, Design3, Design4, and Design5 superimposed in one plot area for comparison, identifying each curve. Draw the Bode magnitude plot 20logio|Hyr (jw)| vs. w for Design2, Design3. Design4, and Design5 superimposed in one plot area, identifying each curve. Compare the resonant peaks M and bandwidths,