Answer Happy

3 Gain and Phase Margins Consider a standard unity-feedback system below with plant and controller transfer functions P(s) = 5 $2 + 4s + 5 and C(s) = = S controller plant u C(s) P(s) (a) (6 points) How many strictly unstable poles does the open-loop system P(s)C(s) have? How many encirclements of -1 does the Nyquist plot make? Is the closed-loop system stable of unstable? Hint: You will need to change the default axis limits used by MATLAB command nyquist. = (b) (2 points) Find the gain y and phase o margins for the closed-loop system and their corre- sponding crossover frequencies wg and wp. You can use the MATLAB command margin. (c) (4 points) The gain margin tell us how much we can change the gain of the plant P(s) before the closed-loop system becomes unstable. Plot the Nyquist plots of L2(s) = yP(s)C(s) and Li(s) = P(s)C(s) on one figure. What does the Nyquist stability criterion says about the closed-loop stability for open-loop transfer function L2(s)? (d) (4 points) To change the phase of the transfer function Li(s) we will need to do more work. Verify that the transfer function below has phase o at the phase-crossover frequency wp i.e. 20 (jwp) = 0. = 0(3) S-P s+p where Wp p= tan(0/2) = (e) (4 points) The phase margin tell us how much we can change the phase of the plant P(s) before the closed-loop system becomes unstable. Plot the Nyquist plots of L3(s) = 0(s)P(s)C(s) and L1(s) = P(s)C(s) on one figure. What does the Nyquist stability criterion says about the closed-loop stability for open-loop transfer function L3(s)? (f) (6 points) Plot the step-response for each of closed-loop system Li(s)/(1+Li(s)). Include the plots in your solution. Which closed-loop systems are strictly stable?