Problem 3. Consider a standard unity-gain negative feedback loop with P(s) 0.1 (s + 1)(8 + 0.1)' C(s) k (s +1) 1. Answer the following questions by sketching (by hand) the Nyquist plot. As always, first sketch the Bode plot; then use that to draw the Nyquist plot. (The Bode/Nyquist plots have to be drawn quite accurately to compute gain and phase margins. In fact, in part (a) even to make the correct conclusion about stability, you have to draw the plots accurately; the asymptotic Bode plot may lead to a wrong answer. I suggest that you draw the Nyquist plot first using the asymptotic Bode plot, but then refine it by using the values of the loop gain L(jw) at a few specific frequencies. Which frequencies? I'd say pick frequency values near the gain and phase crossover frequencies so that you can sketch the parts of the Nyquist plot, where it crosses the real axis and the unit circle, accurately. If you find yourself struggling a lot, do part 2 first, using Matlab, so you already know the answer. Then redraw by hand. The goal is to be able to do a problem like this when you don't have access to Matlab. Also, to compute the margins, don't rely on the Bode plot sketch. Rather, identify the crossover frequencies accurately - you may need a calculator for that - and then compute the corresponding values of L(jw) by again using a calculator.) (a) If k = 10, is the closed loop transfer function from r to y BIBO stable? What are the gain and phase margins of the closed loop? (b) If k = 100, is the closed loop transfer function from r to y BIBO stable? What are the gain and phase margins of the closed loop? 2. Use MATLAB generated Bode plots to answer the following questions: (a) Repeat the questions above (for k = 10 and k = 100) by using Bode/Nyquist plots gener- ated using MATLAB (b) How accurate are the conclusions about stability you made above from the hand drawn Bode/Nyquist plots? What are the errors (in percentage) in the gain and phase margins you estimated above from the hand drawn plots? (c) What is the largest value of k that the closed loop can tolerate without becoming unstable?