Problem 1 (50 points) Consider a system with P(s) — u (s+1)(s+3) and controller K(s). You will design a compensator K(s) that results in a dosed loop system with w„ (natural frequency) = 4 r±, f(damping ratio) =1 using MATLAB (use rlOCUs function). controller E(s) R(s) K(s) plant P(s) C(s) a) In order to have a closed loop system with wn = 4 rasd , 3 = the closed loop poles should be located at s = —3 + jr7, —3 — jr7. Verify these pole locations using the natural frequency and damping ratio given using Pole-Plot (Hint: Check ‘4. Transientresponse’ lecture notes for Pole-Plot) (10points) b) If K(s)=K (K is a constant), can you find a value K that results in the closed loop poles at s = —3 ± j1/7? Explain using root locus. (10points) c) In order to move the closed loop poles to the desired location, you have decided to use lead compensator (K(s) = K, ss÷; (0 < z < p) so that we can move pull the root locus to the left. Choose z and p for this lead compensator by trying different values (It will take some time to find right values that satisfy the condition. There are many possible solutions). Using root locus in MATLAB to determine z and p for your lead compensator. Once you find the values, 1) plot a root locus using MATIAB and 2) mark the point on root locus where it passes through the desired pole location s = —3 ± NI using 'Data Tips' in MATLAB figure and 3) Record the 'Gain' value shown in the 'Data Tips' at this pole location (Hint: Make sure your two desired closed loop poles are dominant. In other words, the desired pole locations should be right to other poles. This allows us to treat our system as rd order system) (20 points). ** Note: You don't need to create lead compensator that makes the closed loop poles passes through right on the desired closed loop pole locations. If your root locus passes near those desired closed loop pole locations, that's enough. d) Using the magnitude condition (Hint: Check '11. Compensator Design' lecture note), find K, value of the compensator for the desired pole locations and compare this value with the gain you recorded in Part (c) (10points)