ECEN 605 - Linear Control Systems Instructor: S.P. Bhattacharyya Homework No: 2 Due: 8th October 2009 1. Stabilizing Set and System Design Choose your own Plant Transfer function P(s) = Np(s) / Dp(s) such that Degree of Dp(s) ≥ 2 The controller C(s) is a PI control as shown in the block diagram below. (a) Use Routh’s Stability Criterion and find the set of Kp and Ki values for which the system is stable. (b) Plot the Stabilizing set For the PI controller set (χ) already determined above impose the following performance specifications and plot the set of values of Kp of Ki for which the system can be designed. i.) Gain Margin ≥X db ii.) Phase Margin ≥ Q⁰ iii.) Over shoot to unit step ≤ h % You can choose your own values for X, Q and h; Hint: Plot the curves for constant GM, PM and Overshoot. Bonus Also impose the following specification along with the above i) Rise time < r seconds ii) Settling time < t seconds + + + - Kp Ki/s P(s)3. Nyquist Meets Bode Let C(s) be the controller’s transfer function and P(s) the Plant’s transfer function. a.) Find the NCCW encirclements around the point -1+j0 in the G(s) plane using the graphical method explained in the class. C(s) = 20 / s+10 P(s) = 1 / (s+2)2 b.) Using the method explained in the class for calculating NCCW encirclements around the -1+j0 in the G(s) plane, calculate the range of values of K for the following systems to be stable. C(s) = K / s+100 P(s) = (s+10)(s+1) / (s+2)3 Reference L.H. Keel and S.P. Bhattacharyya,” Nyquist Meets Bode: Frequency Domain Controller Design” 4. Stabilization of the system by POLE placement Let C(s) be the controller’s transfer function and P(s) the Plant’s transfer function. The Plant’s transfer function is 1 / (s-1)(s-3) . Find the controller’s transfer function for which the system is stable. The Controller should be of form f(s) / (s2)(g(s)) f(s) is a 3rd order function g(s) is a 1st order function P(s) + -- R(s) Y(s) C(s) P(s) + -- R(s) Y(s)