Question

2. Consider the closed-loop system shown below

Here Kp represents the gain of a proportional controller, and the process transfer function is given by .

(a) Sketch the locus of the closed-loop poles as the proportional gain, Kp, varies from 0 to ∞. Be sure to clearly mark poles, zeros, asymptotes, angles of arrival/departure, break-in/away points, and real axis portion of the locus.

(b) Using Routh's array, determine the range of the proportional gain, Kp, for which the closed-loop system is stable.

(c) (6 points) Suppose we want the closed-loop system to have a percentage overshoot less than 45% with a settling time less than 2 second. Can this performance criteria be satisfied by varying the proportional gain, Kp? You may use the fact that the settling time , and that the percent overshoot is related with ζ as $P.O. = 100exp(-\pi \zeta /\sqrt{1 - \zeta ^{2}})$

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