Question

For the following system s+1 R(9) "0_*(*1) R(S) s2 + 64 s2 (32 +81) YM Y(S) S +11 (a) Plot the locus of closed-loop roots with respect to K. (b) Is there a value of K that will cause all complex pairs of closed-loop poles to have a damping ratio greater than 0.5? (c) Find the smallest value of K that yield at least one complex pair of closed-loop poles with the damping ratio 5 = 0.707. (d) Use Matlab to plot the response of the resulting design to a reference step.

Answer 1

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1. Attached below is the root locus plot for the given system:

Root Locus . ....... 15 0.065 0.03 "0.42 0:7 OYO Imaginary Axis (seconds) 0.42 .....0.28.. ............. 0.15 0.065 0.03 404 - 12 Real Axis (seconds)

(b) There is no value of K that will satisfy the given criteria. The reason is that as seen in the below plot, when the gain of the system goes to around 40, one pole becomes comples, whereas the other becomes real:

(c) If the intersections of the root locus with the 0.7 line is seen on the root locus, we get the smallest value of the gain = 28.3 approx, for which a pair of complex poles will have a damping ratio of 0.707:

Root Locus . ....... 15 0.065 0.03 "0.42 0:7 OYO System: sys Gain: 45.7 Pole: -4.2 + 0.6431 Damping: 0.988 Overshoot (%): 0 Frequency (rad/s): 4.25 System: sys Gain: 43.3 Pole: -2.36 + 0.4821 Damping: 0.98 Overshoot (%): 0 Frequency (rad/s): 2.41 Imaginary Axis (seconds) 0.42 .....0.28.. ............. 0.15 0.065 0.03 404 - 12 Real Axis (seconds)

(d) The step response is shown below:

Root Locus . ....... 15 0.065 0.03 "0.42 0:7 System: sys Gain: 28.3 Pole: -1.13 + 1.141 Damping: 0.706 Overshoot (%): 4.37 Frequency (rad/s): 1.61 Imaginary Axis (seconds) 0.42 .....0.28.. ............. 0.15 0.065 0.03 404 - 12 Real Axis (seconds)

Step Response AAAAAAAA Amplitude 0 2 4 6 12 14 16 18 8 10 Time (seconds)

The matlab Code is given below:

sys1 = tf([1 1],[1 11]);
sys2 = tf([1 0 64],[1 0 81 0 0]);
sys = sys1*sys2;
rlocus(sys);
k = 28.3;
figure;
sys3 = feedback(k*sys,1);
step(sys3)

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