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Control Systems ——— 151

Example E3.11: For the closed-loop system defined by

Cs = 1
()
2
Rs s +ζ 1
2 s +
()
(a) plot the unit-step response curves c(t) for æ =0, 0.1, 0.2, 0.4, 0.5, 0.6, 0.8, and 1.0. ù is normalized to 1.
n
(b) plot a three-dimensional plot of (a).
Solution:
>> % Two-dimensional plot and three-dimensional plot of unit-step
>> %response curves for the standard second-order system with wn =1
>> %and zeta = 0, 0.1, 0.2, 0.4, 0.5, 0.6, 0.8, and 1.0
>> t=0:0.2:10;
>> zeta= [0 0.1 0.2 0.4 0.5 0.6 0.8 1.0];
>> for n=1:8;
>> num= [0 0 1];
>> den= [1 2*zeta (n) 1];
>> [y (1:51, n), x, t] = step (num, den, t);
>> end
>> %Two-dimensional diagram with the command plot (t, y)
>> plot (t, y)
>> grid
>> title (‘Plot of unit-step response curves’)
>> xlabel (‘t Sec’)
>> ylabel (‘Response’)
>> text (4.1, 1.86, ‘\zeta=0’)
>> text (3.0, 1.7,‘0.1')
>> text (3.0, 1.5,‘0.2’)
>> text (3.0, 1.22,‘0.4')
>> text (2.9, 1.1,‘0.5’)
>> text (4.0, 1.08,‘0.6')
>> text (3.0, 0.9, ‘0.8’)
>> text (4.0, 0.9,‘1.0')
>> %For three-dimensional plot, we use the command mesh (t, eta, y’)
>> mesh (t, eta, y’)
>> title (‘Three-dimensional plot of unit-step response curves’)
>> xlabel (‘t Sec’)
>> ylabel (‘\zeta’)
>> zlabel (‘Response’)

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