346 Chapter 6 / Control Systems Analysis and Design by the Root-Locus MethodThe open-loop zero at s=0was introduced in the process of modifying the characteristic equa-
tion such that the adjustable variable K=20kwas to appear as a multiplying factor.
We have obtained two different values of kto satisfy the requirement that the damping ratio
of the dominant closed-loop poles be equal to 0.4. The closed-loop transfer function with
k=0.4490is given byThe closed-loop transfer function with k=1.4130 is given byNotice that the system with k=0.4490has a pair of dominant complex-conjugate closed-loop
poles, while in the system with k=1.4130the real closed-loop pole at s=–0.6823is dominant,
and the complex-conjugate closed-loop poles are not dominant. In this case, the response char-
acteristic is primarily determined by the real closed-loop pole.
Let us compare the unit-step responses of both systems. MATLAB Program 6–14 may be
used for plotting the unit-step response curves in one diagram. The resulting unit-step response
curves for Cc 1 (t) k=0.4490andc 2 (t)fork=1.4130Dare shown in Figure 6–63.=
20
(s+2.1589+j4.9652)(s+2.1589-j4.9652)(s+0.6823)C(s)
R(s)=
20
s^3 +5s^2 +32.26s+ 20=
20
(s+1.0490+j2.4065)(s+1.0490-j2.4065)(s+2.9021)C(s)
R(s)=
20
s^3 +5s^2 +12.98s+ 20MATLAB Program 6–14
% ---------- Unit-step response ----------
% ***** Enter numerators and denominators of systems with
% k = 0.4490 and k = 1.4130, respectively. *****
num1 = [20];
den1 = [1 5 12.98 20];
num2 = [20];
den2 = [1 5 32.26 20];
t = 0:0.1:10;
c1 = step(num1,den1,t);
c2 = step(num2,den2,t);
plot(t,c1,t,c2)
text(2.5,1.12,'k = 0.4490')
text(3.7,0.85,'k = 1.4130')
grid
title('Unit-step Responses of Two Systems')
xlabel('t Sec')
ylabel('Outputs c1 and c2')
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