364 Chapter 6 / Control Systems Analysis and Design by the Root-Locus Methodthe points where the root loci cross the imaginary axis can be found by substituting s=jvwith
the characteristic equation as follows:and equating the imaginary part to zero. The result isThus the exact points where the root loci cross the jvaxis are By equating the real
part to zero, we get the gain value Kat the crossing points to be 16.25.A–6–10. Consider a unity-feedback control system with the feed-forward transfer function G(s)given byPlot a root-locus diagram with MATLAB.Solution.The feedforward transfer function G(s)can be written asA possible MATLAB program to plot a root-locus diagram is shown in MATLAB Program 6–18.
The resulting root-locus plot is shown in Figure 6–73.G(s)=K(s+ 1 )
s^4 + 4 s^3 + 11 s^2 + 14 s+ 10G(s)=K(s+ 1 )
(s^2 + 2 s+ 2 )(s^2 + 2 s+ 5 )v=;1.8708.v=;1.8708=(v^4 - 11 v^2 + 10 +K)+j(- 4 v^3 + 14 v)= 0[(jv)^2 + 2 jv+ 2 ][(jv)^2 + 2 jv+ 5 ]+KMATLAB Program 6–18
num = [1 1];
den = [1 4 11 14 10];
K1 = 0:0.1:2;
K2 = 2:0.0.2:2.5;
K3 = 2.5:0.5:10;
K4 = 10:1:50;
K = [K1 K2 K3 K4]
r = rlocus(num,den,K);
plot(r, 'o')
v = [-8 2 -5 5]; axis(v); axis('square')
grid
title('Root-Locus Plot of G(s) = K(s+1)/[(s^2+2s+2)(s^2+2s+5)]')
xlabel('Real Axis')
ylabel('Imag Axis')
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