Page 185 - MATLAB an introduction with applications

170 ——— MATLAB: An Introduction with Applications

From Fig. E3.20(a), we notice that the system is conditionally stable. All critical points for stability lie on the
jω axis.

To obtain the crossing points of the root loci with the jω axis, we substitute s = jω into the
characteristic equation
3
2
5
4
2
s + 9.5s + 28s + 20s + 15s + K(s + 2s + 5) = 0
3
5
or (jω) + 9.5(jω) + 28(jω) + (20 + K)(jω) + (15 + 2K)(jω) + 5K = 0
4
2
or
3
2
5
4
[9.5ω – (20 + K) ω + 5K] + j[ω – 28ω + (15 + 2K) ω] = 0
Equating the real part and imaginary part equal to zero, respectively, we get
4
2
9.5ω – (20 + K) ω + 5K = 0 ...(1)
3
5
ω – 28ω + (15 + 2K) ω = 0 ...(2)
Equation (2) can be written as
ω = 0
2
4
or ω – 28ω + 15 + 2K = 0 ...(3)
−ω + 28ω − 15
4
2
K = ...(4)
2
Substituting Eq.(4) into Eq.(1), we obtain
2
4
2
4
9.5ω – [20 + ½ (–ω + 28ω –15)] ω – 2.5ω + 70ω – 37.5 = 0
2
4
or
6
2
4
0.5ω – 2ω + 57.5ω – 37.5 = 0
The roots of the above equation can be obtained by MATLAB program given below.
% MATLAB Program
a = [0.5 0 – 2 0 57.5 0 –37.5];
roots(a)
MATLAB Output:
ans =
–2.4786 + 2.1157i
–2.4786 – 2.1157i
2.4786 + 2.1157i
2.4786 – 2.1157i
0.8155
–0.8155
The root-locus branch in the upper half plane that goes to infinity crosses the jω axis at ω = 0.8155. The
gain values at these crossing points are given by
– 0.8155 + 28 0.8155 –15
×
4
2
K = = 1.5894 for ω = 0.8155
2

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