Modern Control Engineering

356 Chapter 6 / Control Systems Analysis and Design by the Root-Locus Method

The intersection of the asymptotes and the real axis is found from

The breakaway and break-in points are found from Noting that

we have

from which we get

Point s=–0.467is on a root locus. Therefore, it is an actual breakaway point. The gain values K

corresponding to points are complex quantities. Since the gain values are

not real positive, these points are neither breakaway nor break-in points.

The angle of departure from the complex pole in the upper-half splane is

or

Next we shall find the points where root loci may cross the jvaxis. Since the characteristic

equation is

by substituting s=jvinto it we obtain

or

from which we obtain

The root-locus branches that extend to the right-half splane cross the imaginary axis at

Also, the root-locus branch on the real axis touches the imaginary axis at

Figure 6–68(b) shows a sketch of the root loci for the system. Notice that each root-locus branch

that extends to the right-half splane crosses its own asymptote.

v=;1.6125. v=0.

v=;1.6125, K=37.44 or v=0, K= 0

AK+v^4 - 17 v^2 B+jvA 13 - 5 v^2 B= 0

(jv)^4 +5(jv)^3 +17(jv)^2 +13(jv)+K= 0

s^4 +5s^3 +17s^2 +13s+K= 0

u=-142.13°

u= 180 °-123.69°-108.44°- 90 °

s=-1.642;j2.067

s=-0.467, s=-1.642+j2.067, s=-1.642-j2.067

dK

ds

=-A4s^3 +15s^2 +34s+ 13 B= 0

K=-s(s+1)As^2 +4s+ 13 B=-As^4 +5s^3 +17s^2 +13sB

dKds=0.

s=-

0 + 1 + 2 + 2

4

=-1.25

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