Modern Control Engineering

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

A–6–9. Plot the root loci and asymptotes for a unity-feedback system with the following feedforward

transfer function:

Determine the exact points where the root loci cross the jvaxis

Solution.The feedforward transfer function G(s)can be written as

Note that as sapproaches infinity, can be written as

where we used the following formula:

The expression

gives the equation for the asymptotes.

slimSqG(s)=slimSq

K

(s+ 1 )^4

(s+a)^4 =s^4 + 4 as^3 + 6 a^2 s^2 + 4 a^3 s+a^4

=slimSq

K

(s+ 1 )^4

slimSq

K

s^4 + 4 s^3 + 6 s^2 + 4 s+ 1

slimSqG(s)=slimSq

K

s^4 + 4 s^3 + 11 s^2 + 14 s+ 10

slimSqG(s)

G(s)=

K

s^4 + 4 s^3 + 11 s^2 + 14 s+ 10

G(s)=

K

(s^2 + 2 s+ 2 )(s^2 + 2 s+ 5 )

Root-Locus Plot of G(s)=K/[s(s+1)(s+2)] and Aysmptotes

Imag Axis

4

–4

0

3

2

1

–1

–2

–3

Real Axis

Figure 6–71 –4 –3 –2 –1 041 2 3

Root-locus plot.

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