Modern Control Engineering

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

The value of the gain Kat any point on root locus can be found by applying the magnitude

condition or by use of MATLAB (see Section 6–3). For example, the value of Kat which the

complex-conjugate closed-loop poles have the damping ratio z=0.7can be found by locating the

roots, as shown in Figure 6–10, and computing the value of Kas follows:

Or use MATLAB to find the value of K. (See Section 6–4.)

It is noted that in this system the root locus in the complex plane is a part of a circle. Such a

circular root locus will not occur in most systems. Circular root loci may occur in systems that in-

volve two poles and one zero, two poles and two zeros, or one pole and two zeros. Even in such

systems, whether circular root loci occur depends on the locations of poles and zeros involved.

To show the occurrence of a circular root locus in the present system, we need to derive the

equation for the root locus. For the present system, the angle condition is

Ifs=s+jvis substituted into this last equation, we obtain

which can be written as

or

Taking tangents of both sides of this last equation using the relationship

(6–10)

we obtain

or

which can be simplified to

or

This last equation is equivalent to

v= 0 or (s+2)^2 +v^2 =A 13 B^2

vC(s+2)^2 +v^2 - 3 D= 0

2 v(s+1)

(s+1)^2 - Av^2 - 2 B

=

v

s+ 2

v- 12

s+ 1

+

v+ 12

s+ 1

1 - a

v- 12

s+ 1

ba

v+ 12

s+ 1

b

=

v

s+ 2

; 0

1 <

v

s+ 2

* 0

tan ctan-^1 a

v- 12

s+ 1

b+tan-^1 a

v+ 12

s+ 1

bd=tan ctan-^1 a

v

s+ 2

b; 180 °(2k+1)d

tan (x;y)=

tanx;tany

1 <tanxtany

tan-^1 a

v- 12

s+ 1

b+tan-^1 a

v+ 12

s+ 1

b=tan-^1 a

v

s+ 2

b; 180 °(2k+1)

tan-^1 a

v

s+ 2

b-tan-^1 a

v- 12

s+ 1

b-tan-^1 a

v+ 12

s+ 1

b=; 180 °(2k+1)

/s+^2 +jv- /s+^1 +jv-j^12 - /s+^1 +jv+j^12 =;^180 °(2k+1)

/s+ 2 - /s+ 1 - j 12 - /s+ 1 +j 12 =; 180 °(2k+1)

K=^2

As+ 1 - j 12 BAs+ 1 +j 12 B

s+ 2

2

s=-1.67+j1.70

=1.34

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