Modern Control Engineering

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

R(s) C(s)

(a)

1

s

K

(s+ 1) (s+ 2)

R(s) C(s)

(c)

1

s+ 1

K

s(s+ 1) (s+ 2)

K

s(s+ 2)

s+ 1

H(s)

R(s) C(s)

(b)

+

+

+–

G(s)

+–

Figure 6–14

(a) Control system

with velocity

feedback; (b) and

(c) modified block

diagrams.

Cancellation of Poles of G(s) with Zeros of H(s). It is important to note that

if the denominator of G(s)and the numerator of H(s)involve common factors, then the

corresponding open-loop poles and zeros will cancel each other, reducing the degree of

the characteristic equation by one or more. For example, consider the system shown in

Figure 6–14(a). (This system has velocity feedback.) By modifying the block diagram of

Figure 6–14(a) to that shown in Figure 6–14(b), it is clearly seen that G(s)andH(s)

have a common factor s+1.The closed-loop transfer function C(s)/R(s)is

The characteristic equation is

Because of the cancellation of the terms (s+1)appearing in G(s)andH(s),however,

we have

The reduced characteristic equation is

The root-locus plot of G(s)H(s)does not show all the roots of the characteristic equa-

tion, only the roots of the reduced equation.

To obtain the complete set of closed-loop poles, we must add the canceled pole of

G(s)H(s)to those closed-loop poles obtained from the root-locus plot of G(s)H(s).

The important thing to remember is that the canceled pole of G(s)H(s)is a closed-loop

pole of the system, as seen from Figure 6–14(c).

s(s+2)+K= 0

=

s(s+2)+K

s(s+2)

1 +G(s)H(s)= 1 +

K(s+1)

s(s+1)(s+2)

Cs(s+2)+KD(s+1)= 0

C(s)

R(s)

=

K

s(s+1)(s+2)+K(s+1)

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