Modern Control Engineering

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

Notice that

This network has a dc gain of

From Equation (6–18), we see that this network is a lead network if

or It is a lag network if The pole-zero configurations of this net-

work when and are shown in Figure 6–37(a) and (b),

respectively.

Lead Compensation Techniques Based on the Root-Locus Approach. The

root-locus approach to design is very powerful when the specifications are given in

terms of time-domain quantities, such as the damping ratio and undamped natural

frequency of the desired dominant closed-loop poles, maximum overshoot, rise time,

and settling time.

Consider a design problem in which the original system either is unstable for all val-

ues of gain or is stable but has undesirable transient-response characteristics. In such a

case, the reshaping of the root locus is necessary in the broad neighborhood of the jv

axis and the origin in order that the dominant closed-loop poles be at desired locations

in the complex plane. This problem may be solved by inserting an appropriate lead com-

pensator in cascade with the feedforward transfer function.

The procedures for designing a lead compensator for the system shown in Figure

6–38 by the root-locus method may be stated as follows:

1.From the performance specifications, determine the desired location for the dom-

inant closed-loop poles.

R 1 C 17 R 2 C 2 R 1 C 16 R 2 C 2

a 6 1. R 1 C 16 R 2 C 2.

R 1 C 17 R 2 C 2 ,

Kc a=R 2 R 4 AR 1 R 3 B.

Kc a=

R 4 C 1

R 3 C 2

R 2 C 2

R 1 C 1

=

R 2 R 4

R 1 R 3

, a=

R 2 C 2

R 1 C 1

jv

s

(a)

1

R 2 C 2

-^1
R 1 C 1

jv

s

(b)

1

R 2 C 2

(^1) –
R 1 C 1

00

Figure 6–37

Pole-zero

configurations:

(a) lead network;

(b) lag network.

+– Gc(s) G(s)

Figure 6–38

Control system.

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