Modern Control Engineering

Section 6–6 / Lead Compensation 315

[See Figure 6–40 (b).] In some cases, after the root loci of the original system have been obtained,

the dominant closed-loop poles may be moved to the desired location by simple gain adjustment.

This is, however, not the case for the present system. Therefore, we shall insert a lead compensator

in the feedforward path.

A general procedure for determining the lead compensator is as follows: First, find the sum

of the angles at the desired location of one of the dominant closed-loop poles with the open-loop

poles and zeros of the original system, and determine the necessary angle fto be added so that

the total sum of the angles is equal to The lead compensator must contribute this

anglef. (If the angle fis quite large, then two or more lead networks may be needed rather than

a single one.)

Assume that the lead compensator Gc(s)has the transfer function as follows:

The angle from the pole at the origin to the desired dominant closed-loop pole at s=–1.5+j2.5981

is 120°. The angle from the pole at s=–1 to the desired closed-loop pole is 100.894°. Hence, the

angle deficiency is

Angle deficiency=180°– 120°– 100.894°=–40.894°

Deficit angle 40.894° must be contributed by a lead compensator.

Note that the solution to such a problem is not unique. There are infinitely many solutions.

We shall present two solutions to the problem in what follows.

Method 1. There are many ways to determine the locations of the zero and pole of the lead

compensator. In what follows we shall introduce a procedure to obtain a largest possible value for

a. (Note that a larger value of awill produce a larger value of Kv.In most cases, the larger the Kvis,

the better the system performance.) First, draw a horizontal line passing through point P, the desired

location for one of the dominant closed-loop poles. This is shown as line PAin Figure 6–41. Draw

also a line connecting point Pand the origin. Bisect the angle between the lines PAandPO,as

shown in Figure 6–41. Draw two lines PCandPDthat make angles with the bisector PB.The

intersections of PCandPDwith the negative real axis give the necessary locations for the pole and

zero of the lead network. The compensator thus designed will make point Pa point on the root locus

of the compensated system. The open-loop gain is determined by use of the magnitude condition.

In the present system, the angle of G(s)at the desired closed-loop pole is

n

10

s(s+ 1 )

2

s=-1.5+j2.5981

=-220.894°

;f 2

Gc(s)=Kc a

Ts+ 1

aTs+ 1

=Kc

s+

1

T

s+

1

aT

, (0 6 a 6 1)

; 180 °(2k+1).

jv

O s

A

P

CBD

1

aT

-^1
T

f

(^2) f
2
Figure 6–41
Determination of the
pole and zero of a
lead network.