Modern Control Engineering

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

2.The lag–lead compensator given by Equation (6–23) is modified to

(6–24)

whereb>1.The open-loop transfer function of the compensated system is

Gc(s)G(s).If the static velocity error constant Kvis specified, determine the value

of constant Kcfrom the following equation:

3.To have the dominant closed-loop poles at the desired location, calculate the angle

contributionfneeded from the phase-lead portion of the lag–lead compensator.

4.For the lag–lead compensator, we later choose sufficiently large so that

is approximately unity, where is one of the dominant closed-loop poles. De-

termine the values of and bfrom the magnitude and angle conditions:

5.Using the value of bjust determined, choose so that

The value of the largest time constant of the lag–lead compensator, should not be

too large to be physically realized. (An example of the design of the lag–lead compen-

sator when g=bis given in Example 6–9.)

bT 2 ,

- 5 ° 6

n

s 1 +

1

T 2

s 1 +

1

bT 2

60 °

4

s 1 +

1

T 2

s 1 +

1

bT 2

4  1

T 2

n

s 1 +

1

T 1

s 1 +

b

T 1

=f

4 Kc±

s 1 +

1

T 1

s 1 +

b

T 1

≤GAs 1 B 4 = 1

T 1

s=s 1

4

s 1 +

1

T 2

s 1 +

1

bT 2

4

T 2

=lim

sS 0

sKcG(s)

Kv=lim

sS 0

sGc(s)G(s)

Gc(s)=Kc

AT 1 s+ 1 BAT 2 s+ 1 B

a

T 1

b

s+ 1 bAbT 2 s+ 1 B

=Kc

as+

1

T 1

bas+

1

T 2

b

as+

b

T 1

bas+

1

bT 2

b

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