Modern Control Engineering

Section 6–8 / Lag-Lead Compensation 339

Assuming the specifications are the same as those given in Example 6–8, design a compensator
Gc(s).
The desired locations for the dominant closed-loop poles are at

The open-loop transfer function of the compensated system is

Since the requirement on the static velocity error constant Kvis 80 sec–1,we have

Thus

The time constant and the value of bare determined from

(The angle deficiency of 55° was obtained in Example 6–8.) Referring to Figure 6–58, we can
easily locate points AandBsuch that

(Use a graphical approach or a trigonometric approach.) The result is

or

The phase-lead portion of the lag–lead network thus becomes

For the phase-lag portion, we choose such that it satisfies the conditions

• 5 ° (^6) n
  s+

1

T 2

s+

1

3.503T 2

4

s=-2.50+j4.33

4 60 °

s+

1

T 2

s+

1

3.503T 2

4

s=-2.50+j4.33

1,

T 2

10 a

s+2.38

s+8.34

b

T 1 =

1

2.38

=0.420, b=8.34T 1 =3.503

AO=2.38, BO=8.34

/APB= 55 °,

PA

PB

=

4.77

8

n

s+

1

T 1

s+

b

T 1

4

s=-2.5+j4.33

= 55 °

4

s+

1

T 1

s+

b

T 1

42

40

s(s+0.5)

2

s=-2.5+j4.33

= 4

s+

1

T 1

s+

b

T 1

4

8

4.77

= 1

T 1

Kc= 10

Kv=limsS 0 sGc(s)G(s)=limsS 0 Kc

4

0.5

=8Kc= 80

Gc(s)G(s)=Kc

as+

1

T 1

bas+

1

T 2

b

as+

b

T 1

bas+

1

bT 2

b



4

s(s+0.5)

s=-2.50;j4.33