Modern Control Engineering

Section 6–7 / Lag Compensation 325

The angle contribution of this lag network near a dominant closed-loop pole is about 4°. Because

this angle contribution is not very small, there is a small change in the new root locus near the

desired dominant closed-loop poles.

The open-loop transfer function of the compensated system then becomes

where

The block diagram of the compensated system is shown in Figure 6–49. The root-locus plot for the

compensated system near the dominant closed-loop poles is shown in Figure 6–50(a), together with

the original root-locus plot. Figure 6–50(b) shows the root-locus plot of the compensated system

K=1.06Kˆc

=

K(s+0.05)

s(s+0.005)(s+1)(s+2)

Gc(s)G(s)=Kˆc

s+0.05

s+0.005

1.06

s(s+1)(s+2)

Kcss++ 0.005 0.05

Kc= 0.966

1.06

+– s(s+ 1) (s+ 2)

^

Figure 6–49 ^
Compensated
system.

Figure 6–50
(a) Root-locus plots of the compensated system and uncompensated system; (b) root-locus plot of compensated
system near the origin.

Real Axis

− 3 −2.5 − 2 −1.5 − 1 −0.5 0 0.5 1

(a)

Imag Axis

2

− 2

1.5

− 1

−1.5

1

0

0.5

−0.5

Root-Locus Plots of Compensated and Uncompensated Systems

Uncompensated system

Original closed-loop pole

Compensated system

New closed-

loop pole

−0.4 −0.2 0 0.2 0.4 0.6

Root-Locus Plot of Compensated System near the Origin

Real Axis

Imag Axis−0.1

0.1

0.5

−0.3

−0.4

0.3

0

0.2

−0.2

0.4

−0.5

(b)