332 Chapter 6 / Control Systems Analysis and Design by the Root-Locus Method+– Gc(s) G(s)Figure 6–54
Control system.Then Equation (6–21) becomes
(6–22)
where
Note that gis often chosen to be equal to b.
Lag–lead Compensation Techniques Based on the Root-Locus Approach.
Consider the system shown in Figure 6–54.Assume that we use the lag–lead compensator:
(6–23)
where and (Consider Kcto belong to the lead portion of the lag–lead
compensator.)
In designing lag–lead compensators, we consider two cases where and
Case 1. In this case, the design process is a combination of the design of the
lead compensator and that of the lag compensator. The design procedure for the lag–lead
compensator follows:
1.From the given performance specifications, determine the desired location for the
dominant closed-loop poles.
2.Using the uncompensated open-loop transfer function G(s),determine the angle
deficiency fif the dominant closed-loop poles are to be at the desired location. The
phase-lead portion of the lag–lead compensator must contribute this angle f.
3.Assuming that we later choose sufficiently large so that the magnitude of the lag
portion
4
s 1 +
1
T 2
s 1 +
1
bT 2
4
T 2
gZb.
gZb g=b.
b 71 g 7 1.
Gc(s)=Kc
b
g
AT 1 s+ 1 BAT 2 s+ 1 B
aT 1
g
s+ 1 bAbT 2 s+ 1 B
=Kc±
s+
1
T 1
s+
g
T 1
≤±s+
1
T 2
s+
1
bT 2
≤g=
R 1 +R 3
R 1
7 1, b=
R 2 +R 4
R 2
7 1, Kc=
R 2 R 4 R 6
R 1 R 3 R 5
R 1 +R 3
R 2 +R 4
Eo(s)
Ei(s)
=Kc
b
g£
T 1 s+ 1
T 1
g
s+ 1
≥aT 2 s+ 1
bT 2 s+ 1
b =Kc
as+
1
T 1
bas+
1
T 2
bas+
g
T 1
bas+
1
bT 2
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