Modern Control Engineering

Section 6–6 / Lead Compensation 317

Figure 6–43 shows the root-locus plot for the designed system.

It is worthwhile to check the static velocity error constant Kvfor the system just designed.

Note that the third closed-loop pole of the designed system is found by dividing the charac-

teristic equation by the known factors as follows:

The foregoing compensation method enables us to place the dominant closed-loop poles at

the desired points in the complex plane. The third pole at s =2.65is fairly close to the added

zero at 1.9432. Therefore, the effect of this pole on the transient response is relatively small.

Since no restriction has been imposed on the nondominant pole and no specification has been

given concerning the value of the static velocity error coefficient, we conclude that the present de-

sign is satisfactory.

Method 2. If we choose the zero of the lead compensator at s =1 so that it will cancel the

plant pole at s =1, then the compensator pole must be located at s =3. (See Figure 6–44.)

Hence the lead compensator becomes

The value of Kccan be determined by use of the magnitude condition.

(^2) Kcs+^1
s+ 3

10

s(s+ 1 )

2

s=-1.5+j2.5981

= 1

Gc(s)=Kc

s+ 1

s+ 3

s^3 +5.646s^2 +16.933s+23.875=(s+1.5+j2.5981)(s+1.5-j2.5981)(s+2.65)

=5.139

=limsS 0 sc1.2287

s+1.9432

s+4.6458

10

s(s+ 1 )

d

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

jv

–5 –4 –3 –2 –1 1

j 2

j 1

j 3

• j 3


• j 2


• j 1

s

Figure 6–43
Root-locus plot
of the designed
system.