Modern Control Engineering

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

The closed-loop poles are located at

The damping ratio of the closed-loop poles is. The undamped natural fre-

quency of the closed-loop poles is 3.1623 radsec. Because the damping ratio is small,

this system will have a large overshoot in the step response and is not desirable.

It is desired to design a lead compensator Gc(s)as shown in Figure 6–40(a) so that the dom-

inant closed-loop poles have the damping ratio and the undamped natural frequency

The desired location of the dominant closed-loop poles can be determined from

as follows:

s=-1.5;j2.5981

=(s+1.5+j2.5981)(s+1.5-j 2.5981)

s^2 + 2 zvns+vn^2 =s^2 +3s+ 9

vn=3 radsec.

z=0.5

vn= 210 =

z=( 1  2 ) 210 =0.1581

s=-0.5;j3.1225

R(s) C(s)

(a) (b)

10

s(s+ 1)

G(s)

Closed-loop

pole

jv

− 3 − 2 − 1 1

j 3

j 2

j 1

−j 3

−j 2

−j 1

+

s

Figure 6–39

(a) Control system;

(b) root-locus plot.

(a)

10

s(s+ 1)

G(s)

R(s) C(s)

Gc(s)

(b)

Desired

closed-loop

pole

jv

–3 –1.5 1

j2.5981

j 2

j 1

• j 3


• j 2


• j 1

s

60°

vn = 3

+

Figure 6–40

(a) Compensated

system; (b) desired

closed-loop pole

location.

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