Modern Control Engineering

Example Problems and Solutions 521

Concluding Comments. In the design examples presented in this chapter, we

have been primarily concerned only with the transfer functions of compensators. In ac-

tual design problems, we must choose the hardware. Thus, we must satisfy additional

design constraints such as cost, size, weight, and reliability.

The system designed may meet the specifications under normal operating condi-

tions but may deviate considerably from the specifications when environmental changes

are considerable. Since the changes in the environment affect the gain and time con-

stants of the system, it is necessary to provide automatic or manual means to adjust the

gain to compensate for such environmental changes, for nonlinear effects that were not

taken into account in the design, and also to compensate for manufacturing tolerances

from unit to unit in the production of system components. (The effects of manufactur-

ing tolerances are suppressed in a closed-loop system; therefore, the effects may not be

critical in closed-loop operation but critical in open-loop operation.) In addition to this,

the designer must remember that any system is subject to small variations due mainly

to the normal deterioration of the system.

15

10

5

0

• 5


• 10


• 15

90 °

45 °

0 °

• 45 °


• 90 °
  0.2 0.4 0.6 1 2 4 6 10 20 40
  v in rad/sec

C(

jv

)

R(

jv

)

C(

jv

)

R(

jv

)

in dB

Asymptote

Figure 7–118
Bode diagram for
10(1+jv)C(2+jv)(5+jv)D.

Example Problems and Solutions

A–7–1. Consider a system whose closed-loop transfer function is

Clearly, the closed-loop poles are located at s=–2ands=–5, and the system is not oscillatory.

Show that the closed-loop frequency response of this system will exhibit a resonant peak, al-

though the damping ratio of the closed-loop poles is greater than unity.

Solution.Figure 7–118 shows the Bode diagram for the system. The resonant peak value is ap-

proximately 3.5 dB. (Note that, in the absence of a zero, the second-order system with z>0.7will

not exhibit a resonant peak; however, the presence of a closed-loop zero will cause such a peak.)

C(s)

R(s)

=

10(s+1)

(s+2)(s+5)