Modern Control Engineering

instantaneously transfer energy from the input to the output. In addition, since

noise is almost always present in one form or another, a system with a unity transfer

function is not desirable. A desired control system, in many practical cases, may

have one set of dominant complex-conjugate closed-loop poles with a reasonable

damping ratio and undamped natural frequency. The determination of the significant

part of the closed-loop pole-zero configuration, such as the location of the domi-

nant closed-loop poles, is based on the specifications that give the required system

performance.

Cancellation of Undesirable Complex-Conjugate Poles. If the transfer func-

tion of a plant contains one or more pairs of complex-conjugate poles, then a lead, lag,

or lag–lead compensator may not give satisfactory results. In such a case, a network that

has two zeros and two poles may prove to be useful. If the zeros are chosen so as to

cancel the undesirable complex-conjugate poles of the plant, then we can essentially

replace the undesirable poles by acceptable poles. That is, if the undesirable complex-

conjugate poles are in the left-half splane and are in the form

then the insertion of a compensating network having the transfer function

will result in an effective change of the undesirable complex-conjugate poles to ac-

ceptable poles. Note that even though the cancellation may not be exact, the com-

pensated system will exhibit better response characteristics. (As stated earlier, this

approach cannot be used if the undesirable complex-conjugate poles are in the right-

halfsplane.)

Familiar networks consisting only of RCcomponents whose transfer functions pos-

sess two zeros and two poles are the bridged-Tnetworks. Examples of bridged-Tnet-

works and their transfer functions are shown in Figure 7–117. (The derivations of the

transfer functions of the bridged-Tnetworks were given in Problem A–3–5.)

s^2 + 2 z 1 v 1 s+v^21

s^2 + 2 z 2 v 2 s+v^22

1

s^2 + 2 z 1 v 1 s+v^21

520 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method

C 2

RR

C 1

R 2

CC

ei eo ei R 1 eo

(a) (b)

Eo(s)

Ei(s)

RC 1 RC 2 s^2 + 2 RC 2 s+ 1

RC 1 RC 2 s^2 + (RC 1 + 2 RC 2 )s+ 1

= Eo(s)

Ei(s)

R 1 CR 2 Cs^2 + 2 R 1 Cs+ 1

R 1 CR 2 Cs^2 + (R 2 C+ 2 R 1 C)s+ 1

=

Figure 7–117

Bridged-Tnetworks.

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