Modern Control Engineering

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Example Problems and Solutions 235

It follows that

From the block diagram we have

from which

Therefore, the values of TandKare determined as

A–5–4. Determine the values of Kandkof the closed-loop system shown in Figure 5–53 so that the maximum

overshoot in unit-step response is 25%and the peak time is 2 sec. Assume that J=1kg-m^2.

Solution.The closed-loop transfer function is

By substituting J=1kg-m^2 into this last equation, we have

Note that in this problem

The maximum overshoot Mpis

which is specified as 25%. Hence

from which

zp

21 - z^2

=1.386

e-zp^21 - z

2

=0.25

Mp=e-zp^21 - z

2

vn= 1 K, 2 zvn=Kk

C(s)

R(s)

=

K

s^2 +Kks+K

C(s)

R(s)

=

K

Js^2 +Kks+K

K =v^2 n T=1.14^2 *1.09=1.42

T =

1

2 zvn

=

1

2 0.41.14

=1.09

vn=

A

K

T

, 2 zvn=

1

T

C(s)

R(s)

=

K

Ts^2 +s+K

vn=1.14

+

• +–

R(s) C(s)

k

1

s

K

Js

Figure 5–53
Closed-loop system.