Modern Control Engineering

A–7–2. Consider the system defined by

Obtain the sinusoidal transfer functions and

In deriving and we assume that Simi-

larly, in obtaining and we assume that

Solution.The transfer matrix expression for the system defined by

is given by

whereG(s)is the transfer matrix and is given by

For the system considered here, the transfer matrix becomes

Hence

Assuming that U 2 (jv)=0, we find and as follows:

Similarly, assuming that U 1 (jv)=0, we find and as follows:

Notice that Y 2 (jv)U 2 (jv)is a nonminimum-phase transfer function.

Y 2 (jv)

U 2 (jv)

=

jv- 25

(jv)^2 +4jv+ 25

Y 1 (jv)

U 2 (jv)

=

jv+ 5

(jv)^2 +4jv+ 25

Y 1 (jv)U 2 (jv) Y 2 (jv)U 2 (jv)

Y 2 (jv)

U 1 (jv)

=

- 25

(jv)^2 +4jv+ 25

Y 1 (jv)

U 1 (jv)

=

jv+ 4

(jv)^2 +4jv+ 25

Y 1 (jv)U 1 (jv) Y 2 (jv)U 1 (jv)

B

Y 1 (s)

Y 2 (s)

R= D

s+ 4

s^2 +4s+ 25

• 25
  s^2 +4s+ 25

s+ 5

s^2 +4s+ 25

s- 25

s^2 +4s+ 25

TB

U 1 (s)

U 2 (s)

R

= D

s+ 4

s^2 +4s+ 25

• 25
  s^2 +4s+ 25

s+ 5

s^2 +4s+ 25

s- 25

s^2 +4s+ 25

T

=

1

s^2 +4s+ 25

B

s+ 4

• 25

1

s

RB

1

0

1

1

R

C(s I-A)-^1 B+D= B

1

0

0

1

RB

s

25

- 1

s+ 4

R

• 1
  B

1

0

1

1

R

G(s)=C(s I-A)-^1 B+D

Y(s)=G(s)U(s)

y#=Cx+Du

x#=Ax+Bu

Y 1 (jv)U 2 (jv) Y 2 (jv)U 2 (jv), U 1 (jv)=0.

Y 2 (jv)U 2 (jv). Y 1 (jv)U 1 (jv) Y 2 (jv)U 1 (jv), U 2 (jv)=0.

Y 1 (jv)U 1 (jv),Y 2 (jv)U 1 (jv),Y 1 (jv)U 2 (jv),

B

y 1

y 2

R= B

1

0

0

1

RB

x 1

x 2

R

B

x# 1

x# 2

R= B

0

- 25

1

- 4

RB

x 1

x 2

R+ B

1

0

1

1

RB

u 1

u 2

R

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

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