Example Problems and Solutions 831

where

The transfer function G(s)for the system defined by Equations (10–155) and (10–156) is

Noting that

we have

Note that D=b 0 .Since

we have

Hence,

Thus, we have shown that

Note that what we have derived here can be easily extended to the case when nis any positive

integer.

A–10–9. Consider a system defined by

y =Cx

x# =Ax+Bu

Q-^1 B=C

g 3

g 2

g 1

S= C

b 3 - a 3 b 0

b 2 - a 2 b 0

b 1 - a 1 b 0

S

g 1 =b 1 - a 1 b 0 , g 2 =b 2 - a 2 b 0 , g 3 =b 3 - a 3 b 0

=

b 0 s^3 +b 1 s^2 +b 2 s+b 3

s^3 +a 1 s^2 +a 2 s+a 3

=

b 0 s^3 +Ag 1 +a 1 b 0 Bs^2 +Ag 2 +a 2 b 0 Bs+g 3 +a 3 b 0

s^3 +a 1 s^2 +a 2 s+a 3

=

g 1 s^2 +g 2 s+g 3

s^3 +a 1 s^2 +a 2 s+a 3

+b 0

G(s)=

1

s^3 +a 1 s^2 +a 2 s+a 3

C 1 s s^2 DC

g 3

g 2

g 1

S +D

C

s

• 1
  0

0

s

• 1

a 3

a 2

s+a 1

S

• 
  

  1

  
  

1

s^3 +a 1 s^2 +a 2 s+a 3

C

s^2 +a 1 s+a 2

s+a 1

1

• a 3
  s^2 +a 1 s
  s


• a 3 s


• a 2 s-a 3
  s^2

S

G(s)=[ 0 0 1 ]C

s

• 1
  0

0

s

• 1

a 3

a 2

s+a 1

S

• 1
  C

g 3

g 2

g 1

S+D

CQ=[0 0 1]

G(s)=CQAs I-Q-^1 AQB-^1 Q-^1 B+D

C

g 3

g 2

g 1

S =Q-^1 B