Modern Control Engineering

Example Problems and Solutions 829

Show that

where the bk’s (k=0, 1, 2,p,n)are those coefficients appearing in the numerator of the transfer
function when C(sI-A)–1B+Dis written as follows:

whereD=b 0.

Solution.Let us consider the case where n=3.We shall show that

(10–153)

Note that, by referring to Problem A–10–7, we have

Hence, we need to show that

or

WC (10–154)

0

0

• a 3

1

0

• a 2

0

1

• a 1

S =C

0

1

0

0

0

1

• a 3


• a 2


• a 1

SW

WC

0

0

• a 3

1

0

• a 2

0

1

• a 1

S W-^1 = C

0

1

0

0

0

1

• a 3


• a 2


• a 1

S

(WN) A(WN)-^1 =WCN A(N)-^1 D W-^1 =WC

0

0

• a 3

1

0

• a 2

0

1

• a 1

S W-^1

Q-^1 AQ=(WN) A(WN)-^1 = C

0

1

0

0

0

1

• a 3


• a 2


• a 1

S

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

b 0 sn+b 1 sn-^1 +p+bn- 1 s+bn

sn+a 1 sn-^1 +p+an- 1 s+an

Q-^1 B= F

bn-an b 0

bn- 1 - an- 1 b 0







b 1 - a 1 b 0

V

CQ=[ 0 0 p 0 1 ]

Q-^1 AQ= G

0 1 0    0

0 0 1    0

p

p

p

p

0 0 0    1

• an


• an- 1


• an- 2
  
  
  


• a 1

W