Modern Control Engineering

Example Problems and Solutions 691

A–9–2. Consider the following transfer-function system:

(9–77)

Derive the following observable canonical form of the state-space representation for this transfer-

function system:

(9–78)

(9–79)

Solution.Equation (9–77) can be modified into the following form:

By dividing the entire equation by snand rearranging, we obtain

(9–80)

Now define state variables as follows:

(9–81)

X 1 (s)=

1

s

Cbn U(s)-an Y(s)D

X 2 (s)=

1

s

Cbn- 1 U(s)-an- 1 Y(s)+X 1 (s)D







Xn- 1 (s)=

1

s

Cb 2 U(s)-a 2 Y(s)+Xn- 2 (s)D

Xn (s)=

1

s

Cb 1 U(s)-a 1 Y(s)+Xn- 1 (s)D

+

1

sn-^1

Cbn- 1 U(s)-an- 1 Y(s)D+

1

sn

Cbn U(s)-an Y(s)D

Y(s)=b 0 U(s)+

1

s

Cb 1 U(s)-a 1 Y(s)D+p

+sCan- 1 Y(s)-bn- 1 U(s)D+an Y(s)-bn U(s)= 0

snCY(s)-b 0 U(s)D+sn-^1 Ca 1 Y(s)-b 1 U(s)D+p

y=[ 0 0 p 0 1 ]G

x 1

x 2







xn- 1

xn

W+b 0 u

F

x# 1

x

2







x#n

V =F

0 1    0

0 0    0

p

p

p

0 0    1

• an


• an- 1
  
  
  


• a 1

VF

x 1

x 2







xn

V + F

bn-an b 0

bn- 1 - an- 1 b 0







b 1 - a 1 b 0

Vu

Y(s)

U(s)

=

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

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