Modern Control Engineering

Example Problems and Solutions 693

A–9–3. Consider the transfer-function system defined by

(9–83)

where Derive the state-space representation of this system in the following diagonal

canonical form:

(9–84)

(9–85)

Solution.Equation (9–83) may be written as

(9–86)

Define the state variables as follows:

which may be rewritten as

sXn(s)=-pn Xn(s)+U(s)







sX 2 (s)=-p 2 X 2 (s)+U(s)

sX 1 (s)=-p 1 X 1 (s)+U(s)

Xn(s)=

1

s+pn

U(s)







X 2 (s)=

1

s+p 2

U(s)

X 1 (s)=

1

s+p 1

U(s)

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

c 1

s+p 1

U(s)+

c 2

s+p 2

U(s)+p+

cn

s+pn

U(s)

y=Cc 1 c 2 p cnDF

x 1

x 2







xn

V+b 0 u

F

x# 1

x

2







x#n

V = F

• p 1

0

• p 2
  
  
  

0

• pn

VF

x 1

x 2







xn

V+ F

1 1    1

Vu

piZpj.

=b 0 +

c 1

s+p 1

+

c 2

s+p 2

+p+

cn

s+pn

Y(s)

U(s)

=

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

As+p 1 BAs+p 2 B p As+pnB