Modern Control Engineering

Example Problems and Solutions 697

The inverse Laplace transforms of the preceding nequations give

The output equation, Equation (9–92), can be rewritten as

The inverse Laplace transform of this output equation is

Thus, the state-space representation of the system for the case when the denominator polynomial

involves a triple root –p 1 can be given as follows:

(9–93)

(9–94)

The state-space representation in the form given by Equations (9–93) and (9–94) is said to be in

the Jordan canonical form. Figure 9–4 shows a block diagram representation of the system given

by Equations (9–93) and (9–94).

A–9–5. Consider the transfer-function system

Obtain a state-space representation of this system with MATLAB.

Y(s)

U(s)

=

25.04s+5.008

s^3 +5.03247s^2 +25.1026s+5.008

y = Cc 1 c 2 p cnDF

x 1

x 2







xn

V +b 0 u

H

x

1

x# 2

x# 3

x# 4







x#n

X= H

• p 1
  0 0 0    0

1

• p 1
  0
  p

p

0

1

• p 1
  0
  
  
  
  0

0



0

• p 4

0

p

p







0



0

• pn

XH

x 1

x 2

x 3

x 4







xn

X + H

0 0 1 1    1

Xu

y=c 1 x 1 +c 2 x 2 +c 3 x 3 +c 4 x 4 +p+cn xn+b 0 u

Y(s)=b 0 U(s)+c 1 X 1 (s)+c 2 X 2 (s)+c 3 X 3 (s)+c 4 X 4 (s)+p+cn Xn(s)

x#n=-pn xn+u







x# 4 =-p 4 x 4 +u

x# 3 =-p 1 x 3 +u

x# 2 =-p 1 x 2 +x 3

x# 1 =-p 1 x 1 +x 2