Modern Control Engineering

Example Problems and Solutions 695

then we get a slightly different state-space representation. This choice of state variables gives

from which we obtain

(9–89)

Referring to Equation (9–86), the output equation becomes

from which we get

(9–90)

Equations (9–89) and (9–90) give the following state-space representation for the system:

A–9–4. Consider the system defined by

(9–91)

where the system involves a triple pole at s=–p 1. (We assume that, except for the first three

pi’s being equal, the pi’s are different from one another.) Obtain the Jordan canonical form of the

state-space representation for this system.

Y(s)

U(s)

=

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

As+p 1 B^3 As+p 4 BAs+p 5 BpAs+pnB

y=[ 1 1 p 1 ]F

xˆ 1

xˆ 2







xˆn

V +b 0 u

F

xˆ

1

xˆ

2







xˆ

n

V = F

• p 1

0

• p 2
  
  
  

0

• pn

VF

xˆ 1

xˆ 2







xˆn

V + F

c 1

c 2







cn

Vu

y=xˆ 1 +xˆ 2 +p+xˆn+b 0 u

Y(s)=b 0 U(s)+Xˆ 1 (s)+Xˆ 2 (s)+p+Xˆn(s)

xˆ

n=-pn^ xˆn+cn^ u







xˆ

2 =-p 2 xˆ 2 +c 2 u

xˆ

1 =-p 1 xˆ 1 +c 1 u

sXˆn(s)=-pn Xˆn(s)+cn U(s)







sXˆ 2 (s)=-p 2 Xˆ 2 (s)+c 2 U(s)

sXˆ 1 (s)=-p 1 Xˆ 1 (s)+c 1 U(s)