Modern Control Engineering

832 Chapter 10 / Control Systems Design in State Space

where

The rank of the observability matrix N,

is 2. Hence, the system is completely observable. Transform the system equations into the ob-

servable canonical form.

Solution.Since

we have

Define

where

Then

and

Define

Then the state equation becomes

or

(10–157)

The output equation becomes

y=CQxˆ

= B

0

1

- 1

- 2

RB

xˆ 1

xˆ 2

R + B

0

2

Ru

B

xˆ

1

xˆ

2

R = B

- 1

1

0

1

RB

1

- 4

1

- 3

RB

- 1

1

0

1

RB

xˆ 1

xˆ 2

R + B

- 1

1

0

1

RB

0

2

Ru

xˆ

=Q-^1 AQxˆ +Q-^1 Bu

x=Qxˆ

Q-^1 = B

- 1

1

0

1

R

Q= bB

2

1

1

0

RB

1

- 3

1

- 2

Rr

• 1
  = B

- 1

1

0

1

R

• 1
  = B

- 1

1

0

1

R

N= B

1

1

- 3

- 2

R, W= B

a 1

1

1

0

R = B

2

1

1

0

R

Q=(WN*)-^1

a 1 =2, a 2 = 1

∑s I-A∑=s^2 +2s+ 1 =s^2 +a 1 s+a 2

N= CCA C*D= B

1

1

- 3

- 2

R

A=B

1

- 4

1

- 3

R, B= B

0

2

R, C=[ 1 1 ]

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