Modern Control Engineering

824 Chapter 10 / Control Systems Design in State Space

we have

Define

where

Then

and

Define

Then the state equation becomes

Since

and

we have

which is in the controllable canonical form.

A–10–5. Consider a system defined by

where

A=B

0

- 2

1

- 3

R, B= B

0

2

R, C=[ 1 0 ]

y =Cx

x

=Ax+Bu

B

xˆ

1

xˆ

2

R = B

0

- 1

1

- 2

RB

xˆ 1

xˆ 2

R +B

0

1

Ru

T-^1 B= B

0.5

0.5

0

0.5

RB

0

2

R = B

0

1

R

T-^1 AT= B

0.5

0.5

0

0.5

RB

1

- 4

1

- 3

RB

2

- 2

0

2

R = B

0

- 1

1

- 2

R

xˆ

=T-^1 ATxˆ+T-^1 Bu

x=Txˆ

T-^1 = B

0.5

0.5

0

0.5

R

T= B

0

2

2

- 6

RB

2

1

1

0

R = B

2

- 2

0

2

R

M= B

0

2

2

- 6

R, W=B

2

1

1

0

R

T=MW

a 1 =2, a 2 = 1

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