State-Variable Representation - Control of Color Imaging Systems: Analysis and Design - page 185

for state feedback and state estimator design. The first one is controllable canonical

form and the second one is observable canonical form.

4.4.1 C ONTROLLABLE C ANONICAL F ORM
The controllable canonical form is given by

2
4

3
5

2
4

3
5

2
4

3
5

2
4

3
5

x 1 ( t )
x 2 ( t )
.
x N 1 ( t )
x N ( t )

x 1 ( t )
x 2 ( t )
.
x N 1 ( t )
x N ( t )

0 1 0 0 0

0
.
0
1

0 0 1 0

. . . . .

¼ þ u ( t )( 4 : 47 )

0 0 0 1

a N a N 1 a N 2 a 1

with the output equation

2 3

x 1 ( t )
x 2 ( t )
.
x N 1 ( t )
x N ( t )

4 5

y ( t ) ¼ b N a N b 0 b N 1 a N 1 b 0 b N 2 a N 2 b 0 ½ b 1 a 1 b 0 þ b 0 u ( t )

( 4 : 48 )

4.4.2 O BSERVABLE C ANONICAL F ORM
The observable canonical form is given by

2
4

3
5

2
4

3
5

2
4

3
5

2
4

3
5

x 1 ( t )
x 2 ( t )
.
x N 1 ( t )
x N ( t )

00 0 a N x 1 ( t )

x 2 ( t )
.
x N 1 ( t )
x N ( t )

b N a N b 0
b N 1 a N 1 b 0
b N 2 a N 2 b 0
.
b 1 a 1 b 0

10 0 a N 1

01 0 a N 3

¼ þ u ( t )( 4 : 49 )

. . . .

00 1 a 1

with the output equation

2 3

x 1 ( t )
x 2 ( t )
.
x N 1 ( t )
x N ( t )

4 5

y ( t ) ¼ ½ 000 1 þ b 0 u ( t ) ( 4 : 50 )