State-Variable Representation - Control of Color Imaging Systems: Analysis and Design - page 195

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

2 3 2 3 2 3 2 3

x 1 ( k þ 1 ) 00 0 a N x 1 ( k )

x 2 ( k )
.
x N 1 ( k )
x N ( k )

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

4 5 4 5 4 5 4 5

x 2 ( k þ 1 ) 10 0 a N 1

.
x N 1 ( k þ

01 0 a N 3

¼ þ u ( k )( 4 : 93 )

. . . . .

)

1

x N ( k þ 1 ) 00 1 a 1

with the output equation given as

2 3

x 1 ( k )
x 2 ( k )
.
x N 1 ( k )
x N ( k )

4 5

y ( k ) ¼ ½ 000 1 þ b 0 u ( k ) ( 4 : 94 )

Example 4.10

Consider the following SISO discrete-time LTI system

y(k þ 2) þ 0 : 75y(k þ 1) þ 0 : 25y(k) ¼ u(k þ 2) þ 3u(k þ 1) 4u(k)

Obtain a state-space representation of this system in (a) controllable and (b)
observable canonical forms.

S OLUTION

(a) Controllable canonical form: The state equations are

x 1 (k)
x 2 (k)

u(k)

x 1 (k þ 1) 0 1 0

1

¼ þ

x 2 (k þ 1) 0 : 25 0 : 75

and the output equation is

x 1 (k)
x 2 (k)

y(k) ¼ ½ 4 : 25 2 : 25 þ u(k)

(b) Observable canonical form: The state equations and the output equation are

x 1 (k)
x 2 (k)

u(k)

x 1 (k þ 1) 0 0 : 25 4 : 25

¼ þ and

x 2 (k þ 1) 1 0 : 75 2 : 25

x 1 (k)
x 2 (k)

y(k) ¼ ½ 01 þ u(k)