State-Variable Representation - Control of Color Imaging Systems: Analysis and Design - page 194

D

x ( k +1)

x ( k )

+ +

Unit
delay

y ( k )

B

u ( k ) + C

+

A

FIGURE 4.6 Block diagram of a discrete LTI system in state space.

4.7 STATE-SPACE REPRESENTATION OF DISCRETE-TIME
LTI SYSTEMS
Consider the LTI discrete-time dynamic system described by the difference equation:

X X

N M

y ( k ) ¼ a i y ( k i ) þ b i u ( k i ) ( 4 : 90 )

i ¼ 1 i ¼ 0

The state-space realizations of the above system in two most important standard
forms, i.e., the controllable and observable canonical forms, are discussed below.

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

2 3 2 3

2
4

3
5

2
4

3
5

x 1 ( k þ 1 ) x 1 ( k )

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

0 1 0 0 0

0
.
0
1

4 5 4 5

x 2 ( k þ 1 )

0 0 1 0

. . . . .

.
x N 1 ( k þ

¼ þ u ( k )( 4 : 91 )

) 0 0 0 1

1

x N ( k þ

) a N a N 1 a N 2 a 1

1

with the output equation

2
4

3
5

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

y ( k ) 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 ( k )

( 4 : 92 )