Closed-Loop System Analysis and Design - Control of Color Imaging Systems: Analysis and Design - page 219

Equation 5.21 can be rewritten as

P ( A c ) ¼ a N I þ a N 1 ( A BK ) þ a N 2 ( A 2

ABK BKA c )

þ a N 3 ( A 3 A 2 BK ABKA c BKA 2

c )

þþ( A N A N 1 BK A N 2 BKA c BKA N 1 ) ( 5 : 22 )

c

Simplifying further, this results in

P ( A c ) ¼ a N I þ a N 1 A þ a N 2 A 2 þþ A N

a N 1 BK a N 2 ABK

a N 2 BKA c BKA N 1
c
¼ P ( A ) B (a N 1 K þ a N 2 KA c þþ KA N 1 )

c

AB (a N 2 K þ a N 3 KA c þþ KA N 2 ) A N 1 BK ( 5 : 23 )

c

Equation 5.23 can be written in matrix form as

2
4

3
5

a N 1 K þ a N 2 KA c þ KA N 1

c
a N 2 K þ a N 3 KA c þ KA N 2
c
a N 3 K þ a N 4 KA c þ KA N 3
c P ( A c ) ¼ P ( A ) BABA 2 B

A N 1 B

.
K

( 5 : 24 )

Since P ( A c ) ¼ 0, we have

2 3

a N 1 K þ a N 2 KA c þ KA N 1

c
a N 2 K þ a N 3 KA c þ KA N 2

4 5

c
a N 3 K þ a N 4 KA c þ KA N 3

P ( A ) ¼ BABA 2 B A N 1 B

c

.
K

( 5 : 25 )

Since the system is completely state controllable, the controllability matrix

is nonsingular and its inverse exists, therefore we have

BABA 2 B A N 1 B

2
4

3
5

a N 1 K þ a N 2 KA c þ KA N 1

c
a N 2 K þ a N 3 KA c þ KA N 2

c
a N 3 K þ a N 4 KA c þ KA N 3

1 P ( A )

¼ BABA 2 B

A N 1 B

c

.
K

( : )

5 26