Internal Process Controls - Control of Color Imaging Systems: Analysis and Design - page 526

v ( k )

g

z- 1

x 1 ( k ) x 2 ( k ) x 3 ( k )

x μ+2 ( k )

_

x μ+1 ( k )

g

+

1 1 1

y ( k ) = t c ( k )

u ( k )

z- 1

FIGURE 9.39 Block diagram of open-loop TC model with time delay.

ned as quantities related to
the toner mass at discrete delay cycle. From the block diagram shown in Figure 9.39,
the state equations can be written as

In physical terms, for the TC system, states can be de

x 1 ( k þ 1 ) ¼ x 1 ( k ) þ gu ( k )

x 2 ( k þ 1 ) ¼ x 1 ( k )

.
x mþ 1 ( k þ

( 9 : 99 a)

1 ) ¼ x m ( k )

x mþ 2 ( k þ 1 ) ¼ x mþ 2 ( k ) þ gv ( k )

and the output equation is given by

y ( k ) ¼ t c ( k ) ¼ x mþ 1 ( k ) x mþ 2 ( k ) ( 9 : 99 b)

It is important to note that, in Equation 9.99, the states represent TC at a discrete
delay cycle, which in turn is related to the toner mass.
Total number of states depends on the delay cycles. If the time delay is zero,
that is, when the toner dispense is instantaneous, number of states will be reduced
to two (one due to the mass dispensed and another due to the mass developed) as
seen from the state equation below. The above equations can be grouped into
matrix form as

x ( k þ 1 ) ¼ Ax ( k ) þ Bu ( k )

x mþ 2 ( k þ ) ¼ x mþ 2 ( k ) þ gv ( k )

y ( k ) ¼ Cx ( k ) x mþ 2 ( k )

1 ( 9 : 100 )

where x ( k ) 2 R mþ 1 , x mþ 2 ( k ) 2 R 1 , u ( k ) 2 R 1 , A 2 R (mþ 1 )(mþ 1 ) , B 2 R mþ 1 ,

and

C 2 R mþ 1 . These vectors and matrices are