Internal Process Controls - Control of Color Imaging Systems: Analysis and Design - page 518

or in a more compact form as

u ( k ) ¼ Ky ( k ) ( 9 : 74 )

where
K ¼ t K p t K i ] is the gain matrix

y(k) is the output vector for the system of Figure 9.31

[

The vector y(k) can be written in following form:

t c ( k )
w ( k )

t d ( k )

e s ( k )
w ( k )

10
01

1
0

y ( k ) ¼ ¼ þ

¼ Cx ( k ) þ Fr ( k ) ( 9 : 75 )

, , F ¼ ,

10
01

t c ( k )
w ( k )

1
0

where C ¼ and r ( k ) ¼ t d ( k )

Equations 9.71 and 9.75 represent the TC control system in state variable form and
Equation 9.74 the output feedback equation. Figure 9.32 shows these equations in
block diagram form after the equations are transformed to z-domain. Note that
although the original system was SISO, the control system is SIMO (single input
multiple output). A systematic approach is required to design the controller gains.
The controller gains can be designed using pole placement or linear quadratic
regulator (LQR) techniques (Section 5.2 or 5.3) [13]. Below we show the pole-
placement design.
Substituting Equation 9.74 in Equation 9.71 and assuming r(k þ

x ( k ) ¼

1) ¼ r(k),

following closed-loop state equation is obtained:

x ( k þ 1 ) ¼ ( A BKC ) x ( k ) þ ( E BKF ) r ( k ) ( 9 : 76 )

The matrix, A c ¼ A BKC in Equation 9.76 represents the closed-loop system matrix
of Figure 9.32. Hence, the closed-loop characteristic matrix is zI A c ¼

0. Recall that

F

x ( z )

r ( z ) y ( z )

z -1 I +

E + + A

A

B

u ( z )

-K

FIGURE 9.32 Block diagram of the feedback system with PI controller.