Internal Process Controls - Control of Color Imaging Systems: Analysis and Design - page 519

the gain matrix K places the poles (or roots) of the characteristic equation. In other
words, K is chosen by assigning the closed-loop poles. See Example 9.10 for
more details.

Example 9.10

State equations for a closed-loop TC control system are given below. Design the
gain matrix for placing the poles at (a) l 1 ¼ 0.2, l 2 ¼ 0.25 and (b) l 1 ¼ 0.6,

l 2 ¼ 0.65. Assume TC target in percent. Carrier mass ¼ 800 g. The state equation is

x(k þ 1) ¼ Ax(k) þ Bu(k) þ Er(k þ 1) (9 : 77a)

And the output equation is

y(k) ¼ Cx(k) þ Fr(k) (9 : 77b)

The actuator with output linear feedback is given by

u(k) ¼Ky(k) (9 : 78)

where

, A ¼ , B ¼ , ,

t c (k)
w(k)

10 g

g

0
1

x(k) ¼ E ¼ r(k) ¼ t d (k),

11

, , and K ¼ tK p

10
01

1
0

C ¼ F ¼ tK i

S OLUTION
We use pole-placement design for SISO system discussed in Section 5.2.2.

Case (a): Since TC target is to be calculated as a percentage and the carrier
mass ¼ 800 g, we can write g¼ 100 = 800 ¼ 1 = 8. The open-loop TC equation is

given by

"#
u(k)

x(k)

1 r

1
8

10 0

x(k þ 1) ¼ þ þ (9 : 79)

11

1
8

where r is the constant value of TC expressed in percent. The actuator equation
can be written as follows:

u(k) ¼Ky(k) ¼K[Cx(k) þ Fr(k)] ¼KCx(k) KFr ¼Kx(k) KFr (9 : 80)

Substituting Equation 9.80 in Equation 9.77, we get

x(k þ 1) ¼ (A BK)x(k) BKFr þ Er (9 : 81)