Internal Process Controls - Control of Color Imaging Systems: Analysis and Design - page 521

4.4 8

Poles at [0.2 0.25]
Poles at [0.6 0.65]

Poles at [0.2 0.25]
Poles at [0.6 0.65]

7

4.2

6

4

5

3.8

4

3

3.6

2

3.4

1

3.2

0

-1

3

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

Iteration # Iteration #

FIGURE 9.33 TC convergence plot for a step input (left), corresponding actuator signal

(right).

" # 1

1
8

1
8

T 1 K ¼ a 2 b 2 ½ a 1 b 1 ¼ ½ 0 : 61 0 : 75 ¼ [4 : 88 1 : 12]

1
8

0

(9 : 88)

For this gain vector, clearly the eigenvalues of (ABK) are 0.65.

Figure 9.33 shows the TC convergence plot and the corresponding actuator
curve for a step change in TC from 3% to 4% for case (a) and (b). Actuator limits
have not been considered in this calculation. Hence, the actuator signal shows a
very high dispense magnitude to meet the demand. The desired eigenvalues can
be adjusted to reduce the magnitude if it violates the actuation limits.

l 1 ¼ 0.6, l 2 ¼

9.11.3 D ESIGN OF A TC C ONTROL L OOP WITH A T IME D ELAY
U SING A PI C ONTROLLER
For a TC system with a dispenser lag, the time delay component

0. This delay
can affect the stability and control of the TC system. It can induce oscillations, and
eventually destabilize the system even though the closed-loop poles are assigned to
ensure stability. It can also lead to chaotic behavior when the maximum dispenser
rate is limited by the admix of the material package (i.e., when the actuator,
dispenser rate rails at its maximum limit). Actuator saturation can make the TC
system nonlinear. All of these aspects motivate the need to compensate for the
delay effects.
There are many methods and techniques for the analysis and control of dynamic
systems in the presence of time delay [37]. The Smith predictor-based control
scheme is the most commonly used technique in industrial systems [38

m 6¼

- 40]. Here

we present the design of a Smith predictor for a fixed time delay. For systems with a

varying time delay, underestimating or overestimating the time delay signi

cantly
degrades the control quality. Reference [39] shows a design method of the Smith
predictor with varying time delay.
Assuming zero consumption, that is, v(k) 0, if G pi (z) is the transfer function of

the PI controller shown in Figure 9.31, and G(z)

¼

¼ G p (z)z m is the transfer function