Internal Process Controls - Control of Color Imaging Systems: Analysis and Design - page 512

300 300

Reference TRC
Samples
TRC
Inverse TRC

250 250

200 200

150 150

100 100

50 50

0 0

1 2 3 4 5 6 7 8 9 10 0 50 100 150 200 250 300

(a) Iteration # (b) Gray levels

FIGURE 9.29 (a) Convergence plot (poles ¼ 0.4) for the inverse tone values. (b) Inverse
TRCs for 256 input gray levels shown for reference TRC 3.

9.10 DEAD BEAT RESPONSE
Dead beat response is the response of a control loop in which the error is dead in one
beat. This implies that the desired response of the control system is obtained in a
minimum number of measurement-process-actuation update steps [13]. In other
words, the target is reached in the fewest number of time steps. In the linear discrete
case such as for level 1, 2, or 3 controls, the dead beat response can be achieved by
making the closed-loop transfer function have poles at the origin (i.e., zero). In
this section we will determine the minimum number of steps required to achieve
dead beat response for the both SISO and MIMO systems. There is no such thing as
a dead beat response for continuous control systems. Hence, the theory is developed
only for a discrete system.
Consider the state variable description of Equation 9.48 and the feedback
equation with gain matrix of Equation 9.49. The closed-loop state Equation is
equal to

x ( k þ 1 ) ¼ ( I BK ) x ( k ) þ BKx d

¼ Bx ( k ) þ BKx d

( 9 : 52 )

From Chapter 4, we can write the solution to the above equation as

x ( k ) ¼ B k x 0 þ BKx d ( 9 : 53 )

We know that the feedback control law that assigns all the closed-loop poles to the
origin is a deadbeat control law. Therefore, all eigenvalues of the closed-loop
system (Equation 9.52) will be at zero for deadbeat control. That is, the closed-
loop system matrix B will have all eigenvalues equal to zero. Let 0.

For the SISO case (e.g., Section 9.9.2), the B matrix will be diagonal with all
elements zero for deadbeat control and K ¼ B 1 . Thus, for a SISO system,

l 1 ¼l 2 ¼¼