Internal Process Controls - Control of Color Imaging Systems: Analysis and Design - page 495

Using final value property, we have

e ss ( k )j open ¼ X d x 0 ( 9 : 44 )

We find that the steady-state error is a function of the state vector, x 0 and is not equal

to zero unless u 0 creates the same value as the desired vector, X d , which is hardly the
case since there is always drift = media change happening during system operation.

9.8 DESIGN OF THE GAIN MATRIX
The design of the gain matrix is important for two reasons: (1) for controlling the
convergence rate (or number of iterations) for the states to a steady-state value and
(2) for staying within the stability bounds so that even when the toner mass deviates
due to various disturbances (e.g., toner usage change, media change, developer
aging, dark decay, etc.) in the print engine, the closed-loop system should move
the states to the desired steady-state value. In this section, the design of gain matrix
using a limited pole-placement design technique is discussed.
There is no simple generalized procedure for pole-placement design of MIMO
systems. Since the degree of freedom available to choose the right kind of gains is
suf ciently large, one can assign performance measures to pick the gains (e.g., to

improve the output states in the presence of noise), to assign eigenvectors that would
respond according to the natural modes of the system. Such techniques are more
elaborate and dif

cult. Optimal controls and various robust pole assignment tech-
niques are readily available in the literature [13,24] and discussed in Chapter 5 to
perform those calculations using the state variable formulation described above. For
a simple pole-placement design for level 2 system, let us choose the gain matrix in
such a way that the eigenvalues of the closed-loop system matrix, A-BK, shown in
Equation 9.36 has the desired closed-loop poles. To select desired poles for a discrete
system, see Refs. [13,24,25]. The location of the closed-loop poles dictates the
required number of iterations for driving the error states toward zero. Since A ¼ I
for the level 1 and 2 system, the gain matrix can be easily written in terms of closed-
loop poles as

K ¼ B 1 s ( 9 : 45 )

where 3 matrix determined by the closed-loop poles. For a stable system,

all the poles should be assigned values between 0 and 1. For a three-input three-
output control system,

s is a 3

s is written in terms of three poles (p 1 , p 2 , p 3 ) as follows:

2
4

3
5

1 p 1 0 0

s ¼ p 2 ( : )

0 1 0 9 46

p 3

0 0 1

Theoretically, for the system with A ¼ I, we can show that when all three closed-loop
poles are chosen real and equal, the gain matrix will result in a more robust closed-
loop performance.