Internal Process Controls - Control of Color Imaging Systems: Analysis and Design - page 497

This is true even for the level 1 controller gain matrix. Also, notice the changes
between two gain matrices. They are in the last column of the gain matrix which is
due to change in p 3 . The third pole p 3 controls how the DMA transients are
affected due to the actuator, V bias .
Case (ii): Step responses for the DMA states were created by changing the
DMA vector from [0.0384 0.1301 0.3398] to [0.0384 0.1301 0.3738] for two
different gain matrices shown in case (i). Level 2 controller has sensing-processing-
actuation updates at every 30 pitch, whereas the inner level 1 loop is updated
every pitch. DMA vector shown in Figure 9.15a converges to the desired steady-
state value after few updates. The actuator response is shown in Figure 9.15b.
Figure 9.15c shows the response of the inner loop when the level 2 control loop is
in its last iteration pass.

Example 9.7

The cost of the electrostatic sensing system can be reduced by removing the
charge sensor (ESV) and associated level 1 control loop. It is now required to
design a three-input three-output developability control loop using three process
actuators, the grid voltage, U g (k)

¼V g (k), the exposure intensity, U l (k) ¼X(k), the

development bias, U b (k) ¼V bias (k), and measurements from the DMA sensor. Let

D l (k), D m (k), and D h (k) represent the three different DMA measurements shown in
the state vector, x(k), measured every pitch, indicated by the parameter, k. The linear
state variable description of the control system is characterized by the Jacobian
matrix at

cm 2 ,

the nominal operating point {U go ¼ 600 V, U lo ¼ 8 erg =

U bo ¼ 300 V} as shown below:

2
4

3
5 x(k)

2
4

3
5 v(k)

100
010
001

0 : 2789 0 : 2777 8 : 7889

10 3 x(k þ 1) ¼ þ 0 : 9761 1 : 0703 29 : 7035

0 : 4225 1 : 2727 16 : 8863

2
4

3
5x(k)

100
010
001

y(k) ¼

2
4

3
5 ; v(k)

2
4

3
5 ; y(k)

D l (k)
D m (k)
D h (k)

DU g (k)
DU l (k)
DU b (k)

where x(k) ¼ ¼ ¼ x(k). The controller uses a gain

matrix and the integrator modeled by following equation:

e(k þ 1) ¼ e(k) Bv(k)

v(k) ¼þKe(k)

where e(k) ¼ x d x(k), and x d is the desired state vector.

i. Find the gain matrix for placing the poles at location [0.3, 0.3, 0.3] using

Equation 9.45.

ii. Show the time evolution of the states for a step response as a function of

pitch number using recursive solution of equation (Section 4.8.3), the meas-
urement-actuation updates are executed at every pitch.