1Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Robust Control and Diagnostic Strategies forXerographic Printing

Perry Y. LiDepartment of Mechanical Engineering

University of MinnesotaMinneapolis MN 55455

pli@me.umn.eduhttp://www.me.umn.edu/~pli/

2Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Outline

• Background and Motivation

• Robust TRC control problem

• Bayesian Network Modeling

• Conclusions

3Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Acknowledgements

• Collaborative work with• S. Dianat, Rochester Institute of Technology• Leon Zhong, U. of Minnesota

• Support from Document Company, Xerox• through the Xerox University Affair Council Grant

4Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Xerographic Process

• Charge

• Expose

• Develop

• Transfer

• Fuse

Electrostatics Mechanical Electrical Optics Software

Interdisciplinary:

5Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Robustness IssuesXerographic Image Output Terminal

• is complicated• involves many physical processes in various disciplines

• physical processes e.g. electrostatics based ones, are affected by• materials: toner and P/R• environment: temperature and humidity• machine variations: critical dimensions• cleanliness: dirt accumulation on charging wire

• larger scale variations• system faults• component degradation

6Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Objectives for Control Engineers• Manage printing process so that printer works well despite

• disturbances and faults

Imageprocessing

PhysicalMarking

disturbance

output

• Adjust image processing unit

• Adjust physical marking process

7Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Strategy 1: StabilizationMinor Disturbances and Plant Variations

• Stabilization strategy• No need to know source

of disturbance• Does not explicitly

construct system’s health• Printer remains on-line

• Challenges:• Identify actuators and sensors• Formulate control problem• Design / analyze control

algorithm

Imageprocessing

PhysicalMarking

disturbance

output

Controller

Stabilization of Tone Reproduction Curves

8Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Strategy #2: Diagnostics BasedFaults and Component Degradations

Explicitly diagnose faults / system’s state of degradations• Active testing• Reconfigure for degraded operation• Repair• Temporarily off-line

Benefits• Extend system’s use

despite faults• otherwise machine just

shuts down• Shorter / no service calls

Imageprocessing

PhysicalMarking

disturbance

Controller

Diagnostics

TestingReconfigure

Service/RepairBayesian Network Based DiagnosticsBayesian Network Based Diagnostics

output input

9Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Rest of this talk ….

• Stabilization of tone reproduction curves• For details, see: Li and Dianat, Proceedings of IEEE International

Conference on Controls Applications (Sept. 1998). Also to appear in theIEEE Transactions on Control Systems Technology.

• Bayesian network based diagnostics• For details, see: Zhong and Li, Proceedings of the ASME IMECE,

Dynamic Systems and Control Division - 2000. Volume 1.

10Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Tone Reproduction Curves

• TRC: Tone-in Tone-out; Ideally: TRC(tone-in) = tone-in

Color image separation

Half-toning

Half-toning

Half-toning

Xerographicmarking

Register and FuseHalf-toning

Tone Reproduction

Curve

Tone Reproduction

Curve

Tonein Half-toning

Xerographicmarking

Filter(A/C)

Toneout

half-tone half-toneCon-tone

Con-tone !

Con-tone

11Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Identifying Sensors and Actuators

Actuators• Grid voltage• Laser power• Development bias voltages

Sensors• Monochrome sensorpatches• Area coveragesensors

12Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

• TRC is a transformation

• Space of all TRC is potentially infinite dimension:• Discretize tones into 8 bits, i.e. p=256 tones• TRC is a p-dimensional

• Few actuators about 3 to 5• Few sensors ( less than 5 sensor patches / belt cycle)

• samples TRC

TRC Stabilization - Problem Formulation

]1,0[]1,0[: →TRC

Tonein Half-toning

Xerographicmarking

Filter(A/C)

Toneout

Ton

e -o

u t

actual TRC

desired TRC

measured tones

Tone -in

13Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

TRC Control Problem

Features:• Nonlinear• Disturbances• Uncertain models

- actuatorsd(k) - disturbances

- measured tones

Manipulate actuators settings based on TAC sensor measurements so that TRC (as a function) is close to the identity / nominal

Manipulate actuators settings based on TAC sensor measurements so that TRC (as a function) is close to the identity / nominal

))(),(( kdkuTRC Φ=

�

������

�

�

=

)),(),((

)),(),(()),(),((

2

1

ptonekdku

tonekdkutonekdku

φ

φφ

�

�

))(),(( kdkuCy Φ⋅=

)(ku

)(ky

14Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

• Use same number of inputs and measurements• Design cares only about measured tones• Theoretically can ensure measured tones to conform perfectly

Issues:• Accurate control at measured tones• Poor response at unmeasured tones• Model uncertainty can cause

stability issues

Integral Control method (not recommended)

tone

area

cov

erag

e

actual TRC

desired TRC

measured tones

Plantcontroller+

-

r(k) y(k)

Make y(k) close to r(k)

15Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

“Curve fitting” approach• “Select” actuator settings such that the actual TRC is close to the

desired TRC all ALL tones:

• Unmeasured tones must be filled in from model information

• Incorporate possible effects of• disturbances and plant uncertainty

[ ]2

1)( )()),(),(())(( min

=

−=p

iidiku toneTRCtonekdkuTRCkuJ

Ton

e -o

u t

actual TRC

desired TRC

measured tones

Tone -in

16Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Modeling

• Essentially static system• Nominal linear plant model - factorial experiments• Effects of unknown disturbances and uncertainty• Nonlinearity captured by uncertainty

[ ] )()(ˆ)()( kuWkIkdke ud ⋅∆+⋅+⋅= φφ

)()( keCky ⋅=)(k∆

P11 P12 P13

P21 P22 P23

P33P32P31

K(z)

d(k)e(k)

y(k)

• k=1,2,…., time sample index• e(k) = TRC error (p-dim), p-large• y(k) = TRC error at measured tones

(n-dim), n-small• u(k) = control deviation from nominal

(m-dim), m-small

Plant uncertainty

controller

u(k)

17Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Nominal Model

18Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Robust TRC Control• Minimize effects of d(k) on e(k)

• since d(k) and arbitrary,minimize worst case

• equivalent to minimizing inducednorm

• Robust static performance• disturbance and uncertainty

slowly varying

• What is a good controller ?• Define steady state performance:

• Design D.C. gain of controller tominimize

)(k∆

P11 P12 P13

P21 P22 P23

P33P32P31

K(z)

d(k)e(k)

y(k) u(k)

∞∞∞∞ ⋅∆=⋅ dKFeWe ),(

��

�

��

��

�

∆= ∞∞

≤∆

∞

∞

),(:min:)( sup1

KFK

γ

γγ

)( ∞Kγ

Closed loop system

LFT

)(k∆

19Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Two step control design method• Step 1: Find optimal D.C. gain

• Static (relative to dynamic) robust control problem is convex• Unique optimal D.C. gain can be found by solving Linear Matrix

Inequalities

• Step 2: Design controller so that it is causal• y(k) cannot be sensed until u(k) is issued• u(k) must only depend on y(k-1), y(k-2) etc.

• Choose dynamic controller with optimal D.C. gain

• Eigen-values of A chosen for noise immunity

)()( kyKku ⋅= ∞

)()()1( kyBkuAku ⋅+⋅=+

20Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Simulation results: Steady state response

Worst case disturbance for robust controller Worst case disturbance for integral controller

21Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Simulations: Dynamic Responses

Effect of dynamicdisturbances

Effect of measurement

noise

22Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Experimental Result

• Target = nominal - tone* slope• Slope = 0.02

23Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

3 Patch Integral Control• 3-I-3-O Integral Control• Measurements at

tt = 5, 17, 30• blows up after 1st cycle

Diagnosis:• Sensitivity matrix is very ill

conditioned:-0.2702 0.1402 0.1822-0.3818 0.1991 0.2837

-0.2420 0.1292 0.2323

svd = [ 0.7205 0.0436 0.0002]

24Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Bayesian Network Based Diagnostics• Explicit determination of

fault conditions and state ofmachine

• Inference based onparsimonious sensors

• Diagnostic problem:• Determine a set of fault

state / state of componentdegradation that canexplain observations

Imageprocessing

PhysicalMarking

disturbance

Controller

Diagnostics

TestingReconfigure

Service/Repair

25Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

M

C2 C1

C3

C0 C a*

C2 C1 C3

b*

Perceptually Non-Uniform

Perceptually nearly uniform CIELAB

Reconfiguration example:Color Management

Scenario:• Cyan capability is reduced

• e.g. due to reduced Cyanlaser power

• Problem diagnosed

Reconfiguration:• Compute optimal color

projection• Device specific!

desired

26Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Desired = [150, 150, 125]

Uncompesnated = [135, 150, 125]

Optimized = [135,142, 118]

Desired = [220, 80, 100]

Uncompensated = [210, 80, 100]

Optimized = [210, 74, 92],

Desired UnchangedAdjusted

Uncompensated

Desired UncompensatedOptimized

Optimized

Desired

Optimized

Uncompensated1.20=∆E

02.0=∆E

4.17=∆E

01.0=∆E

27Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Other utilities of Diagnostics• Optimize diagnostic tests

• Which tests to perform ?• Will new information be

important?

• Repair:• Which component is the culprit?

• Do I have enough information?

• Rank competing diagnosis

Imageprocessing

PhysicalMarking

disturbance

Controller

Diagnostics

TestingReconfigure

Service/Repair

28Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Bayesian Belief Network (BBN) ApproachProbabilistic approach• Captures and make inference

based on joint probabilities ofvariables that describe machine

• Ai = variables of components,• e.g. voltages, parameters etc.

• If Obs={observed variables} isobserved, what is the state of Bj?

• Compute P(Bj | Obs)

• Bayes rule

Imageprocessing

PhysicalMarking

disturbance

Controller

Diagnostics

TestingReconfigure

Service/Repair

),,,( 321 �AAAP

29Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Benefits of Probabilistic Approach• Rank non-unique competing

diagnosis

• Provides confidence level (50%probability or 98% probability)

• Test utility of diagnostics tests:

• I(T) = information of test• W(T) = expected performance

improvements• C(T) = cost of test

Imageprocessing

PhysicalMarking

disturbance

Controller

Diagnostics

TestingReconfigure

Service/Repair

)()()()( TCTWTITV −+=

30Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Issues in using probabilistic approach• Complexity in computation and storage

• is high dimensional

• n - variables, each has ‘p’ possible states: dimensional

),,,( 321 �AAAP

np

• Bayesian network:• compact representation of joint representation• exploit causal structure of problems• store and compute based on conditional probabilities (that involves few

nodes)• manageable in large problems

31Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Bayesian Network Model• Directed acyclic graph (DAG) of

variables

• To each Variable A with Parents B1,B2,..., Bn , associate a conditionalprobability table P( A | B1 ...Bn )

• Ancestor node: Prior probabilities• P(A), P(C)

• Joint probability can be obtainedfrom conditional probabilities:

P ( A1, A2, … , As ) ))(|(1 ii

s

iAparentsAP

=Π=

A

B

C

D

P(D | A, B,C)

P(B|A)

Hence, given BBN, one can do everything as if joint probability is known

32Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

BBN Example: Wet grass• Holmes grass is wet.• Is it due to

• rain (R)• or sprinkler (S) ?

• Storage• BBN - 12 numbers• P(R, S, H, W) - 16

R

HW

S

P(R) = (0.2, 0.8), P(S)= (0.1,0.9)

R = y R = nW = y 1 0.2W = n 0 0.8

P( W | R )

R = y R = nS = y ( 1, 0 ) ( 0.9, 0.1 )S = n ( 1, 0 ) ( 0, 1 )

P( H | R, S )

Binary nodes : true or false

• Evidence: H = y• P( R=y | H=y ) = 0.736 ,

P( S=y|H=y ) = 0.339

• More evidence:• Watson’s grass is also wet (W = y)• P( R=y | H=y,W=y) = 0.93 ,

P( S=y|H=y,W=y ) = 0.161

33Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Simple BBN Model of Xerographic Process• Simple model to illustrate principles• Monochrome solid color

• Vs - charged voltage• Vp - exposed voltage• DMA - developed mass density

• Di - desired input tone• Do - output print density

• Vs, Vp, DMA - observed variables• Vg, Po, Vb - control variables

34Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Color Printing Process

Ci Co

Mi Mo

Yi Yo

Ki Ko

Fuse

Cyan

Magenta

Yellow PrintingSubsystem

Black PrintingSubsystem

InputOutput

A series of monochromatic printing sub-processes• concatenate several monochrome BBN

Color Printing Process

35Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

BBN Model construction procedure• Define continuous BBN model based on physical model

• e.g.

• Assign probability distributions based on uncertainties in mathematicalmodels

• Define prior probabilities of ancestor nodes• Actuator variables have narrow distributions

• Discretization of BBN model• e.g.

• Implementation in HUGIN software

]8.2,0[∈A

{ }highmediumlowA ,,∈

36Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

• Physical model (Schein 92):

• Uncertainty in mathematical model:

where

• N(mean , variance) = Gaussian distribution

• Model from experiments• Failure data from field

Charging Subsystem Example

v

Vs

Vi CI0Vg

)/exp()]/exp(1[ 00 CvVIVCvVIVV giggs −+−−=

]),,,,,([),,,,|( 00 sgigis ICVVfNICVVVp σνν =

)/exp()]/exp(1[),,,,( 000 CvVIVCvVIVICVVf gigggi −+−−=ν

37Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

-50 -40 -30 -20 -10 0 10 20 30 40 500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Discretization of BBNWhy?Why?• Nodes can take on continuous range

of values• Cannot implement continuous BBNs

with arbitrary conditional probabilities• High computational cost

How?How?• Divide value range into “n” intervals

• How to partition ?

Prior probability, P(A), of node A

A

P(A

)

38Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Maximum Entropy Discretization• Minimize number of intervals• Each interval must provide maximum information

• View each discretized node as an information channel:• Variable A can be: Sym-1, Sym-2, ….. Sym-n• Prior probability of each symbol: p1, p2, …. , pn

• Maximize the entropy:

• Each interval should have equal probabilities:=

−=n

iii ppSH

1)log()(

nppp n /1...21 ==== -50 -40 -30 -20 -10 0 10 20 30 40 500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Prior probability, P(A), of node A

39Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Discretization flowchartGet the node

to be discretized

Parent nodes discretized?

Yes

Segmentation

Joint Probability of parents

End

Conditional Probability

Parent nodesindependent?

Yes

NoMarginalization

Has parents?

Yes

No

Has parents?Yes

No

No

BA

C E

D

• Recursive algorithm• Checks independence• Group nodes to increase

efficiency

• Implemented in Matlab

40Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Discretization ProcedureDiscretization(Node node)

If node is top-level ancestor node

Segment node’s range and get the controids

Else if node is a node with parents

For each node, pr_node in the list of its parent nodes

If pr_node has not been discretized

Discretization (pr_node)

EndEnd If node is a node with parents

Acquire the joint probability table of node’s parent nodes

Approximate the continuous probability distribution of node

Segment node’s range and get the controids

Calculate the discretized conditional probability, P(node | node’s parents )

EndEnd

BA

C E

D

41Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Implementation on HUGIN• HUGIN - commercial software for BBNs• Performs inferences and probability updates as evidence is introduced• Uses discrete nodes (restricted continuous nodes)• Efficient updates (junction tree method)• Discretization algorithm writes to HUGIN readable format

42Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

BBN Prediction Example: TRC• Run BBN forward• Provide actuator values and various input tones• Observe probability distribution of output tones• Probabilistic Tone Reproduction Curve

L ML M MH HL

MH-

0.2000

0.4000

0.6000

0.8000

1.0000

Probability

Input Density

Output Desity

Input and Output Density

LMLMMHH

43Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Inference example: Print quality diagnostics•• Run BBN backwards (inference)Run BBN backwards (inference)SenarioSenario

• An image with moderate density ( Di=”M” ) is to be printed• Actual print is denser than desired, ( DMA = “H”)

DiagnosisDiagnosis• Evidence: Di=”M”, DMA = “H”:

• 1) Toner charge (Q) is low with probability 0.61• 2) Laser power (Po) is low with probability 0.69

• More test: check laser power• Evidence: Laser power is normal,

• Tribo (Q/M) is low with probability of 0.98• check developer

44Mechatronics and Intelligent Machines Laboratory

M..E., University of MinnesotaRobust Control and Diagnostic Strategies for Xerographic Printing

Conclusions• Two strategies to manage disturbances / faults• Stabilization of Tone Reproduction Curves

• potentially high dimension• small number of actuators and sensors• uncertainties, disturbances etc.• “curve fitting” approach

• Diagnostic framework for Xerographic printing:• Probabilistic framework• Bayesian Belief Network (BBN) approach

• compact representation• Continuous BBN from physics• Maximum entropy automatic discretization• Demonstrated approach

• more realistic model, field data etc. neededT

one

-ou t

actual TRC

desired TRC

measured tones

Tone -in