5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Control of Manufacturing Control of Manufacturing ProcessesProcesses

Subject 2.830 Spring 2004Spring 2004Lecture #23Lecture #23

““Variance Reduction in DiscreteVariance Reduction in DiscreteClosedClosed--Loop SystemsLoop Systems””

May 6, 2004May 6, 2004

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

OutlineOutline•• Integral Control: a zIntegral Control: a z--plane Design plane Design

ExampleExample•• How to Model Variation for Cycle to How to Model Variation for Cycle to

Cycle ControlCycle Control•• Impact of Independence and Impact of Independence and

DependenceDependence•• Correlation and Variance ReductionCorrelation and Variance Reduction•• Effect of Controller DesignEffect of Controller Design

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

A A zz-- plane Design plane Design ExampleExample

R(z)Gc(z) Gp(z)-

Y(z)

p

z

Gc(z) = Kc

Problems:

•Poor SS error

•low “loop” gain

+1

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Error: Try an IntegratorError: Try an Integratorui = Kc ei

j=1

i

∑ running sum of all errors

recursive formui+1 = ui + Kcei+1

Z[ ]

Gc (z) = Kcz

z −1zU = U + KczE

U(z )E(z)

= Kcz

z −1pole at 1 and zero at 0

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

New Root LocusNew Root Locus

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Real Axis

Imag

Axi

s

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Design Points?Design Points?

•• Minimum Settling Times (Minimum Minimum Settling Times (Minimum Radius)Radius)

•• Zero Overshoot regions?Zero Overshoot regions?•• Maximum Gain Points?Maximum Gain Points?

–– Without OvershootWithout Overshoot–– With “acceptable” OvershootWith “acceptable” Overshoot

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Design ExtremesDesign Extremes

All points on this circle have the same settling times

Fastest Settling w/o oscillation

Highest Gain with

ζ=0.1

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Real Axis

Imag

Axi

s

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Gain Gain -- Root LocationsRoot Locations

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Real Axis

Imag

Axi

s Kc = 0.086

Kc = 0.35Kc = 2.9

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Expected ResponseExpected Response

Minimum Settling:radius r~0.7

ts = −4T

ln(r)for us, T=1 (the cycle time)

Thus the system should settle in 4/0.36 = 11.2

which is 11 or 12 cycles for all three cases!

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Low Gain ResponseLow Gain Response

Time (samples)

Am

plitu

de

TextEnd

Step Response

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Kc= 0.086

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Medium Gain ResponseMedium Gain Response

TextEnd

Step Response

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

Kc= 0.35

Am

plitu

de

Time (samples)

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

High Gain ResponseHigh Gain Response

Time (samples)

Am

plitu

de

TextEnd

Step Response

0 5 10 15 20 25

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Kc= 2.9

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Add a Pure Delay to the IntegratorAdd a Pure Delay to the Integrator

Gc (z) = Kcz

z −1z −1 = Kc

1z − 1

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Real Axis

Imag

Axi

s

Gain for ζ=0.1

“Fastest”

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

FastestFastest

Time (samples)

Am

plitu

de

TextEnd

Step Response

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Prediction = 14 steps to settle

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Highest Gain ResponseHighest Gain Response

Time (samples)

Am

plitu

de

TextEnd

Step Response

0 20 40 60 800

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Prediction = 55 steps to settle

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Cycle to Cycle Control Cycle to Cycle Control and Process Variationand Process Variation

•• Recall the Output of a “Real Process”Recall the Output of a “Real Process”

•• Random even with machine and Random even with machine and material feedback controlmaterial feedback control

1.940

1.945

1.950

1.955

1.960

Run Number

Cooling TimeChange

Injection Velocity Change

ShiftChange

ShiftChange

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Output Disturbance ModelOutput Disturbance Model

d(t) TS

Kp

(τ ps +1)y(t)u(t) yi

y(t) yi

iTS

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Or In Cycle to Cycle Control TermsOr In Cycle to Cycle Control Terms

d

Gp(z)-r y

Gc(z)

Where:

d(t) is a sequence of random numbers governed by

a stationary normal distribution function

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Gaussian White NoiseGaussian White Noise

•• A continuous random variable that at A continuous random variable that at any instant is governed by a normal any instant is governed by a normal distributiondistribution

•• From instant to instant there is no From instant to instant there is no correlation correlation

•• Therefore if we sample this process we Therefore if we sample this process we get:get:

•• A NIDI random numberA NIDI random numberNormal Identically Distributed Independent

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

The Gaussian “Process”The Gaussian “Process”

i

i+1

i+2

...

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Expected Disturbance RejectionExpected Disturbance Rejectiondi

Gp(z)-r y

Gc(z)

Y(z)D(z)

=1

1 + GcGp

if di = constant, then we can look at steady - state behavior:

limz→1

(z − 1)Y(z) = (z −1)1

1+ GcGp

*z

(z −1)

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

And For ExampleAnd For Examplelimz→1

(z − 1)Y(z) = (z −1)1

1+ GcGp

*z

(z −1)

GcGp =Kc

z − p

limz→1

(z − 1)Y(z) = (z −1)1

1+Kc

z − p

*z

(z −1)

1

1Y (z)| D |

=1

1+ Kc

1 − pHigher gain KHigher gain Kcc improves “rejectionimproves “rejection”

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

But But

•• ddii is defined as a NIDI sequenceis defined as a NIDI sequence•• Therefore:Therefore:

–– Each successive value of the sequence is Each successive value of the sequence is probably differentprobably different

–– Knowing the prior values: Knowing the prior values: ddii--11, d, dii--22, d, dii--33,… ,… will not help in predicting the next valuewill not help in predicting the next value

di ≠ a1di−1 + a2 di− 2 + a3di−3 +

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

ThusThus

dir

Kp/z-y

Kc

This implies that with our cycle to cycle process model under proportional control:

xue

The output of the plant xi will at best represent the errorfrom the previous value of di-1

xi = −KcKpdi−1will not cancel will not cancel ddii

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

But What if we use EWMA?But What if we use EWMA?di

Kp/z-xr y

Kcue

r/(z-p)

pAi = ryi−1 + (1− r)Ai−1

X(z) =KcKp

zr

(z − p)D(z ) =

Kz(z − p)

D(z)

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

EWMA Filter EffectEWMA Filter Effect

X(z) =KcKp

zr

(z − p)D(z ) =

Kz(z − p)

D(z)

X(z)(z 2 − pz) + KD(z)

xi+ 2 = pxi+1 + Kdi ⇒ xi = pxi−1 + Kdi−2

Again, the compensating output is based on Again, the compensating output is based on “old” and uncorrelated information about “old” and uncorrelated information about dd and and will not cancel the current will not cancel the current ddii

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Effect of ControllerEffect of Controller

di

Gc (z) = KC' (z − α )

(z −1)

Kp/z-xr y

Gc(z) ue

X(z)(z 2 − z) = (z − α )KD(z)

xi+ 2 = xi+1 + Kdi+1 + αKdi

SAME DEALSAME DEAL

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Simulation:Simulation:Effect of Controller DesignEffect of Controller Design

K s2y /s2

n

0 1

0.1 1.03

0.2 1.07

0.5 1.33

1.0 2.28

1z

Process

z+.5z-.5

Controller

k

Gain

n

To Workspace1

y

To Workspace

++

Sum1

RandomNumber

+-

Sum

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Variance Change with GainVariance Change with Gain

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 0.2 0.4 0.6 0.8 1 1.2Gain

s/so

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Kc s2y /s2

n

0 1

0.1 1.02

0.2 1.06

0.5 1.4

0.9 5.5

Kc

Gain

d

disturbanceNIDI Sequence

y

output

++

Sum11/z

Process

+-

Sum

Simulations: Feedback Simulations: Feedback of Run dataof Run data

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

0.5z-0.5

EWMA

Kc

Gain

d

disturbanceNIDI Sequence

y

output

++

Sum11/z

Process

+-

Sum

Simulations: Simulations: EWMA FeedbackEWMA Feedback

Kc s2y /s2

n

0 1

0.125 1.006

0.5 1.09

1.0 1.44

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

What if the Disturbance is not NIDI?What if the Disturbance is not NIDI?

NIDI Sequence

az-a

Filter(Correlator)

cdidi

CD(Z )(z − a) = aD(z)

cdi+1 = a(di − cdi )

unknown known

expect some correlation, expect some correlation, therefore ability to therefore ability to counteract some of the counteract some of the disturbancesdisturbances

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

What if the Disturbance What if the Disturbance is not NIDI?is not NIDI?

++

Sum1

y

Ouput

+-

Sum

0.7z+0.7

Filter(Correlator)

NIDI Sequence

n

Disturbance

1/z

Process

Kc

Gain

Kc s2y /s2

n

0 1

0.1 0.89

0.25 0.77

0.5 0.69

0.9 1.39

Proportional ControlProportional Control

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Gain Gain -- Variance ReductionVariance Reduction

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 0.2 0.4 0.6 0.8 1

s/so

Gain

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Gain Gain -- Variance ReductionVariance Reduction

s/so

Loop Gain

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 0.2 0.4 0.6 0.8 1

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

What if the Disturbance is What if the Disturbance is not NIDI?not NIDI?

0.5z+0.5

EWMA

++

Sum1z

y

Ouput

+-

Sum

0.7z-0.7

Filter(Correlator)

NIDI Sequence

n

Disturbance

1/z

Process

Kc

Gain

Kc s2y /s2

n

0 1

0.125 0.94

0.5 0.85

1.0 0.90

Proportional Control Proportional Control with EWMA Feedbackwith EWMA Feedback

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Conclusions from Conclusions from Cycle to Cycle ControlCycle to Cycle Control

•• Can be Modeled as a Discrete ProcessCan be Modeled as a Discrete Process•• Simple Plant Model of a Delay of (at Simple Plant Model of a Delay of (at

least) one Cycleleast) one Cycle•• We Can Use Feedback Control to We Can Use Feedback Control to

“Center” the Process“Center” the Process–– PI Controller examplePI Controller example

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

Conclusions from Conclusions from Cycle to Cycle ControlCycle to Cycle Control

•• Feedback Control of NIDI Disturbance Feedback Control of NIDI Disturbance will Increase Variancewill Increase Variance–– Variance Increases with GainVariance Increases with Gain

•• BUT: If Disturbance is BUT: If Disturbance is NIDNID but not but not II; We ; We CAN Decrease VarianceCAN Decrease Variance–– Higher Gains Higher Gains --> Lower Variance> Lower Variance–– Design Problem: Low Error and Low Design Problem: Low Error and Low

VarianceVariance

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

How to Tell if DisturbanceHow to Tell if Disturbanceis Independentis Independent

•• Correlation of output dataCorrelation of output data–– Look at the Autocorrelation Look at the Autocorrelation –– Effect of Filter on AutocorrelationEffect of Filter on Autocorrelation

•• Reaction of Process to FeedbackReaction of Process to Feedback–– If variance decreases data has dependenceIf variance decreases data has dependence

5/6/04 Lecture 23 © D.E. Hardt, all rights reserved

But Does It Really Work?But Does It Really Work?–– Let’s Look at Bending and Injection Let’s Look at Bending and Injection

MoldingMolding