Lecture 15 - Distributed Control · Lecture 15 - Distributed Control • Spatially distributed systems • Motivation • Paper machine application • Feedback control with regularization

EE392m - Winter 2003 Control Engineering 15-1

Lecture 15 - Distributed Control

• Spatially distributed systems• Motivation• Paper machine application• Feedback control with regularization• Optical network application• Few words on good stuff that was left out


Sensors

Actuators

Distributed Array Control• Sensors and actuators are organized in large arrays distributed

in space.• Controlling spatial distributions of physical variables• Problem simplification: the process and the arrays are

uniform in spatial coordinate• Problems:

– modeling– identification– control


Distributed Control Motivation• Sensors and actuators are becoming cheaper

– electronics almost free

• Integration density increases• MEMS sensors and actuators• Control of spatially distributed systems increasingly common• Applications:

– paper machines– fiberoptic networks– adaptive and active optics– semiconductor processes– flow control– image processing

EE392m - Winter 2003 Control Engineering

Scanning gauge

MDmachinedirection

CDcross

direction

Paper Machine Process

• Control objective: flat profiles in the cross-direction• The same control technology for different actuator types: flow

uniformity control, thermal control of deformations, and others

© Honeywell


Headbox with Slice Lip CD Actuators

© Honeywell


Profile Control System

© Honeywell


Biaxial Plastic Line Control

© Honeywell


CD da ta box/actua tor number

mea surement gj

actua tor s e tpoint

Model Structure

• Process-independent model structure

• G - spatial response matrix withcolumns gj

• Known parametric form of the spatialresponse (noncausal FIR)

• Green Function of the distributedsystem

nmnm GUYUGY

,,, ℜ∈ℜ∈ℜ∈∆=∆

)(, jkkj cxgg −= ϕ


Measured profileresponse, Y(t)

Actuator setpointarray, U(t)

CD

MD

Process Model Identification

• Extract noncausal FIR model• Fit parameterized response shape



Simple I control

• Compare to Lecture 4, Slide 5• Step to step update:

• Closed-loop dynamics

• Steady state: z = 1

[ ]dYtYktUtUtDtUGtY

−−−−=+⋅=

)1()1()()()()(

( ) [ ]DzkGYkGIzY d )1()1( 1 −++−= −

)(, 1 DYGUYY dd −== −

I control


Simple I control

Issues with simple I control• G not square positive definite

– use GT as a spatial pre-filter

• For ill-conditioned G get verylarge control, picketing– use regularized inverse

• Slowly growing instability– control not robust– regularization helps again

.

0 50 100 150 2000

2

4ERROR NORM

0 20 40 60 80 100-0.5

0

0.5ERROR PROFILE

0 20 40 60 80 100-20

0

20CONTROL PROFILE

DGDYGYtDtUGGtY

TG

TG

GT

G

==

+⋅=

,

)()()(


LTIPlant

Frequency Domain - Time

• LTI system is a convenient engineering model• LTI system as an input/output operator• Causal• Can be diagonalized by harmonic functions• For each frequency, the response is defined by amplitude

and phase


LSIPlant

Frequency Domain - Space

• Linear Spatially Invariant (LSI) system• LSI system is a convenient engineering model• LSI system as an input/output operator• Noncausal• Can be diagonalized by harmonic functions• Diagonalization = modal analysis; spatial

modes are harmonic functions


( ) )1()1()( −−−−−=∆ tSUYtYKtU d

Control with Regularization

• Add integrator leakage term

• Feedback operator K– spatial loopshaping

• KG ≈ 1 at low spatial frequencies• KG ≈ 0 at high spatial frequencies

• Smoothing operator S– regularization

• S ≈ 0 at low spatial frequencies• S ≈ s0 at high spatial frequencies - regularization


Spatial Frequency Analysis

• Matrix G → convolution operator g (noncausal FIR) →spatial frequency domain (Fourier) g(v)

• Similarly: K → k(v) and S → s(v)• Each spatial frequency - mode - evolves independently

dkgz

szykgsz

kgy d )()(1)(1

)()()(1)()()(

ννν

νννννν

+−+−+

++−=

• Steady state

dkgs

sykgs

kgy d )()()()(

)()()()()()(

νννν

νννννν

++

+=

( ))()()()()(

)()( ννννν

νν dykgs

ku d −+

=


Sample Controller Design

• Spatial domain loopshaping is easy - it is noncausal• Example controller with regularization

0 100 200 300 400 5000

1

2

3ERROR NORM

0 20 40 60 80 100-0.5

0

0.5ERROR PROFILE

0 20 40 60 80 100-2

0

2CONTROL PROFILE

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.5

1PLANT

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.10.20.30.4

FEEDBACK

-5 -4 -3 -2 -1 0 1 2 3 4 5-0.01

0

0.01

0.02

0.03SMOOTHING

0 1 2 3

0.5

1

1.52

0 1 2 30.50.50.50.50.5

0 1 2 3

0.010.020.030.04

G

K

S

For more depth and references, see: Gorinevsky, Boyd, Stein, ACC 2003


WDM network equalization• WDM (Wave Division Multiplexing) networks

– multiple (say 40) independent laser signals with closely spacewavelength packed (multiplexed) into a single fiber

– each wavelength is independently modulated– in the end the signals are unpacked (de-mux) and demodulated– increases bandwidth 40 times without laying new fiber


WDM network equalization• Analog optical amplifiers (EDFA)

amplify all channels• Attenuation and amplification distort

carrier intensity profile• The profile can be flattened through

active control

See more detail at:www126.nortelnetworks.com/news/papers_pdf/electronicast_1030011.pdf

-40

-35

-30

-25

-20

-15

-10

-5

0

5

1525 1535 1545 1555 1565


WDM network equalization

• Logarithmic (dB)attenuation for asequence of notchfilters

)()( 01

ckwaN

kk −−=∑

=

λλϕλ

∑=

=

⋅⋅=N

kk

N

AA

AAA

1

1

loglog

K

Notch filter shapeAttenuation gain - control handle

WDM


Good stuff that was left out

• Estimation and Kalman filtering– navigation systems– data fusion and inferential sensing in fault tolerant systems

• Adaptive control– adaptive feedforward, noise cancellation, LMS– industrial processes– thermostats– bio-med applications, anesthesia control– flight control

• System-level logic• Integrated system/vehicle control

Documents

Lecture 15 - Distributed Control · Lecture 15 - Distributed Control • Spatially distributed systems • Motivation • Paper machine application • Feedback control with regularization