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
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Lecture 15 - Distributed Control · Lecture 15 - Distributed Control • Spatially distributed systems • Motivation • Paper machine application • Feedback control with regularization
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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
EE392m - Winter 2003 Control Engineering 15-2
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
EE392m - Winter 2003 Control Engineering 15-3
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
• Known parametric form of the spatialresponse (noncausal FIR)
• Green Function of the distributedsystem
nmnm GUYUGY
,,, ℜ∈ℜ∈ℜ∈∆=∆
)(, jkkj cxgg −= ϕ
EE392m - Winter 2003 Control Engineering 15-9
Measured profileresponse, Y(t)
Actuator setpointarray, U(t)
CD
MD
Process Model Identification
• Extract noncausal FIR model• Fit parameterized response shape
EE392m - Winter 2003 Control Engineering 15-10
EE392m - Winter 2003 Control Engineering 15-11
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
EE392m - Winter 2003 Control Engineering 15-12
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
==
+⋅=
,
)()()(
EE392m - Winter 2003 Control Engineering 15-13
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
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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
EE392m - Winter 2003 Control Engineering 15-15
( ) )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
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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 −+
=
EE392m - Winter 2003 Control Engineering 15-17
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
– 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
EE392m - Winter 2003 Control Engineering 15-19
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
EE392m - Winter 2003 Control Engineering 15-20
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
EE392m - Winter 2003 Control Engineering 15-21
Good stuff that was left out
• Estimation and Kalman filtering– navigation systems– data fusion and inferential sensing in fault tolerant systems