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 15-1 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 Sensors

Actuators

EE392m - Winter 2003 Control Engineering 15-2 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 15-3 Paper Machine Process

MD CD machine cross direction direction

© Honeywell Scanning gauge • 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

EE392m - Winter 2003 Control Engineering Headbox with Slice Lip CD Actuators

EE392m - Winter 2003 Control Engineering 15-5 © Honeywell Profile Control System

EE392m - Winter 2003 Control Engineering Model Structure

• Process-independent model structure measurement g ∆Y = G∆U j m n m,n Y ∈ℜ ,U ∈ℜ ,G ∈ℜ actuator s etpoint

• G - spatial response matrix with

columns gj • Known parametric form of the spatial CD data box/actuator number response (noncausal FIR) g = gϕ(x − c ) • Green Function of the distributed j,k k j system

EE392m - Winter 2003 Control Engineering 15-8 Process Model Identification CD Actuator setpoint array, U(t)

Measured profile response, Y(t)

• Extract noncausal FIR model • Fit parameterized response shape

EE392m - Winter 2003 Control Engineering 15-9 EE392m - Winter 2003 Control Engineering 15-10 Simple I control

• Compare to Lecture 4, Slide 5 I control • Step to step update: Y (t) = G ⋅U (t) + D(t) = − − []− − U(t) U(t 1) k Y (t 1) Yd • Closed-loop dynamics = ()− + −1[]+ − Y (z 1)I kG kGYd (z 1)D

• Steady state: z = 1 = = −1 − Y Yd , U G (Yd D)

EE392m - Winter 2003 Control Engineering 15-11 Simple I control

. ERROR NORM Issues with simple I control 4 • G not square positive definite 2

T 0 –use G as a spatial pre-filter 0 50 ERROR100 PROFILE 150 200 0.5 = T ⋅ + YG (t) G G U(t) DG (t) = T = T 0 YG G Y , DG G D -0.5 0 20 CONTROL40 PROFILE60 80 100 • For ill-conditioned G get very 20

large control, picketing 0 – use regularized inverse -20 • Slowly growing instability 0 20 40 60 80 100 – control not robust – regularization helps again

EE392m - Winter 2003 Control Engineering 15-12 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

LTI Plant

EE392m - Winter 2003 Control Engineering 15-13 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 LSI Plant

EE392m - Winter 2003 Control Engineering 15-14 Control with Regularization

• Add integrator leakage term ∆ = − ()− − − − U(t) K Y (t 1) Yd SU(t 1) • 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

EE392m - Winter 2003 Control Engineering 15-15 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 g(νν )k(ν ) z −1+ s(ν ) y(ν ) = ν y + ν d z −1+ s( ) + g( )k(ν ) d z −1+ g( )k(ν ) • Steady state ν ν ν ν ν ν = g( )k( ) + s( ) y( ) ν ν yd ν d s( ) + g(ν)k(ν ) s( ) + g( )k(ν ) ν ν k(ν ) u(ν ) = ()y ( ) − d(ν ) s( ) + g( )k( ) d

EE392m - Winter 2003 Control Engineering 15-16 Sample Controller Design

• Spatial domain loopshaping is easy - it is noncausal • Example controller with regularization ERROR NORM PLANT 1 3 2 2 1.5 0.5 G 1 1 0.5 0 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 0 1 2 3 0 100 ERROR200 PROFILE300 400 500 FEEDBACK 0.5 0.5 0.4 0.5 0.3 0.5 0 K 0.2 0.5 0.1 0.5 0 -0.5 -5 -4 -3 -2 -1 0 1 2 3 4 5 0 1 2 3 0 20 40 60 80 100 SMOOTHING CONTROL PROFILE 0.03 2 0.02 0.04 0.03 0.01 0 S 0.02 0 0.01 -0.01 -2 -5 -4 -3 -2 -1 0 1 2 3 4 5 0 1 2 3 0 20 40 60 80 100 For more depth and references, see: Gorinevsky, Boyd, Stein, ACC 2003 EE392m - Winter 2003 Control Engineering 15-17 WDM network equalization

• WDM (Wave Division Multiplexing) networks – multiple (say 40) independent laser signals with closely space wavelength 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 15-18 WDM network equalization

• Analog optical amplifiers (EDFA) 5 0 amplify all channels -5 -10 • Attenuation and amplification distort -15 -20 carrier intensity profile -25 -30 -35 • The profile can be flattened through -40 active control 1525 1535 1545 1555 1565

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

EE392m - Winter 2003 Control Engineering 15-19 WDM network equalization

• Logarithmic (dB) attenuation for a sequence of notch filters = ⋅ ⋅ A A1 K AN N = log A ∑log Ak λ k=1 N ϕ λ = − λ − a( ) ∑ wk ( 0 ck) WDM k=1 Attenuation gain - Notch filter shape control handle

EE392m - Winter 2003 Control Engineering 15-20 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

EE392m - Winter 2003 Control Engineering 15-21