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Adaptive Control Loops for Advanced LIGO

Brett Shapiro25 February 2011

2

Control Loops Keep LIGO RunningEvolving seismic noise from:• weather

• people

… adaptive control also makes a very good thesis topic…

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How are Adaptive Loops Useful?Simple example: Push on the pendulum to maintain a constant L.

Control Law

L

0 2 4 6 8 100

0.5

1

1.5

2Length Signal

0 2 4 6 8 10-2

-1

0

1

2Control Signal SignalApplied Force

0 2 4 6 8 10-2

-1

0

1

2Control Signal SignalApplied Force

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Adaptive Control for Isolation Systems

Suspensions:• Angular mirror control• Damping (modal or classical)• Length control?

Seismic Control• Isolation loops• sensor blending?

Can be used to optimize in real time:• aLIGO noise budget amplification (from sensor noise, barkhausen noise, etc)• actuator forces• RMS error signals• just about anything else

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LASTI Quad-Triple Cavity

Quad Pendulum Triple Pendulum

16 meters

Adaptive Control MEDM Screen

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Questions to Answer

1. What is being controlled in this experiment?2. What is the adaptation optimizing?3. What parameters are being adapted?4. How are they being adapted?

8

Triple

LASTI Experimental Adaptive Setup

Control Law

Quad

Optimization Goals:1. Cavity Length RMS2. Avoid saturating the actuators

- Penultimate mass- Test mass

Solution: Least squares adaptation

Test mass (TM)

Penultimate mass (PUM)

Top mass (Top)

Length measurement:AKA ‘Error signal’

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Control Block Diagram Diagram

3 primary components to adaptive control:

1. Control

Triple

+

-

Control

0 2 4 6 8 100

0.5

1

1.5

2Length Signal

Cavity Noise

Error signal

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Adaptive Control Architecture

3 primary components to adaptive control:

1. Control – parameterized in terms of adapting parameters2. Adaptation algorithm – updates control parameters

Triple

+

-

Control

Adapt0 2 4 6 8 10

0

0.5

1

1.5

2Length Signal

Cavity Noise

Error signal

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Adaptive Control Architecture

3 primary components to adaptive control:

1. Control – parameterized in terms of adapting parameters2. Adaptation algorithm – updates control parameters3. Costs – variables that are optimized with adaptation

Triple

+

-

Control

Cost Adapt0 2 4 6 8 10

0

0.5

1

1.5

2Length Signal

Cavity Noise

Error signal

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Adaptive Control Architecture

3 primary components to adaptive control:

1. Control – parameterized in terms of adapting parameters2. Adaptation algorithm – updates control parameters

- uses a least squares optimization routine3. Costs – variables that are optimized with adaptation

Triple

+

-

Control

Cost Adapt0 2 4 6 8 10

0

0.5

1

1.5

2Length Signal

Cavity Noise

Error signal

Cost Box: Summation of CostsTriple

+

-

Control

Cost Adapt0 2 4 6 8 10

0

0.5

1

1.5

2Length Signal

Cavity Noise

Error signal

The cost box measures the performance values we care about and scales them to get the costs:

1. Error RMS2. PUM force RMS3. Test mass force RMS

Control BoxTriple

+

-

Control

Cost Adapt0 2 4 6 8 10

0

0.5

1

1.5

2Length Signal

Cavity Noise

Error signal

10-10

100

Mag

nitu

de (

abs)

10-2

100

102-540

-360

-180

0

180

Pha

se (

deg)

Hierarchical Loop Gains.

Frequency (Hz)

PUMTM

Control filters are parameterized in terms of:• PUM crossover frequencies• Test mass crossover frequency

Adaptation BlockTriple

+

-

Control

Cost Adapt0 2 4 6 8 10

0

0.5

1

1.5

2Length Signal

Cavity Noise

Error signal

Adaptation Algorithm: Least Squares Minimization Approach

cccv T

ii

2

1

2

1 2

ccv

T

• Total system cost

• Cost gradient

c

J• System Jacobian matrix

Definitions for optimization

v = total costc = list of costs we want to optimizeθ = list of adjustable control parametersJ = System Jacobian matrix

Variable list

JT yields gradient descent (1st order). J-1 yields Gauss-Newton (2nd order).

Optimization routine

Adaptation Algorithm: quadratic minimization approach

v = total costc = list of costs we want to optimizeθ = list of adjustable control parametersJ = system Jacobian matrixα = user defined scalar step size

Variable list

Adaptation Problem

• Question: What is J?

• Answer: We do not know. It depends on θ and c, and other unknown or unmodeled parameters.

• Good news: we can estimate it in real-time. A recursive least squares algorithm (RLS) is used.

Results

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Simulated Results

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Results

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Results

0 20 40 60 80 100 120 140 160 180 20010

-10

10-9

Measured Error Signal RMS and Parameter Performance

Cav

ity E

rror

(m

rms)

Cavity RMSGoal

0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300

TM

UG

F (

Hz)

Time (minutes)

Penultimate Mass Unity Gain Frequency

0 20 40 60 80 100 120 140 160 180 2006

7

8

9

10

11

12

PU

M U

GF

(H

z)

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Conclusions• Adaptive control is a powerful real-time self-tuning

method for many aLIGO loops.– Can target arbitrary performance requirements:

• Avoiding actuator saturations• Minimizing noise amplification

• Real-time RLS estimation of system response compensates for unknowns adequately.

• Adaptation speed is limited by RMS averaging• Complexities beyond this talk exist

– Achieving good estimates with the Jacobian– Setting good stopping and starting conditions

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Backups

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Feedback Filter Box

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Transfer Function Canonical forms

Laplace Transfer function

State space transfer function in observer canonical form

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Parametric Transfer Functions

2233

)()(

)(

ugfugf

ugf

ugfugfugf

ugf

ugf fpspfsp

sK

fspsp

sK

su

sy

)(01)(

)(1

)(0

1)()( 2

txty

tup

txfp

pftx ugf

ugf

ugf

Test mass feedback filter: Laplace form

Test mass feedback filter: State space form

ωugf = unity gain freq.p -> gives phase margin around ugff -> f*ωugf is the freq. of a low pass poleK is a constant scaling factor which compensates for the plant gain

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Parametric Filter in LIGO’s RCG Simulink Environment

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Parametric Controller TFs

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Parametric Controller TFs

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Cost Box Details

RMSGoal

RMSGoalRMSRMSScaled E

EEE

,

,,

)025.0/(tan2

21

,

RMS

RMS

RMSScaled

U

UU

Error RMS scaling Actuator RMS scaling

The error RMS is scaled relative to a target value (such as the design value).

The actuator RMS is scaled relative to its saturation limit. In this case, the scaled value is set to ‘explode’ when the RMS reaches the saturation limit.

Adaptation Algorithm: quadratic minimization approach

cJ Ttt

1

1st order algorithm: gradient descent method

v = total costc = list of costs we want to optimizeθ = list of adjustable control parametersJ = system Jacobian matrixα = user defined scalar step sizet = current time step

Variable list

cJtt

1

1

2nd order algorithm: Gauss-Newton method

cJtt

1 For noninvertible J(+ -> pseudoinverse)

For invertible J

Cost Reduction Choices

Adaptation Algorithm: quadratic minimization approach

v = total costc = list of costs we want to optimizeθ = list of adjustable control parametersJ = system Jacobian matrixα = user defined scalar step sizet = current time step

Variable list

Jacobian definition

Adaptation Problem

K = Jacobian update gain

c

J

0

Jce

)(1

ttt JcKJJ

Approximate error

RLS updating

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RLS for Jacobian Estimation

)( ,,1, tiTtttiti JcKJJ

ttTt

tt

P

PK

tTtt PKIP

11

Definitions:i = row index of the Jacobian matrixλ = exponential forgetting factor.0 < λ ≤ 1

t

nni

Tnn

nt JcV1

2, )(

2

1)(

Cost function optimized by RLS

Iterative RLS algorithm

1.

2.

3.

0. Initialize J and P

4. Go back to 1.

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Convergence of Jacobian

t

n

Tnn

1

A necessary condition for the convergence of RLS algorithm to the true J is that this matrix is nonsingular.

A real-time estimate of the invertiblility of this matrix is to calculate this matrix over a finite number of time steps and then calculate its condition number. The condition number is the smallest eigenvalue divided by the largest.

Condition number R = (max eigenvalue of Θ)/(min eigenvalue Θ)

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Results

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Results

0 20 40 60 80 100 120 140 160 180 20010

-4

10-3

10-2

10-1

100

Jacobian Singular Values

Time (minutes)

Mag

nitu

de

S1

S2

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Thesis Contributions• The literature has very little on generating real-time

estimates for how controller parameters influence the statistics of linear system performance.

• Similarly there is little in the literature on using real-time function minimization techniques to optimize linear system performance.

• Optimization of Jacobian estimation accuracy.• Optimizing adaptation rate using the measured

statistics of the stochastic system performance.• Use of singular value decomposition to quantify

behavior of control adaption and system Jacobian.