System Identification for X-dynamics

Trento_19_20_02_2007 S. Vitale 1

LTPLTP

System Identification for X-dynamics

Data Analysis 4


LTPLTPLTP dynamicsDegree of freedom

Measurement Channel Resolution Control Actuation

x1-X GRS/Ifo High DFACS Electrostatics/Thrusters

x2-x1 GRS/Ifo High Suspension Electrostatics

y1-Y GRS Low DFACS Electrostatics/Thrusters

y2-Y GRS Low DFACS Electrostatics/Thrusters

z1-Y GRS Low DFACS Electrostatics/Thrusters

z2-Y GRS Low DFACS Electrostatics/Thrusters

1-H GRS/Ifo High Suspension Electrostatics

2-H GRS/Ifo High Suspension Electrostatics

1- GRS/Ifo High Suspension Electrostatics

2- GRS/Ifo High Suspension Electrostatics

1- GRS Low DFACS Electrostatics/Thrusters

2- GRS Low Suspension Electrostatics

X Ranging Very low Orbit Thrusters/low freq

Y Ranging Very low Orbit Thrusters/low freq

Z Ranging Very low Orbit Thrusters/low freq

H ST Very low AOCS Thrusters/low freq

ST Very low AOCS Thrusters/low freq

ST Very low AOCS Thrusters/low freq


LTPLTP

LTP dynamics within MBW

• 2 measured and controlled Degrees of Freedom within Measurement Bandwidth

• 3 Actuators (1 redundant)• 4 Signals (2 redundant)• A 2 Input-2 Output system with redundant

sensing and actuation• 4 Measurable transfer functions• If signals are used as stimuli, separates from

rest of DOF (cross-talks shows as excess noise


LTPLTP

2 2p1 1 o 2 1 n,1

2 2p2 2 o 2 1 n,2 lfs n, i,

2 21 p1 1 2 p2 2

1 n,1 2 n,2 lfs n, i, df 1 n,1 i,1

s x x x A g

s x x x A g h o o o

A x x

g g h o o o G H o o o

The starting x-dynamics

Signals

Dynamical variables•In the absence of imperfections

•o1 = x1

•o∆ = x2-x1


LTPLTP

2 2p1 1 o 2 1 n,1

2 2p2 2 o 2 1 n,2 lfs n, i,

2 21 p1 1 2 p2 2

1 n,1 2 n,2 lfs n, i, df 1 n,1 i,1

s x x x A g

s x x x A g h o o o

A x x


An example, the x-dynamics

Force noise

Read-out noise


LTPLTP

Control forces

2 2p1 1 o 2 1 n,1

2 2p2 2 o 2 1 n,2 lfs n, i,

2 21 p1 1 2 p2 2

1 n,1 2 n,2 lfs n, i, df 1 n,1 i,1

s x x x A g

s x x x A g h o o o

A x x



Force inputs


LTPLTP

2 2p1 1 o 2 1 n,1

2 2p2 2 o 2 1 n,2 lfs n, i,

2 21 p1 1 2 p2 2

1 n,1 2 n,2 lfs n, i, df 1 n,1 i,1

s x x x A g

s x x x A g h o o o

A x x



2 2 2 21 p1 2 p2 1 2 p2 o

n,1 1 n,1 2 n,2 lfs n, i, df 1 n,1 i,1

2 2 2 2p2 p1 1 p2 o n,2 n,1 lfs n, i,

s 1 x x

g g g h o o o G H o o o

x s 2 x g g h o o o

The unmeasured variable A disappears, x1, x2, x1o1, ∆x = x2-x1 o∆

2 inputs2 outputs


LTPLTP


2 2 2 21 p1 2 p2 1 2 p2 o

n,1 1 n,1 2 n,2 lfs n, i, df 1 n,1 i,1

2 2 2 2p2 p1 1 p2 o n,2 n,1 lfs n, i,

s 1 x x


x s 2 x g g h o o o

1 1,1 1 1,

,1 1 ,

o S x S x

o S x S x

21

n,1 n,1 n,2

2 2 21 p1 2 p2 2 p2 o

1 2 lfs df

2 2 2p2 p1 p2 o lfs

n, i, 1 n,1 i,1

21 n,2 n,1 n, i,

s 1 x x

g g g o o o G o o o

x s x g g o o o

h H

2 h

1,1 1

,

1 ,1

1,1

o x x

o x

S S

xS S

(Frequency dependent) parameters to be measured


LTPLTP

2 2 2 21 p1 2 p2 1 2 p2 o

n,1 1 n,1 2 n,2 lfs n, i, df 1 n,1 i,1

2 2 2 2p2 p1 1 p2 o n,2 n,1 lfs n, i,

s 1 x x


x s 2 x g g h o o o

D q g �

1 1,1 1 1,

,1 1 ,

o S x S x

o S x S x

Maximum likelihood estimator

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LTPLTP

2 2 2 21 p1 2 p2 1 2 p2 o

n,1 1 n,1 2 n,2 lfs n, i, df 1 n,1 i,1

2 2 2 2p2 p1 1 p2 o n,2 n,1 lfs n, i,

s 1 x x


x s 2 x g g h o o o

D q g �

1 1,1 1 1,

,1 1 ,

o S x S x

o S x S x

n ig g C o

�


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LTPLTP

2 2 2 21 p1 2 p2 1 2 p2 o

n,1 1 n,1 2 n,2 lfs n, i, df 1 n,1 i,1

2 2 2 2p2 p1 1 p2 o n,2 n,1 lfs n, i,

s 1 x x


x s 2 x g g h o o o

D q g �

1 1,1 1 1,

,1 1 ,

o S x S x

o S x S x

n ig g C o

�

no S q o �


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LTPLTP

D q g �

n ig g C o o �

no S q o � Sign

als on

ly

na D,S,C a ��

g C

1 1n n iD S C o g D S o C o

��


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LTPLTP

The nominal response

o o i ia D C o C o ��

g

2 2 2 21 1 p1 2 p2 df 2 p2 o lfs 1

2 2 2 2p2 p1 p2 o lfs

a s 1 H h x

a s 2 h x

df i,1 lfs i,

lfs i,

H o h o

h o

•- open loop force on S/C

•open loop difference of force on test-masses

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LTPLTP

The noise

n ng D o �

n,1 1 n,1 2 n,2

n,2 n,1

g g g G

g g

2 2 2 2n,11 p1 2 p2 2 p2 o

2 2 2 2n,p2 p1 p2 o

Readout Noise Equivalent Force

os 1

os 2

Channels are correlated

•- open loop difference of force between test-mass 1 and S/C

•open loop difference of force on test-masses

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LTPLTP

The noise

x

x1

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LTPLTP

na D,S,C a ��

g C

Log

dataN

,k, j k k j j

k, j 1

1a D,S,C, t C, t a D,S,C, t C, t

2

��

g g

Pick matrices values that maximize

n , n ,

1,

k, j a ,a sampleR k j T


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LTPLTP

Maximum Likelihood

• Requires Inversion of bigN x bigN matrix

• Requires non linear minimization tool

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LTPLTP

An alternative approach:linearisation

• Good for studying the problem

• Allows simplified theory to be applied

• Allows quick estimation of Fisher Matrix and parameter resolution

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LTPLTP

D q g �

n ig g C o o �

no S q o �

1

1

n n i

D D I S C C o

g D D I S o C C o

��

��

Expand matrices as function of imperfections

D D q g ��

n ig g C C o o ��

no I S q o ��

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LTPLTP

1

1

n n i

D D I S C C o

g D D I S o C C o

��

��

LinearizationTo first order in:

#

•noise

1

o

1

n n i

D D I S C C o o

g D D I S o C C o

��

��

o io iD Ca C o o � � �

g o o i ia D C o C o ��

g

11

i

n n

D C o D S D C D C C C

g D

a

o

��

�

g 11

i

n n

a D C o D S D C D C C C

g D o

��

�

g

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LTPLTP

Imperfections 1/4

- 1 11 and 2 21

- 2 2p1 p1 11

- 2 2p2 p2 21

- o o 1

- lfs lfsh h 1 h

- df dfH H 1 H

- 1 n,1 1,1 1 1,o o 1 S x S x

n, , ,1 1o o 1 S x S x

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LTPLTP

Imperfections 2/4

2 2p1 p1

s1 i

1 1 1 12p1

i

o 10 111.4 10 s

and 5

2p1 0.1Hz

6.7 10

o 910

1 1 12p1

i

o 2 6 2p1 1.3 10 s

o 2 6 2p2 2 10 s

o 9 2o 5 10 s

o 35 10

o 410

2 2p1 1 1 p2 2 2 o

2 2 2p2 2 p1 1 2 1 p2 2 2

i iD

i i

�

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LTPLTP

Imperfections 3/4

H H H-sT 1+ T s

df,o df ,o Hdf H H

H H H H

H 1+ H e H e sH 1 H sT T

1 s 1 1 s 1 s

An elementary model for delays and roll-off

H hdf H H H lfs h h h

H h

hlfs h h h

h

s sH H sT T h h sT T

1 s 1 sC

s0 h h sT T

1 s

�

1

1 1 1

1

S1,1 S S S 1,

S

S,1 , S S S

S

sS sT T S

1 sS

sS S sT T

1 s

�

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LTPLTP

Imperfections 4/4

• 20 Imperfection Parameters

• Each parameter generates a signal

11

ia D S D C D C C C

�� g iP

� g

imperfections imperfectionsN N

kk 1 k 1

P

�

kg g

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LTPLTP

Example: Swept-sine input

0.03 mHz to 30 mHz in 104s

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LTPLTP

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LTPLTP

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LTPLTP

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LTPLTP

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LTPLTP

Extracting amplitudes

2

k k,n nn 1

ˆ h t a t dt

Find h’s with no bias and minimal variance

1

21

i, j k j l,i l, i,l,i 1 k, j

1ˆ ˆ S g g d2

imperfectionsN2

1 * *k,n n,m k, j j

m 1 j 1

h S g

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LTPLTP

Playing with a very simplified model looking for G. Very large signal

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LTPLTP

Conclusion

• System identification requires– Vector pre-processing (filter and linear

combination)– Multiple and correlated series Wiener filter/

Likelihood estimator– Noise model parameterization from best

measurement– Dynamics pre-modeling– Assessment of signals that can be uploaded– …….

Documents

System Identification for X-dynamics