mirnov array signal processing using eigspec: beyond ffts and … · 2015. 2. 18. · extended...

Mirnov array signal processing using eigspec: beyond FFTs and SVDs

ByK. Erik J. Olofssona

In collaboration with J.M. Hansona, D. Shirakia, F.A. Volpea,D.A. Humphreysb, R.J. La Hayeb,M.J. Lanctotb, E.J. Straitb, A.S. Welanderb,E. Kolemenc, M. Okabayashic, andF. Turcoa, J. Ferronb

aColumbia University, APAM, bGeneral Atomics, cPPPL

Presented atMHD control workshop, Santa Fe, New Mexico, USANovember 18-20, 2013

Olofsson et al., MHDWS, November 2013 2

Outline of presentation

• Motivation– Why develop new signal processing methods for Mirnov array analysis?

• Algorithm introduction and outline– Example comparisons: FFT, SVD– Subspace identification and low rank signals– Feature extraction

• DIII-D analysis examples– NTM detection in the presence of sawtooth precursors– Fishbone burst signal decomposition

• Summary


Array signal processing methods can significantly aid the interpretation of Mirnov magnetics data

• MHD activity can be inferred from kHz-range fluctuations

– Dynamic, transient and time-varying signals

Mirnov array probes measurepoloidal field fluctuations

• Fluctuations dominated by a handful of frequencies

– Specialized methods seem applicable

TASK: explicit decomposition of the full array signal into source terms

RESULT:for each block of the time-series, inventory obtained (freq./amp., modal shape)


- 2 . 5 - 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5- 2

- 1

0

1

2

3

4

l o g1 0 ω

cros

s-po

wer

log

10P

y 1y 2

( ω)

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 0- 0 . 5

0

0 . 5

ω1

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 0- 0 . 5

0

0 . 5

ω2

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 0- 0 . 1

- 0 . 0 5

0

0 . 0 5

0 . 1

nois

e

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 0- 1

- 0 . 5

0

0 . 5

1

Mirn

ov p

robe

pai

r [a

.u.]

t [ s a m p l e s ]

y1

y2

“Subspace” method can resolve frequencies that are closely spaced; where DFT (FFT) fails

• The “subspace” method to be defined later, is “parametric”.

– These tend to produce better estimates for shorter time-series than “nonparametric” (DFT) methods

• The “subspace” method has documented applicability for operational modal analysis

– line-spectrum + damping rates

+

+

=


y [c

hann

el in

dex]

t [ s a m p l e i n d e x ]

M i r n o v b l o c k d a t a m a t r i x Y

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0

5

1 0

1 5

2 0

“Subspace” method designed for multivariate linear system estimation; direct SVD is not

• Direct SVD is not a modal analysis “operation” in itself– It has more to do with data compression

tru

e m

od

e #

1

0 5 0 1 0 0 1 5 0

5

1 0

1 5

2 0

tru

e m

od

e #

2

0 5 0 1 0 0 1 5 0

5

1 0

1 5

2 0

no

ise

0 5 0 1 0 0 1 5 0

5

1 0

1 5

2 0

+

TASK: inverse problem of the mode summation


y [c

hann

el in

dex]

t [ s a m p l e i n d e x ]

M i r n o v b l o c k d a t a m a t r i x Y

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0

5

1 0

1 5

2 0

“Subspace” method may decompose an array data matrix into sinusoidal modes when SVD cannot

“subspace” method SVD “modes”


Flowchart of the feature-extraction stage in the eigspec Mirnov array code

Input data

Output data

Mirnov data block Y

Dimension-reductionvia randomprojection

"Compressed"data block Z

Time-seriesmodellingusing SSI

Model 1

Model 2

Order-MACmodal subsystemselection

Frequency shortlist

Shape-vectorestimation usingMirnov data block Y

Shape vectors

modalrepresentation



Input data

Output data

Mirnov data block Y




Model 1

Model 2


Frequency shortlist


Shape vectors

modalrepresentation

Key algorithm is here


Stochastic subspace identification (SSI) is a technique that estimates a time-series model on state-space form

• Sequence of m-channel Mirnov data vector samples y(k)

x (k+1)=A x (k)+K e (k)y (k )=C x(k )+e(k )

x (k+1)=AK x (k )+K y (k )y (k)=C x (k )+e(k)

AK=A−KC

{⋯ y(k−1) y(k ) y (k+1) ⋯} time-indices k=1..N

• Stochastic time-series model on state-space form

• Task is to find the system matrices A, C (and K) from data { y(k) }

It is assumed that { e(k) } is a sequence of random vectors


SSI can be seen as a reduced-rank regression approach that exploits the shift-structure of key matrices

Y f (k )=[ CCACA2⋮CA f −1

]x (k)+[ I 0 0 ⋯ 0CK I 0 ⋯ 0⋮ ⋮ ⋮ ⋮ ⋮CA f −2 K ⋯ ⋯ CK I ]E f (k ) =Γ x(k )+FE f (k )State-to-future mapping

Past-to-state mapping

x (k )=[AKp−1 K ⋯ AK2 K AK K K ]Y p(k )+O (AKp )≈LY p(k )

Y f (k )=[ y (k )y (k +1)y (k+2)⋮y (k + f −1)

]Y p(k )=[ y (k−p)⋮y (k−3)y(k−2)y (k−1)

] E f (k )=[e (k )

e(k+1)e (k+2)

⋮e (k + f −1)

]Using extended array data vectors

Iteration of themodel equations

Y f (k )≈Γ LY p(k)+FE f (k )


Model reduction step using SVD on a (possibly) weighted version of the future-past covariance matrix

Y f (k )≈Γ LY p(k)+FE f (k )

Y f (k )≈Γ LY p(k)+FE f (k )=[ CCACA2⋮CA f −1

][AKp−1K ⋯ AK2 K AK K K ]Y p(k )+FE f (k )R fp=Ε [Y f (k )Y pT (k )]=Γ LΕ [Y p(k )Y pT (k )]=ΓL R pp

R fp≈U r S rV rT

Γ̂=U r S r1/2

Expectation over k

Truncated SVD of covariance matrix

Extended observability matrix estimate

• A and C now inferred from the shift-structure of Γ– Many many variations of the SSI algorithms– eigspec implements the above, but also a more elaborate SVD-weighting (CCA)



Input data

Output data

Mirnov data block Y




Model 1

Model 2


Frequency shortlist


Shape vectors

modalrepresentation

This was covered



Input data

Output data

Mirnov data block Y




Model 1

Model 2


Frequency shortlist


Shape vectors

modalrepresentation

This was covered Now this.


m o d e f r o m s y s t e m r2

mod

e fr

om s

yste

m r

1

M A C ( i , j)

1 2 3 4 5 6 7 8 9 1 0

1

2

3

4

5

6

0

0 . 2

0 . 4

0 . 6

0 . 8

1


mod

e fr

om s

yste

m r

1

D S T ( i , j)

1 2 3 4 5 6 7 8 9 1 0

1

2

3

4

5

6

0

0 . 2

0 . 4

0 . 6

0 . 8

1

Modal subsystem selection using an “Order-MAC” criteria yields a frequency shortlist

x̃ (k+1)= Ã x̃ (k )+ K̃ e (k)y (k )=C̃ x̃(k )+e(k )

x̃=W−1 x

Modal form: A-matrix diagonalized

Shape-vector is a column of C̃

MAC (v , w)=(vH w)(wH v)

(vH v)(wHw)

Shape correlation “metric”

DST (λi ,λ j)=max(0,1−∣λi−λ j∣∣λi∣ )Eigenvalue “stability”



mod

e fr

om s

yste

m r

1

M A C ( i , j)

1 2 3 4 5 6 7 8 9 1 0

1

2

3

4

5

6

0

0 . 2

0 . 4

0 . 6

0 . 8

1


mod

e fr

om s

yste

m r

1

D S T ( i , j )

1 2 3 4 5 6 7 8 9 1 0

1

2

3

4

5

6

0

0 . 2

0 . 4

0 . 6

0 . 8

1

Modal subsystem selection using an “Order-MAC” criteria yields a frequency shortlist

MAC (v , w)=(vH w)(wH v)

(vH v)(wHw)

Shape correlation “metric”

DST (λi ,λ j)=max(0,1−∣λi−λ j∣∣λi∣ )Eigenvalue “stability”

Example: modes 1,4,5 in system 1have matching mode insystem 2



Input data

Output data

Mirnov data block Y




Model 1

Model 2


Frequency shortlist


Shape vectors

modalrepresentation

This was covered Also coveredNow this.


The Mirnov array data Y is decomposed into a sum of products of shape-vectors and sinusoidal vectors

• S-matrix constructed from frequencies of the selected modal subsystems– Contains sinusoidal columns– Least-squares estimation to obtain shape-vectors (columns of D)– This minimizes the Frobenius norm of residual E

Y=DST+ED̂=YS (ST S )−1


DIII-D Mirnov array analysis: example eigspec applications

• Decomposition of a fishbone burst signal– Amplitude envelope, frequency sweep, modal shape

• The problem of mixing up sawtooth precursor oscillations with m/n=2/1 NTM signatures

– Shot #154986 (found by Rob) shows an example case where the precursor has almost identical frequency to the 2/1 that emerges just after the ST crash


Detection of m/n=2/1 NTM signal triggered by sawtooth crash; mode-locking event, to be analyzed here

• Experiment #154986– Current profile effects on TM stability– ITER-like plasma with high torque– q95 ~ 3.25, βN~1.9 @ t=3300ms

3 3 0 0 3 3 5 0 3 4 0 0 3 4 5 0 3 5 0 0 3 5 5 0 3 6 0 0- 3 0 0

- 2 0 0

- 1 0 0

0

1 0 0

2 0 0

Mirn

ov p

robe

[T/s

]

t [ m s ]

M P I 6 6 M 3 1 2 D

Array-averaged FFT spectrogramfor 39 Mirnov channels

shot #154986

What are these?

39x


FFT-based frequency analysis (with n-number assignment) is less clear than eigspec's SSI-based method

#154986eigspec

#154986modespec

• Probe-pair coherence method

– Diffuseness of short-time DFT

– Possible clean-up: average many pairs

• Subspace method in eigspec

– Frequencies not restricted to any “FFT-bin”

– SSI natively incorporates the entire array


Post-processing extracted modes using shape-coherence to reference point reveals n=1 trace “event”

#154986eigspec

Reference vectorEvent changescharacter of n=1 trace

Co

lorin

g is th

e coh

erence valu

e [0,1]to

the referen

ce shap

e vector


Post-processing with shape-coherence as a similarity metric allows automatic clustering of extracted modes

#154986eigspec

cluster index label

n=1 trace split in two groups!


“Gaussian Process Regression” technique used to estimate smooth modal patterns from shape-vectors

#154986eigspec

θ

φ

0 1 2 3 4 5 6

- 3

- 2

- 1

0

1

2

3

θ

φ

0 1 2 3 4 5 6

- 3

- 2

- 1

0

1

2

3

θφ

0 1 2 3 4 5 6

- 3

- 2

- 1

0

1

2

3

3/2 “3/1” 2/1

Typical pattern for sawtooth precursor!“gaps” are at up-down locations

Ou

tbo

ard m

idp

lane at th

eta=0


Fishbone signal array decomposition; example of transient activity analysis using eigspec

3 7 6 0 3 7 7 0 3 7 8 0 3 7 9 0 3 8 0 0 3 8 1 0 3 8 2 0- 3 0 0

- 2 0 0

- 1 0 0

0

1 0 0

2 0 0

3 0 0

Mirn

ov p

robe

[T/s

]

t [ m s ]

M P I 6 6 M 0 6 7 D

shot #155279

Fishbonetime-trace forsingle probe

Array-averagedspectrogram


Fishbone signal array decomposition; example of transient activity analysis using eigspec

3 7 6 0 3 7 7 0 3 7 8 0 3 7 9 0 3 8 0 0 3 8 1 0 3 8 2 0- 3 0 0

- 2 0 0

- 1 0 0

0

1 0 0

2 0 0

3 0 0

Mirn

ov p

robe

[T/s

]

t [ m s ]

M P I 6 6 M 0 6 7 D

shot #155279

Fishbonetime-trace forsingle probe

Array-averagedspectrogramwith overlayedeigspec features


Fishbone signal array decomposition shows a bundle of n-numbers (1,2,3,4) per burst; analysis window is 0.5 ms

Amplitudeenvelopes

Frequencysweeps

eigspec analysis time-frameshort compared to burst-length

shot #155279


θ

φ

0 1 2 3 4 5 6

- 3

- 2

- 1

0

1

2

3

θ

φ

0 1 2 3 4 5 6- 3

- 2

- 1

0

1

2

3

Fishbone signal array decomposition reveals a “phase-folded” dominant modal pattern; similar to sawteeth

Amplitudeenvelopes

dominantmodal patterns

shot #155279

n=2n=1


θ

φ

0 1 2 3 4 5 6- 3

- 2

- 1

0

1

2

3

Fishbone signal array decomposition reveals a “phase-folded” dominant modal pattern; similar to sawteeth

Amplitudeenvelopes

shot #155279

n=1

“opposite”Inboard helicity


Conclusions

• Improved signal processing may be required to implement NTM detectors for real-time control

– Sawtooth precursors and fishbones exhibit n=1 modal patterns on the Mirnov array– Sawtooth precursors can have the same frequency as 2/1 NTMs

• Improved signal processing enables more detailed offline data analysis – The eigspec code generates “clean” spectra that may be easier to relate to MHD

modelling.

• Magnetics fluctuation frequencies could be paired with other diagnostics– The eigspec code (or similar methods) appears to be applicable also to the integrations

of several diagnostic systems, such as SXR, BES and ECE

PowerPoint Presentation##TitleSlide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29

mirnov array signal processing using eigspec: beyond ffts and … · 2015. 2. 18. · extended...

Documents