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
Olofsson et al., MHDWS, November 2013 3
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)
Olofsson et al., MHDWS, November 2013 4
- 2 . 5 - 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5- 2
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Mirn
ov p
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“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
+
+
=
Olofsson et al., MHDWS, November 2013 5
y [c
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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
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“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
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TASK: inverse problem of the mode summation
Olofsson et al., MHDWS, November 2013 6
y [c
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t [ s a m p l e i n d e x ]
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“Subspace” method may decompose an array data matrix into sinusoidal modes when SVD cannot
“subspace” method SVD “modes”
Olofsson et al., MHDWS, November 2013 7
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
Olofsson et al., MHDWS, November 2013 8
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
Key algorithm is here
Olofsson et al., MHDWS, November 2013 9
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
Olofsson et al., MHDWS, November 2013 10
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 )
Olofsson et al., MHDWS, November 2013 11
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)
Olofsson et al., MHDWS, November 2013 12
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
This was covered
Olofsson et al., MHDWS, November 2013 13
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
This was covered Now this.
Olofsson et al., MHDWS, November 2013 14
m o d e f r o m s y s t e m r2
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M A C ( i , j)
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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”
Olofsson et al., MHDWS, November 2013 15
m o d e f r o m s y s t e m r2
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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
Olofsson et al., MHDWS, November 2013 16
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
This was covered Also coveredNow this.
Olofsson et al., MHDWS, November 2013 17
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
Olofsson et al., MHDWS, November 2013 18
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
Olofsson et al., MHDWS, November 2013 19
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
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Mirn
ov p
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[T/s
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M P I 6 6 M 3 1 2 D
Array-averaged FFT spectrogramfor 39 Mirnov channels
shot #154986
What are these?
39x
Olofsson et al., MHDWS, November 2013 20
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
Olofsson et al., MHDWS, November 2013 21
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
Olofsson et al., MHDWS, November 2013 22
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!
Olofsson et al., MHDWS, November 2013 23
“Gaussian Process Regression” technique used to estimate smooth modal patterns from shape-vectors
#154986eigspec
θ
φ
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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
Olofsson et al., MHDWS, November 2013 24
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
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M P I 6 6 M 0 6 7 D
shot #155279
Fishbonetime-trace forsingle probe
Array-averagedspectrogram
Olofsson et al., MHDWS, November 2013 25
Fishbone signal array decomposition; example of transient activity analysis using eigspec
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shot #155279
Fishbonetime-trace forsingle probe
Array-averagedspectrogramwith overlayedeigspec features
Olofsson et al., MHDWS, November 2013 26
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
Olofsson et al., MHDWS, November 2013 27
θ
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Fishbone signal array decomposition reveals a “phase-folded” dominant modal pattern; similar to sawteeth
Amplitudeenvelopes
dominantmodal patterns
shot #155279
n=2n=1
Olofsson et al., MHDWS, November 2013 28
θ
φ
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- 2
- 1
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Fishbone signal array decomposition reveals a “phase-folded” dominant modal pattern; similar to sawteeth
Amplitudeenvelopes
shot #155279
n=1
“opposite”Inboard helicity
Olofsson et al., MHDWS, November 2013 29
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
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