whatcansystemidentiﬁcationoﬀerto impedancespectroscopy? · 0 10 20 30 40-100-50 0 frequency...

What can System Identification Offer toImpedance Spectroscopy?

Rik PintelonEbrahim Louarroudi, and John Lataire

Vrije Universiteit Brussel, Department ELEC

Outline

I Motivating ExamplesI System Identification in a NutshellI Nonparametric Impedance SpectroscopyI Parametric Impedance SpectroscopyI Take Home Message

Outline

Motivating Examples

Example 1: myocardial electrical impedance measurements(EBI, Universitat Politecnica de Catalunya, Spain)

i t( )

v t( )Z jw t,( )

Electrodes

• goal: distinguish normal, ischemic and infarcted heart tissue• observation: periodic change impedance• reason: heartbeat

Motivating Examples

Example 1: myocardial electrical impedance measurements(EBI, Universitat Politecnica de Catalunya, Spain)

i t( )

v t( )Z jw t,( )

Electrodes

• goal: distinguish normal, ischemic and infarcted heart tissue• observation: periodic change impedance• reason: heartbeat

Motivating Examples

Example 2: non-invasive lung impedance measurements(EeSA, Universiteit Gent, Belgium)

p t( ) p0+

q t( )

• goal: detect pulmonary diseases• observation: periodic change impedance• reason: breathing

Motivating Examples

Example 2: non-invasive lung impedance measurements(EeSA, Universiteit Gent, Belgium)

p t( ) p0+

q t( )

• goal: detect pulmonary diseases• observation: periodic change impedance• reason: breathing

Motivating Examples

Example 3: electrochemical impedance spectroscopy(SURF, Vrije Universiteit Brussel, Belgium)

counter electrodereference electrodei(t)

working electrode

electrolyte

• goal: understanding/monitoring pit corrosion• observation: impedance evolves in time• reason: pitting

Motivating Examples

Example 3: electrochemical impedance spectroscopy(SURF, Vrije Universiteit Brussel, Belgium)

counter electrodereference electrodei(t)

working electrode

electrolyte

• goal: understanding/monitoring pit corrosion• observation: impedance evolves in time• reason: pitting

Motivating ExamplesConclusions

Observation: real life examples are oftenI time-variantI nonlinearI non-repeatable

Ultimate goal:I estimate a nonlinear time-variant model from

a single experiment

Does system identification offer a solution?I not yet

a single experiment

Outline

I Motivating ExamplesI System Identification in a Nutshell

• Excitation Signal• Experimental Setup• Model• Cost Function

I Nonparametric Impedance SpectroscopyI Parametric Impedance SpectroscopyI Take Home Message

Outline

System Identification in a NutshellChoice of the Excitation Signal

Single sine excitation

Broadband excitations:I binary signalsI random noiseI periodic noiseI random phase multisine

System Identification in a NutshellChoice of the Excitation Signal

Single sine excitation

Broadband excitations:I binary signalsI random noiseI periodic noiseI random phase multisine

System Identification in a NutshellChoice of the Excitation Signal: single sine versus broadband

For example, 3 sinewaves

0 100 200 300 400−1.0

−0.5

1.0single sine (ss)

0 100 200 300 400−2.0

−1.0

2.0multisine (ms)

with same total experiment time, and same power

rmsms (f ) = rmsss (f ) /√3

std noisems (f ) 6 std noisess (f ) /√3

⇓SNRms (f ) > SNRss (f )

System Identification in a NutshellChoice of the Excitation Signal: single sine versus broadband

For example, 3 sinewaves

0 100 200 300 400−1.0

−0.5

1.0single sine (ss)

0 100 200 300 400−2.0

−1.0

2.0multisine (ms)

with same total experiment time, and same power

rmsms (f ) = rmsss (f ) /√3

std noisems (f ) 6 std noisess (f ) /√3

⇓SNRms (f ) > SNRss (f )

System Identification in a NutshellChoice of the Excitation Signal: broadband excitations

signals

spectra

Two periods or more ⇒ noise variance estimate

0 50 100 150 200

−1.0

noisy sinewave

0 50 100 150 200

−1.0

residual sinewave

I true noise standard deviation σ = 0.1I estimated noise standard deviation σ = 0.103

Two periods or more ⇒ noise variance estimate

0 50 100 150 200

−1.0

noisy sinewave

0 50 100 150 200

−1.0

residual sinewave

I true noise standard deviation σ = 0.1I estimated noise standard deviation σ = 0.103

System Identification in a NutshellChoice of the Excitation Signal: conclusions

Advantages multisine excitations1. SNR(multisine) > SNR(single sine)2. guaranteed amplitude spectrum3. logarithmic frequency distribution possible4. measure 2 periods or more ⇒ quantify the noise variance and

the variance of the nonlinear distortions5. no leakage errors

Disadvantage multisine excitations1. loss in frequency resolution of a factor 2 w.r.t.

a random excitation2. synchronisation between generator and acquisition needed

System Identification in a NutshellChoice of the Excitation Signal: conclusions

Advantages multisine excitations1. SNR(multisine) > SNR(single sine)2. guaranteed amplitude spectrum3. logarithmic frequency distribution possible4. measure 2 periods or more ⇒ quantify the noise variance and

the variance of the nonlinear distortions5. no leakage errors

Disadvantage multisine excitations1. loss in frequency resolution of a factor 2 w.r.t.

a random excitation2. synchronisation between generator and acquisition needed

Outline

System Identification in a NutshellChoice of the Experimental Setup: zero-order-hold

v kTs( )

i0 t( )+ZOH Actuator

r kTs( )

Acquisition

process noise

measurementnoise

Z s( )v0 t( )

Sensor+Generator/controller

I discrete-time model from r (kTs) to v (kTs)I known input, noisy outputI well suited for prediction and control

System Identification in a NutshellChoice of the Experimental Setup: zero-order-hold

v kTs( )

i0 t( )+ZOH Actuator

r kTs( )

Acquisition

process noise

measurementnoise

Z s( )v0 t( )

Sensor+Generator/controller

I discrete-time model from r (kTs) to v (kTs)I known input, noisy outputI well suited for prediction and control

System Identification in a NutshellChoice of the Experimental Setup: band-limited

i kTs( )

Acquisition

measurementnoise

Sensor++

v kTs( )

i0 t( )+Generator/ ZOHcontroller

r kTs( ) process noise

measurementnoise

Z s( )v0 t( )

Actuator

(anti-aliasfilter)

Acquisition

Sensor+

(anti-aliasfilter)

I continuous-time model from i (kTs) to v (kTs)I noisy input, noisy outputI well suited for physical interpretation

System Identification in a NutshellChoice of the Experimental Setup: band-limited

i kTs( )

Acquisition

measurementnoise

Sensor++

v kTs( )

i0 t( )+Generator/ ZOHcontroller

r kTs( ) process noise

measurementnoise

Z s( )v0 t( )

Actuator

(anti-aliasfilter)

Acquisition

Sensor+

(anti-aliasfilter)

I continuous-time model from i (kTs) to v (kTs)I noisy input, noisy outputI well suited for physical interpretation

System Identification in a NutshellChoice of the Experimental Setup: calibration

Gv s( )

vcal t( )

i0 t( )Generator/ ZOHcontroller Actuator

r kTs( )

Zcal s( )v0 t( )

Gi s( )

ical t( )

Vcal (jω)Ical (jω)

=Gv (jω)Gi (jω)

Zcal (jω)

System Identification in a NutshellChoice of the Experimental Setup: calibration

Gv s( )

vcal t( )

i0 t( )Generator/ ZOHcontroller Actuator

r kTs( )

Zcal s( )v0 t( )

Gi s( )

ical t( )

Vcal (jω)Ical (jω)

=Gv (jω)Gi (jω)

Zcal (jω)

System Identification in a NutshellChoice of the Experimental Setup: summary

Impedance spectroscopyI use a band-limited measurement setupI calibrate the setup (sensors and acquisition) with

a known impedanceI use continuous-time models

Outline

System Identification in a NutshellChoice of the Model

I ZOH measurement setup ⇒ discrete-time model

I BL measurement setup ⇒ continuous-time model

System Identification in a NutshellChoice of the Model: continuous-time

I linear time-invariant: (fractional) differential equationdifficulty: synthesis equivalent electrical network

• non-uniqueness• non-identifiability• lacking theory

I linear time-variant: differential equation with (periodically)time-varying coefficientsdifficulty: synthesis equivalent electrical network

I nonlinear: block oriented, nonlinear state space, neural nets,Gaussian processesdifficulty: no unique description, physical interpretation

I nonlinear time-varying: not mature yet

Outline

System Identification in a NutshellChoice of the Cost Function

The cost function

• measures the distance between the data and the model• sets the statistical properties of the estimated model• weighted nonlinear least squares, for example,

|Z (jωk)− Z (jωk , θ)|2

var (Z (jωk))

Result

• estimate θ• Cov(θ) due to noise and nonlinear distortion

System Identification in a NutshellChoice of the Cost Function

The cost function

• measures the distance between the data and the model• sets the statistical properties of the estimated model• weighted nonlinear least squares, for example,

|Z (jωk)− Z (jωk , θ)|2

var (Z (jωk))

Result

• estimate θ• Cov(θ) due to noise and nonlinear distortion

Outline

I Motivating ExamplesI System Identification in a NutshellI Nonparametric Impedance Spectroscopy

• Linear Time-Invariant• Linear Time-Variant• Summary

I Parametric Impedance SpectroscopyI Take Home Message

Outline

Nonparametric Impedance SpectroscopyLTI Framework: definition of the best linear time-invariant approximation

Nonlinear time-variant (NLTV) system

excited by a random phase multisine u(t)

Definition

GBLTI (jωk) =E

Y (k)U (k)

E|U (k)|2

Y (k)U (k)

with U(k) = DFT(u(t)), and E taken w.r.t. the randomrealisation of u(t)

Nonparametric Impedance SpectroscopyLTI Framework: definition of the best linear time-invariant approximation

Nonlinear time-variant (NLTV) system

excited by a random phase multisine u(t)

Definition

GBLTI (jωk) =E

Y (k)U (k)

E|U (k)|2

Y (k)U (k)

with U(k) = DFT(u(t)), and E taken w.r.t. the randomrealisation of u(t)

Nonparametric Impedance SpectroscopyLTI Framework: properties of the best linear time-invariant approximation

Property

is the best linear time-invariant (BLTI) approximation of the NLTVsystem

• Yres(k) is uncorrelated with – but not independent of – U(k)• Yres(k) has zero mean value

Conclusion

• Yres(k) acts as noise on the nonparametricfrequency response function (impedance) estimate

Property

is the best linear time-invariant (BLTI) approximation of the NLTVsystem

• Yres(k) is uncorrelated with – but not independent of – U(k)• Yres(k) has zero mean value

Conclusion

• Yres(k) acts as noise on the nonparametricfrequency response function (impedance) estimate

Assumption: the output is the sum of

• a nonlinear time-invariant part• a linear time-variant part

Assumption: the output is the sum of

• a nonlinear time-invariant part• a linear time-variant part

Properties

• YTV(k) and YS(k) are uncorrelated with – but notindependent of – U(k)

• YTV(k) and YS(k) are mutually uncorrelated• YTV(k) and YS(k) have zero mean value

Conclusion

• YTV(k) and YS(k) act as (frequency correlated) noiseon the FRF estimate

Properties

• YTV(k) and YS(k) are uncorrelated with – but notindependent of – U(k)

• YTV(k) and YS(k) are mutually uncorrelated• YTV(k) and YS(k) have zero mean value

Conclusion

• YTV(k) and YS(k) act as (frequency correlated) noiseon the FRF estimate

Nonparametric Impedance SpectroscopyLTI Framework: measuring the best linear time-invariant approximation

How to distinguish

• noise NY (k)• nonlinear distortion YS(k)• time-variation YTV(k)

for random phase multisines u(t)?

How to distinguish

• noise NY (k)• nonlinear distortion YS(k)• time-variation YTV(k)

for random phase multisines u(t)?

Key properties• ys(t) has the same periodicity as u(t)• yTV(t) depends linearly on u(t)• ys(t) and yTV(t) are uncorrelated

Nonparametric Impedance SpectroscopyLTI Framework: electrical circuit operating in open loop

Time-variant second order bandpass filter

u(t): input, random phase multisine, 100 mVrms,522 sinewaves in [230 Hz, 40 kHz]

y(t): outputp(t): gate voltage

Time-variant second order bandpass filter

u(t): input, random phase multisine, 100 mVrms,522 sinewaves in [230 Hz, 40 kHz]

y(t): outputp(t): gate voltage

Variation gate voltage over 2 multisine periods

0 5000 10000 15000

−830

−820

−810

−800

Samples

Scheduling signal

1 LTV experiment (black)3 LTI experiments (gray)

0 10 20 30 40−100

Frequency (kHz)

(jωk)

0 10 20 30 40−8.5

−8.4

−8.3

−8.2

Frequency (kHz)

Input BLTI Model

0 10 20 30 40−120

Frequency (kHz)

Output BLTI Model

Y (k) = GBLTI (jωk)U (k)+YTV(k)+YS(k)+NY (k)

Comparison with time-invariant experiments

0 10 20 30 40−100

Frequency (kHz)

Comparison LTV and LTI

0 10 20 30 40

−100

Frequency (kHz)

Comparison LTV and LTI

YS(k) NY (k)

Outline

Nonparametric Impedance SpectroscopyLTV Framework: time-variant FRF

y (t) =ˆ +∞

−∞g (t, τ) u (τ) dτ

y (t) =ˆ +∞

−∞g (t, τ) u (τ) dτ

Define

G (jω, t) =´ +∞−∞ g (t, t − τ) e−jωτdτ

Properties (Zadeh, 1950)

• Steady state response to u (t) = sin (ω0t)

y (t) = |G (jω0, t)| sin (ω0t + ∠G (jω0, t))

• Transient response

y (t) = L−1 G (s, t)U (s)

Define

G (jω, t) =´ +∞−∞ g (t, t − τ) e−jωτdτ

y (t) = L−1 G (s, t)U (s)

Define

G (jω, t) =´ +∞−∞ g (t, t − τ) e−jωτdτ

y (t) = L−1 G (s, t)U (s)

Nonparametric Impedance SpectroscopyLTV Framework: slowly time-varying

Series expansion time-variant FRF

G (jω, t) =∞∑

Gp (jω) fp (t)

with fp(t) a complete set of basis functions over [0,T ], for example,

• sines and cosines• polynomials

Constraints

f0(t) = 1 and 1T

´ T0 fp(t)dt = 0

Definition slowly time-varying

G (jω, t) =Nb∑p=0

Gp (jω) fp (t)

G (jω, t) =∞∑

Gp (jω) fp (t)

Constraints

f0(t) = 1 and 1T

´ T0 fp(t)dt = 0

Gp (jω) fp (t)

G (jω, t) =∞∑

Gp (jω) fp (t)

Constraints

f0(t) = 1 and 1T

´ T0 fp(t)dt = 0

Gp (jω) fp (t)

y (t) = L−1 G (s, t)U (s) =Nb∑p=0

L−1 Gp (s)U (s) fp (t)

... ...

y (t) = L−1 G (s, t)U (s) =Nb∑p=0

L−1 Gp (s)U (s) fp (t)

... ...

Nonparametric Impedance SpectroscopyLTV Framework: nonlinear slowly time-varying

... ...

linear slowly TV

... ...

NL Slowly TV

... ... ... ...

• ys(t) has the same periodicity as u(t)• yTV(t) depends linearly on u(t)

... ... ... ...

• ys(t) has the same periodicity as u(t)• yTV(t) depends linearly on u(t)

... ... ... ...• ys(t) has the same periodicity as u(t)• yTV(t) depends linearly on u(t)

Nonparametric Impedance SpectroscopyLTV Framework: estimating the time-variant FRF

... ... ... ...

u (t)→ y (t)

u (t)u1 (t)

...uNb (t)

→ y (t)

SISO LTV MISO LTI

Nonparametric Impedance SpectroscopyLTV Framework: estimating the time-variant FRF

... ... ... ...

u (t)→ y (t)

u (t)u1 (t)

...uNb (t)

→ y (t)

SISO LTV MISO LTI

Nonparametric Impedance SpectroscopyLTV Framework: measurement electrical circuit

Measurement time: 0.7 sSampling frequency: 156 kHzBandwidth excitation: [200 Hz, 40 kHz]Frequency resolution: 1.36 Hz

Estimate time-variant FRF with Nb = 9

... ... ... ...

Time-variant FRF G (jωk , t)

Top view time-variant FRF

0 0.2 0.4 0.6 0.8−2.2

−2.0

−1.8

−1.6

−1.4

Time (s)G

Nonparametric Impedance SpectroscopyLTV Framework: myocardial electrical impedance measurements

Random phase multisine current with 26 harmonics logarithmicallydistributed in the band [1 kHz, 939 kHz]

i t( )

v t( )Z jw t,( )

Electrodes

Zoom voltage and current spectra

Nonparametric estimate time-variant impedance

(frequency (Hz))

Z(jωk, t)

time (s)

Outline

Nonparametric Impedance SpectroscopySummary

Using 2 periods of the (transient) response to a random phasemultisine excitation

I BLTI approximation impedance• noise variance• variance nonlinear distortions• time-variant effects

I time-variant impedance• noise variance• variance nonlinear distortions

and this for periodic and arbitrary time-variations

Outline

I Motivating ExamplesI System Identification in a NutshellI Nonparametric Impedance SpectroscopyI Parametric Impedance Spectroscopy

• Linear Time-Invariant• Linear Time-Variant

I Take Home Message

Outline

I Take Home Message

Parametric Impedance SpectroscopyLTI Framework: cost functions

I Current controlled (galvanostatic) ⇒ impedance modeling∑k

|Z (jωk)− Z (jωk , θ)|2

var (Z (jωk))

I Voltage controlled (potentiostatic) ⇒ admittance modeling∑k

|Y (jωk)− Y (jωk , θ)|2

var (Y (jωk))

I Other ⇒ use the current and voltage spectra∑k

|V (k)− Z (jωk , θ) I (k)|2

var (V (k)− Z (jωk , θ) I (k))

|Z (jωk)− Z (jωk , θ)|2

var (Z (jωk))

|Y (jωk)− Y (jωk , θ)|2

var (Y (jωk))

|V (k)− Z (jωk , θ) I (k)|2

|Z (jωk)− Z (jωk , θ)|2

var (Z (jωk))

|Y (jωk)− Y (jωk , θ)|2

var (Y (jωk))

|V (k)− Z (jωk , θ) I (k)|2

Outline

I Take Home Message

Parametric Impedance SpectroscopyLTV Framework: cost function

Differential equation with time-varying coefficientsI piecewise polynomialI sum of sines and cosines

Cost function

|e (k , θ)|2

var (e (k , θ))

with e (k , θ) the DFT of the equation errorof the differential equation

Parametric Impedance SpectroscopyLTV Framework: cost function

Differential equation with time-varying coefficientsI piecewise polynomialI sum of sines and cosines

Cost function

|e (k , θ)|2

var (e (k , θ))

with e (k , θ) the DFT of the equation errorof the differential equation

Parametric Impedance SpectroscopyLTV Framework: myocardial electrical impedance measurements

i t( )

v t( )Z jw t,( )

Electrodes

Intracellular liquid

Extracellular

liquid

i t( )

v t( )Z jw t,( )

Electrodes

Intracellular liquid

Extracellular

liquid

(frequency (Hz))

Z(jωk, t)

time (s)

Synthesize for each t the RC -network

Black: actual circuit parameters; Red: periodic reconstruction

0 5 10105

time (s)

0 5 10

time (s)

0 5 106

time (s)

Outline

Take Home Message

Use random phase multisine excitations

I detect and quantify• the noise• the nonlinear behaviour• the time-variation

in impedance measurements

I measure the time-variant impedance for• periodic• arbitrary

time-variation and quantify• the noise• the nonlinear behaviour

Take Home Message

What can System Identification Offer toImpedance Spectroscopy?

References

Papers on the Properties of Linear Time-Variant SystemsL. A. Zadeh, “The determination of the impulsive response of variable networks,”Journal of Applied Physics, vol. 21, pp. 642-645, 1950.L. A. Zadeh, “Frequency analysis of variable networks,” Proceedings of the I.R.E., vol.38, pp. 291-299, 1950.L. A. Zadeh, “On stability of linear varying-parameter systems,” Journal of AppliedPhysics, vol. 22, no. 4, pp. 402-405, 1951.

Papers on the Nonparametric Estimation of the BLTI ApproximationJ. Lataire, E. Louarroudi, and R. Pintelon, “Detecting a time-varying behavior infrequency response function measurements,” IEEE Trans. Instrum. and Meas., vol.61, no. 8, pp. 2132-2143, 2012.R. Pintelon, E. Louarroudi, and J. Lataire, “Detection and quantification of theinfluence of time variation in frequency response function measurements using arbitraryexcitations,” IEEE Trans. Instrum. and Meas., vol. 61, no. 12, pp. 3387-3395, 2012.R. Pintelon, E. Louarroudi, and J. Lataire, “Detection and quantification of theinfluence of time-variation in closed loop frequency response function measurements,”IEEE Trans. Instrum. and Meas., vol. 62, no. 4, pp. 853-863, 2013.R. Pintelon, E. Louarroudi, and J. Lataire, “Detecting and quantifying the nonlinearand time-variant effects in FRF measurements using periodic excitations,” IEEE Trans.Instrum. and Meas., vol. 62, no. 12, pp. 3361-3373, 2013.

References

Paper on the Parametric Estimation of the BLTI ApproximationL. Ljung, “Estimating linear time-invariant models of nonlinear time-varying systems,”European Journal of Control , vol. 7, no. 2-3, pp. 203-219, 2001.

Papers on the Nonparametric Estimation of the Time-Variant FRFJ. Lataire, E. Louarroudi, and R. Pintelon, “Non-parametric estimate of the systemfunction of a time-varying system,” Automatica, vol. 48, no. 4, pp. 666-672, 2012.E. Louarroudi, R. Pintelon, and J. Lataire, “Nonparametric tracking of the time-varyingdynamics of weakly nonlinear periodically time-varying systems using periodic inputs,”IEEE Trans. Instrum. and Meas., vol. 61, no. 5, pp. 1384-1394, 2012.

Papers on the Parametric Estimation of the Time-Variant Transfer FunctionJ. Lataire, and R. Pintelon, “Frequency domain weighted nonlinear least squaresestimation of continuous-time, time-varying systems,” IET Control Theory &Applications, vol. 5, no. 7, pp. 923-933, 2011.E. Louarroudi et al., “Frequency domain, parametric estimation of the evolution of thetime-varying dynamics of periodically time-varying systems from noisy input–outputobservations ,” Mech. Systems and Sign. Proc., doi 10.1016/j.ymssp.2013.03.013.

whatcansystemidentiﬁcationoﬀerto impedancespectroscopy? · 0 10 20 30 40-100-50 0 frequency...

Documents