A posteriori identifiability of the maxwell slip model of hysteresis
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A-Posteriori Identifiability of the Maxwell Slip Model of Hysteresis. Demosthenes D. Rizos EMPA, Swiss Federal Laboratory of Material Testing and Research Duebendorf , Switzerland Spilios D. Fassois Department of Mechanical Engineering and Aeronautics University of Patras , Greece

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A-Posteriori Identifiability of the Maxwell Slip Model of Hysteresis

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A posteriori identifiability of the maxwell slip model of hysteresis

A-Posteriori Identifiability of the Maxwell Slip Model of Hysteresis

Demosthenes D. Rizos

EMPA, Swiss Federal Laboratory of Material Testing and Research Duebendorf, Switzerland

Spilios D. Fassois

Department of Mechanical Engineering and Aeronautics

University of Patras, Greece

Milano, 2011


Talk outline

Talk outline

The Maxwell Slip Model Structure

The General Identification Problem

A-posteriori Identifiability

Discussion on the Conditions

Results

Conclusions


A posteriori identifiability of the maxwell slip model of hysteresis

State Equations ( i=1,…,M):

1. Maxwell Slip Model Structure

Model parameters

Output Equation

  • Advantages

  • Simplicity

  • Physical Interpretation

  • Hysteresis with nonlocal memory

  • Applications

  • Friction (Lampaert et al. 2002; Parlitz et al. 2004, Rizos and Fassois 2004, Worden et al. 2007, Padthe et al. 2008)

  • PZT stack actuators (Goldfarb and Celanovic 1997, Choi et al. 2002, Georgiou and Ben Mrad, 2006)

  • Characterization of materials (Zhang et al. 2011)


A posteriori identifiability of the maxwell slip model of hysteresis

2. The General Identification Problem

ε(t) (Noisy data)

ε(t) = 0 (Noise free data)

x(t)

Cost function (Mo known):

Cost function :

Identification Stages

Stages 1+2+3 Qualitative Experimental Design

[Rizos and Fassois, 2004]

1st Stage:ε(t) = 0 , Moknown  A – priori global identifiability

Paper Contribution

2nd Stage:ε(t) = 0, Moknown Conditions on  “Persistence” of excitation

[Rizos and Fassois, 2004]

3rd Stage:ε(t) = 0,  Conditions for  A – priori global distiguisability

[to be submitted, 2011]

Stages 1+2+3 Qualitative Experimental Design

[Paper contribution]

4th Stage:Moknown Consistency:  A – posteriori global identifiability

6th Stage:Both unknown + noisy data  A – posteriori global disguishability

5th Stage:Moknown Asymptotic variance and normality of the postulated estimator

[to be submitted, 2011]

[to be submitted, 2011]


A posteriori identifiability of the maxwell slip model of hysteresis

3. A – posteriori identifiability

?

Is the postulated estimator consistent?:

[Pötcher and Prucha, 1997]

[Ljung, 1997]

[Bauer and Ninness, 2002]

  • Framework:

  • Uniform of Law of Large Numbers (ULLN)

  • is the identifiably unique minimizer of

E: the Expectation operator


A posteriori identifiability of the maxwell slip model of hysteresis

Framework

  • Identifiable uniqueness

A – priori identifiability conditions

D.D. Rizos and S.D. Fassois, Chaos 2004

2. “Persistence” of excitation

D.D. Rizos and S.D. Fassois, Chaos 2004,

D.D. Rizos and S.D. Fassois, TAC 2011 – to be submitted


A posteriori identifiability of the maxwell slip model of hysteresis

Framework

  • Uniform of Law of Large Numbers (ULLN)

(Newey, Econometrica 1991)

Compact parameter space

Pointwise Law of Large Numbers (LLN):

Lipschitz condition


A posteriori identifiability of the maxwell slip model of hysteresis

Novel Contribution

Proposition: Assume that the noise is subject to:

Also, let the model structure be known, the

parameter space be compact and the actual

system be subject to:

1.

2.

Also the excitation is “persistent”.

Then:

ULLN proved

Theorem 2.3

Ljung, 1997

Newey

Econometrica 1991

LLN

+

Identifiablyuniqueness

proved

, and bounded forth moments

ULLN

Identifiableuniqueness

Lipschitz condition

Lemma 3.1 - Pötcher and Prucha, 1997


A posteriori identifiability of the maxwell slip model of hysteresis

4. Discussion on the Conditions

1. Compactness (not necessary condition)

2. ,

(necessary condition – lost of the a-priori identifiability)

3. Noise assumptions (not necessary condition – but rather mild)

4. “Persistence” of excitation (The excitation should invoke the following):

2nd: Stick slip transitions

(necessary condition)

1st: Remove Transient effects

(necessary condition)

Δ4

Δ3

Δ2

Δ1


A posteriori identifiability of the maxwell slip model of hysteresis

5. Results

Noise Free Monte Carlo Estimations


A posteriori identifiability of the maxwell slip model of hysteresis

6. Conclusions

  • The consistency of a postulated output-error estimator for identifying the Maxwell Slip model has been addressed.

  • The Maxwell Slip model is a – posteriori global identifiable under

  • “almost minimal” and mild conditions.

Thank you for your attention!


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