# 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 - PowerPoint PPT Presentation

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A-Posteriori Identifiability of the Maxwell Slip Model of HysteresisDemosthenes D. RizosEMPA, Swiss Federal Laboratory of Material Testing and Research Duebendorf, Switzerland

Spilios D. FassoisDepartment of Mechanical Engineering and AeronauticsUniversity of Patras, Greece

Milano, 2011

1Talk outlineThe Maxwell Slip Model Structure

The General Identification Problem

A-posteriori Identifiability

Discussion on the Conditions

Results

Conclusions

1. Maxwell Slip Model Structure

State Equations ( i=1,,M):Output Equation

Advantages Simplicity Physical Interpretation Hysteresis with nonlocal memoryApplications 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)

Model parameters

Stages 1+2+3 Qualitative Experimental Design 2. The General Identification ProblemCost function :

Cost function (Mo known):

Paper Contribution1st Stage: (t) = 0 , Mo known A priori global identifiability

(t) = 0 (Noise free data)[Rizos and Fassois, 2004]2nd Stage: (t) = 0, Mo known Conditions on Persistence of excitation x(t)

[Rizos and Fassois, 2004]3rd Stage: (t) = 0, Conditions for A priori global distiguisability

[to be submitted, 2011]Identification Stages

4th Stage: Mo known Consistency: A posteriori global identifiability [Paper contribution](t) (Noisy data)Stages 1+2+3 Qualitative Experimental Design 5th Stage: Mo known Asymptotic variance and normality of the postulated estimator[to be submitted, 2011]

6th Stage: Both unknown + noisy data A posteriori global disguishability [to be submitted, 2011]3. A posteriori identifiability

Is the postulated estimator consistent?: ?Framework :

Uniform of Law of Large Numbers (ULLN)

is the identifiably unique minimizer of

E: the Expectation operator[Ptcher and Prucha, 1997][Ljung, 1997][Bauer and Ninness, 2002]Identifiable uniqueness

FrameworkA 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

6Uniform of Law of Large Numbers (ULLN)

Compact parameter space

Pointwise Law of Large Numbers (LLN):

Lipschitz condition

(Newey, Econometrica 1991)FrameworkProposition: Assume that the noise is subject to:

Also, let the model structure be known, the parameter space be compact and the actualsystem be subject to:

1.

2.

Also the excitation is persistent.

Then:

, and bounded forth moments

Identifiably uniqueness proved+Lemma 3.1 - Ptcher and Prucha, 1997Identifiableuniqueness ULLN provedNewey Econometrica 1991ULLNLipschitz conditionLLN

Theorem 2.3Ljung, 1997

Novel Contribution4. Discussion on the Conditions1. 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):12341st: Remove Transient effects(necessary condition)

2nd: Stick slip transitions(necessary condition)

5. Results

Noise Free Monte Carlo Estimations6. 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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