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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

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Talk outline 1.The Maxwell Slip Model Structure 2.The General Identification Problem 3.A-posteriori Identifiability 4.Discussion on the Conditions 5.Results 6.Conclusions

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1. Maxwell Slip Model Structure State Equations ( i=1,…,M): 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) Model parameters

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Stages Qualitative Experimental Design 2. The General Identification Problem Cost function : Cost function (M o known): Paper Contribution 1 st Stage: ε(t) = 0, M o known A – priori global identifiability ε(t) = 0 (Noise free data) [Rizos and Fassois, 2004] 2 nd Stage: ε(t) = 0, M o known Conditions on “Persistence” of excitation x(t) [Rizos and Fassois, 2004] 3 rd Stage: ε(t) = 0, Conditions for A – priori global distiguisability [to be submitted, 2011] Identification Stages 4 th Stage: M o known Consistency: A – posteriori global identifiability [Paper contribution] ε(t) (Noisy data) Stages Qualitative Experimental Design 5 th Stage: M o known Asymptotic variance and normality of the postulated estimator [to be submitted, 2011] 6 th Stage: Both unknown + noisy data A – posteriori global disguishability [to be submitted, 2011]

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3. A – posteriori identifiability Is the postulated estimator consistent?: ? Framework : 1.Uniform of Law of Large Numbers (ULLN) 2. is the identifiably unique minimizer of E: the Expectation operator [Pötcher and Prucha, 1997] [Ljung, 1997] [Bauer and Ninness, 2002]

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Identifiable uniqueness Framework 1.A – priori identifiability conditions D.D. Rizos and S.D. Fassois, Chaos “ Persistence” of excitation D.D. Rizos and S.D. Fassois, Chaos 2004, D.D. Rizos and S.D. Fassois, TAC 2011 – to be submitted

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Uniform of Law of Large Numbers (ULLN) 1.Compact parameter space 2.Pointwise Law of Large Numbers (LLN): 3.Lipschitz condition (Newey, Econometrica 1991) Framework

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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: Also the excitation is “persistent”. Then:, and bounded forth moments Identifiably uniqueness proved + Lemma Pötcher and Prucha, 1997 Identifiable uniqueness ULLN proved Newey Econometrica 1991 ULLN Lipschitz condition LLN Theorem 2.3 Ljung, 1997 Novel Contribution

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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): Δ1Δ1 Δ2Δ2 Δ3Δ3 Δ4Δ4 1 st : Remove Transient effects (necessary condition) 2 nd : Stick slip transitions (necessary condition)

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5. Results Noise Free Monte Carlo Estimations

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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.

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