A-Posteriori Identifiability of the Maxwell Slip Model of Hysteresis

1. 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

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

3. 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)

4. 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]

5. 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

6. 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

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

8. 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

9. 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

10. 5. Results Noise Free Monte Carlo Estimations

11. 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!