A-Posteriori Identifiability of the Maxwell Slip Model
of HysteresisDemosthenes D. Rizos
EMPA, Swiss Federal Laboratory of Material Testing and Research Duebendorf, Switzerland
Spilios D. FassoisDepartment of Mechanical Engineering and Aeronautics
University of Patras, Greece
Milano, 2011
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
1. Maxwell Slip Model Structure1. Maxwell Slip Model StructureState 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
Stages 1+2+3 Qualitative Experimental Design
2. The General Identification Problem2. The General Identification Problem
Cost function :
1st Stage: ε(t) = 0 , Mo known A – priori global identifiability
ε(t) = 0 (Noise free data)
[Rizos and Fassois, 2004]
2nd Stage: ε(t) = 0, Mo known Conditions on “Persistence” of excitation
x(t)
[Rizos and Fassois, 2004]
3rd Stage: ε(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]
ε(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 identifiability3. A – posteriori identifiabilityIs 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]
Identifiable uniquenessFramework
1. 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
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
Proposition: 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 momentsIdentifiably uniqueness
proved
+
Lemma 3.1 - Pötcher and Prucha, 1997
Identifiableuniqueness
ULLN proved
Newey Econometrica 1991
ULLN
Lipschitz condition
LLN
Theorem 2.3Ljung, 1997
Novel Contribution
4. Discussion on the Conditions4. 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):
Δ1
Δ2
Δ3
Δ4
1st: Remove Transient effects(necessary condition)
2nd: Stick slip transitions(necessary condition)
5. Results5. Results
Noise Free Monte Carlo Estimations
6. Conclusions6. 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!
