A general framework to approach the challenge of uncertainty propagation in model based prognostics is presented in this work. It is shown how the so-called Point Estimate Methods (PEMs) are ideally suited for this purpose because of the following reasons: 1) A credible propagation and representation of Gaussian (normally distributed) uncertainty can be done with a minimum of computational effort for non-linear applications. 2) Also non-Gaussian uncertainties can be propagated by evaluating suitable transfer functions inherently. 3) Confidence intervals of simulation results can be derived which do not have to be symmetrically distributed around the mean value by applying PEM in conjunction with the Cornish-Fisher expansion. 4) Moreover, the entire probability function of simulation results can be reconstructed efficiently by the proposed framework. The joint evaluation of PEM with the Polynomial Chaos expansion methodology is likely to provide good approximation results. Thus, non-Gaussian probability density functions can be derived as well. 5) The presented framework of uncertainty propagation is derivativefree, i.e. even non-smooth (non-differentiable) propagation problems can be tackled in principle. 6) Although the PEM is sample-based the overall method is deterministic. Computational results are reproducible which might be important to safety critical applications. - Consequently, the proposed approach may play an essential part in contributing to render the prognostics and health management into a more credible process. A given study of a generic uncertainty propagation problem supports this issue illustratively.
How to Cite
unscented transform, RUL uncertainty, uncertainty propagation, point estimate method
Breipohl, A. M. (1970). Probabilistic system analysis. JOHN WILEY & SONS, INC.
Daigle, M., & Goebel, K. (2010). Improving computational efficiency of prediction in model-based prognsotics using the unscented transformation. Procceedings of the Annual Conference of the Prognostics and Health Management Society.
Daigle, M., & Sankararaman, S. (2013). Advanced methods for determining prediction uncertainty in model-based prognostics with application to planetary rovers. Annual Conference of the Prognostics and Health Management Society.
Daigle, M., Saxena, A., & Goebel, K. (2012). An efficient deterministic approach to model-based prediction uncertainty estimation. Annual Conference of the Prognostics and Health Management Society.
Evans, D. H. (1967). An application of numerical integration techniques to statistical tolerancing. Technometrics, 9, 441-456.
Evans, D. H. (1974). Statistical tolerancing: The state of the art. Journal of Quality Technology, 6, 188-195.
Hines, W. W., Montgomery, D. C., Goldsman, D. M., & Borror, C. M. (2003). Probability and statistics in engineering. JOHN WILEY & SONS, INC.
Isukapalli, S. S. (1999). Uncertainty analysis of transporttransformation models. Unpublished doctoral dissertation, Rutgers, The State University of New Jersey.
Julier, S. J., & Uhlmann, J. K. (1994). A general method for approximating nonlinear transformations of probability distributions (Tech. Rep.). Dept. of Engineering Science, University of Oxford.
Julier, S. J., & Uhlmann, J. K. (2004). Unscented filtering and nonliner estimation. Proceedings of the IEEE, 92, 401-422.
Kay, S. M. (1993). Fundamentals of statistical signal processing: Estimation theory. Prentice Hall PTR.
Kulkarni, C. S., Biswas, G., Celaya, J. R., & Goebel, K. (2013). Physics based degradation models for electronic capacitor prognostics under thermal overstress conditions. International Journal of Prognostics and Health Management.
Lapira, E., Brisset, D., Davari, H., Siegel, D., & Lee, J. (2012). Wind turbine performance assessment using multi-regime modeling approach. Renewable Energy, 45, 86-95.
Lee, S. H., & Chen, W. (2007). A comparative study of uncertainty propagation methods for black-box type functions. In Asme 2007 international design engineering technical conferences & computers and information in engineering conference.
Lerner, U. N. (2002). Hybrid bayesian networks for reasoning about complex systems. Unpublished doctoral dissertation, Stanford University.
Maitre, O. P. L., & Knio, O. M. (2010). Spectral methods for uncertainty quantification. Springer.
Mandur, J., & Budman, H. (2012). A polynomial-chaos based algorithm for robust optimization in the presence of bayesian uncertainty. In 8th ifac symposium on advanced control of chemical processes.
Mattson, C. A., Anderson, T. V., Larson, B. J., & Fullwood, D. T. (2012). Efficient propagation of error through system models for functions common in engineering. Journal of Mechanical Design, 134.
Mekid, S., & Vaja, D. (2008). Propagation of uncertainty: Expressions of second and third order uncertainty with third and fourth moments. Measurement, 41, 600-609.
Saha, B., Goeble, K., Poll, S., & Christophersen, J. (2009). Prognostics methods for battery health monitoring using a bayesian framework. IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, 52(2).
Saltelli, A., Ratto, M., Tarantola, S., & Campolongo, F. (2005). Sensititivity analysis for chemical models. Chemical Reviews, 105, 28112828.
Sankararaman, S., Daigle, M., Saxena, A., & Goebel, K. (2013). Analytical algorithms to quantify the uncertainty in remaining useful life prediction. In Aerospace conference.
Sankararaman, S., & Goebel, K. (2013). Uncertainty quantification in remaining useful life of aerospace components using state space models and inverse form (Tech. Rep.). NASA Ames Research Center; Moffett Field, CA, United States.
Schenkendorf, R. (2014). Optimal experimental design for parameter identification and model selection. Unpublished doctoral dissertation, Otto-von-Guericke University, Magdeburg, Germany.
Sch¨oniger, A., Nowak, W., & Franssen, H.-J. H. (2012). Parameter estimation by ensemble kalman filters with transformed data: Approach and application to hydraulic tomography. Water Resources Research, 48.
Sobol’, I. M. (1993). Sensitivity analysis for nonlinear mathematical models. Mathematical Modeling and Computational Experiment, 1, 407–414.
Sobol’, I. M. (2001). Global sensitivity indices for nonlinear mathematical models and the monte carlo estimates. Ma, 55, 271–280.
Stengel, R. F. (1994). Optimal control and estimation. Dover Publications.
Templeton, B. A. (2009). A polynomial chaos approach to control. Unpublished master’s thesis, Virginia Polytechnic Institute ans State University.
Tyler, G. W. (1953). Numerical integration of functions of several variables. Canadian Jn. Math., 5, 393-412.
Usaola, J. (2009). Probabilistic load flow with wind production uncertainty using cumulants and cornish-fisher expansion. Electrical Power and Energy Systems, 31, 474-481.
Williard, N., He, W., Osterman, M., & Pecht, M. (2013). Comperative analysis of features for determining state of health in lithium-ion batteries. International Journal of Prognostics and Health Management.
Xue, J., & Ma, J. (2012). A comparative study of several taylor expansion methods on error propagation. In Geoinformatics, 2012 20th international conference on geoinformatics.
Zhang, J. (2006). The calculating formulae, and experimental methods in error propagation analysis. IEEE Transactions on Reliability, 55, 169-181.
Zhang, X., & Pisu, P. (2014). Prognostic-oriented fuel cell catalyst aging modeling and its application to healthmonitoring and prognostics of a pem fuel cell. International Journal of Prognostics and Health Management, 5.
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