zbMATH — the first resource for mathematics

Parameter estimation in stochastic grey-box models. (English) Zbl 1048.93088
Summary: An efficient and flexible parameter estimation scheme for grey-box models in the sense of discretely, partially observed Itô stochastic differential equations with measurement noise is presented along with a corresponding software implementation. The estimation scheme is based on the extended Kalman filter and features maximum likelihood as well as maximum a posteriori estimation on multiple independent data sets, including irregularly sampled data sets and data sets with occasional outliers and missing observations. The software implementation is compared to an existing software tool and proves to have better performance both in terms of quality of estimates for nonlinear systems with significant diffusion and in terms of reproducibility. In particular, the new tool provides more accurate and more consistent estimates of the parameters of the diffusion term.

93E10 Estimation and detection in stochastic control theory
93A30 Mathematical modelling of systems (MSC2010)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Full Text: DOI
[1] Allgöwer, F., & Zheng, A. (Eds.) (2000). Nonlinear model predictive control, Progress in systems & control theory, Vol. 26. Switzerland: Birkhauser Verlag.
[2] Åström, K.J., Introduction to stochastic control theory, (1970), Academic Press New York, USA · Zbl 0537.93048
[3] Bak, J.; Madsen, H.; Nielsen, H.A., Goodness of fit of stochastic differential equations, ()
[4] Bitmead, R.R.; Gevers, M.; Wertz, V., Adaptive optimal control—the thinking Man’s GPC, (1990), Prentice-Hall New York, USA · Zbl 0751.93052
[5] Bohlin, T. (2001). A grey-box process identification tool: Theory and practice. Technical Report IR-S3-REG-0103, Department of Signals, Sensors and Systems, Royal Institute of Technology, Stockholm, Sweden.
[6] Bohlin, T.; Graebe, S.F., Issues in nonlinear stochastic grey-box identification, International journal of adaptive control and signal processing, 9, 465-490, (1995)
[7] Dennis, J.E.; Schnabel, R.B., Numerical methods for unconstrained optimization and nonlinear equations, (1983), Prentice-Hall Englewood Cliffs, USA · Zbl 0579.65058
[8] Holst, J., Holst, U., Madsen, H., & Melgaard, H. (1992). Validation of grey box models. In L. Dugard, M. M’Saad, & I. D. Landau (Eds.), Selected papers from the fourth IFAC symposium on adaptive systems in control and signal processing (pp. 407-414). Oxford: Pergamon Press.
[9] Huber, P.J., Robust statistics, (1981), Wiley New York, USA · Zbl 0536.62025
[10] Jacobsen, J.L.; Madsen, H., Grey box modelling of oxygen levels in a small stream, Environmetrics, 7, 1, 109-121, (1996)
[11] Jazwinski, A.H., Stochastic processes and filtering theory, (1970), Academic Press New York, USA · Zbl 0203.50101
[12] Kloeden, P.E.; Platen, E., Numerical solution of stochastic differential equations, (1992), Springer Berlin, Germany · Zbl 0925.65261
[13] Kristensen, N. R., & Madsen, H. (2003). Continuous time stochastic modelling—CTSM 2.2. Technical University of Denmark, Lyngby, Denmark.
[14] Kristensen, N. R., Madsen, H., & Jørgensen, S. B. (2001). Computer aided continuous time stochastic process modelling. In R. Gani, & S. B. Jørgensen (Eds.), European symposium on computer aided process engineering, Vol. 11. Amsterdam: Elsevier.
[15] Kristensen, N. R., Madsen, H., & Jørgensen, S. B. (2002). Using continuous time stochastic modelling and nonparametric statistics to improve the quality of first principles models. In J. Grievink, & J. van Schijndel (Eds.), European symposium on computer aided process engineering, Vol. 12. Amsterdam: Elsevier.
[16] Kristensen, N. R., Madsen, H., & Jørgensen, S. B. (2004). A method for systematic improvement of stochastic grey-box models. Computers and Chemical Engineering (in press).
[17] Ljung, L., System identification: theory for the user, (1987), Prentice-Hall New York, USA · Zbl 0615.93004
[18] Madsen, H.; Holst, J., Estimation of continuous-time models for the heat dynamics of a building, Energy and buildings, 22, 67-79, (1995)
[19] Moler, C.; van Loan, C.F., Nineteen dubious ways to compute the exponential of a matrix, SIAM review, 20, 4, 801-836, (1978) · Zbl 0395.65012
[20] Nielsen, J.N.; Madsen, H., Applying the EKF to stochastic differential equations with level effects, Automatica, 37, 107-112, (2001) · Zbl 0982.93070
[21] Nielsen, J.N.; Madsen, H.; Young, P.C., Parameter estimation in stochastic differential equationsan overview, Annual reviews in control, 24, 83-94, (2000)
[22] Söderström, T.; Stoica, P., System identification, (1989), Prentice-Hall New York, USA · Zbl 0714.93056
[23] Unbehauen, H.; Rao, G.P., Continuous-time approaches to system identification—A survey, Automatica, 26, 1, 23-35, (1990) · Zbl 0714.93007
[24] Unbehauen, H.; Rao, G.P., A review of identification in continuous-time systems, Annual reviews in control, 22, 145-171, (1998) · Zbl 0934.93018
[25] van Loan, C.F., Computing integrals involving the matrix exponential, IEEE transactions on automatic control, 23, 3, 395-404, (1978) · Zbl 0387.65013
[26] Young, P.C., Parameter estimation for continuous-time models—A survey, Automatica, 17, 1, 23-39, (1981) · Zbl 0451.93052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.