Skip to main content

Least Squares Parameter Estimation for Dynamic Processes

  • Chapter
  • First Online:
Identification of Dynamic Systems

Abstract

The application of the method of least squares to static models has been described in the previous chapter and is well known to scientists for a long time already. The application of the method of least squares to the identification of dynamic processes has been tackled with much later in time. First works on the parameter estimation of AR models have been reported in the analysis of time series of economic data (Koopmans, 1937; Mann and Wald, 1943) and for the difference equations of linear dynamic processes (Kalman, 1958; Durbin, 1960; Levin, 1960; Lee, 1964).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Albert R, Sittler RW (1965) A method for computing least squares estimators that keep up with the data. SIAM J Control Optim 3(3):384–417

    MATH  MathSciNet  Google Scholar 

  • Åström KJ, Bohlin T (1965) Numerical identification of linear dynamic systems from normal operating records. In: Proceedings of the IFAC Symposium Theory of Self-Adaptive Control Systems, Teddington

    Google Scholar 

  • Åström KJ, Eykhoff P (1971) System identification – a survey. Automatica 7(2):123–162

    Article  MATH  Google Scholar 

  • Baur U (1976) On-Line Parameterschätzverfahren zur Identifikation linearer, dynamischer Prozesse mit Prozeßrechnern: Entwicklung, Vergleich, Erprobung: KfK-PDV-Bericht Nr. 65. Kernforschungszentrum Karlsruhe, Karlsruhe

    Google Scholar 

  • Becker HP (1990) Beiträge zur rekursiven Parameterschätzung zeitvarianter Prozesse. Fortschr.-Ber. VDI Reihe 8 Nr. 203. VDI Verlag, Düsseldorf

    Google Scholar 

  • Bellmann R, Åström KJ (1970) On structural identifiability. Math Biosci 7(3–4):329–339

    Article  Google Scholar 

  • Bombois X, Anderson BDO, Gevers M (2005) Quantification of frequency domain error bounds with guaranteed confidence level in prediction error identification. Syst Control Lett 54(11):471–482

    Article  MATH  MathSciNet  Google Scholar 

  • Box GEP, Jenkins GM, Reinsel GC (2008) Time series analysis: Forecasting and control, 4th edn. Wiley Series in Probability and Statistics, John Wiley, Hoboken, NJ

    MATH  Google Scholar 

  • Burg JP (1968) A new analysis technique for time series data. In: Proceedings of NATO Advanced Study Institute on Signal Processing, Enschede

    Google Scholar 

  • Campi MC, Weyer E (2002) Finite sample properties of system identification methods. IEEE Trans Autom Control 47(8):1329–1334

    Article  MathSciNet  Google Scholar 

  • Deutsch R (1965) Estimation theory. Prentice-Hall, Englewood Cliffs, NJ

    MATH  Google Scholar 

  • van Doren JFM, Douma SG, van den Hof PMJ, Jansen JD, Bosgra OH (2009) Identifiability: From qualitative analysis to model structure approximation. In: Proceedings of the 15th IFAC Symposium on System Identification, Saint-Malo, France

    Google Scholar 

  • Durbin J (1960) Estimation of parameters in time-series regression models. J Roy Statistical Society B 22(1):139–153

    MATH  MathSciNet  Google Scholar 

  • Edward J, Fitelson M (1973) Notes on maximum-entropy processing (Corresp.). IEEE Trans Inf Theory 19(2):232–234

    Article  MATH  Google Scholar 

  • Eykhoff P (1974) System identification: Parameter and state estimation. Wiley-Interscience, London

    Google Scholar 

  • Gauss KF (1809) Theory of the motion of the heavenly bodies moving about the sun in conic sections: Reprint 2004. Dover phoenix editions, Dover, Mineola, NY

    Google Scholar 

  • Genin Y (1968) A note on linear minimum variance estimation problems. IEEE Trans Autom Control 13(1):103–103

    Article  Google Scholar 

  • Gevers M (2005) Identification for control: From early achievements to the revival of experimental design. Eur J Cont 2005(11):1–18

    MathSciNet  Google Scholar 

  • Goodwin GC, Sin KS (1984) Adaptive filtering, prediction and control. Prentice-Hall information and system sciences series, Prentice-Hall, Englewood Cliffs, NJ

    MATH  Google Scholar 

  • Heij C, Ran A, Schagen F (2007) Introduction to mathematical systems theory : linear systems, identification and control. Birkhäuser Verlag, Basel

    MATH  Google Scholar 

  • Hoerl AE, Kennard RW (1970a) Ridge regression: Application to nonorthogonal problems. Technometrics 12(1):69–82

    Article  MATH  MathSciNet  Google Scholar 

  • Hoerl AE, Kennard RW (1970b) Ridge regression: Biased estimation for nonorthogonal problems. Technometrics 12(1):55–67

    Article  MATH  MathSciNet  Google Scholar 

  • Isermann R (1974) Prozessidentifikation: Identifikation und Parameterschätzung dynamischer Prozesse mit diskreten Signalen. Springer, Heidelberg

    Google Scholar 

  • Isermann R (1987) Digitale Regelsysteme Band 1 und 2. Springer, Heidelberg

    Google Scholar 

  • Isermann R (1991) Digital control systems, 2nd edn. Springer, Berlin

    Google Scholar 

  • Isermann R (2005) Mechatronic Systems: Fundamentals. Springer, London

    Google Scholar 

  • Isermann R (2006) Fault-diagnosis systems: An introduction from fault detection to fault tolerance. Springer, Berlin

    Google Scholar 

  • Isermann R, Baur U (1974) Two-step process identification with correlation analysis and least-squares parameter estimation. J Dyn Syst Meas Contr 96:426–432

    Article  MATH  Google Scholar 

  • Johnston J, DiNardo J (1997) Econometric Methods: Economics Series, 4th edn. McGraw-Hill, New York, NY

    Google Scholar 

  • Kalman RE (1958) Design of a self-optimizing control system. Trans ASME 80:468–478

    Google Scholar 

  • Kendall MG, Stuart A (1977a) The advanced theory of statistics: Design and analysis, and time-series (vol. 3). Charles Griffin, London

    Google Scholar 

  • Kendall MG, Stuart A (1977b) The advanced theory of statistics: Inference and relationship (vol. 2). Charles Griffin, London

    Google Scholar 

  • Klinger A (1968) Prior information and bias in sequential estimation. IEEE Trans Autom Control 13(1):102–103

    Article  Google Scholar 

  • Koopmans TC (1937) Linear regression analysis of economic time series. Netherlands Economic Institute, Haarlem

    MATH  Google Scholar 

  • Lee KI (1964) Optimal estimation, identification, and control, Massachusetts Institute of Technology research monographs, vol 28. MIT Press, Cambridge, MA

    Google Scholar 

  • Levin MJ (1960) Optimum estimation of impulse response in the presence of noise. IRE Trans Circuit Theory 7(1):50–56

    Google Scholar 

  • Ljung L (1999) System identification: Theory for the user, 2nd edn. Prentice Hall Information and System Sciences Series, Prentice Hall PTR, Upper Saddle River, NJ

    Google Scholar 

  • Makhoul J (1975) Linear prediction: A tutorial review. Proc IEEE 63(4):561–580

    Article  Google Scholar 

  • Makhoul J (1976) Correction to “Linear prediction : A tutorial review”. Proc IEEE 64(2):285

    Article  Google Scholar 

  • Mann HB, Wald W (1943) On the statistical treatment of linear stochastic difference equations. Econometrica 11(3/4):173–220

    Article  MATH  MathSciNet  Google Scholar 

  • Mendel JM (1973) Discrete techniques of parameter estimation: The equation error formulation, Control Theory, vol 1. Marcel Dekker, New York

    Google Scholar 

  • Neumann D (1991) Fault diagnosis of machine-tools by estimation of signal spectra. In: Proceedings of the IFAC/IMACS Sympsoium on Fault Detection, Supervision, and Safety for Technical Processes SAFEPROCESS’91, Baden-Baden, Germany

    Google Scholar 

  • Neumann D, Janik W (1990) Fehlerdiagnose an spanenden Werkzeugmaschinen mit parametrischen Signalmodellen von Schwingungen. In: VDI-Schwingungstagung Mannheim, VDI-Verlag, Düsseldorf, Germany

    Google Scholar 

  • Ninness B, Goodwin GC (1995) Estimation of model quality. Automatica 31(12):32–74

    Article  MathSciNet  Google Scholar 

  • Pandit SM, Wu SM (1983) Time series and system analysis with applications. Wiley, New York

    MATH  Google Scholar 

  • Panuska V (1969) An adaptive recursive least squares identification algorithm. In: Proceedings of the IEEE Symposium in Adaptive Processes, Decision and Control

    Google Scholar 

  • Pintelon R, Schoukens J (2001) System identification: A frequency domain approach. IEEE Press, Piscataway, NJ

    Book  Google Scholar 

  • Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2007) Numerical recipes: The art of scientific computing, 3rd edn. Cambridge University Press, Cambridge, UK

    MATH  Google Scholar 

  • Sagara S, Wada K, Gotanda H (1979) On asymptotic bias of linear least squares estimator. In: Proceedings of the 5th IFAC Symposium on Identification and System Parameter Estimation Darmstadt, Pergamon Press, Darmstadt, Germany

    Google Scholar 

  • Scheuer HG (1973) Ein für den Prozessrechnereinsatz geeignetes Identifikationsverfahren auf der Grundlage von Korrelationsfunktionen. Dissertation. Universität Trier, Trier

    Google Scholar 

  • Staley RM, Yue PC (1970) On system parameter identifability. Inf Sci 2(2):127–138

    Article  MATH  MathSciNet  Google Scholar 

  • Stuart TA, Ord JK, Kendall MG (1987) Kendalls advanced theory of statistics: Distribution theory (vol. 1). Charles: Griffin Book

    Google Scholar 

  • Tikhonov AN (1995) Numerical methods for the solution of ill-posed problems, Mathematics and its applications, vol 328. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  • Tikhonov AN, Arsenin VY (1977) Solutions of ill-posed problems. Scripta series in mathematics, Winston, Washington, D.C.

    Google Scholar 

  • Tong H (1975) Autoregressive model fitting with noisy data by Akaike’s information criterion. IEEE Trans Inf Theory 21(4):476–480

    Article  MATH  Google Scholar 

  • Tong H (1977) More on autoregressive model fitting with noisy data by Akaike’s information criterion. IEEE Trans Inf Theory 23(3):409–410

    Article  MATH  Google Scholar 

  • Tse E, Anton J (1972) On the identifiability of parameters. IEEE Trans Autom Control 17(5):637–646

    Article  MATH  MathSciNet  Google Scholar 

  • Ulrych TJ, Bishop TN (1975) Maximum entropy spectral analysis and autoregressive decomposition. Rev Geophys 13(1):183–200

    Article  Google Scholar 

  • Vuerinckx R, Pintelon R, Schoukens J, Rolain Y (2001) Obtaining accurate confidence regions for the estimated zeros and poles in system identification problems. IEEE Trans Autom Control 46(4):656–659

    Article  MATH  MathSciNet  Google Scholar 

  • Weyer E, Campi MC (2002) Non-asymptotic confidence ellipsoids for the leastsquares estimate. Automatica 38(9):1539–1547

    Article  MATH  MathSciNet  Google Scholar 

  • Young P (1984) Recursive estimation and time-series analysis: An introduction. Communications and control engineering series, Springer, Berlin

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rolf Isermann .

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Berlin Heidelberg

About this chapter

Cite this chapter

Isermann, R., Münchhof, M. (2011). Least Squares Parameter Estimation for Dynamic Processes. In: Identification of Dynamic Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78879-9_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-78879-9_9

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-78878-2

  • Online ISBN: 978-3-540-78879-9

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics