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Static Systems Identification

  • Karel J. Keesman
Part of the Advanced Textbooks in Control and Signal Processing book series (C&SP)

Abstract

In Chap. 5 we start with the identification of static linear systems, that is, no dynamics are involved. The output of a static system depends only on the input at the same instant and thus shows instantaneous responses. In particular, the so-called least-squares method is introduced. As will be seen in Chaps. 5 and  6, the least-squares method for the static linear case forms the basis for solving nonlinear and dynamic estimation problems. For the analysis of the resulting estimates, properties like bias and accuracy are treated. Special attention is paid to errors-in-variables problems, which allow noise in both input and output variables, to maximum likelihood estimation as a unified approach to estimation, in particular well-defined in the case of normal distributions, and to bounded-noise problems for cases with small data sets.

Keywords

Singular Value Decomposition Sensitivity Matrix Total Little Square Nonlinear Regression Model Parameter Estimation Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [ADSC98]
    S. Audoly, L. D’Angiò, M.P. Saccomani, C. Cobelli, Global identifiability of linear compartmental models—a computer algebra algorithm. IEEE Trans. Biomed. Eng. 45(1), 36–47 (1998) CrossRefGoogle Scholar
  2. [And85]
    B.D.O. Anderson, Identification of scalar errors-in-variables models with dynamics. Automatica 21(6), 709–716 (1985) MathSciNetMATHCrossRefGoogle Scholar
  3. [Bar74]
    Y. Bard, Nonlinear Parameter Estimation (Academic Press, San Diego, 1974) MATHGoogle Scholar
  4. [Bjo96]
    A. Bjork, Numerical Methods for Least Squares Problems (SIAM, Philadelphia, 1996) Google Scholar
  5. [BK70]
    R. Bellman, K.J. Åström, On structural identifiability. Math. Biosci. 7, 329–339 (1970) CrossRefGoogle Scholar
  6. [Box71]
    M.J. Box, Bias in nonlinear estimation. J. R. Stat. Soc., Ser. B, Stat. Methodol. 33(2), 171–201 (1971) MathSciNetMATHGoogle Scholar
  7. [CGCE03]
    M.J. Chapman, K.R. Godfrey, M.J. Chappell, N.D. Evans, Structural identifiability for a class of non-linear compartmental systems using linear/non-linear splitting and symbolic computation. Math. Biosci. 183(1), 1–14 (2003) MathSciNetMATHCrossRefGoogle Scholar
  8. [CM78]
    F.L. Chernousko, A.A. Melikyan, Game Problems of Control and Search (Nauka, Moscow, 1978) (in Russian) Google Scholar
  9. [DK09]
    T.G. Doeswijk, K.J. Keesman, Linear parameter estimation of rational biokinetic functions. Water Res. 43(1), 107–116 (2009) CrossRefGoogle Scholar
  10. [DS98]
    N.R. Draper, H. Smith, Introduction to Linear Regression Analysis, 4th edn. Wiley Series in Probability and Statistics (Wiley, New York, 1998) Google Scholar
  11. [DvdH96]
    H.G.M. Dötsch, P.M.J. van den Hof, Test for local structural identifiability of high-order non-linearly parametrized state space models. Automatica 32(6), 875–883 (1996) MathSciNetMATHCrossRefGoogle Scholar
  12. [ECCG02]
    N.D. Evans, M.J. Chapman, M.J. Chappell, K.R. Godfrey, Identifiability of uncontrolled nonlinear rational systems. Automatica 38(10), 1799–1805 (2002) MathSciNetMATHCrossRefGoogle Scholar
  13. [FH82]
    E. Fogel, Y.F. Huang, On the value of information in system identification-bounded noise case. Automatica 18, 229–238 (1982) MathSciNetMATHCrossRefGoogle Scholar
  14. [GVL80]
    G.H. Golub, C.F. Van Loan, An analysis of the total least squares problem. SIAM J. Numer. Anal. 17(6), 883–893 (1980) MathSciNetMATHCrossRefGoogle Scholar
  15. [GVL89]
    G.H. Golub, C.F. Van Loan, Matrix Computations, 2nd edn. (Johns Hopkins University Press, Baltimore, 1989) MATHGoogle Scholar
  16. [GW74]
    K. Glover, J.C. Willems, Parametrizations of linear dynamical systems: canonical forms and identifiability. IEEE Trans. Autom. Control AC-19(6), 640–646 (1974) MathSciNetCrossRefGoogle Scholar
  17. [HS09]
    M. Hong, T. Söderström, Relations between bias-eliminating least squares, the Frisch scheme and extended compensated least squares methods for identifying errors-in-variables systems. Automatica 45(1), 277–282 (2009) MathSciNetMATHCrossRefGoogle Scholar
  18. [HSZ07]
    M. Hong, T. Söderström, W.X. Zheng, A simplified form of the bias-eliminating least squares method for errors-in-variables identification. IEEE Trans. Autom. Control 52(9), 1754–1756 (2007) CrossRefGoogle Scholar
  19. [Ips09]
    I. Ipsen, Numerical Matrix Analysis: Linear Systems and Least Squares (SIAM, Philadelphia, 2009) MATHGoogle Scholar
  20. [KD09]
    K.J. Keesman, T.G. Doeswijk, Direct least-squares estimation and prediction of rational systems: application to food storage. J. Process Control 19, 340–348 (2009) MATHCrossRefGoogle Scholar
  21. [Kee90]
    K.J. Keesman, Membership-set estimation using random scanning and principal component analysis. Math. Comput. Simul. 32(5–6), 535–544 (1990) MathSciNetCrossRefGoogle Scholar
  22. [Kee97]
    K.J. Keesman, Weighted least-squares set estimation from l norm bounded-noise data. IEEE Trans. Autom. Control AC 42(10), 1456–1459 (1997) MathSciNetCrossRefGoogle Scholar
  23. [Kee03]
    K.J. Keesman, Bound-based identification: nonlinear-model case, in Encyclopedia of Life Science Systems Article 6.43.11.2, ed. by H. Unbehauen. UNESCO EOLSS (2003) Google Scholar
  24. [KMVH03]
    A. Kukush, I. Markovsky, S. Van Huffel, Consistent estimation in the bilinear multivariate errors-in-variables model. Metrika 57(3), 253–285 (2003) MathSciNetGoogle Scholar
  25. [Koo37]
    T.J. Koopmans, Linear regression analysis of economic time series. The Netherlands (1937) Google Scholar
  26. [KS04]
    K.J. Keesman, R. Stappers, Nonlinear set-membership estimation: a support vector machine approach. J. Inverse Ill-Posed Probl. 12(1), 27–41 (2004) MathSciNetMATHCrossRefGoogle Scholar
  27. [Kur77]
    A.B. Kurzhanski, Control and Observation Under Uncertainty (Nauka, Moscow, 1977) (in Russian) Google Scholar
  28. [Lev64]
    M.J. Levin, Estimation of a system pulse transfer function in the presence of noise. IEEE Trans. Autom. Control 9, 229–335 (1964) CrossRefGoogle Scholar
  29. [MB82]
    M. Milanese, G. Belforte, Estimation theory and uncertainty intervals evaluation in presence of unknown but bounded errors. IEEE Trans. Autom. Control AC 27(2), 408–414 (1982) MathSciNetCrossRefGoogle Scholar
  30. [Mil95]
    M. Milanese, Properties of least-squares estimates in set membership identification. Automatica 31, 327–332 (1995) MathSciNetMATHCrossRefGoogle Scholar
  31. [MNPLE96]
    M. Milanese, J.P. Norton, H. Piet-Lahanier, E. Walter (eds.), Bounding Approaches to System Identification (Plenum, New York, 1996) MATHGoogle Scholar
  32. [MPV06]
    D.C. Montgomery, E.A. Peck, G.G. Vining, Introduction to Linear Regression Analysis, 4th edn. Wiley Series in Probability and Statistics (Wiley, New York, 2006) Google Scholar
  33. [MRCW01]
    G. Margaria, E. Riccomagno, M.J. Chappell, H.P. Wynn, Differential algebra methods for the study of the structural identifiability of rational function state-space models in the biosciences. Math. Biosci. 174(1), 1–26 (2001) MathSciNetMATHCrossRefGoogle Scholar
  34. [MV91a]
    M. Milanese, A. Vicino, Optimal estimation theory for dynamic systems with set membership uncertainty: an overview. Automatica 27(6), 997–1009 (1991) MathSciNetMATHCrossRefGoogle Scholar
  35. [MWDM02]
    I. Markovsky, J.C. Willems, B. De Moor, Continuous-time errors-in-variables filtering, in Proceedings of the IEEE Conference on Decision and Control, vol. 3 (2002), pp. 2576–2581 Google Scholar
  36. [Nor86]
    J.P. Norton, An Introduction to Identification (Academic Press, San Diego, 1986) MATHGoogle Scholar
  37. [Nor87]
    J.P. Norton, Identification and application of bounded-parameter models. Automatica 23(4), 497–507 (1987) MathSciNetMATHCrossRefGoogle Scholar
  38. [Nor03]
    J.P. Norton, Bound-based Identification: linear-model case, in Encyclopedia of Life Science Systems Article 6.43.11.2, ed. by H. Unbehauen. UNESCO EOLSS (2003) Google Scholar
  39. [NW82]
    V.V. Nguyen, E.F. Wood, Review and unification of linear identifiability concepts. SIAM Rev. 24(1), 34–51 (1982) MathSciNetCrossRefGoogle Scholar
  40. [OWG04]
    S. Ognier, C. Wisniewski, A. Grasmick, Membrane bioreactor fouling in sub-critical filtration conditions: a local critical flux concept. J. Membr. Sci. 229, 171–177 (2004) CrossRefGoogle Scholar
  41. [PC07]
    G. Pillonetto, C. Cobelli, Identifiability of the stochastic semi-blind deconvolution problem for a class of time-invariant linear systems. Automatica 43(4), 647–654 (2007) MathSciNetMATHCrossRefGoogle Scholar
  42. [PH05]
    R.L.M. Peeters, B. Hanzon, Identifiability of homogeneous systems using the state isomorphism approach. Automatica 41(3), 513–529 (2005) MathSciNetMATHCrossRefGoogle Scholar
  43. [SAD03]
    M.P. Saccomani, S. Audoly, L. D’Angiò, Parameter identifiability of nonlinear systems: The role of initial conditions. Automatica 39(4), 619–632 (2003) MathSciNetMATHCrossRefGoogle Scholar
  44. [Sch73]
    F.C. Schweppe, Uncertain Dynamic Systems (Prentice-Hall, New York, 1973) Google Scholar
  45. [SD98]
    W. Scherrer, M. Deistler, A structure theory for linear dynamic errors-in-variables models. SIAM J. Control Optim. 36(6), 2148–2175 (1998) MathSciNetMATHCrossRefGoogle Scholar
  46. [Söd07]
    T. Söderström, Errors-in-variables methods in system identification. Automatica 43(6), 939–958 (2007) MathSciNetMATHCrossRefGoogle Scholar
  47. [Söd08]
    T. Söderström, Extending the Frisch scheme for errors-in-variables identification to correlated output noise. Int. J. Adapt. Control Signal Process. 22(1), 55–73 (2008) MATHCrossRefGoogle Scholar
  48. [Sor80]
    H.W. Sorenson, Parameter Estimation (Dekker, New York, 1980) MATHGoogle Scholar
  49. [SSM02]
    T. Söderström, U. Soverini, K. Mahata, Perspectives on errors-in-variables estimation for dynamic systems. Signal Process. 82(8), 1139–1154 (2002) MATHCrossRefGoogle Scholar
  50. [vdH98]
    J.M. van den Hof, Structural identifiability of linear compartmental systems. IEEE Trans. Autom. Control 43(6), 800–818 (1998) MATHCrossRefGoogle Scholar
  51. [VGR89]
    S. Vajda, K.R. Godfrey, H. Rabitz, Similarity transformation approach to identifiability analysis of nonlinear compartmental models. Math. Biosci. 93(2), 217–248 (1989) MathSciNetMATHCrossRefGoogle Scholar
  52. [VHMVS07]
    S. Van Huffel, I. Markovsky, R.J. Vaccaro, T. Söderström, Total least squares and errors-in-variables modeling. Signal Process. 87(10), 2281–2282 (2007) MATHCrossRefGoogle Scholar
  53. [vS94]
    J.H. van Schuppen, Stochastic realization of a Gaussian stochastic control system. J. Acta Appl. Math. 35(1–2), 193–212 (1994) MATHCrossRefGoogle Scholar
  54. [Wal82]
    E. Walter, Identifiability of State Space Models. Lecture Notes in Biomathematics, vol. 46. (Springer, Berlin, 1982) MATHGoogle Scholar
  55. [Wal03]
    E. Walter, Bound-based Identification, in Encyclopedia of Life Science Systems Article 6.43.11.2, ed. by H. Unbehauen. UNESCO EOLSS (2003) Google Scholar
  56. [WP90]
    E. Walter, L. Pronzato, Qualitative and quantitative experiment design for phenomenological models—a survey. Automatica 26(2), 195–213 (1990) MathSciNetMATHCrossRefGoogle Scholar
  57. [You84]
    P.C. Young, Recursive Estimation and Time-series Analysis: An Introduction. Communications and Control Engineering (Springer, Berlin, 1984) MATHGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Systems and Control GroupWageningen UniversityWageningenNetherlands

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