×

Markov chain approach to identifying Wiener systems. (English) Zbl 1245.93134

Summary: Identification of the Wiener system composed of an infinite impulse response (IIR) linear subsystem followed by a static nonlinearity is considered. The recursive estimates for unknown coefficients of the linear subsystem and for the values of the nonlinear function at any fixed points are given by the stochastic approximation algorithms with expanding truncations (SAAWET). With the help of properties of the Markov chain connected with the linear subsystem, all estimates derived in the paper are proved to be strongly consistent. In comparison with the existing results on the topic, the method presented in the paper simplifies the convergence analysis and requires weaker conditions. A numerical example is given, and the simulation results are consistent with the theoretical analysis.

MSC:

93E12 Identification in stochastic control theory
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Zhu Y. Distillation column identification for control using Wiener model. In: Proceedings of American Control Conference, San Diego, 1999. 55: 3462–3466
[2] Kalafatis A, Arifin N, Wang L, et al. A new approach to the identification of pH processes based on the Wiener model. Chem Eng Sci, 1995, 50: 3693–3701 · doi:10.1016/0009-2509(95)00214-P
[3] Hunter I W, Korenberg M J. The identification of nonlinear biological systems: Wiener and Hammerstein cascade models. Bio Cybern, 1986, 55: 136–144 · Zbl 0611.92002
[4] Bai E W. Frequency domain identification of Wiener models. Automatica, 2003, 39: 1521–1530 · Zbl 1032.93010 · doi:10.1016/S0005-1098(03)00149-3
[5] Boyd S, Chua L O. Fading memory and the problem of approximating nonlinear operators with Volterra series. IEEE Trans Circ Syst, 1985, 32: 1150–1161 · Zbl 0587.93028 · doi:10.1109/TCS.1985.1085649
[6] Chen H F. Recursive identification for Wiener model with discontinuous piece-wise linear function. IEEE Trans Autom Control, 2006, 51: 390–400 · Zbl 1366.93670 · doi:10.1109/TAC.2005.864183
[7] Greblicki W. Nonparametric approach to Wiener system identification. IEEE Trans Circuits Syst-I: Fundam Theory Appl, 1997, 44: 538–545 · doi:10.1109/81.586027
[8] Hagenblad A, Ljung L, Wills A. Maximum likelihood identification of Wiener models. Automatica, 2008, 44: 2697–2705 · Zbl 1152.93508 · doi:10.1016/j.automatica.2008.02.016
[9] Hu X L, Chen H F. Strong consistence of recursive identification forWiener systems. Automatica, 2005, 41: 1905–1916 · Zbl 1087.93057 · doi:10.1016/j.automatica.2005.06.006
[10] Hu X L, Chen H F. Identification for Wiener systems with RTF subsystems. European J Control, 2006, 6: 581–594 · Zbl 1293.93752 · doi:10.3166/ejc.12.581-594
[11] Nordsjö A E, Zetterberg L H. Identification of certain time-varying nonlinear Wiener and Hammerstein systems. IEEE Trans Signal Process, 2001, 49: 577–592 · doi:10.1109/78.905884
[12] Verhaegen M, Westwick D. Identifying MIMO Wiener systems in the context of subspace model identificatin methods. Int J Control, 1996, 63: 331–349 · Zbl 0848.93014 · doi:10.1080/00207179608921846
[13] Vörös J. Parameter identification of Wiener systems with discontinuous nonlinearities. Syst Control Lett, 2001, 44: 363–372 · Zbl 0986.93020 · doi:10.1016/S0167-6911(01)00155-4
[14] Wigren T. Convergence analysis of recursive algorithms based on the nonlinear Wiener model. IEEE Trans Autom Control, 1994, 39: 2191–2206 · Zbl 0814.93074 · doi:10.1109/9.333765
[15] Chen H F, Guo L. Identification and Stochastic Adaptive Control. Boston: Birkhäuser, 1991 · Zbl 0747.93002
[16] Fan J Q, Yao Q. Nonlinear Time Series: Nonparametric and Parametric Approach. New York: Springer-Verlag, 2003 · Zbl 1014.62103
[17] Ljung L. System Identification: Theory for Users. Upper Saddle River: Prentice Hall, 1987 · Zbl 0615.93004
[18] Zhao W X, Chen H F, Zheng W X. Recursive identification for nonlinear ARX systems based on stochastic approximation algorithm. IEEE Trans Autom Control, 2010, 55: 1287–1299 · Zbl 1368.93754 · doi:10.1109/TAC.2010.2042236
[19] Bussgang J J. Crosscorrelation functions of amplitude-distorted Gaussian signals. Technical Report 216. MIT Research Laboratory of Electronics, 1952
[20] Song Q J, Chen H F. Identification of errors-in-variables systems with ARMA observation noise. Syst Control Lett, 2008, 57: 420–424 · Zbl 1139.93034 · doi:10.1016/j.sysconle.2007.10.010
[21] Ciarlet P G. Introduction to Numerical Linear Algebra and Optimisation. Cambridge: Cambridge University Press, 1989
[22] Meyn S P, Tweedie R L. Markov Chains and Stochastic Stability. London: Springer-Verlag, 1993
[23] Davydov Yu A. Mixing conditions for Markov chains. SIAM Probability Appl, 1973, 18: 312–328 · Zbl 0297.60031
[24] Nummelin E. General Irreducible Markov Chains and Non-negative Operators. Cambridge: Cambridge University Press, 1984 · Zbl 0551.60066
[25] Tong H. Nonlinear Time Series. Oxford: Oxford University Press, 1990 · Zbl 0716.62085
[26] Masry E, Györfi L. Strong consistency and rates for recursive probability density estimators of stationary processes. J Multivariate Analysis, 1987, 22: 79–93 · Zbl 0619.62079 · doi:10.1016/0047-259X(87)90077-7
[27] Chen H F. Stochastic Approximation and Its Applications. Dordrecht: Kluwer, 2002 · Zbl 1008.62071
[28] Song Q J, Chen H F. Identification of Wiener systems with internal noise. J Syst Sci Complex, 2008, 21: 378–393 · Zbl 1173.93034 · doi:10.1007/s11424-008-9120-z
[29] Hirschman I I, Widder D V. The Convolution Transform. Princeton, NJ: Princeton University Press, 1955 · Zbl 0065.09301
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.