×

Structure and parameter identification for Bayesian Hammerstein system. (English) Zbl 1331.93215

Summary: In this paper, we consider the structure and parameter identification problem for Bayesian Hammerstein system. A structure identification algorithm is proposed, in which the system order, system parameters and regularization parameters are all unknown in the considered system. The joint posterior distribution of system parameters and the value of basis functions \(k\) are obtained via sampling theory. The proposed identification algorithm is based on the reversible jump Markov chain Monte Carlo method. There are two main characteristics of the algorithm: (i) By using the birth move and death move strategy, the parameter \(k\) is searched quickly in the number of basis functions space until the suitable value of \(k\) is found. (ii) The distributions of system parameters are changed with the value of \(k\) in the Bayesian framework, and the parameters are successfully found after the value of \(k\) is stable. Two examples are provided to show the effectiveness of the proposed algorithm. The performances of the algorithm are validated with the results of statistical analyses including parameter estimate error, MSE, NRMSE, MAD, Theils inequality coefficient, etc.

MSC:

93E12 Identification in stochastic control theory
93C10 Nonlinear systems in control theory
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ding, F., Peter Liu, X.P., Liu, G.J.: Identification methods for Hammerstein nonlinear systems. Dig. Signal Process. 21, 215-238 (2011) · doi:10.1016/j.dsp.2010.06.006
[2] Ding, F., Shi, Y., Chen, T.W.: Gradient-based identification methods for Hammerstein nonlinear ARMAX models. Nonlinear Dyn. 45(1), 31-43 (2005) · Zbl 1134.93321
[3] Li, J.H., Ding, F., Hua, L.: Maximum likelihood Newton recursive and the Newton iterative estimation algorithms for Hammerstein CARAR systems. Nonlinear Dyn. 75(1), 235-245 (2014) · Zbl 1281.93099 · doi:10.1007/s11071-013-1061-y
[4] Ding, J., Ding, F., Peter Liu, X.P., Liu, G.J.: Hierarchical least squares identification for linear SISO systems with dual-rate sampled-data. IEEE Trans. Autom. Control 56(11), 2677-2683 (2011) · Zbl 1368.93744 · doi:10.1109/TAC.2011.2158137
[5] Liu, X.G.: Least squares based iterative identification for a class of multirate systems. Automatica 46(3), 549-554 (2010) · Zbl 1194.93079 · doi:10.1016/j.automatica.2010.01.007
[6] Han, X.Q., Xie, L., Ding, F., Liu, X.G.: Hierarchical least-squares based iterative identification for multivariable systems with moving average noises. Math. Comput. Model. 51(9-10), 1213-1220 (2010) · Zbl 1198.93216 · doi:10.1016/j.mcm.2010.01.003
[7] Zhang, Z.N., Ding, F., Liu, X.G.: Hierarchical gradient based iterative parameter estimation algorithm for multivariable output error moving average systems. Comput. Math. Appl. 61(3), 672-682 (2011) · Zbl 1217.15022 · doi:10.1016/j.camwa.2010.12.014
[8] Greblicki, W.: Nonlinearity estimation in Hammerstein systems based on ordered observations. IEEE Trans. Signal Process. 44(5), 1224-1233 (1996) · doi:10.1109/78.502334
[9] Haber, R., Unbehauen, H.: Structure identification of nonlinear dynamic systems—a survey of input/output approaches. Automatica 26(4), 651-677 (1990) · Zbl 0721.93023 · doi:10.1016/0005-1098(90)90044-I
[10] Prakriya, S., Hatzinakos, D.: Blind identification of linear subsystems of LTI-ZMNL-LTI models with cyclostationary inputs. IEEE Trans. Signal Process. 45(8), 2023-2036 (1997) · doi:10.1109/78.611201
[11] Chaudhary, N.I., Raja, M.A.Z., Khan, J.A., Aslam, M.S.: Identification of input nonlinear control auto-regressive systems using fractional signal processing approach. Sci World J. doi:10.1155/2013/467276 (2013) · Zbl 1214.93115
[12] Sun, J.L., Liu, X.G.: A novel APSO-aided maximum likelihood identification method for Hammerstein systems. Nonlinear Dyn. 73(1-2), 449-462 (2013) · Zbl 1281.93035 · doi:10.1007/s11071-013-0800-4
[13] Narendra, K.S., Gallman, P.G.: An iterative method for the identification of nonlinear systems using a Hammerstein model. IEEE Trans. Autom. Control 11(3), 546-550 (1966) · doi:10.1109/TAC.1966.1098387
[14] Chang, F., Luus, R.: A noniterative method for identification using Hammerstein model. IEEE Trans. Autom. Control 16, 464-468 (1971) · doi:10.1109/TAC.1971.1099787
[15] Bai, E.: An optimal two-stage identification algorithm for Hammerstein-Wiener nonlinear systems. Automatica 34(3), 333-338 (1998) · Zbl 0915.93018 · doi:10.1016/S0005-1098(97)00198-2
[16] Chang, F., Luus, R.: A noniterative method for identification using Hammerstein model. IEEE Trans. Automat. Control 16(5), 464-468 (1971) · doi:10.1109/TAC.1971.1099787
[17] Bilings, S.A., Fakhouri, S.Y.: Identification of a class of nonlinear systems using correlation analysis. Proc. Inst. Elect. Eng. 125(7), 691-697 (1978) · doi:10.1049/piee.1978.0161
[18] Bai, E.W., Fu, M.: A blind approach to Hammerstein model identification. IEEE Trans. Signal Process. 50(7), 1610-1619 (2002) · doi:10.1109/TSP.2002.1011202
[19] Zahoor, R.M.A., Qureshi, I.M.: A modified least mean square algorithm using fractional derivative and its application to system identification. Eur. J. Sci. Res 35(1), 14-21 (2009)
[20] Aslam, M.S., Zahoor, R.M.A.: A new adaptive strategy to improve online secondary path modeling in active noise control systems using fractional signal processing approach. Signal Process. J. doi:10.1016/j.sigpro.2014.04.012 · Zbl 1364.93821
[21] Shah, S.M., Samar, R., Raja, M.A.Z., Chambers, J.A.: Fractional normalized filtered-error least mean squares algorithm for application in active noise control systems. Electron. Lett. 50(14), 973-975 (2014) · doi:10.1049/el.2014.1275
[22] Zahoor, R.M.A., Chaudhary, N.I.: Two-stage fractional least mean square identification algorithm for parameter estimation of CARMA systems. Signal Process. J. doi:10.1016/j.sigpro.2014.06.015
[23] Akaike, H.: A new look at the statistical model identification. IEEE Trans. Autom. Control 19, 716-723 (1974) · Zbl 0314.62039 · doi:10.1109/TAC.1974.1100705
[24] Rissanen, J.: Modeling by shortest data description. Automatica 14, 465-478 (1978) · Zbl 0418.93079 · doi:10.1016/0005-1098(78)90005-5
[25] Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6, 461-467 (1985) · Zbl 0379.62005 · doi:10.1214/aos/1176344136
[26] Ninness, B., Henriksen, S.: Bayesian system identification via Markov chain Monte Carlo techniques. Automatica 46, 40-51 (2010) · Zbl 1214.93115 · doi:10.1016/j.automatica.2009.10.015
[27] Baldacchino, T., Anderson, S.R., Kadirkamanathan, V.: Computational system identification for Bayesian NARMAX modelling. Automatica 49, 2641-2651 (2013) · Zbl 1364.93821 · doi:10.1016/j.automatica.2013.05.023
[28] redrik Lindstena, F., Schon, T.B., Jordanb, M.I.: Bayesian semiparametric Wiener system identification. Automatica 49, 2053-2063 (2013) · Zbl 1364.93831 · doi:10.1016/j.automatica.2013.03.021
[29] Robert, C.P., Ryde, T., Titterington, D.M.: Bayesian inference in hidden Markov models through jump Markov chain Monte Carlo. J. R. Stat. Soc. Ser. B 62(1), 57-75 (2000) · Zbl 0941.62090 · doi:10.1111/1467-9868.00219
[30] Bernardo, J.M., Smith, A.F.M.: Bayesian Theory. Wiley, New York (1994) · Zbl 0796.62002 · doi:10.1002/9780470316870
[31] Vermaak, J., Andrieu, C., Doucet, A., Godsill, S.J.: Reversible jump Markov chain Monte Carlo strategies for Bayesian model selection in autoregressive processes. J. Time Ser. Anal. 25(6), 785-809 (2004) · Zbl 1062.62206 · doi:10.1111/j.1467-9892.2004.00380.x
[32] Hastings, W.K.: Monte Carlo sampling methods using Markov Chains and their applications. Biometrika 57(1), 97-109 (1970) · Zbl 0219.65008 · doi:10.1093/biomet/57.1.97
[33] Ninness, B., Henriksen, S.: Bayesian system identification via Markov chain Monte Carlo techniques. Automatica 46(1), 40-51 (2010) · Zbl 1214.93115 · doi:10.1016/j.automatica.2009.10.015
[34] Vermaak, J., Andrieu, C., Doucet, A., Godsill, S.J.: Bayesian model selection of autoregressive processes. Technical Report. University of Cambridge, Department of Engineering, Cambridge, UK (2000) · Zbl 1062.62206
[35] Green, P.J.: Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82(4), 711-732 (1995) · Zbl 0861.62023
[36] Besag, J.: A candidate’s formula—a curious result in Bayesian prediction. Biometrika 76, 183 (1989) · Zbl 0664.62028 · doi:10.1093/biomet/76.1.183
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.