Filtering based recursive least squares algorithm for Hammerstein FIR-MA systems. (English) Zbl 1281.93102

Summary: We consider the parameter estimation problem for Hammerstein finite impulse response (FIR) systems. An estimated noise transfer function is used to filter the input-output data of the Hammerstein system. By combining the key-term separation principle and the filtering theory, a recursive least squares algorithm and a filtering-based recursive least squares algorithm are presented. The proposed filtering-based recursive least squares algorithm can estimate the noise and system models. The given examples confirm that the proposed algorithm can generate more accurate parameter estimates and has a higher computational efficiency than the recursive least squares algorithm.


93E11 Filtering in stochastic control theory
93E24 Least squares and related methods for stochastic control systems
93E10 Estimation and detection in stochastic control theory
Full Text: DOI


[1] Li, J.H., Ding, F.: Maximum likelihood stochastic gradient estimation for Hammerstein systems with colored noise based on the key term separation technique. Comput. Math. Appl. 62(11), 4170-4177 (2011) · Zbl 1236.93150
[2] Li, J.H., Ding, F., Yang, G.W.: Maximum likelihood least squares identification method for input nonlinear finite impulse response moving average systems. Math. Comput. Model. 55(3-4), 442-450 (2012) · Zbl 1255.93147
[3] Hu, H.Y., Ding, F.: An iterative least squares estimation algorithm for controlled moving average systems based on matrix decomposition. Appl. Math. Lett. 25(12), 2332-2338 (2012) · Zbl 1252.65105
[4] Ding, F., Chen, T.: Identification of dual-rate systems based on finite impulse response models. Int. J. Adapt. Control Signal Process. 18(7), 589-598 (2004) · Zbl 1055.93018
[5] Ho, W.H., Chou, J.H., Guo, C.Y.: Parameter identification of chaotic systems using improved differential evolution algorithm. Nonlinear Dyn. 61(1-2), 29-41 (2010) · Zbl 1204.93034
[6] Chen, J., Zhang, Y., Ding, R.F.: Gradient-based parameter estimation for input nonlinear systems with ARMA noises based on the auxiliary model. Nonlinear Dyn. (2013). doi: 10.1007/s11071-013-0758-2 · Zbl 1284.93224
[7] Hu, P.P., Ding, F.: Multistage least squares based iterative estimation for feedback nonlinear systems with moving average noises using the hierarchical identification principle. Nonlinear Dyn. (2013). doi:10.1007/s11071-013-0812-0 · Zbl 1281.93033
[8] Narayanan, M.D., Narayanan, S., Padmanabhan, C.: Parametric identification of nonlinear systems using multiple trials. Nonlinear Dyn. 48(4), 341-360 (2007) · Zbl 1177.93031
[9] Silva, W.: Identification of nonlinear aeroelastic systems based on the Volterra theory: progress and opportunities. Nonlinear Dyn. 39(1-2), 25-62 (2005) · Zbl 1104.70014
[10] Dehghan, M., Hajarian, M.: Fourth-order variants of Newton’s method without second derivatives for solving non-linear equations. Eng. Comput. 29(4), 356-365 (2012)
[11] Dehghan, M., Hajarian, M.: Iterative algorithms for the generalized centro-symmetric and central anti-symmetric solutions of general coupled matrix equations. Eng. Comput. 29(5), 528-560 (2012)
[12] Li, J.H.: Parameter estimation for Hammerstein CARARMA systems based on the Newton iteration. Appl. Math. Lett. 26(1), 91-96 (2013) · Zbl 1255.65119
[13] Ding, F., Liu, X.P., Liu, G.: Identification methods for Hammerstein nonlinear systems. Digit. Signal Process. 21(2), 215-238 (2011)
[14] Chen, J., Wang, X.P., Ding, R.F.: Gradient based estimation algorithm for Hammerstein systems with saturation and dead-zone nonlinearities. Appl. Math. Model. 36(1), 238-243 (2012) · Zbl 1236.93142
[15] Wang, D.Q., Ding, F.: Least squares based and gradient based iterative identification for Wiener nonlinear systems. Signal Process. 91(5), 1182-1189 (2011) · Zbl 1219.94052
[16] Vazquez Feijoo, J.A., Worden, K., Rodriguez, N.J., Pozos Osorio, J., Matadamas Ortiz, P.: Analysis and control of nonlinear systems with DC terms. Nonlinear Dyn. 58(4), 753-775 (2009) · Zbl 1183.70069
[17] Fan, D., Lo, K.: Identification for disturbed MIMO Wiener systems. Nonlinear Dyn. 55(1-2), 31-42 (2009) · Zbl 1170.93375
[18] Thothadri, M., Casas, R.A., Moon, F.C., D’Andrea, R., Johnson, C.R. Jr.: Nonlinear system identification of multi-degree-of-freedom systems. Nonlinear Dyn. 32(3), 307-322 (2003) · Zbl 1062.70601
[19] Ding, F., Shi, Y., Chen, T.: Gradient-based identification methods for Hammerstein nonlinear ARMAX models. Nonlinear Dyn. 45(1-2), 31-43 (2006) · Zbl 1134.93321
[20] Ding, F., Shi, Y., Chen, T.: Auxiliary model based least-squares identification methods for Hammerstein output-error systems. Syst. Control Lett. 56(5), 373-380 (2007) · Zbl 1130.93055
[21] Bai, E.: An optimal two-stage identification algorithm for Hammerstein-Wiener nonlinear systems. Automatica 34(3), 333-338 (1998) · Zbl 0915.93018
[22] Ding, F., Chen, T.: Identification of Hammerstein nonlinear ARMAX systems. Automatica 41(9), 1479-1489 (2005) · Zbl 1086.93063
[23] Wang, D.Q., Chu, Y.Y., Yanga, G., Ding, F.: Auxiliary model-based recursive generalized least squares parameter estimation for Hammerstein OEAR systems. Math. Comput. Model. 52(1-2), 309-317 (2010) · Zbl 1201.93134
[24] Wang, D.Q., Ding, F.: Input-output data filtering based recursive least squares parameter estimation for CARARMA systems. Digit. Signal Process. 20(4), 991-999 (2010)
[25] Ahmad, M.S., Kukrer, O., Hocanin, A.: Recursive inverse adaptive filtering algorithm. Digit. Signal Process. 21(4), 491-496 (2011)
[26] Wang, D.Q.: Least squares-based recursive and iterative estimation for output error moving average systems using data filtering. IET Control Theory Appl. 5(14), 1648-1657 (2011)
[27] Xie, L., Yang, H.Z., Ding, F.: Recursive least squares parameter estimation for non-uniformly sampled systems based on the data filtering. Math. Comput. Model. 54(1-2), 315-324 (2011) · Zbl 1225.62120
[28] Yu, B., Shi, Y., Huang, H.: H_2 and H_infinity filtering for multirate systems using lifted models. Circuits Syst. Signal Process. 27(5), 699-711 (2008) · Zbl 1173.93360
[29] Xiao, Y.S., Yue, N.: Parameter estimation for nonlinear dynamical adjustment models. Math. Comput. Model. 54(5-6), 1561-1568 (2011) · Zbl 1228.93116
[30] Vörös, J.: Iterative algorithm for parameter identification of Hammerstein systems with two-segment nonlinearities. IEEE Trans. Autom. Control 44(11), 2145-2149 (1999) · Zbl 1136.93446
[31] Xiao, Y.S., Ding, F., Zhou, Y., Li, M., Dai, J.: On consistency of recursive least squares identification algorithms for controlled auto-regression models. Appl. Math. Model. 32(11), 2207-2215 (2008) · Zbl 1156.93411
[32] Gu, Y., Ding, F.: Auxiliary model based least squares identification method for a state space model with a unit time-delay. Appl. Math. Model. 36(12), 5773-5779 (2012) · Zbl 1349.93394
[33] Ding, F., Ding, J.: Least squares parameter estimation with irregularly missing data. Int. J. Adapt. Control Signal Process. 24(7), 540-553 (2010) · Zbl 1200.93130
[34] Shi, Y., Yu, B.: Robust mixed H2/H∞ control of networked control systems with random time delays in both forward and backward communication links. Automatica 47(4), 754-760 (2011) · Zbl 1215.93045
[35] Shi, Y., Yu, B.: Output feedback stabilization of networked control systems with random delays modeled by Markov chains. IEEE Trans. Autom. Control 54(7), 1668-1674 (2009) · Zbl 1367.93538
[36] Wang, W., Ding, F., Dai, J.Y.: Maximum likelihood least squares identification for systems with autoregressive moving average noise. Appl. Math. Model. 36(5), 1842-1853 (2012) · Zbl 1242.62105
[37] Liu, Y.J., Xie, L., Ding, F.: An auxiliary model based on a recursive least-squares parameter estimation algorithm for non-uniformly sampled multirate systems. Proc. Inst. Mech. Eng., Part I, J. Syst. Control Eng. 223(4), 445-454 (2009)
[38] Ding, F., Qiu, L., Chen, T.: Reconstruction of continuous-time systems from their non-uniformly sampled discrete-time systems. Automatica 45(2), 324-332 (2009) · Zbl 1158.93365
[39] Ding, F.: Coupled-least-squares identification for multivariable systems. IET Control Theory Appl. 7(1), 68-79 (2013)
[40] Ding, F., Liu, X.G., Chu, J.: Gradient-based and least-squares-based iterative algorithms for Hammerstein systems using the hierarchical identification principle. IET Control Theory Appl. 7(x) (2013). doi:10.1049/iet-cta.2012.0313 · Zbl 1170.93375
[41] Ding, F.: Two-stage least squares based iterative estimation algorithm for CARARMA system modeling. Appl. Math. Model. 37(7), 4798-4808 (2013) · Zbl 1438.93228
[42] Ding, F.: Decomposition based fast least squares algorithm for output error systems. Signal Process. 93(5), 1235-1242 (2013)
[43] Liu, Y.J., Sheng, J., Ding, R.F.: Convergence of stochastic gradient estimation algorithm for multivariable ARX-like systems. Comput. Math. Appl. 59(8), 2615-2627 (2010) · Zbl 1193.60057
[44] Ding, F., Liu, Y.J., Bao, B.: Gradient based and least squares based iterative estimation algorithms for multi-input multi-output systems. Proc. Inst. Mech. Eng., Part I, J. Syst. Control Eng. 226(1), 43-55 (2012)
[45] Ding, F., Chen, T.: Performance analysis of multi-innovation gradient type identification methods. Automatica 43(1), 1-14 (2007) · Zbl 1140.93488
[46] Liu, Y.J., Xiao, Y.S., Zhao, X.L.: Multi-innovation stochastic gradient algorithm for multiple-input single-output systems using the auxiliary model. Appl. Math. Comput. 215(4), 1477-1483 (2009) · Zbl 1177.65095
[47] Ding, F., Liu, X.P., Liu, G.: Auxiliary model based multi-innovation extended stochastic gradient parameter estimation with colored measurement noises. Signal Process. 89(10), 1883-1890 (2009) · Zbl 1178.94137
[48] Han, L.L., Ding, F.: Multi-innovation stochastic gradient algorithms for multi-input multi-output systems. Digit. Signal Process. 19(4), 545-554 (2009)
[49] Ding, F.: Several multi-innovation identification methods. Digit. Signal Process. 20(4), 1027-1039 (2010)
[50] Ding, F., Liu, X.P., Liu, G.: Multi-innovation least squares identification for linear and pseudo-linear regression models. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 40(3), 767-778 (2010)
[51] Ding, F., Liu, G., Liu, X.P.: Parameter estimation with scarce measurements. Automatica 47(8), 1646-1655 (2011) · Zbl 1232.62043
[52] Ding, F.: Hierarchical multi-innovation stochastic gradient algorithm for Hammerstein nonlinear system modeling. Appl. Math. Model. 37(4), 1694-1704 (2013) · Zbl 1349.93391
[53] Ding, F., Chen, T.: Hierarchical gradient-based identification of multivariable discrete-time systems. Automatica 41(2), 315-325 (2005) · Zbl 1073.93012
[54] Ding, F., Chen, T.: Hierarchical least squares identification methods for multivariable systems. IEEE Trans. Autom. Control 50(3), 397-402 (2005) · Zbl 1365.93551
[55] Ding, F., Chen, T.: Hierarchical identification of lifted state-space models for general dual-rate systems. IEEE Trans. Circuits Syst. I, Regul. Pap. 52(6), 1179-1187 (2005) · Zbl 1374.93342
[56] Han, H.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
[57] 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
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.