##
**Least-squares-based iterative identification algorithm for Wiener nonlinear systems.**
*(English)*
Zbl 1266.93032

Summary: This paper focuses on the identification problem of Wiener nonlinear systems. The application of the key-term separation principle provides a simplified form of the estimated parameter model. To solve the identification problem of Wiener nonlinear systems with the unmeasurable variables in the information vector, the least-squares-based iterative algorithm is presented by replacing the unmeasurable variables in the information vector with their corresponding iterative estimates. The simulation results indicate that the proposed algorithm is effective.

### MSC:

93B30 | System identification |

PDF
BibTeX
XML
Cite

\textit{L. Zhou} et al., J. Appl. Math. 2013, Article ID 565841, 6 p. (2013; Zbl 1266.93032)

Full Text:
DOI

### References:

[1] | F. Ding, System Identification-New Theory and Methods, Science Press, Beijing, China, 2013. |

[2] | L. C. Zhou, X. L. Li, and F. Pan, “Gradient-based iterative identification for MISO Wiener nonlinear systems: application to a glutamate fermentation process,” Applied Mathematics Letters, 2013. · Zbl 1307.93438 |

[3] | D. Q. Wang and F. Ding, “Hierarchical least squares estimation algorithm for Hammerstein-Wiener systems,” IEEE Signal Processing Letters, vol. 19, no. 12, pp. 825-828, 2012. |

[4] | D. Q. Wang and F. Ding, “Least squares based and gradient based iterative identification for Wiener nonlinear systems,” Signal Processing, vol. 91, no. 5, pp. 1182-1189, 2011. · Zbl 1219.94052 |

[5] | J. Chen, “Gradient-based iterative algorithm for Wiener systems with saturation and dead-zone nonlinearities,” Journal of Vibration and Control, 2012. · Zbl 1236.93142 |

[6] | L. C. Zhou, X. L. Li, and F. Pan, “Gradient based iterative parameter identification for Wiener nonlinear systems,” Applied Mathematical Modelling, vol. 37, 2013. · Zbl 1307.93438 |

[7] | D. Q. Wang, “Least squares-based recursive and iterative estimation for output error moving average systems using data filtering,” IET Control Theory & Applications, vol. 5, no. 14, pp. 1648-1657, 2011. |

[8] | Y. Liu, Y. Xiao, and X. Zhao, “Multi-innovation stochastic gradient algorithm for multiple-input single-output systems using the auxiliary model,” Applied Mathematics and Computation, vol. 215, no. 4, pp. 1477-1483, 2009. · Zbl 1177.65095 |

[9] | Y. Liu, J. Sheng, and R. Ding, “Convergence of stochastic gradient estimation algorithm for multivariable ARX-like systems,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2615-2627, 2010. · Zbl 1193.60057 |

[10] | H. Q. Han, L. Xie, F. Ding, and X. Liub, “Hierarchical least squares based iterative identification for multivariable systems with moving average noises,” Mathematical and Computer Modelling, vol. 51, no. 9-10, pp. 1213-1220, 2010. · Zbl 1198.93216 |

[11] | Z. Zhang, F. Ding, and X. Liu, “Hierarchical gradient based iterative parameter estimation algorithm for multivariable output error moving average systems,” Computers & Mathematics with Applications, vol. 61, no. 3, pp. 672-682, 2011. · Zbl 1217.15022 |

[12] | Y. Liu and F. Ding, “Convergence properties of the least squares estimation algorithm for multivariable systems,” Applied Mathematical Modelling, vol. 37, no. 1-2, pp. 476-483, 2013. · Zbl 1349.74043 |

[13] | F. Ding, L. Qiu, and T. Chen, “Reconstruction of continuous-time systems from their non-uniformly sampled discrete-time systems,” Automatica, vol. 45, no. 2, pp. 324-332, 2009. · Zbl 1158.93365 |

[14] | J. Ding, F. Ding, X. P. Liu, and G. Liu, “Hierarchical least squares identification for linear SISO systems with dual-rate sampled-data,” IEEE Transactions on Automatic Control, vol. 56, no. 11, pp. 2677-2683, 2011. · Zbl 1368.93744 |

[15] | X. Liu and J. Lu, “Least squares based iterative identification for a class of multirate systems,” Automatica, vol. 46, no. 3, pp. 549-554, 2010. · Zbl 1194.93079 |

[16] | Y. Liu, F. Ding, and Y. Shi, “Least squares estimation for a class of non-uniformly sampled systems based on the hierarchical identification principle,” Circuits, Systems, and Signal Processing, vol. 31, no. 6, pp. 1985-2000, 2012. · Zbl 1269.93127 |

[17] | Y. J. Liu, D. Q. Wang, and F. Ding, “Least-squares based iterative algorithms for identifying Box-Jenkins models with finite measurement data,” Digital Signal Processing, vol. 20, no. 5, pp. 1458-1467, 2010. |

[18] | F. Ding, P. X. Liu, and G. Liu, “Identification methods for Hammerstein nonlinear systems,” Digital Signal Processing, vol. 21, no. 2, pp. 215-238, 2011. |

[19] | F. Ding, “Decomposition based fast least squares algorithm for output error systems,” Signal Processing, vol. 93, no. 5, pp. 1235-1242, 2013. |

[20] | F. Ding, X. G. Liu, and J. Chu, “Gradient-based and least-squares-based iterative algorithms for Hammerstein systems using the hierarchical identification principle,” IET Control Theory and Applications, vol. 7, 2013. |

[21] | D. Wang, R. Ding, and X. Dong, “Iterative parameter estimation for a class of multivariable systems based on the hierarchical identification principle and the gradient search,” Circuits, Systems, and Signal Processing, vol. 31, no. 6, pp. 2167-2177, 2012. |

[22] | F. Ding, “Two-stage least squares based iterative estimation algorithm for CARARMA system modeling,” Applied Mathematical Modelling, vol. 37, no. 7, pp. 4798-4808, 2013. · Zbl 1438.93228 |

[23] | F. Ding, Y. J. Liu, and B. Bao, “Gradient based and least squares based iterative estimation algorithms for multi-input multi-output systems,” Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, vol. 226, no. 1, pp. 43-55, 2012. |

[24] | F. Ding, P. X. Liu, and G. Liu, “Gradient based and least-squares based iterative identification methods for OE and OEMA systems,” Digital Signal Processing, vol. 20, no. 3, pp. 664-677, 2010. |

[25] | W. L. Xiong, J. X. Ma, and R. F. Ding, “An iterative numerical algorithm for modeling a class of Wiener nonlinear systems,” Applied Mathematics Letters, vol. 26, no. 4, pp. 487-493, 2013. · Zbl 1261.65068 |

[26] | M. M. Liu, Y. S. Xiao, and R. F. Ding, “Iterative identification algorithm for Wiener nonlinear systems using the Newton method,” Applied Mathematical Modelling, vol. 37, no. 9, pp. 6584-6591, 2013. · Zbl 1438.93229 |

[27] | F. Ding, G. Liu, and X. P. Liu, “Partially coupled stochastic gradient identification methods for non-uniformly sampled systems,” IEEE Transactions on Automatic Control, vol. 55, no. 8, pp. 1976-1981, 2010. · Zbl 1368.93121 |

[28] | F. Ding, G. Liu, and X. P. Liu, “Parameter estimation with scarce measurements,” Automatica, vol. 47, no. 8, pp. 1646-1655, 2011. · Zbl 1232.62043 |

[29] | J. Vörös, “Identification of nonlinear cascade systems with time-varying backlash,” Journal of Electrical Engineering, vol. 62, no. 2, pp. 87-92, 2011. |

[30] | F. Ding, Y. Shi, and T. Chen, “Auxiliary model-based least-squares identification methods for Hammerstein output-error systems,” Systems & Control Letters, vol. 56, no. 5, pp. 373-380, 2007. · Zbl 1130.93055 |

[31] | F. Ding and T. Chen, “Identification of Hammerstein nonlinear ARMAX systems,” Automatica, vol. 41, no. 9, pp. 1479-1489, 2005. · Zbl 1086.93063 |

[32] | F. Ding, Y. Shi, and T. Chen, “Gradient-based identification methods for Hammerstein nonlinear ARMAX models,” Nonlinear Dynamics, vol. 45, no. 1-2, pp. 31-43, 2006. · Zbl 1134.93321 |

[33] | F. Ding and T. Chen, “Combined parameter and output estimation of dual-rate systems using an auxiliary model,” Automatica, vol. 40, no. 10, pp. 1739-1748, 2004. · Zbl 1162.93376 |

[34] | F. Ding and T. Chen, “Identification of dual-rate systems based on finite impulse response models,” International Journal of Adaptive Control and Signal Processing, vol. 18, no. 7, pp. 589-598, 2004. · Zbl 1055.93018 |

[35] | F. Ding and J. Ding, “Least-squares parameter estimation for systems with irregularly missing data,” International Journal of Adaptive Control and Signal Processing, vol. 24, no. 7, pp. 540-553, 2010. · Zbl 1200.93130 |

[36] | F. Ding and T. Chen, “Parameter estimation of dual-rate stochastic systems by using an output error method,” IEEE Transactions on Automatic Control, vol. 50, no. 9, pp. 1436-1441, 2005. · Zbl 1365.93480 |

[37] | F. Ding, “Hierarchical multi-innovation stochastic gradient algorithm for Hammerstein nonlinear system modeling,” Applied Mathematical Modelling, vol. 37, no. 4, pp. 1694-1704, 2013. · Zbl 1349.93391 |

[38] | F. Ding and T. Chen, “Performance analysis of multi-innovation gradient type identification methods,” Automatica, vol. 43, no. 1, pp. 1-14, 2007. · Zbl 1140.93488 |

[39] | F. Ding, X. P. Liu, and G. Liu, “Auxiliary model based multi-innovation extended stochastic gradient parameter estimation with colored measurement noises,” Signal Processing, vol. 89, no. 10, pp. 1883-1890, 2009. · Zbl 1178.94137 |

[40] | F. Ding, “Several multi-innovation identification methods,” Digital Signal Processing, vol. 20, no. 4, pp. 1027-1039, 2010. |

[41] | Y. Liu, L. Yu, and F. Ding, “Multi-innovation extended stochastic gradient algorithm and its performance analysis,” Circuits, Systems, and Signal Processing, vol. 29, no. 4, pp. 649-667, 2010. · Zbl 1196.94026 |

[42] | D. Q. Wang and F. Ding, “Performance analysis of the auxiliary models based multi-innovation stochastic gradient estimation algorithm for output error systems,” Digital Signal Processing, vol. 20, no. 3, pp. 750-762, 2010. |

[43] | F. Ding, X. P. Liu, and G. Liu, “Multi-innovation least squares identification for linear and pseudo-linear regression models,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 40, no. 3, pp. 767-778, 2010. |

[44] | L. Xie, Y. J. Liu, H. Z. Yang, and F. Ding, “Modelling and identification for non-uniformly periodically sampled-data systems,” IET Control Theory & Applications, vol. 4, no. 5, pp. 784-794, 2010. |

[45] | Y. Zhang and G. Cui, “Bias compensation methods for stochastic systems with colored noise,” Applied Mathematical Modelling, vol. 35, no. 4, pp. 1709-1716, 2011. · Zbl 1217.93163 |

[46] | Y. Zhang, “Unbiased identification of a class of multi-input single-output systems with correlated disturbances using bias compensation methods,” Mathematical and Computer Modelling, vol. 53, no. 9-10, pp. 1810-1819, 2011. · Zbl 1219.93141 |

[47] | F. Ding, T. Chen, and L. Qiu, “Bias compensation based recursive least squares identification algorithm for MISO systems,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 53, no. 5, pp. 349-353, 2006. |

[48] | J. Ding and F. Ding, “Bias compensation-based parameter estimation for output error moving average systems,” International Journal of Adaptive Control and Signal Processing, vol. 25, no. 12, pp. 1100-1111, 2011. · Zbl 1263.93215 |

[49] | W. Wang, F. Ding, and J. Dai, “Maximum likelihood least squares identification for systems with autoregressive moving average noise,” Applied Mathematical Modelling, vol. 36, no. 5, pp. 1842-1853, 2012. · Zbl 1242.62105 |

[50] | J. Li and F. Ding, “Maximum likelihood stochastic gradient estimation for Hammerstein systems with colored noise based on the key term separation technique,” Computers & Mathematics with Applications, vol. 62, no. 11, pp. 4170-4177, 2011. · Zbl 1236.93150 |

[51] | J. Li, F. Ding, and G. Yang, “Maximum likelihood least squares identification method for input nonlinear finite impulse response moving average systems,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 442-450, 2012. · Zbl 1255.93147 |

[52] | J. Li, “Parameter estimation for Hammerstein CARARMA systems based on the Newton iteration,” Applied Mathematics Letters of Rapid Publication, vol. 26, no. 1, pp. 91-96, 2013. · Zbl 1255.65119 |

[53] | J. Li, R. Ding, and Y. Yang, “Iterative parameter identification methods for nonlinear functions,” Applied Mathematical Modelling, vol. 36, no. 6, pp. 2739-2750, 2012. · Zbl 1246.93114 |

[54] | F. Ding and T. Chen, “Hierarchical identification of lifted state-space models for general dual-rate systems,” IEEE Transactions on Circuits and Systems. I. Regular Papers, vol. 52, no. 6, pp. 1179-1187, 2005. · Zbl 1374.93342 |

[55] | F. Ding and T. Chen, “Hierarchical least squares identification methods for multivariable systems,” IEEE Transactions on Automatic Control, vol. 50, no. 3, pp. 397-402, 2005. · Zbl 1365.93551 |

[56] | W. Xiong, W. Fan, and R. Ding, “Least-squares parameter estimation algorithm for a class of input nonlinear systems,” Journal of Applied Mathematics, vol. 2012, Article ID 684074, 14 pages, 2012. · Zbl 1251.62036 |

[57] | X. L. Li, R. F. Ding, and L. C. Zhou, “Least squares based iterative identification algorithm for Hammerstein nonlinear systems with non-uniform sampling,” International Journal of Computer Mathematics, 2013. · Zbl 1272.93082 |

[58] | P. P. Hu and F. Ding, “Multistage least squares based iterative estimation for feedback nonlinear systems with moving average noises using the hierarchical identification principle,” Nonlinear Dynamics, vol. 52, 2013. · Zbl 1281.93033 |

[59] | F. Ding, “Coupled-least-squares identification for multivariable systems,” IET Control Theory and Applications, vol. 7, no. 1, pp. 68-79, 2013. |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.