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**Least-squares parameter estimation algorithm for a class of input nonlinear systems.**
*(English)*
Zbl 1251.62036

Summary: We study least-squares parameter estimation algorithms for input nonlinear systems, including the input nonlinear controlled autoregressive (IN-CAR) model and the input nonlinear controlled autoregressive autoregressive moving average (IN-CARARMA) model. The basic idea is to obtain linear-in-parameters models by overparameterizing such nonlinear systems and to use the least-squares algorithm to estimate the unknown parameter vectors. It is proved that the parameter estimates consistently converge to their true values under the persistent excitation condition. A simulation example is provided.

### MSC:

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

62F12 | Asymptotic properties of parametric estimators |

65C60 | Computational problems in statistics (MSC2010) |

### Keywords:

input nonlinear controlled autoregressive model; input nonlinear controlled autoregressive autoregressive moving average model
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\textit{W. Xiong} et al., J. Appl. Math. 2012, Article ID 684074, 14 p. (2012; Zbl 1251.62036)

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### References:

[1] | M. R. Zakerzadeh, M. Firouzi, H. Sayyaadi, and S. B. Shouraki, “Hysteresis nonlinearity identification using new Preisach model-based artificial neural network approach,” Journal of Applied Mathematics, Article ID 458768, 22 pages, 2011. · Zbl 1215.93037 |

[2] | X.-X. Li, H. Z. Guo, S. M. Wan, and F. Yang, “Inverse source identification by the modified regularization method on poisson equation,” Journal of Applied Mathematics, vol. 2012, Article ID 971952, 13 pages, 2012. · Zbl 1234.35313 |

[3] | Y. Shi and H. Fang, “Kalman filter-based identification for systems with randomly missing measurements in a network environment,” International Journal of Control, vol. 83, no. 3, pp. 538-551, 2010. · Zbl 1222.93228 |

[4] | 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 |

[5] | 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 |

[6] | 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 |

[7] | 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 |

[8] | J. Ding and F. Ding, “The residual based extended least squares identification method for dual-rate systems,” Computers & Mathematics with Applications, vol. 56, no. 6, pp. 1479-1487, 2008. · Zbl 1155.93435 |

[9] | L. Han and F. Ding, “Identification for multirate multi-input systems using the multi-innovation identification theory,” Computers & Mathematics with Applications, vol. 57, no. 9, pp. 1438-1449, 2009. · Zbl 1186.93076 |

[10] | 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 |

[11] | F. Ding, T. Chen, and Z. Iwai, “Adaptive digital control of Hammerstein nonlinear systems with limited output sampling,” SIAM Journal on Control and Optimization, vol. 45, no. 6, pp. 2257-2276, 2007. · Zbl 1126.93034 |

[12] | 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 |

[13] | 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 |

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

[15] | 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 |

[16] | D. Wang and F. Ding, “Extended stochastic gradient identification algorithms for Hammerstein-Wiener ARMAX systems,” Computers & Mathematics with Applications, vol. 56, no. 12, pp. 3157-3164, 2008. · Zbl 1165.65308 |

[17] | D. Wang, Y. Chu, G. Yang, and F. Ding, “Auxiliary model based recursive generalized least squares parameter estimation for Hammerstein OEAR systems,” Mathematical and Computer Modelling, vol. 52, no. 1-2, pp. 309-317, 2010. · Zbl 1201.93134 |

[18] | D. Wang, Y. Chu, and F. Ding, “Auxiliary model-based RELS and MI-ELS algorithm for Hammerstein OEMA systems,” Computers & Mathematics with Applications, vol. 59, no. 9, pp. 3092-3098, 2010. · Zbl 1193.93170 |

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

[20] | D. 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 |

[21] | W. Fan, F. Ding, and Y. Shi, “Parameter estimation for Hammerstein nonlinear controlled auto-regression models,” in Proceedings of the IEEE International Conference on Automation and Logistics, pp. 1007-1012, Jinan, China, August 2007. |

[22] | L. Wang, F. Ding, and P. X. Liu, “Convergence of HLS estimation algorithms for multivariable ARX-like systems,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1081-1093, 2007. · Zbl 1117.93332 |

[23] | G. C. Goodwin and K. S. Sin, Adaptive Filtering, Prediction and Control, Prentice-Hall, Englewood Cliffs, NJ, USA, 1984. · Zbl 0653.93001 |

[24] | 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 |

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

[26] | 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 |

[27] | L. Han and F. Ding, “Multi-innovation stochastic gradient algorithms for multi-input multi-output systems,” Digital Signal Processing, vol. 19, no. 4, pp. 545-554, 2009. |

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

[29] | D. 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. |

[30] | J. Zhang, F. Ding, and Y. Shi, “Self-tuning control based on multi-innovation stochastic gradient parameter estimation,” Systems & Control Letters, vol. 58, no. 1, pp. 69-75, 2009. · Zbl 1154.93040 |

[31] | F. Ding, H. Chen, and M. Li, “Multi-innovation least squares identification methods based on the auxiliary model for MISO systems,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 658-668, 2007. · Zbl 1114.93101 |

[32] | 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. |

[33] | F. Ding, P. X. Liu, and G. Liu, “Multiinnovation least-squares identification for system modeling,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 40, no. 3, Article ID 5299173, pp. 767-778, 2010. |

[34] | J. Ding, Y. Shi, H. Wang, and F. Ding, “A modified stochastic gradient based parameter estimation algorithm for dual-rate sampled-data systems,” Digital Signal Processing, vol. 20, no. 4, pp. 1238-1247, 2010. |

[35] | F. Ding, P. X. Liu, and H. Yang, “Parameter identification and intersample output estimation for dual-rate systems,” IEEE Transactions on Systems, Man, and Cybernetics A, vol. 38, no. 4, pp. 966-975, 2008. |

[36] | Y. Liu, D. 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. |

[37] | D. Wang and F. Ding, “Input-output data filtering based recursive least squares identification for CARARMA systems,” Digital Signal Processing, vol. 20, no. 4, pp. 991-999, 2010. |

[38] | 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. |

[39] | D. Wang, G. Yang, and R. Ding, “Gradient-based iterative parameter estimation for Box-Jenkins systems,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1200-1208, 2010. · Zbl 1201.94046 |

[40] | L. Xie, H. Yang, and F. Ding, “Recursive least squares parameter estimation for non-uniformly sampled systems based on the data filtering,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 315-324, 2011. · Zbl 1225.62120 |

[41] | F. Ding, Y. 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. |

[42] | F. Ding, “Hierarchical multi-innovation stochastic gradient algorithm for Hammerstein nonlinear system modeling,” Applied Mathematical Modelling. In press. · Zbl 1349.93391 |

[43] | 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 |

[44] | Y. Liu, L. Xie, and F. Ding, “An auxiliary model based on a recursive least-squares parameter estimation algorithm for non-uniformly sampled multirate systems,” Proceedings of the Institution of Mechanical Engineers. Part I: Journal of Systems and Control Engineering, vol. 223, no. 4, pp. 445-454, 2009. |

[45] | 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 |

[46] | 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 |

[47] | 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 |

[48] | F. Ding and T. Chen, “Performance bounds of forgetting factor least-squares algorithms for time-varying systems with finite meaurement data,” IEEE Transactions on Circuits and Systems. I. Regular Papers, vol. 52, no. 3, pp. 555-566, 2005. · Zbl 1374.93390 |

[49] | 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 |

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