Parameter estimation with scarce measurements. (English) Zbl 1232.62043

Ding, Feng; Liu, Guangjun; Liu, Xiaoping P.

Automatica 47, No. 8, 1646-1655 (2011).

Summary: The problems of parameter estimation are addressed for systems with scarce measurements. A gradient-based algorithm is derived to estimate the parameters of the input-output representation with scarce measurements, and the convergence properties of the parameter estimation and unavailable output estimation are established using the Kronecker lemma and a deterministic version of the martingale convergence theorem. Finally, an example is provided to demonstrate the effectiveness of the proposed algorithm.

MSC:

62F10	Point estimation
93B30	System identification
62H12	Estimation in multivariate analysis
93E10	Estimation and detection in stochastic control theory
65C60	Computational problems in statistics (MSC2010)

Keywords:

recursive identification; stochastic gradient; missing data; multi-innovation identification

Software:

astsa

References:

[1]	Albertos, P.; Sanchis, R.; Sala, A., Output prediction under scarce data operation: control applications, Automatica, 35, 10, 1671-1681 (1999) · Zbl 0935.93058
[2]	Broersen, P. M.T.; de Waele, S.; Bos, R., Autoregressive spectral analysis when observations are missing, Automatica, 40, 9, 1495-1504 (2004) · Zbl 1055.93553
[3]	Ding, F., Several multi-innovation identification methods, Digital Signal Processing, 20, 4, 1027-1039 (2010)
[4]	Ding, F.; Chen, T., Parameter estimation of dual-rate stochastic systems by using an output error method, IEEE Transactions on Automatic Control, 50, 9, 1436-1441 (2005) · Zbl 1365.93480
[5]	Ding, F.; Chen, T., Performance bounds of forgetting factor least squares algorithm for time-varying systems with finite measurement data, IEEE Transactions on Circuits and Systems—I: Regular Papers, 52, 3, 555-566 (2005) · Zbl 1374.93390
[6]	Ding, F.; Chen, T., Performance analysis of multi-innovation gradient type identification methods, Automatica, 43, 1, 1-14 (2007) · Zbl 1140.93488
[7]	Ding, F.; Ding, J., Least squares parameter estimation for systems with irregularly missing data, International Journal of Adaptive Control and Signal Processing, 24, 7, 540-553 (2010) · Zbl 1200.93130
[8]	Ding, F.; Liu, X. P.; Liu, G., Auxiliary model based multi-innovation extended stochastic gradient parameter estimation with colored measurement noises, Signal Processing, 89, 10, 1883-1890 (2009) · Zbl 1178.94137
[9]	Ding, F.; Liu, G.; Liu, X. P., Partially coupled stochastic gradient identification methods for non-uniformly sampled systems, IEEE Transactions on Automatic Control, 55, 8, 1976-1981 (2010) · Zbl 1368.93121
[10]	Ding, F.; Liu, X. P.; Liu, G., Multi-innovation least squares identification for linear and pseudo-linear regression models, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 40, 3, 767-778 (2010)
[11]	Ding, F.; Liu, X. P.; Liu, G., Gradient based and least-squares based iterative identification methods for OE and OEMA systems, Digital Signal Processing, 20, 3, 664-677 (2010)
[12]	Ding, F.; Liu, X. P.; Liu, G., Identification methods for Hammerstein nonlinear systems, Digital Signal Processing, 21, 2, 215-238 (2011)
[13]	Ding, F.; Liu, X. P.; Yang, H. Z., Parameter identification and intersample output estimation for dual-rate systems, IEEE Transactions on Systems, Man & Cybernetics, Part A (Systems & Humans), 38, 4, 966-975 (2008)
[14]	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
[15]	Ding, J.; Shi, Y.; Wang, H. G.; Ding, F., A modified stochastic gradient based parameter estimation algorithm for dual-rate sampled-data systems, Digital Signal Processing, 20, 4, 1238-1249 (2010)
[16]	Gibson, S.; Ninness, B., Robust maximum-likelihood estimation of multivariable dynamic systems, Automatica, 41, 10, 1667-1682 (2005) · Zbl 1087.93054
[17]	Goodwin, G. C.; Sin, K. S., Adaptive filtering, prediction and control (1984), Prentice-Hall: Prentice-Hall Englewood Cliffs, New Jersey · Zbl 0653.93001
[18]	Gopaluni, R. B., A particle filter approach to identification of nonlinear processes under missing observations, Canadian Journal of Chemical Engineering, 86, 6, 1081-1092 (2008)
[19]	Han, L. L.; Ding, F., Multi-innovation stochastic gradient algorithms for multi-input multi-output systems, Digital Signal Processing, 19, 4, 545-554 (2009)
[20]	Isaksson, A. J., Identification of ARX-models subject to missing data, IEEE Transactions on Automatic Control, 38, 5, 813-819 (1993) · Zbl 0785.93028
[21]	Kim, J.; Stoffer, D. S., Fitting stochastic volatility models in the presence of irregular sampling via particle methods and the EM algorithm, Journal of Time Series Analysis, 29, 5, 811-833 (2008) · Zbl 1199.62042
[22]	Li, D.; Shah, S. L.; Chen, T., Identification of fast-rate models from multirate data, International Journal of Control, 74, 7, 680-689 (2001) · Zbl 1038.93017
[23]	Li, D.; Shah, S. L.; Chen, T.; Qi, K. Z., Application of dual-rate modeling to CCR octane quality inferential control, IEEE Transactions on Control Systems Technology, 11, 1, 43-51 (2003)
[24]	Liu, X. G.; Lu, J., Least squares based iterative identification for a class of multirate systems, Automatica, 46, 3, 549-554 (2010) · Zbl 1194.93079
[25]	Liu, Y. J.; Sheng, J.; Ding, R. F., Convergence of stochastic gradient algorithm for multivariable ARX-like systems, Computers & Mathematics with Applications, 59, 8, 2615-2627 (2010) · Zbl 1193.60057
[26]	Liu, Y. J.; Wang, D. Q.; Ding, F., Least-squares based iterative algorithms for identifying Box-Jenkins models with finite measurement data, Digital Signal Processing, 20, 5, 1458-1467 (2010)
[27]	Liu, Y. J.; Xiao, Y. S.; Zhao, X. L., Multi-innovation stochastic gradient algorithm for multiple-input single-output systems using the auxiliary model, Applied Mathematics and Computation, 215, 4, 1477-1483 (2009) · Zbl 1177.65095
[28]	Liu, Y. J.; Xie, L.; Ding, F., An auxiliary model 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), 223, 4, 445-454 (2009)
[29]	Liu, Y. J.; Yu, L.; Ding, F., Multi-innovation extended stochastic gradient algorithm and its performance analysis, Circuits, Systems, and Signal Processing, 29, 4, 649-667 (2010) · Zbl 1196.94026
[30]	Ljung, L., System identification: theory for the user (1999), Prentice-Hall: Prentice-Hall Englewood Cliffs, New Jersey
[31]	Pintelon, R.; Schoukens, J., Frequency domain system identification with missing data, IEEE Transactions on Automatic Control, 45, 2, 364-369 (2000) · Zbl 0970.93514
[32]	Raghavan, H.; Tangirala, A. K.; Gopaluni, R. B.; Shah, S. L., Identification of chemical processes with irregular output sampling, Control Engineering Practice, 14, 4, 467-480 (2006)
[33]	Sanchis, R.; Albertos, P., Recursive identification under scarce measurements: convergence analysis, Automatica, 38, 3, 535-544 (2002) · Zbl 1001.93084
[34]	Sanchis, R.; Peñarrocha, I.; Albertos, P., Design of robust output predictors under scarce measurements with time-varying delays, Automatica, 43, 2, 281-289 (2007) · Zbl 1111.93021
[35]	Shumway, R. H.; Stoffer, D. S., An approach to time series smoothing and forecasting using the EM algorithm, Journal of Time Series Analysis, 3, 4, 253-264 (1982) · Zbl 0502.62085
[36]	Shumway, R. H.; Stoffer, D. S., Time series analysis and its applications (2000), Springer · Zbl 0502.62085
[37]	Solo, V., The convergence of AML, IEEE Transactions on Automatic Control, 24, 6, 958-962 (1979) · Zbl 0422.62083
[38]	Wallin, R., Isaksson, A. J., & Ljung, L. (2000). An iterative method for identification of ARX models from incomplete data. In Proceedings of 39th IEEE conference on control and decision. CDC2000. 12-15 December 2000. Sydney, Australia; Wallin, R., Isaksson, A. J., & Ljung, L. (2000). An iterative method for identification of ARX models from incomplete data. In Proceedings of 39th IEEE conference on control and decision. CDC2000. 12-15 December 2000. Sydney, Australia
[39]	Wallin, R.; Isaksson, A. J.; Noréus, O., Extensions to output prediction under scarce data operation: control applications, Automatica, 37, 12, 2069-2071 (2001) · Zbl 1031.93145
[40]	Wang, D. Q.; Ding, F., Performance analysis of the auxiliary models based multi-innovation stochastic gradient estimation algorithm for output error systems, Digital Signal Processing, 20, 3, 750-762 (2010)
[41]	Wang, D. Q.; Ding, F., Input-output data filtering based recursive least squares parameter estimation for CARARMA systems, Digital Signal Processing, 20, 4, 991-999 (2010)
[42]	Wang, D. Q.; Ding, F., Least squares based and gradient based iterative identification for Wiener nonlinear systems, Signal Processing, 91, 5, 1182-1189 (2011) · Zbl 1219.94052
[43]	Wang, Z. D.; Ho, D. W.C.; Liu, Y. R.; Liu, X. H., Robust \(H\)-infinity control for a class of nonlinear discrete time-delay stochastic systems with missing measurements, Automatica, 45, 3, 684-691 (2009) · Zbl 1166.93319
[44]	Wei, G. L.; Wang, Z. D.; Shu, H. S., Robust filtering with stochastic nonlinearities and multiple missing measurements, Automatica, 45, 3, 836-841 (2009) · Zbl 1168.93407
[45]	Xie, L.; Liu, Y. J.; Yang, H. Z.; Ding, F., Modeling and identification for non-uniformly periodically sampled-data systems, IET Control Theory & Applications, 4, 5, 784-794 (2010)
[46]	Zhang, J. B.; Ding, F.; Shi, Y., Self-tuning control based on multi-innovation stochastic gradient parameter estimation, Systems & Control Letters, 58, 1, 69-75 (2009) · Zbl 1154.93040