## Parameter estimation with scarce measurements.(English)Zbl 1232.62043

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)

astsa
Full Text:

### 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 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 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 (pp. 203-208). [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
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.