Nonparametric instrumental variables for identification of block-oriented systems

Grzegorz Mzyk

International Journal of Applied Mathematics and Computer Science, Tome 23 (2013), p. 521-537 / Harvested from The Polish Digital Mathematics Library

Résumé

A combined, parametric-nonparametric identification algorithm for a special case of NARMAX systems is proposed. The parameters of individual blocks are aggregated in one matrix (including mixed products of parameters). The matrix is estimated by an instrumental variables technique with the instruments generated by a nonparametric kernel method. Finally, the result is decomposed to obtain parameters of the system elements. The consistency of the proposed estimate is proved and the rate of convergence is analyzed. Also, the form of optimal instrumental variables is established and the method of their approximate generation is proposed. The idea of nonparametric generation of instrumental variables guarantees that the I.V. estimate is well defined, improves the behaviour of the least-squares method and allows reducing the estimation error. The method is simple in implementation and robust to the correlated noise.

Détail

Publié le : 2013-01-01

Zbl 1279.93103

EUDML-ID : urn:eudml:doc:262481

Bibliographie

[000] Bai, E. (1998). An optimal two-stage identification algorithm for Hammerstein-Wiener nonlinear systems, Automatica 34(3): 333-338. | Zbl 0915.93018

[001] Chen, S. and Billings, S. (1989). Representations of non-linear systems: The NARMAX model, International Journal of Control 49(3): 1013-1032. | Zbl 0674.93009

[002] Chow, Y. and Teicher, H. (2003). Probability Theory: Independence, Interchangeability, Martingales, Springer-Verlag, New York, NY. | Zbl 1049.60001

[003] Findeisen, W., Bailey, F., Brdyś, M., Malinowski, K., Tatjewski, P. and Woźniak, A. (1980). Control and Coordination in Hierarchical Systems, J. Wiley, Chichester/New York, NY. | Zbl 0534.93002

[004] Finigan, B. and Rowe, I. (1974). Strongly consistent parameter estimation by the introduction of strong instrumental variables, IEEE Transactions on Automatic Control 19(6): 825-830. | Zbl 0291.93060

[005] Giri, F. and Bai, E.W. (2010). Block-Oriented Nonlinear System Identification, Lecture Notes in Control and Information Sciences, Vol. 404, Springer, Berlin. | Zbl 1201.93004

[006] Greblicki, W. and Pawlak, M. (2008). Nonparametric System Identification, Cambridge University Press, New York, NY. | Zbl 1172.93001

[007] Haber, R. and Keviczky, L. (1999). Nonlinear System Identification: Input-Output Modeling Approach, Kluwer Academic Publishers, Dordrecht. | Zbl 0934.93004

[008] Hannan, E. and Deistler, M. (1988). The Statistical Theory of Linear Systems, John Wiley and Sons, New York, NY. | Zbl 0641.93002

[009] Hansen, L. and Singleton, K. (1982). Generalized instrumental variables estimation of nonlinear rational expectations models, Econometrica: Journal of the Econometric Society 50(5): 1269-1286. | Zbl 0497.62098

[010] Hasiewicz, Z. (1989). Applicability of least-squares to the parameter estimation of large-scale no-memory linear composite systems, International Journal of Systems Science 20(12): 2427-2449. | Zbl 0686.93087

[011] Hasiewicz, Z. and Mzyk, G. (2009). Hammerstein system identification by non-parametric instrumental variables, International Journal of Control 82(3): 440-455. | Zbl 1168.93412

[012] Hill, D. and Chong, C. (1989). Lyapunov functions of Lur'e-Postnikov form for structure preserving models of power systems, Automatica 25(3): 453-460. | Zbl 0684.93056

[013] Hill, D. and Mareels, I. (1990). Stability theory for differential/algebraic systems with application to power systems, IEEE Transactions on Circuits and Systems 37(11): 1416-1423. | Zbl 0717.93044

[014] Kincaid, D. and Cheney, E. (2002). Numerical Analysis: Mathematics of Scientific Computing, Vol. 2, American Mathematical Society, Pacific Grove, CA. | Zbl 0877.65002

[015] Kowalczuk, Z. and Kozłowski, J. (2000). Continuous-time approaches to identification of continuous-time systems, Automatica 36(8): 1229-1236. | Zbl 0953.93503

[016] Kudrewicz, J. (1976). Functional Analysis for Control and Electronics Engineers, PWN, Warsaw, (in Polish). | Zbl 0377.46005

[017] Lu, J. and Hill, D. (2007). Impulsive synchronization of chaotic Lur'e systems by linear static measurement feedback: An LMI approach, IEEE Transactions on Circuits and Systems II: Express Briefs 54(8): 710-714.

[018] Mzyk, G. (2007). Generalized kernel regression estimate for the identification of Hammerstein systems, International Journal of Applied Mathematics and Computer Science 17(2): 189-197, DOI: 10.2478/v10006-007-0018-z. | Zbl 1119.93416

[019] Mzyk, G. (2009). Nonlinearity recovering in Hammerstein system from short measurement sequence, IEEE Signal Processing Letters 16(9): 762-765.

[020] Mzyk, G. (2013). Instrumental variables for nonlinearity recovering in block-oriented systems driven by correlated signal, International Journal of Systems Science, DOI: 10.1080/00207721.2013.775682. | Zbl 1317.93259

[021] Rao, C. (1973). Linear Statistical Inference and Its Applications, Wiley, New York, NY. | Zbl 0256.62002

[022] Sagara, S. and Zhao, Z.-Y. (1990). Numerical integration approach to on-line identification of continuous-time systems, Automatica 26(1): 63-74. | Zbl 0715.93026

[023] Sastry, S. (1999). Nonlinear Systems: Analysis, Stability, and Control, Interdisciplinary Applied Mathematics, Vol. 10, Springer, New York, NY. | Zbl 0924.93001

[024] Söderström, T. and Stoica, P. (1983). Instrumental Variable Methods for System Identification, Vol. 161, Springer-Verlag, Berlin. | Zbl 0522.93003

[025] Söderström, T. and Stoica, P. (1989). System Identification, Prentice Hall, Englewood Cliffs, NJ. | Zbl 0695.93108

[026] Söderström, T. and Stoica, P. (2002). Instrumental variable methods for system identification, Circuits, Systems, and Signal Processing 21(1): 1-9. | Zbl 0522.93003

[027] Stoica, P. and Söderström, T. (1982). Instrumental-variable methods for identification of Hammerstein systems, International Journal of Control 35(3): 459-476. | Zbl 0479.93071

[028] Suykens, J., Yang, T. and Chua, L. (1998). Impulsive synchronization of chaotic Lur'e systems by measurement feedback, International Journal of Bifurcation and Chaos 8(06): 1371-1381. | Zbl 0936.93027

[029] Ward, R. (1977). Notes on the instrumental variable method, IEEE Transactions on Automatic Control 22(3): 482-484.

[030] Wong, K. and Polak, E. (1967). Identification of linear discrete time systems using the instrumental variable method, IEEE Transactions on Automatic Control 12(6): 707-718.

[031] Zhang, Y., Bai, E., Libra, R., Rowden, R. and Liu, H. (1996). Simulation of spring discharge from a limestone aquifer in Iowa, USA, Hydrogeology Journal 4(4): 41-54.

[032] Zhao, Z.-Y., Sagara, S. and Wada, K. (1991). Bias-compensating least squares method for identification of continuous-time systems from sampled data, International Journal of Control 53(2): 445-461. | Zbl 0725.93088