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1

Nonparametric instrumental variables for identification of block-oriented systems

Mzyk G.

International Journal of Applied Mathematics and Computer Science

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2013

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Vol. 23, no. 3

521--537

EN

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.

2

Generalized kernel regression estimate for the identification of Hammerstein systems

Mzyk G.

International Journal of Applied Mathematics and Computer Science

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2007

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Vol. 17, no 2

189-197

EN

A modified version of the classical kernel nonparametric identification algorithm for nonlinearity recovering in a Hammerstein system under the existence of random noise is proposed. The assumptions imposed on the unknown characteristic are weak. The generalized kernel method proposed in the paper provides more accurate results in comparison with the classical kernel nonparametric estimate, regardless of the number of measurements. The convergence in probability of the proposed estimate to the unknown characteristic is proved and the question of the convergence rate is discussed. Illustrative simulation examples are included.

3

Zastosowanie metody zmiennych instrumentalnych do identyfikacji systemów Hammersteina-Wienera

Mzyk G.

Pomiary Automatyka Kontrola

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2001

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R. 47, nr 7/8

35--39

PL

W pracy rozpatruje się zastosowanie metody najmniejszych kwadratów i metody zmiennych instrumentalnych do estymacji parametrów blokowo zorientowanych, nieliniowych systemów dynamicznych. Dla systemu o strukturze Hammersteina-Wienera zaproponowano algorytm identyfikacji metodą zmiennych instrumentalnych i porównano go z algorytmem najmniejszych kwadratów. Udowodniono zgodność zaproponowanego estymatora, nawet dla przypadku występowania skorelowanych zakłóceń. Rozwiązano problem optymalnej generacji zmiennych instrumentalnych.

EN

Application of least squares and instrumental variables to recovering parameters of nonlinear complex dynamic block-oriented systems is examined. For a system with the Hammerstein-Wiener structure the instrumental variable algorithm is designed and compared with the least squares algorithm for estimating system parameters. The advantages of the proposed estimator are discussed and in particular its weak consistency, even in the presence of correlated noise is shown. The problem of generating optimal values of instruments is analysed.