Combined Parametric-Nonparametric Identification of by Grzegorz Mzyk

By Grzegorz Mzyk

This e-book considers an issue of block-oriented nonlinear dynamic approach identity within the presence of random disturbances. This classification of platforms contains a number of interconnections of linear dynamic blocks and static nonlinear components, e.g., Hammerstein method, Wiener method, Wiener-Hammerstein ("sandwich") process and additive NARMAX structures with suggestions. Interconnecting indications will not be available for dimension. The mixed parametric-nonparametric algorithms, proposed within the ebook, may be chosen dependently at the past wisdom of the procedure and indications. such a lot of them are in keeping with the decomposition of the complicated process identity job into less complicated neighborhood sub-problems through the use of non-parametric (kernel or orthogonal) regression estimation. within the parametric degree, the generalized least squares or the instrumental variables process is usually utilized to deal with correlated excitations. restrict houses of the algorithms were proven analytically and illustrated in basic experiments.

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For further discussion we refer to [91] and [92]. , provides the estimate cN0 ,M which converges (as M → ∞) to the true parameter vector c∗ provided that RM (u) is a consistent estimate of the regression function R(u). The following theorem refers to this property. 8. Assume that the computed cN0 ,M is unique and for each M , cN0 ,M ∈ C. 53) as M → ∞. Proof. 10. e. the estimate cN0 ,M converges to c∗ in the same sense and with the same guaranteed speed as RM (u) to R(u). 8. We confined ourselves to the convergence in probability as such particular type of convergence has been widely examined in the literature concerning non-parametric estimation of nonlinearities (regression functions) for Hammerstein systems (see [40], [48], [49]).

5). 9). e. the instruments ψk . The following theorem can be proved. 7. , vk−p are noise∗ free outputs of the system (Fig. e. 38) Proof. 9. 1 From now on the Euclidean norm will be denoted by · 2 to avoid ambiguity. 7 is only of theoretical value because of inaccessibility in the system of {wk } and {vk }. However it provides a guideline concerning the best choice of instruments, which can be used as a starting point for setting up a ’practical’ routine for synthesis of the instruments. 50) of wk ’s and vk,M ’s are nonparametric estimates of vk ’s calculated as F vk,M = γi,M wk−i,M i=0 with (see [48]) γi,M = κi,M /κ0,M , y= 1 M κi,M = M yk , k=1 u= 1 M 1 M M−i (yk+i − y)(uk − u), k=1 M uk , k=1 and F being a chosen ”cut-off level” of the infinite length impulse response {γi } of the linear dynamics in the Hammerstein system.

E. ∞ ωi = 2−i i=0 (cf. 4)). Random input and white noise processes {uk } and {εk } were generated according to the uniform distributions uk ∼ U [−5; 5] and εk ∼ U [−εmax; εmax ] (cf. 4) and vmax = wmax i=0 |γi | = 3wmax with wmax = maxuk ∈[−5;5] φT (uk )c (cf. 5)) is the magnitude of the noiseless output signal; in our experiment vmax = 165. 006ε2max , computed according to the rule recommended in [49] (see Section 8, p. 145 there). 75. 36 = (2M ) . e. N = (2M ) each number M of data the experiment was repeated P = 10 times, and accuracy of the estimates cN0 ,M and γN,M was evaluated using the average relative estimation error δθ (N, M ) = (1/P ) P p=1 (p) θˆN,M − θ 2 / θ 2 2 2 ·100%, where (p) θˆN,M is the estimate of θ ∈ {c, γ} obtained in the pth run, and · 2 is the Euclidean vector norm.

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