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Download Combined Parametric-Nonparametric Identification of by Grzegorz Mzyk PDF

By Grzegorz Mzyk

This booklet considers an issue of block-oriented nonlinear dynamic method id within the presence of random disturbances. This type of platforms contains a variety of interconnections of linear dynamic blocks and static nonlinear parts, e.g., Hammerstein process, Wiener process, Wiener-Hammerstein ("sandwich") method and additive NARMAX platforms with suggestions. Interconnecting signs will not be available for dimension. The mixed parametric-nonparametric algorithms, proposed within the booklet, might be chosen dependently at the previous wisdom of the approach and indications. so much of them are in accordance with the decomposition of the complicated procedure 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 method is usually utilized to deal with correlated excitations. restrict homes of the algorithms were proven analytically and illustrated in uncomplicated experiments.

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Additional resources for Combined Parametric-Nonparametric Identification of Block-Oriented Systems

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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. Exemplary results of two-stage identification of the nonlinear static characteristic for M = 100 and M = 500 measurement data and N SR = 5%, along with the true characteristic and the ’data’ points 0 =4 {(un , wn,M )}N n=1 (bold-faced) computed in Stage 1 by the kernel method, are visualized for a single trial in Fig.

In Stage 2 (parametric), using the obtained estimates wk of wk , we identify the linear dynamic subsystem. The instrumental variables based method is introduced and analyzed for estimation of parameters of ARMA model. Both internal signals (inputs of the linear subsystem) and instruments are generated by the nonparametric regression estimation. The convergence rate is strictly proved and the problem of optimal, in the minimax sense, generation of instrumental variables is solved. 3, the effectiveness of the approach under incomplete a priori knowledge of the static nonlinearity is also illustrated in simulation examples.

8. 6. 33) as N, M → ∞, provided that N M −τ → 0. 34) Proof. 4 with obvious substitutions, and hence the proof is here omitted. 32) in general we have 0 < τ < 1/2, hence to fulfil the conditions (a’) and (b’) far more input-output data, M , must be used in Stage 1 for nonparametric estimation of interactions {wk } than the inputs (IV ) and instruments, N , for computation of the estimate θN,M in Stage 2. This can be explained by the necessity of proper reduction of estimation errors of wk and slower convergence of nonparametric estimates.

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