By Grzegorz Mzyk
This booklet considers an issue of block-oriented nonlinear dynamic procedure id within the presence of random disturbances. This classification of platforms contains numerous interconnections of linear dynamic blocks and static nonlinear components, e.g., Hammerstein approach, Wiener approach, Wiener-Hammerstein ("sandwich") approach and additive NARMAX structures with suggestions. Interconnecting indications are usually not obtainable for size. The mixed parametric-nonparametric algorithms, proposed within the ebook, will be chosen dependently at the earlier wisdom of the procedure and signs. so much of them are in line with the decomposition of the advanced process identity job into easier neighborhood sub-problems through the use of non-parametric (kernel or orthogonal) regression estimation. within the parametric level, the generalized least squares or the instrumental variables procedure is often utilized to deal with correlated excitations. restrict homes of the algorithms were proven analytically and illustrated in basic experiments.
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Additional resources for Combined Parametric-Nonparametric Identification of Block-Oriented Systems
Example text
44], [90], [108]), admitting arbitrary input density, nonivertible characteristic and IIR linear block. Nevertheless, the rate of convergence of estimates is still not satisfying. 3 Sandwich (Wiener-Hammerstein) System In the Wiener-Hammerstein (sandwich) system, presented in Fig. 9, the nonlinear block μ() is surrounded by two linear dynamics {λj }∞ j=0 and {γi }∞ i=0 . j 0 yk Fig. 9 Wiener-Hammerstein (sandwich) system ∞ yk = ⎛ ∞ γi vk−i + zk , where vk = μ ⎝ i=0 ⎞ λj uk−j ⎠ . 49) j=0 Since both xk and vk cannot be measured, the system as a whole cannot be distinguished with the system composed with the elements { c11 λj }, c2 μ(c1 x), { c12 γi }, and the nonlinear characteristic can be identified only up to c1 and c2 .
20)). 1). Problem decomposition by application of mixed parametric-nonparametric approach allows to solve the identification task for a broad class of nonlinear characteristics including functions, which are not linear in the parameters. 1, where the combined parametric-nonparametric algorithm has been proposed for the identification of parameters appearing linearly in the static nonlinear element. In Stage 2 of the procedure, using the nonparametric estimates wn of wn , computed in Stage 1, we identify parameters of nonlinearity by minimization of appropriate empirical quadratic criterion function.
D. umax , some umax > 0, and there exists a bounded random process |uk | probability density of uk , say ν(u). 7. e. 17) where wmax is some unknown positive constant. 8. 18) i=0 with the infinite and unknown impulse response {γi }∞ i=0 (such that ∞). 9. , is a bounded stationary zero-mean white noise εmax ), independent of the input signal {uk }, and {ωi }∞ (Eεk = 0, |εk | i=0 ∞ is unknown; i=0 |ωi | < ∞. Hence the noise {zk } is a stationary zero-mean ∞ and bounded process |zk | zmax , where zmax = εmax i=0 |ωi |.