Nonparametric identification of nonlinearities in block-oriented systems by orthogonal wavelets with compact support

On-line wavelet estimation of Hammerstein system nonlinearity

“…Therefore, the algorithm replicates the asymptotic behavior of the aforementioned on-line nonparametric algorithms (and hence possesses the same limit properties as their off-line prototypes, cf. the results of Greblicki and Pawlak (2008), Hasiewicz et al (2005)), offering additionally-practically important-advantages (collected in Table 1), cf. the works of ) andŚliwiński et al (2007.…”

“…The first subroutine is performed after each arrival of the new measurement pair (x k , y k ) and is based on the observation that, under Assumptions 1-4, the coefficients α Mn , a Mn , β mn and b mn are equal to the respective expectations Hasiewicz et al, 2005):…”

“…From the asymptotic properties viewpoint (Appendix), the initial value of M can be an arbitrary natural number and in applications one can set it using the off-line scale selection rule, M = 1 2ν+1 log 2 k (Hasiewicz et al, 2005). If, for instance, the estimation routine starts with the first measurement pair (i.e., for k = 1), the rule yields M = 0.…”

“…This makes them biased estimates (see the formula (12) in Appendix) but simultaneously contributes to computational simplicity of the algorithm (in particular, the same current value k is used at the same time in all active empirical coefficients calculations). Zero-initialization can also be motivated by 516 P.Śliwiński the observation that wavelet coefficients decay with the growing scale m as fast as O(2 −m/2 ) for any nonlinearities (and even faster for smoother ones, e.g., for differentiable nonlinearities the vanishing rate raises to the order O(2 −3m/2 ) (Hasiewicz et al, 2005;Mallat, 1998, Chapter 9).…”

“…The proposed type of on-line algorithm has not yet been explored in the literature (a similar, order statistics, algorithm used byŚliwiński et al (2009) is also semirecursive and based on the estimation of the expansion coefficients but requires all measurements to be stored and thus remains inapplicable in on-line conditions) and can be seen as a direct on-line counterpart of off-line wavelet algorithms examined, e.g., byŚliwiński and Hasiewicz (2008) as well as Hasiewicz et al (2005). Limit properties of the algorithm, viz.…”