The unknown state dependent additive nonlinearity influencing the input output description is modeled as an absolutely bounded "time varying perturbation".
All the unknown additive nonlinearities and exogenous perturbations are lumped into an absolutely bounded, unstructured, vector of time signals whose components may be locally on-line estimated by means of a high gain Generalized Proportional Integral (GPI) observer.
It is shown that unlike Fourier series, a wavelet expansion has a summation kernel which is absolutely bounded by dilations of a radial decreasing L1 convolution kernel H(|x − y|).
Finally, Open image in new window is a smooth function of X hence it is also a semimartingale, which is special if its associated jump process Open image in new window is absolutely bounded; cf. Jacod and Shiryaev (2003, Lem. I.4.24).
By (ell_{p}), (ell_{infty }), c and (c_{0}), we denote the spaces of p-absolutely summable, bounded, convergent and null sequences, respectively, where (1leq p<infty).
By (ell_{p}), (ell_{infty }), c and (c_{0}), we denote the spaces of absolutely p-summable, bounded, convergent and null sequences, respectively, where (1leq p<infty).
Lemma 2.6 [2] Suppose that f1 x) and f2 y) are real, bounded, absolutely continuous functions on R with f ′ 1 ( x ) ≤ C 1 and f ′ 2 ( y ) ≤ C 2. Then for any random variables X and Y, C o v f 1 ( X ), f 2 ( Y ) ≤ C 1 C 2 - C o v ( X, Y ) + 8 p - ( X, Y ) X 2, 1 Y 2, 1, where X 2, 1 = ∫ 0 ∞ P 1 2 X > x d x.
Throughout w, ℓ ∞, ℓ p, l 1, c and c 0 denote the spaces of all, bounded, p-absolutely summable, absolutely summable, convergent and null sequences x = ( x k ) with complex terms, respectively.
Also, by bs, cs, (l_{1}), and (l_{p}) ((1< p<infty)), we denote the spaces of all bounded, convergent, absolutely, and p-absolutely convergent series, respectively.
Also by bs, cs, ℓ 1, and ℓ p, we denote the spaces of all bounded, convergent, absolutely, and p-absolutely convergent series, respectively.
Also by bs, cs, l 1, and l ∞, we denote the spaces of all bounded, convergent, absolutely convergent and p-absolutely convergent series, respectively; where 1 < p < ∞.
