The Anisotropic Norm of Signals: Towards Possible Definitions

Anisotropy-based robust performance criteria for statistically uncertain linear continuous time invariant stochastic systems

“…Here, for simplicity, the shaping filter is also assumed to be a stable system with a finite-dimensional state ξ in (10), (11), so that ( 17) is a rational function…”

“…Theorem 1. Suppose the system (4), ( 7) and the shaping filter (10) are stable. Then the RMS gain (28) holds as a limit and is computed as…”

“…is the convolution of the impulse responses of the lowpass filter (13) and the shaping filter (10): φ (τ) := √ 2Ωe −Ωτ , ψ(τ) := ce τα β + dδ (τ), τ 0 (with δ the Dirac delta function [41]). Since V is a standard Wiener process in R m , application of the Ito isometry to (33)…”

“…Suppose the system (4), ( 7) is stable. Then for any µ satisfying inf ω∈R λ min (Λ(ω)) µ F ∞ in (39) and any ε > 0, there exists a rational spectral density S (associated with a stable shaping filter (10)) such that ||||F||| S − µ| < ε.…”

“…Subsequently (in the 2000s and more recently), the theory was being developed in the form of a suboptimal guaranteed approach to systems with internal perturbations [27], towards convexification of the anisotropy-based control synthesis [30,37,38] and suboptimal observer design [40,39], and also taking into account nonzero-mean input disturbances [28] and multiplicative noise [29] in the system, to mention some of the developments. The results of the anisotropy-based theory have been adapted to descriptor systems in [2] 2 Alternative proximity measures (such as the relative Renyi entropy and Hellinger distance) instead of the Kullback-Leibler informational divergence in application to the anisotropy functional are discussed in [10]. 3 1 in the sense of statistical correlations between the entries of a multivariable noise at the same or different moments of time 2 where there are mathematical and bibliographic errors and inaccuracies: for example, [2, Definition 3.1 on p. 61] introduces erroneous dimensions of vectors and corresponding spaces in the definition of anisotropy and a wrong sign of the differential entropy; [2, p. 62] specifies a wrong analyticity domain for transfer functions; the order of noncommuting matrix factors in the factorization of spectral densities in [2, Eq.…”