In this paper, we introduce the concept of x-maximal flatness. A control system is x-maximally flat if the number of new states gained by each successive derivation of the flat output is the largest possible. Firstly, we show that the only control-linear systems that are xmaximally flat are those that are static feedback equivalent to the m-chained form. Secondly, we generalize that result from...

In the present paper, the problem of multi-pumping all Raman amplifier has been investigated to obtain a maximum flatness gain and increases the bandwidth of the cascaded Raman amplifiers over a wide range of optical signal wavelengths for long-haul ultra-wide wavelength division multiplexing (UW-WDM) transmission systems due to multi-pumping wavelengths and pumping power . Three cases are anal...

The average concentration of tracers advected from a point source by a multivariate normal velocity field is shown to deviate from a Gaussian profile. The flatness ~kurtosis! is calculated using an asymptotic series expansion valid for velocity fields with short correlation times or weak space dependence. An explicit formula for the excess flatness at first order demonstrates maximum deviation ...