5,744 research outputs found
Randomized exponential integrators for modulated nonlinear Schr\"odinger equations
We consider the nonlinear Schr\"odinger equation with dispersion modulated by
a (formal) derivative of a time-dependent function with fractional Sobolev
regularity of class for some . Due to the loss
of smoothness in the problem classical numerical methods face severe order
reduction. In this work, we develop and analyze a new randomized exponential
integrator based on a stratified Monte Carlo approximation. The new
discretization technique averages the high oscillations in the solution
allowing for improved convergence rates of order . In addition, the
new approach allows us to treat a far more general class of modulations than
the available literature. Numerical results underline our theoretical findings
and show the favorable error behavior of our new scheme compared to classical
methods
Beyond the Waterbed Effect: Development of Fractional Order CRONE Control with Non-Linear Reset
In this paper a novel reset control synthesis method is proposed: CRONE reset
control, combining a robust fractional CRONE controller with non-linear reset
control to overcome waterbed effect. In CRONE control, robustness is achieved
by creation of constant phase behaviour around bandwidth with the use of
fractional operators, also allowing more freedom in shaping the open-loop
frequency response. However, being a linear controller it suffers from the
inevitable trade-off between robustness and performance as a result of the
waterbed effect. Here reset control is introduced in the CRONE design to
overcome the fundamental limitations. In the new controller design, reset phase
advantage is approximated using describing function analysis and used to
achieve better open-loop shape. Sufficient quadratic stability conditions are
shown for the designed CRONE reset controllers and the control design is
validated on a Lorentz-actuated nanometre precision stage. It is shown that for
similar phase margin, better performance in terms of reference-tracking and
noise attenuation can be achieved.Comment: American Control Conference 201
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