5,275 research outputs found
Contraction analysis of switched Filippov systems via regularization
We study incremental stability and convergence of switched (bimodal) Filippov
systems via contraction analysis. In particular, by using results on
regularization of switched dynamical systems, we derive sufficient conditions
for convergence of any two trajectories of the Filippov system between each
other within some region of interest. We then apply these conditions to the
study of different classes of Filippov systems including piecewise smooth (PWS)
systems, piecewise affine (PWA) systems and relay feedback systems. We show
that contrary to previous approaches, our conditions allow the system to be
studied in metrics other than the Euclidean norm. The theoretical results are
illustrated by numerical simulations on a set of representative examples that
confirm their effectiveness and ease of application.Comment: Preprint submitted to Automatic
Piecewise smooth systems near a co-dimension 2 discontinuity manifold: can one say what should happen?
We consider a piecewise smooth system in the neighborhood of a co-dimension 2
discontinuity manifold . Within the class of Filippov solutions, if
is attractive, one should expect solution trajectories to slide on
. It is well known, however, that the classical Filippov
convexification methodology is ambiguous on . The situation is further
complicated by the possibility that, regardless of how sliding on is
taking place, during sliding motion a trajectory encounters so-called generic
first order exit points, where ceases to be attractive.
In this work, we attempt to understand what behavior one should expect of a
solution trajectory near when is attractive, what to expect
when ceases to be attractive (at least, at generic exit points), and
finally we also contrast and compare the behavior of some regularizations
proposed in the literature.
Through analysis and experiments we will confirm some known facts, and
provide some important insight: (i) when is attractive, a solution
trajectory indeed does remain near , viz. sliding on is an
appropriate idealization (of course, in general, one cannot predict which
sliding vector field should be selected); (ii) when loses attractivity
(at first order exit conditions), a typical solution trajectory leaves a
neighborhood of ; (iii) there is no obvious way to regularize the
system so that the regularized trajectory will remain near as long as
is attractive, and so that it will be leaving (a neighborhood of)
when looses attractivity.
We reach the above conclusions by considering exclusively the given piecewise
smooth system, without superimposing any assumption on what kind of dynamics
near (or sliding motion on ) should have been taking place.Comment: 19 figure
Smoothing tautologies, hidden dynamics, and sigmoid asymptotics for piecewise smooth systems
Switches in real systems take many forms, such as impacts, electronic relays,
mitosis, and the implementation of decisions or control strategies. To
understand what is lost, and what can be retained, when we model a switch as an
instantaneous event, requires a consideration of so-called hidden terms. These
are asymptotically vanishing outside the switch, but can be encoded in the form
of nonlinear switching terms. A general expression for the switch can be
developed in the form of a series of sigmoid functions. We review the key steps
in extending the Filippov's method of sliding modes to such systems. We show
how even slight nonlinear effects can hugely alter the behaviour of an
electronic control circuit, and lead to `hidden' attractors inside the
switching surface.Comment: 12 page
Quantum Electroweak Symmetry Breaking Through Loop Quadratic Contributions
Based on two postulations that (i) the Higgs boson has a large bare mass GeV at the characteristic energy scale which defines
the standard model (SM) in the ultraviolet region, and (ii) quadratic
contributions of Feynman loop diagrams in quantum field theories are physically
meaningful, we show that the SM electroweak symmetry breaking is induced by the
quadratic contributions from loop effects. As the quadratic running of Higgs
mass parameter leads to an additive renormalization, which distinguishes from
the logarithmic running with a multiplicative renormalization, the symmetry
breaking occurs once the sliding energy scale moves from down to a
transition scale at which the additive renormalized Higgs
mass parameter gets to change the sign. With the input of
current experimental data, this symmetry breaking energy scale is found to be
GeV, which provides another basic energy scale for the
SM besides . Studying such a symmetry breaking mechanism could play an
important role in understanding both the hierarchy problem and naturalness
problem. It also provides a possible way to explore the experimental
implications of the quadratic contributions as lies within the
probing reach of the LHC and the future Great Collider.Comment: 10 pages, 2 figures, published versio
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