1,480 research outputs found
A Numerical Slow Manifold Approach to Model Reduction for Optimal Control of Multiple Time Scale ODE
Time scale separation is a natural property of many control systems that can
be ex- ploited, theoretically and numerically. We present a numerical scheme to
solve optimal control problems with considerable time scale separation that is
based on a model reduction approach that does not need the system to be
explicitly stated in singularly perturbed form. We present examples that
highlight the advantages and disadvantages of the method
Slow invariant manifold of heartbeat model
A new approach called Flow Curvature Method has been recently developed in a
book entitled Differential Geometry Applied to Dynamical Systems. It consists
in considering the trajectory curve, integral of any n-dimensional dynamical
system as a curve in Euclidean n-space that enables to analytically compute the
curvature of the trajectory - or the flow. Hence, it has been stated on the one
hand that the location of the points where the curvature of the flow vanishes
defines a manifold called flow curvature manifold and on the other hand that
such a manifold associated with any n-dimensional dynamical system directly
provides its slow manifold analytical equation the invariance of which has been
proved according to Darboux theory. The Flow Curvature Method has been already
applied to many types of autonomous dynamical systems either singularly
perturbed such as Van der Pol Model, FitzHugh-Nagumo Model, Chua's Model, ...)
or non-singularly perturbed such as Pikovskii-Rabinovich-Trakhtengerts Model,
Rikitake Model, Lorenz Model,... More- over, it has been also applied to
non-autonomous dynamical systems such as the Forced Van der Pol Model. In this
article it will be used for the first time to analytically compute the slow
invariant manifold analytical equation of the four-dimensional Unforced and
Forced Heartbeat Model. Its slow invariant manifold equation which can be
considered as a "state equation" linking all variables could then be used in
heart prediction and control according to the strong correspondence between the
model and the physiological cardiovascular system behavior.Comment: arXiv admin note: substantial text overlap with arXiv:1408.171
Canards from Chua's circuit
The aim of this work is to extend Beno\^it's theorem for the generic
existence of "canards" solutions in singularly perturbed dynamical systems of
dimension three with one fast variable to those of dimension four. Then, it is
established that this result can be found according to the Flow Curvature
Method. Applications to Chua's cubic model of dimension three and four enable
to state the existence of "canards" solutions in such systems.Comment: arXiv admin note: text overlap with arXiv:1408.489
Slow invariant manifolds as curvature of the flow of dynamical systems
Considering trajectory curves, integral of n-dimensional dynamical systems,
within the framework of Differential Geometry as curves in Euclidean n-space,
it will be established in this article that the curvature of the flow, i.e. the
curvature of the trajectory curves of any n-dimensional dynamical system
directly provides its slow manifold analytical equation the invariance of which
will be then proved according to Darboux theory. Thus, it will be stated that
the flow curvature method, which uses neither eigenvectors nor asymptotic
expansions but only involves time derivatives of the velocity vector field,
constitutes a general method simplifying and improving the slow invariant
manifold analytical equation determination of high-dimensional dynamical
systems. Moreover, it will be shown that this method generalizes the Tangent
Linear System Approximation and encompasses the so-called Geometric Singular
Perturbation Theory. Then, slow invariant manifolds analytical equation of
paradigmatic Chua's piecewise linear and cubic models of dimensions three, four
and five will be provided as tutorial examples exemplifying this method as well
as those of high-dimensional dynamical systems
Singularly Perturbed Monotone Systems and an Application to Double Phosphorylation Cycles
The theory of monotone dynamical systems has been found very useful in the
modeling of some gene, protein, and signaling networks. In monotone systems,
every net feedback loop is positive. On the other hand, negative feedback loops
are important features of many systems, since they are required for adaptation
and precision. This paper shows that, provided that these negative loops act at
a comparatively fast time scale, the main dynamical property of (strongly)
monotone systems, convergence to steady states, is still valid. An application
is worked out to a double-phosphorylation ``futile cycle'' motif which plays a
central role in eukaryotic cell signaling.Comment: 21 pages, 3 figures, corrected typos, references remove
Formation of Multiple Groups of Mobile Robots Using Sliding Mode Control
Formation control of multiple groups of agents finds application in large
area navigation by generating different geometric patterns and shapes, and also
in carrying large objects. In this paper, Centroid Based Transformation (CBT)
\cite{c39}, has been applied to decompose the combined dynamics of wheeled
mobile robots (WMRs) into three subsystems: intra and inter group shape
dynamics, and the dynamics of the centroid. Separate controllers have been
designed for each subsystem. The gains of the controllers are such chosen that
the overall system becomes singularly perturbed system. Then sliding mode
controllers are designed on the singularly perturbed system to drive the
subsystems on sliding surfaces in finite time. Negative gradient of a potential
based function has been added to the sliding surface to ensure collision
avoidance among the robots in finite time. The efficacy of the proposed
controller is established through simulation results.Comment: 8 pages, 5 figure
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