7,770 research outputs found
Model Reduction Near Periodic Orbits of Hybrid Dynamical Systems
We show that, near periodic orbits, a class of hybrid models can be reduced
to or approximated by smooth continuous-time dynamical systems. Specifically,
near an exponentially stable periodic orbit undergoing isolated transitions in
a hybrid dynamical system, nearby executions generically contract
superexponentially to a constant-dimensional subsystem. Under a non-degeneracy
condition on the rank deficiency of the associated Poincare map, the
contraction occurs in finite time regardless of the stability properties of the
orbit. Hybrid transitions may be removed from the resulting subsystem via a
topological quotient that admits a smooth structure to yield an equivalent
smooth dynamical system. We demonstrate reduction of a high-dimensional
underactuated mechanical model for terrestrial locomotion, assess structural
stability of deadbeat controllers for rhythmic locomotion and manipulation, and
derive a normal form for the stability basin of a hybrid oscillator. These
applications illustrate the utility of our theoretical results for synthesis
and analysis of feedback control laws for rhythmic hybrid behavior
Model Reduction by Moment Matching for Linear Switched Systems
Two moment-matching methods for model reduction of linear switched systems
(LSSs) are presented. The methods are similar to the Krylov subspace methods
used for moment matching for linear systems. The more general one of the two
methods, is based on the so called "nice selection" of some vectors in the
reachability or observability space of the LSS. The underlying theory is
closely related to the (partial) realization theory of LSSs. In this paper, the
connection of the methods to the realization theory of LSSs is provided, and
algorithms are developed for the purpose of model reduction. Conditions for
applicability of the methods for model reduction are stated and finally the
results are illustrated on numerical examples.Comment: Sent for publication in IEEE TAC, on October 201
Adiabatic reduction of models of stochastic gene expression with bursting
This paper considers adiabatic reduction in both discrete and continuous
models of stochastic gene expression. In gene expression models, the concept of
bursting is a production of several molecules simultaneously and is generally
represented as a compound Poisson process of random size. In a general
two-dimensional birth and death discrete model, we prove that under specific
assumptions and scaling (that are characteristics of the mRNA-protein system)
an adiabatic reduction leads to a one-dimensional discrete-state space model
with bursting production. The burst term appears through the reduction of the
first variable. In a two-dimensional continuous model, we also prove that an
adiabatic reduction can be performed in a stochastic slow/fast system. In this
gene expression model, the production of mRNA (the fast variable) is assumed to
be bursty and the production of protein (the slow variable) is linear as a
function of mRNA. When the dynamics of mRNA is assumed to be faster than the
protein dynamics (due to a mRNA degradation rate larger than for the protein)
we prove that, with the appropriate scaling, the bursting phenomena can be
transmitted to the slow variable. We show that the reduced equation is either a
stochastic differential equation with a jump Markov process or a deterministic
ordinary differential equation depending on the scaling that is appropriate.
These results are significant because adiabatic reduction techniques seem to
have not been applied to a stochastic differential system containing a jump
Markov process. Last but not least, for our particular system, the adiabatic
reduction allows us to understand what are the necessary conditions for the
bursting production-like of protein to occur.Comment: 24 page
Reachability in Biochemical Dynamical Systems by Quantitative Discrete Approximation (extended abstract)
In this paper, a novel computational technique for finite discrete
approximation of continuous dynamical systems suitable for a significant class
of biochemical dynamical systems is introduced. The method is parameterized in
order to affect the imposed level of approximation provided that with
increasing parameter value the approximation converges to the original
continuous system. By employing this approximation technique, we present
algorithms solving the reachability problem for biochemical dynamical systems.
The presented method and algorithms are evaluated on several exemplary
biological models and on a real case study.Comment: In Proceedings CompMod 2011, arXiv:1109.104
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