6,583 research outputs found
Exact eigenvalue assignment of linear scalar systems with single delay using Lambert W function
Eigenvalue assignment problem of a linear scalar system with a single
discrete delay is analytically and exactly solved. The existence condition of
the desired eigenvalue is established when the current and delay states are
present in the feedback loop. Design of the feedback controller is then
followed. Furthermore, eigenvalue assignment for the input-delay system is also
obtained as well. Numerical examples illustrate the procedure of assigning the
desired eigenvalue
Optimal linear stability condition for scalar differential equations with distributed delay
Linear scalar differential equations with distributed delays appear in the
study of the local stability of nonlinear differential equations with feedback,
which are common in biology and physics. Negative feedback loops tend to
promote oscillations around steady states, and their stability depends on the
particular shape of the delay distribution. Since in applications the mean
delay is often the only reliable information available about the distribution,
it is desirable to find conditions for stability that are independent from the
shape of the distribution. We show here that for a given mean delay, the linear
equation with distributed delay is asymptotically stable if the associated
differential equation with a discrete delay is asymptotically stable. We
illustrate this criterion on a compartment model of hematopoietic cell dynamics
to obtain sufficient conditions for stability
Distributed Decision Through Self-Synchronizing Sensor Networks in the Presence of Propagation Delays and Asymmetric Channels
In this paper we propose and analyze a distributed algorithm for achieving
globally optimal decisions, either estimation or detection, through a
self-synchronization mechanism among linearly coupled integrators initialized
with local measurements. We model the interaction among the nodes as a directed
graph with weights (possibly) dependent on the radio channels and we pose
special attention to the effect of the propagation delay occurring in the
exchange of data among sensors, as a function of the network geometry. We
derive necessary and sufficient conditions for the proposed system to reach a
consensus on globally optimal decision statistics. One of the major results
proved in this work is that a consensus is reached with exponential convergence
speed for any bounded delay condition if and only if the directed graph is
quasi-strongly connected. We provide a closed form expression for the global
consensus, showing that the effect of delays is, in general, the introduction
of a bias in the final decision. Finally, we exploit our closed form expression
to devise a double-step consensus mechanism able to provide an unbiased
estimate with minimum extra complexity, without the need to know or estimate
the channel parameters.Comment: To be published on IEEE Transactions on Signal Processin
Variable-delay feedback control of unstable steady states in retarded time-delayed systems
We study the stability of unstable steady states in scalar retarded
time-delayed systems subjected to a variable-delay feedback control. The
important aspect of such a control problem is that time-delayed systems are
already infinite-dimensional before the delayed feedback control is turned on.
When the frequency of the modulation is large compared to the system's
dynamics, the analytic approach consists of relating the stability properties
of the resulting variable-delay system with those of an analogous distributed
delay system. Otherwise, the stability domains are obtained by a numerical
integration of the linearized variable-delay system. The analysis shows that
the control domains are significantly larger than those in the usual
time-delayed feedback control, and that the complexity of the domain structure
depends on the form and the frequency of the delay modulation.Comment: 13 pages, 8 figures, RevTeX, accepted for publication in Physical
Review
Hopf bifurcations in time-delay systems with band-limited feedback
We investigate the steady-state solution and its bifurcations in time-delay
systems with band-limited feedback. This is a first step in a rigorous study
concerning the effects of AC-coupled components in nonlinear devices with
time-delayed feedback. We show that the steady state is globally stable for
small feedback gain and that local stability is lost, generically, through a
Hopf bifurcation for larger feedback gain. We provide simple criteria that
determine whether the Hopf bifurcation is supercritical or subcritical based on
the knowledge of the first three terms in the Taylor-expansion of the
nonlinearity. Furthermore, the presence of double-Hopf bifurcations of the
steady state is shown, which indicates possible quasiperiodic and chaotic
dynamics in these systems. As a result of this investigation, we find that
AC-coupling introduces fundamental differences to systems of Ikeda-type [Ikeda
et al., Physica D 29 (1987) 223-235] already at the level of steady-state
bifurcations, e.g. bifurcations exist in which limit cycles are created with
periods other than the fundamental ``period-2'' mode found in Ikeda-type
systems.Comment: 32 pages, 5 figures, accepted for publication in Physica D: Nonlinear
Phenomen
Characteristic matrices for linear periodic delay differential equations
Szalai et al. (SIAM J. on Sci. Comp. 28(4), 2006) gave a general construction
for characteristic matrices for systems of linear delay-differential equations
with periodic coefficients. First, we show that matrices constructed in this
way can have a discrete set of poles in the complex plane, which may possibly
obstruct their use when determining the stability of the linear system. Then we
modify and generalize the original construction such that the poles get pushed
into a small neighborhood of the origin of the complex plane.Comment: 17 pages, 1 figur
Positive trigonometric polynomials for strong stability of difference equations
We follow a polynomial approach to analyse strong stability of linear
difference equations with rationally independent delays. Upon application of
the Hermite stability criterion on the discrete-time homogeneous characteristic
polynomial, assessing strong stability amounts to deciding positive
definiteness of a multivariate trigonometric polynomial matrix. This latter
problem is addressed with a converging hierarchy of linear matrix inequalities
(LMIs). Numerical experiments indicate that certificates of strong stability
can be obtained at a reasonable computational cost for state dimension and
number of delays not exceeding 4 or 5
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