36,324 research outputs found
On the Positive Effect of Delay on the Rate of Convergence of a Class of Linear Time-Delayed Systems
This paper is a comprehensive study of a long observed phenomenon of increase
in the stability margin and so the rate of convergence of a class of linear
systems due to time delay. We use Lambert W function to determine (a) in what
systems the delay can lead to increase in the rate of convergence, (b) the
exact range of time delay for which the rate of convergence is greater than
that of the delay free system, and (c) an estimate on the value of the delay
that leads to the maximum rate of convergence. For the special case when the
system matrix eigenvalues are all negative real numbers, we expand our results
to show that the rate of convergence in the presence of delay depends only on
the eigenvalues with minimum and maximum real parts. Moreover, we determine the
exact value of the maximum rate of convergence and the corresponding maximizing
time delay. We demonstrate our results through a numerical example on the
practical application in accelerating an agreement algorithm for
networked~systems by use of a delayed feedback
On Robustness Analysis of a Dynamic Average Consensus Algorithm to Communication Delay
This paper studies the robustness of a dynamic average consensus algorithm to
communication delay over strongly connected and weight-balanced (SCWB)
digraphs. Under delay-free communication, the algorithm of interest achieves a
practical asymptotic tracking of the dynamic average of the time-varying
agents' reference signals. For this algorithm, in both its continuous-time and
discrete-time implementations, we characterize the admissible communication
delay range and study the effect of the delay on the rate of convergence and
the tracking error bound. Our study also includes establishing a relationship
between the admissible delay bound and the maximum degree of the SCWB digraphs.
We also show that for delays in the admissible bound, for static signals the
algorithms achieve perfect tracking. Moreover, when the interaction topology is
a connected undirected graph, we show that the discrete-time implementation is
guaranteed to tolerate at least one step delay. Simulations demonstrate our
results
Asymptotic behavior of age-structured and delayed Lotka-Volterra models
In this work we investigate some asymptotic properties of an age-structured
Lotka-Volterra model, where a specific choice of the functional parameters
allows us to formulate it as a delayed problem, for which we prove the
existence of a unique coexistence equilibrium and characterize the existence of
a periodic solution. We also exhibit a Lyapunov functional that enables us to
reduce the attractive set to either the nontrivial equilibrium or to a periodic
solution. We then prove the asymptotic stability of the nontrivial equilibrium
where, depending on the existence of the periodic trajectory, we make explicit
the basin of attraction of the equilibrium. Finally, we prove that these
results can be extended to the initial PDE problem.Comment: 29 page
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