1,507 research outputs found
Positive Forms and Stability of Linear Time-Delay Systems
We consider the problem of constructing Lyapunov functions for linear
differential equations with delays. For such systems it is known that
exponential stability implies the existence of a positive Lyapunov function
which is quadratic on the space of continuous functions. We give an explicit
parametrization of a sequence of finite-dimensional subsets of the cone of
positive Lyapunov functions using positive semidefinite matrices. This allows
stability analysis of linear time-delay systems to be formulated as a
semidefinite program.Comment: journal version, 14 page
Distributed delays stabilize negative feedback loops
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 oscillation 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 the linear equation with
distributed delays is asymptotically stable if the associated differential
equation with a discrete delay of the same mean is asymptotically stable.
Therefore, distributed delays stabilize negative feedback loops
Zero-Delay Rate Distortion via Filtering for Vector-Valued Gaussian Sources
We deal with zero-delay source coding of a vector-valued Gauss-Markov source
subject to a mean-squared error (MSE) fidelity criterion characterized by the
operational zero-delay vector-valued Gaussian rate distortion function (RDF).
We address this problem by considering the nonanticipative RDF (NRDF) which is
a lower bound to the causal optimal performance theoretically attainable (OPTA)
function and operational zero-delay RDF. We recall the realization that
corresponds to the optimal "test-channel" of the Gaussian NRDF, when
considering a vector Gauss-Markov source subject to a MSE distortion in the
finite time horizon. Then, we introduce sufficient conditions to show existence
of solution for this problem in the infinite time horizon. For the asymptotic
regime, we use the asymptotic characterization of the Gaussian NRDF to provide
a new equivalent realization scheme with feedback which is characterized by a
resource allocation (reverse-waterfilling) problem across the dimension of the
vector source. We leverage the new realization to derive a predictive coding
scheme via lattice quantization with subtractive dither and joint memoryless
entropy coding. This coding scheme offers an upper bound to the operational
zero-delay vector-valued Gaussian RDF. When we use scalar quantization, then
for "r" active dimensions of the vector Gauss-Markov source the gap between the
obtained lower and theoretical upper bounds is less than or equal to 0.254r + 1
bits/vector. We further show that it is possible when we use vector
quantization, and assume infinite dimensional Gauss-Markov sources to make the
previous gap to be negligible, i.e., Gaussian NRDF approximates the operational
zero-delay Gaussian RDF. We also extend our results to vector-valued Gaussian
sources of any finite memory under mild conditions. Our theoretical framework
is demonstrated with illustrative numerical experiments.Comment: 32 pages, 9 figures, published in IEEE Journal of Selected Topics in
Signal Processin
Sufficient Stability Conditions for Time-varying Networks of Telegrapher's Equations or Difference Delay Equations
We give a sufficient condition for exponential stability of a network of
lossless telegrapher's equations, coupled by linear time-varying boundary
conditions. The sufficient conditions is in terms of dissipativity of the
couplings, which is natural for instance in the context of microwave circuits.
Exponential stability is with respect to any -norm, .
This also yields a sufficient condition for exponential stability to a special
class of linear time-varying difference delay systems which is quite explicit
and tractable. One ingredient of the proof is that exponential stability
for such difference delay systems is independent of , thereby reproving in a
simpler way some results from [3].Comment: To be published in SIAM Journal on Mathematical Analysis, most
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