3 research outputs found
A criterion of existence and uniqueness of the Lyapunov matrix for a class of time delay systems
The Lyapunov matrix for systems of linear time-delay equations is a matrix-valued function
which is a solution of a special dynamic system with some additional boundary conditions.
This matrix allows to construct the complete type Lyapunov–Krasovskii functionals with a
prescribed derivative, which are used successfully in analysis of behavior of time-delay systems.
It was shown in works of Kharitonov and Chashnikov that the Lyapunov condition, that is
the absence of opposite eigenvalues of the system, guarantees the existence and uniqueness
of the Lyapunov matrix for systems of retarded type with multiple delays and for systems of
neutral type with a single delay. In this contribution, we consider a linear time-invariant system
of neutral type with two delays. It is shown that under a certain constraint the Lyapunov
condition, that is the absence of opposite eigenvalues of the system, is also a criterion of the
existence and uniqueness of the Lyapunov matrix. Refs 14
Performance Improvement in Noisy Linear Consensus Networks with Time-Delay
We analyze performance of a class of time-delay first-order consensus
networks from a graph topological perspective and present methods to improve
it. The performance is measured by network's square of H-2 norm and it is shown
that it is a convex function of Laplacian eigenvalues and the coupling weights
of the underlying graph of the network. First, we propose a tight convex, but
simple, approximation of the performance measure in order to achieve lower
complexity in our design problems by eliminating the need for
eigen-decomposition. The effect of time-delay reincarnates itself in the form
of non-monotonicity, which results in nonintuitive behaviors of the performance
as a function of graph topology. Next, we present three methods to improve the
performance by growing, re-weighting, or sparsifying the underlying graph of
the network. It is shown that our suggested algorithms provide near-optimal
solutions with lower complexity with respect to existing methods in literature.Comment: 16 pages, 11 figure
Computing delay Lyapunov matrices and H2 norms for large-scale problems
A delay Lyapunov matrix corresponding to an exponentially stable system of
linear time-invariant delay differential equations can be characterized as the
solution of a boundary value problem involving a matrix valued delay
differential equation. This boundary value problem can be seen as a natural
generalization of the classical Lyapunov matrix equation. We present a general
approach for computing delay Lyapunov matrices and H2 norms for systems with
multiple discrete delays, whose applicability extends towards problems where
the matrices are large and sparse, and the associated positive semidefinite
matrix (the ``right-hand side' for the standard Lyapunov equation), has a low
rank. In contract to existing methods that are based on solving the boundary
value problem directly, our method is grounded in solving standard Lyapunov
equations of increased dimensions. It combines several ingredients: i) a
spectral discretization of the system of delay equations, ii) a targeted
similarity transformation which induces a desired structure and sparsity
pattern and, at the same time, favors accurate low rank solutions of the
corresponding Lyapunov equation, and iii) a Krylov method for large-scale
matrix Lyapunov equations. The structure of the problem is exploited in such a
way that the final algorithm does not involve a preliminary discretization
step, and provides a fully dynamic construction of approximations of increasing
rank. Interpretations in terms of a projection method directly applied to a
standard linear infinite-dimensional system equivalent to the original
time-delay system are also given. Throughout the paper two didactic examples
are used to illustrate the properties of the problem, the challenges and
methodological choices, while numerical experiments are presented at the end to
illustrate the effectiveness of the algorithm