332 research outputs found
Two combined methods for the global solution of implicit semilinear differential equations with the use of spectral projectors and Taylor expansions
Two combined numerical methods for solving semilinear differential-algebraic
equations (DAEs) are obtained and their convergence is proved. The comparative
analysis of these methods is carried out and conclusions about the
effectiveness of their application in various situations are made. In
comparison with other known methods, the obtained methods require weaker
restrictions for the nonlinear part of the DAE. Also, the obtained methods
enable to compute approximate solutions of the DAEs on any given time interval
and, therefore, enable to carry out the numerical analysis of global dynamics
of mathematical models described by the DAEs. The examples demonstrating the
capabilities of the developed methods are provided. To construct the methods we
use the spectral projectors, Taylor expansions and finite differences. Since
the used spectral projectors can be easily computed, to apply the methods it is
not necessary to carry out additional analytical transformations
A power consensus algorithm for DC microgrids
A novel power consensus algorithm for DC microgrids is proposed and analyzed.
DC microgrids are networks composed of DC sources, loads, and interconnecting
lines. They are represented by differential-algebraic equations connected over
an undirected weighted graph that models the electrical circuit. A second graph
represents the communication network over which the source nodes exchange
information about the instantaneous powers, which is used to adjust the
injected current accordingly. This give rise to a nonlinear consensus-like
system of differential-algebraic equations that is analyzed via Lyapunov
functions inspired by the physics of the system. We establish convergence to
the set of equilibria consisting of weighted consensus power vectors as well as
preservation of the weighted geometric mean of the source voltages. The results
apply to networks with constant impedance, constant current and constant power
loads.Comment: Abridged version submitted to the 20th IFAC World Congress, Toulouse,
Franc
The Waveform Relaxation Method for Systems of Differential/Algebraic Equations
An extension of the waveform relaxation (WR) algorithm to systems of differential/algebraic equations (DAE) is presented. Although this type of application has been explored earlier in relation to VLSI circuits, the algorithm has not been generalized to include the vast array of DAE system structures. The solvability and convergence requirements of the WR algorithm for higher-index systems are established. Many systems in robotics and control applications are modeled with DAE systems having an index greater than two. Computer simulation of these systems has been hampered by numerical integration methods which perform poorly and must be explicitly tailored to the system. The WR algorithm presents a means by which these systems may be more efficiently simulated by breaking them into weakly coupled subsystems, many of which will no longer retain the limiting high-index properties
Differential-Algebraic Equations
Differential-Algebraic Equations (DAE) are today an independent field of research, which is gaining in importance and becoming of increasing interest for applications and mathematics itself. This workshop has drawn the balance after about 25 years investigations of DAEs and the research aims of the future were intensively discussed
Certifying Stability and Performance of Uncertain Differential-Algebraic Systems: A Dissipativity Framework
This paper presents a novel framework for characterizing dissipativity of
uncertain dynamical systems subject to algebraic constraints. The main results
provide sufficient conditions for dissipativity when uncertainties are
characterized by integral quadratic constraints. For polynomial or linear
dynamics, these conditions can be efficiently verified through sum-of-squares
or semidefinite programming. The practical impact of this work is illustrated
through a case study that examines performance of the IEEE 39-bus power network
with uncertainties used to model a set of potential line failures
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