35,384 research outputs found
Lyapunov Criterion for Stochastic Systems and Its Applications in Distributed Computation
This paper presents new sufficient conditions for convergence and asymptotic
or exponential stability of a stochastic discrete-time system, under which the
constructed Lyapunov function always decreases in expectation along the
system's solutions after a finite number of steps, but without necessarily
strict decrease at every step, in contrast to the classical stochastic Lyapunov
theory. As the first application of this new Lyapunov criterion, we look at the
product of any random sequence of stochastic matrices, including those with
zero diagonal entries, and obtain sufficient conditions to ensure the product
almost surely converges to a matrix with identical rows; we also show that the
rate of convergence can be exponential under additional conditions. As the
second application, we study a distributed network algorithm for solving linear
algebraic equations. We relax existing conditions on the network structures,
while still guaranteeing the equations are solved asymptotically.Comment: 14 pages, 1 figur
Recent Advances in Computational Methods for the Power Flow Equations
The power flow equations are at the core of most of the computations for
designing and operating electric power systems. The power flow equations are a
system of multivariate nonlinear equations which relate the power injections
and voltages in a power system. A plethora of methods have been devised to
solve these equations, starting from Newton-based methods to homotopy
continuation and other optimization-based methods. While many of these methods
often efficiently find a high-voltage, stable solution due to its large basin
of attraction, most of the methods struggle to find low-voltage solutions which
play significant role in certain stability-related computations. While we do
not claim to have exhausted the existing literature on all related methods,
this tutorial paper introduces some of the recent advances in methods for
solving power flow equations to the wider power systems community as well as
bringing attention from the computational mathematics and optimization
communities to the power systems problems. After briefly reviewing some of the
traditional computational methods used to solve the power flow equations, we
focus on three emerging methods: the numerical polynomial homotopy continuation
method, Groebner basis techniques, and moment/sum-of-squares relaxations using
semidefinite programming. In passing, we also emphasize the importance of an
upper bound on the number of solutions of the power flow equations and review
the current status of research in this direction.Comment: 13 pages, 2 figures. Submitted to the Tutorial Session at IEEE 2016
American Control Conferenc
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