170 research outputs found
Gradient and Passive Circuit Structure in a Class of Non-linear Dynamics on a Graph
We consider a class of non-linear dynamics on a graph that contains and
generalizes various models from network systems and control and study
convergence to uniform agreement states using gradient methods. In particular,
under the assumption of detailed balance, we provide a method to formulate the
governing ODE system in gradient descent form of sum-separable energy
functions, which thus represent a class of Lyapunov functions; this class
coincides with Csisz\'{a}r's information divergences. Our approach bases on a
transformation of the original problem to a mass-preserving transport problem
and it reflects a little-noticed general structure result for passive network
synthesis obtained by B.D.O. Anderson and P.J. Moylan in 1975. The proposed
gradient formulation extends known gradient results in dynamical systems
obtained recently by M. Erbar and J. Maas in the context of porous medium
equations. Furthermore, we exhibit a novel relationship between inhomogeneous
Markov chains and passive non-linear circuits through gradient systems, and
show that passivity of resistor elements is equivalent to strict convexity of
sum-separable stored energy. Eventually, we discuss our results at the
intersection of Markov chains and network systems under sinusoidal coupling
Integral control of port-Hamiltonian systems: non-passive outputs without coordinate transformation
In this paper we present a method for the addition of integral action to
non-passive outputs of a class of port-Hamiltonian systems. The proposed
integral controller is a dynamic extension, constructed from the open loop
system, such that the closed loop preserves the port-Hamiltonian form. It is
shown that the controller is able to reject the effects of both matched and
unmatched disturbances, preserving the regulation of the non-passive outputs.
Previous solutions to this problem have relied on a change of coordinates
whereas the presented solution is developed using the original state vector
and, therefore, retains its physical interpretation. In addition, the resulting
closed loop dynamics have a natural interpretation as a Control by
Interconnection scheme.Comment: 8 pages, 2 figure
Stabilization of structure-preserving power networks with market dynamics
This paper studies the problem of maximizing the social welfare while
stabilizing both the physical power network as well as the market dynamics. For
the physical power grid a third-order structure-preserving model is considered
involving both frequency and voltage dynamics. By applying the primal-dual
gradient method to the social welfare problem, a distributed dynamic pricing
algorithm in port-Hamiltonian form is obtained. After interconnection with the
physical system a closed-loop port-Hamiltonian system of differential-algebraic
equations is obtained, whose properties are exploited to prove local asymptotic
stability of the optimal points.Comment: IFAC World Congress 2017, accepted, 6 page
Dissipative Linear Stochastic Hamiltonian Systems
This paper is concerned with stochastic Hamiltonian systems which model a
class of open dynamical systems subject to random external forces. Their
dynamics are governed by Ito stochastic differential equations whose structure
is specified by a Hamiltonian, viscous damping parameters and
system-environment coupling functions. We consider energy balance relations for
such systems with an emphasis on linear stochastic Hamiltonian (LSH) systems
with quadratic Hamiltonians and linear coupling. For LSH systems, we also
discuss stability conditions, the structure of the invariant measure and its
relation with stochastic versions of the virial theorem. Using Lyapunov
functions, organised as deformed Hamiltonians, dissipation relations are also
considered for LSH systems driven by statistically uncertain external forces.
An application of these results to feedback connections of LSH systems is
outlined.Comment: 10 pages, 1 figure, submitted to ANZCC 201
Krylov projection methods for linear Hamiltonian systems
We study geometric properties of Krylov projection methods for large and
sparse linear Hamiltonian systems. We consider in particular energy
preservation. We discuss the connection to structure preserving model
reduction. We illustrate the performance of the methods by applying them to
Hamiltonian PDEs.Comment: 16 pages, 17 figure
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