232,229 research outputs found
Decomposition of Nonlinear Dynamical Systems Using Koopman Gramians
In this paper we propose a new Koopman operator approach to the decomposition
of nonlinear dynamical systems using Koopman Gramians. We introduce the notion
of an input-Koopman operator, and show how input-Koopman operators can be used
to cast a nonlinear system into the classical state-space form, and identify
conditions under which input and state observable functions are well separated.
We then extend an existing method of dynamic mode decomposition for learning
Koopman operators from data known as deep dynamic mode decomposition to systems
with controls or disturbances. We illustrate the accuracy of the method in
learning an input-state separable Koopman operator for an example system, even
when the underlying system exhibits mixed state-input terms. We next introduce
a nonlinear decomposition algorithm, based on Koopman Gramians, that maximizes
internal subsystem observability and disturbance rejection from unwanted noise
from other subsystems. We derive a relaxation based on Koopman Gramians and
multi-way partitioning for the resulting NP-hard decomposition problem. We
lastly illustrate the proposed algorithm with the swing dynamics for an IEEE
39-bus system.Comment: 8 pages, submitted to IEEE 2018 AC
Pattern transition in spacecraft formation flying using bifurcating potential field
Many new and exciting space mission concepts have developed around spacecraft formation flying, allowing for autonomous distributed systems that can be robust, scalable and flexible. This paper considers the development of a new methodology for the control of multiple spacecraft. Based on the artificial potential function method, research in this area is extended by considering the new approach of using bifurcation theory as a means of controlling the transition between different formations. For real, safety or mission critical applications it is important to ensure that desired behaviours will occur. Through dynamical systems theory, this paper also aims to provide a step in replacing traditional algorithm validation with mathematical proof, supported through simulation. This is achieved by determining the non-linear stability properties of the system, thus proving the existence or not of desired behaviours. Practical considerations such as the issue of actuator saturation and communication limitations are addressed, with the development of a new bounded control law based on bifurcating potential fields providing the key contribution of this paper. To illustrate spacecraft formation flying using the new methodology formation patterns are considered in low-Earth-orbit utilising the Clohessy-Wiltshire relative linearised equations of motion. It is shown that a formation of spacecraft can be driven safely onto equally spaced projected circular orbits, autonomously reconfiguring between them, whilst satisfying constraints made regarding each spacecraft
Unequal Intra-layer Coupling in a Bilayer Driven Lattice Gas
The system under study is a twin-layered square lattice gas at half-filling,
being driven to non-equilibrium steady states by a large, finite `electric'
field. By making intra-layer couplings unequal we were able to extend the phase
diagram obtained by Hill, Zia and Schmittmann (1996) and found that the
tri-critical point, which separates the phase regions of the stripped (S) phase
(stable at positive interlayer interactions J_3), the filled-empty (FE) phase
(stable at negative J_3) and disorder (D), is shifted even further into the
negative J_3 region as the coupling traverse to the driving field increases.
Many transient phases to the S phase at the S-FE boundary were found to be
long-lived. We also attempted to test whether the universality class of D-FE
transitions under a drive is still Ising. Simulation results suggest a value of
1.75 for the exponent gamma but a value close to 2.0 for the ratio gamma/nu. We
speculate that the D-FE second order transition is different from Ising near
criticality, where observed first-order-like transitions between FE and its
"local minimum" cousin occur during each simulation run.Comment: 29 pages, 19 figure
Data driven problems in elasticity
We consider a new class of problems in elasticity, referred to as Data-Driven
problems, defined on the space of strain-stress field pairs, or phase space.
The problem consists of minimizing the distance between a given material data
set and the subspace of compatible strain fields and stress fields in
equilibrium. We find that the classical solutions are recovered in the case of
linear elasticity. We identify conditions for convergence of Data-Driven
solutions corresponding to sequences of ap- proximating material data sets.
Specialization to constant material data set sequences in turn establishes an
appropriate notion of relaxation. We find that relaxation within this
Data-Driven framework is fundamentally different from the classical relaxation
of energy functions. For instance, we show that in the Data-Driven framework
the relaxation of a bistable material leads to material data sets that are not
graphs.Comment: Result now covers the two well problem in full generality. Proof
simplified. New Figure 9 illustrates geometry of separatio
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