6,283 research outputs found
Generalized Proportional Allocation Policies for Robust Control of Dynamical Flow Networks
We study a robust control problem for dynamical flow networks. In the
considered dynamical models, traffic flows along the links of a transportation
network --modeled as a capacited multigraph-- and queues up at the nodes,
whereby control policies determine which incoming queues at a node are to be
allocated service simultaneously, within some predetermined scheduling
constraints. We first prove a fundamental performance limitation by showing
that for a dynamical flow network to be stabilizable by some control policy it
is necessary that the exogenous inflows belong to a certain stability region,
that is determined by the network topology, link capacities, and scheduling
constraints. Then, we introduce a family of distributed controls, referred to
as Generalized Proportional Allocation (GPA) policies, and prove that they
stabilize a dynamical transportation network whenever the exogenous inflows
belong to such stability region. The proposed GPA control policies are
decentralized and fully scalable as they rely on local feedback information
only. Differently from previously studied maximally stabilizing control
strategies, the GPA control policies do not require any global information
about the network topology, the exogenous inflows, or the routing, which makes
them robust to demand variations and unpredicted changes in the link capacities
or the routing decisions. Moreover, the proposed GPA control policies also take
into account the overhead time while switching between services. Our
theoretical results find one application in the control of urban traffic
networks with signalized intersections, where vehicles have to queue up at
junctions and the traffic signal controls determine the green light allocation
to the different incoming lanes
On resilient control of dynamical flow networks
Resilience has become a key aspect in the design of contemporary
infrastructure networks. This comes as a result of ever-increasing loads,
limited physical capacity, and fast-growing levels of interconnectedness and
complexity due to the recent technological advancements. The problem has
motivated a considerable amount of research within the last few years,
particularly focused on the dynamical aspects of network flows, complementing
more classical static network flow optimization approaches. In this tutorial
paper, a class of single-commodity first-order models of dynamical flow
networks is considered. A few results recently appeared in the literature and
dealing with stability and robustness of dynamical flow networks are gathered
and originally presented in a unified framework. In particular, (differential)
stability properties of monotone dynamical flow networks are treated in some
detail, and the notion of margin of resilience is introduced as a quantitative
measure of their robustness. While emphasizing methodological aspects --
including structural properties, such as monotonicity, that enable tractability
and scalability -- over the specific applications, connections to
well-established road traffic flow models are made.Comment: accepted for publication in Annual Reviews in Control, 201
Optimal pricing control in distribution networks with time-varying supply and demand
This paper studies the problem of optimal flow control in dynamic inventory
systems. A dynamic optimal distribution problem, including time-varying supply
and demand, capacity constraints on the transportation lines, and convex flow
cost functions of Legendre-type, is formalized and solved. The time-varying
optimal flow is characterized in terms of the time-varying dual variables of a
corresponding network optimization problem. A dynamic feedback controller is
proposed that regulates the flows asymptotically to the optimal flows and
achieves in addition a balancing of all storage levels.Comment: Submitted to 21st International Symposium on Mathematical Theory of
Networks and Systems (MTNS) in December 201
Robust Network Routing under Cascading Failures
We propose a dynamical model for cascading failures in single-commodity
network flows. In the proposed model, the network state consists of flows and
activation status of the links. Network dynamics is determined by a, possibly
state-dependent and adversarial, disturbance process that reduces flow capacity
on the links, and routing policies at the nodes that have access to the network
state, but are oblivious to the presence of disturbance. Under the proposed
dynamics, a link becomes irreversibly inactive either due to overload condition
on itself or on all of its immediate downstream links. The coupling between
link activation and flow dynamics implies that links to become inactive
successively are not necessarily adjacent to each other, and hence the pattern
of cascading failure under our model is qualitatively different than standard
cascade models. The magnitude of a disturbance process is defined as the sum of
cumulative capacity reductions across time and links of the network, and the
margin of resilience of the network is defined as the infimum over the
magnitude of all disturbance processes under which the links at the origin node
become inactive. We propose an algorithm to compute an upper bound on the
margin of resilience for the setting where the routing policy only has access
to information about the local state of the network. For the limiting case when
the routing policies update their action as fast as network dynamics, we
identify sufficient conditions on network parameters under which the upper
bound is tight under an appropriate routing policy. Our analysis relies on
making connections between network parameters and monotonicity in network state
evolution under proposed dynamics
Robust Distributed Routing in Dynamical Flow Networks - Part I: Locally Responsive Policies and Weak Resilience
Robustness of distributed routing policies is studied for dynamical flow
networks, with respect to adversarial disturbances that reduce the link flow
capacities. A dynamical flow network is modeled as a system of ordinary
differential equations derived from mass conservation laws on a directed
acyclic graph with a single origin-destination pair and a constant inflow at
the origin. Routing policies regulate the way the inflow at a non-destination
node gets split among its outgoing links as a function of the current particle
density, while the outflow of a link is modeled to depend on the current
particle density on that link through a flow function. The dynamical flow
network is called partially transferring if the total inflow at the destination
node is asymptotically bounded away from zero, and its weak resilience is
measured as the minimum sum of the link-wise magnitude of all disturbances that
make it not partially transferring. The weak resilience of a dynamical flow
network with arbitrary routing policy is shown to be upper-bounded by the
network's min-cut capacity, independently of the initial flow conditions.
Moreover, a class of distributed routing policies that rely exclusively on
local information on the particle densities, and are locally responsive to
that, is shown to yield such maximal weak resilience. These results imply that
locality constraints on the information available to the routing policies do
not cause loss of weak resilience. Some fundamental properties of dynamical
flow networks driven by locally responsive distributed policies are analyzed in
detail, including global convergence to a unique limit flow.Comment: 32 pages, 5 figures, journal submissio
Discrete analogue computing with rotor-routers
Rotor-routing is a procedure for routing tokens through a network that can
implement certain kinds of computation. These computations are inherently
asynchronous (the order in which tokens are routed makes no difference) and
distributed (information is spread throughout the system). It is also possible
to efficiently check that a computation has been carried out correctly in less
time than the computation itself required, provided one has a certificate that
can itself be computed by the rotor-router network. Rotor-router networks can
be viewed as both discrete analogues of continuous linear systems and
deterministic analogues of stochastic processes.Comment: To appear in Chaos Special Focus Issue on Intrinsic and Designed
Computatio
Resilience of Locally Routed Network Flows: More Capacity is Not Always Better
In this paper, we are concerned with the resilience of locally routed network
flows with finite link capacities. In this setting, an external inflow is
injected to the so-called origin nodes. The total inflow arriving at each node
is routed locally such that none of the outgoing links are overloaded unless
the node receives an inflow greater than its total outgoing capacity. A link
irreversibly fails if it is overloaded or if there is no operational link in
its immediate downstream to carry its flow. For such systems, resilience is
defined as the minimum amount of reduction in the link capacities that would
result in the failure of all the outgoing links of an origin node. We show that
such networks do not necessarily become more resilient as additional capacity
is built in the network. Moreover, when the external inflow does not exceed the
network capacity, selective reductions of capacity at certain links can
actually help averting the cascading failures, without requiring any change in
the local routing policies. This is an attractive feature as it is often easier
in practice to reduce the available capacity of some critical links than to add
physical capacity or to alter routing policies, e.g., when such policies are
determined by social behavior, as in the case of road traffic networks. The
results can thus be used for real-time monitoring of distance-to-failure in
such networks and devising a feasible course of actions to avert systemic
failures.Comment: Accepted to the IEEE Conference on Decision and Control (CDC), 201
Convexity and Robustness of Dynamic Traffic Assignment and Freeway Network Control
We study the use of the System Optimum (SO) Dynamic Traffic Assignment (DTA)
problem to design optimal traffic flow controls for freeway networks as modeled
by the Cell Transmission Model, using variable speed limit, ramp metering, and
routing. We consider two optimal control problems: the DTA problem, where
turning ratios are part of the control inputs, and the Freeway Network Control
(FNC), where turning ratios are instead assigned exogenous parameters. It is
known that relaxation of the supply and demand constraints in the cell-based
formulations of the DTA problem results in a linear program. However, solutions
to the relaxed problem can be infeasible with respect to traffic dynamics.
Previous work has shown that such solutions can be made feasible by proper
choice of ramp metering and variable speed limit control for specific traffic
networks. We extend this procedure to arbitrary networks and provide insight
into the structure and robustness of the proposed optimal controllers. For a
network consisting only of ordinary, merge, and diverge junctions, where the
cells have linear demand functions and affine supply functions with identical
slopes, and the cost is the total traffic volume, we show, using the maximum
principle, that variable speed limits are not needed in order to achieve
optimality in the FNC problem, and ramp metering is sufficient. We also prove
bounds on perturbation of the controlled system trajectory in terms of
perturbations in initial traffic volume and exogenous inflows. These bounds,
which leverage monotonicity properties of the controlled trajectory, are shown
to be in close agreement with numerical simulation results
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