184 research outputs found
Boundary Observer for Congested Freeway Traffic State Estimation via Aw-Rascle-Zhang model
This paper develops boundary observer for estimation of congested freeway
traffic states based on Aw-Rascle-Zhang(ARZ) partial differential equations
(PDE) model. Traffic state estimation refers to acquisition of traffic state
information from partially observed traffic data. This problem is relevant for
freeway due to its limited accessibility to real-time traffic information. We
propose a boundary observer design so that estimates of aggregated traffic
states in a freeway segment are obtained simply from boundary measurement of
flow and velocity. The macroscopic traffic dynamics is represented by the ARZ
model, consisting of coupled nonlinear hyperbolic PDEs for traffic
density and velocity. Analysis of the linearized ARZ model leads to the study
of a hetero-directional hyperbolic PDE model for congested traffic regime.
Using spatial transformation and PDE backstepping method, we construct a
boundary observer with a copy of the nonlinear plant and output injection of
boundary measurement errors. The output injection gains are designed for the
error system of the linearized ARZ model so that the exponential stability of
error system in the norm and finite-time convergence to zero are
guaranteed. Simulations are conducted to validate the boundary observer design
for nonlinear ARZ model without knowledge of initial conditions
Learning Nash Equilibria in Congestion Games
We study the repeated congestion game, in which multiple populations of
players share resources, and make, at each iteration, a decentralized decision
on which resources to utilize. We investigate the following question: given a
model of how individual players update their strategies, does the resulting
dynamics of strategy profiles converge to the set of Nash equilibria of the
one-shot game? We consider in particular a model in which players update their
strategies using algorithms with sublinear discounted regret. We show that the
resulting sequence of strategy profiles converges to the set of Nash equilibria
in the sense of Ces\`aro means. However, strong convergence is not guaranteed
in general. We show that strong convergence can be guaranteed for a class of
algorithms with a vanishing upper bound on discounted regret, and which satisfy
an additional condition. We call such algorithms AREP algorithms, for
Approximate REPlicator, as they can be interpreted as a discrete-time
approximation of the replicator equation, which models the continuous-time
evolution of population strategies, and which is known to converge for the
class of congestion games. In particular, we show that the discounted Hedge
algorithm belongs to the AREP class, which guarantees its strong convergence
Computing the log-determinant of symmetric, diagonally dominant matrices in near-linear time
We present new algorithms for computing the log-determinant of symmetric,
diagonally dominant matrices. Existing algorithms run with cubic complexity
with respect to the size of the matrix in the worst case. Our algorithm
computes an approximation of the log-determinant in time near-linear with
respect to the number of non-zero entries and with high probability. This
algorithm builds upon the utra-sparsifiers introduced by Spielman and Teng for
Laplacian matrices and ultimately uses their refined versions introduced by
Koutis, Miller and Peng in the context of solving linear systems. We also
present simpler algorithms that compute upper and lower bounds and that may be
of more immediate practical interest.Comment: Submitted to the SIAM Journal on Computing (SICOMP
Modeling and Estimation of the Humans' Effect on the CO2 Dynamics Inside a Conference Room
We develop a data-driven, {\em Partial Differential Equation-Ordinary
Differential Equation} (PDE-ODE) model that describes the response of the {\em
Carbon Dioxide} (\cotwon) dynamics inside a conference room, due to the
presence of humans, or of a user-controlled exogenous source of \cotwon. We
conduct two controlled experiments in order to develop and tune a model whose
output matches the measured output concentration of \cotwo inside the room,
when known inputs are applied to the model. In the first experiment, a
controlled amount of \cotwo gas is released inside the room from a regulated
supply, and in the second, a known number of humans produce a certain amount of
\cotwo inside the room. For the estimation of the exogenous inputs, we design
an observer, based on our model, using measurements of \cotwo concentrations at
two locations inside the room. Parameter identifiers are also designed, based
on our model, for the online estimation of the parameters of the model. We
perform several simulation studies for the illustration of our designs
Embarrassingly Parallel Time Series Analysis for Large Scale Weak Memory Systems
Second order stationary models in time series analysis are based on the
analysis of essential statistics whose computations follow a common pattern. In
particular, with a map-reduce nomenclature, most of these operations can be
modeled as mapping a kernel that only depends on short windows of consecutive
data and reducing the results produced by each computation. This computational
pattern stems from the ergodicity of the model under consideration and is often
referred to as weak or short memory when it comes to data indexed with respect
to time. In the following we will show how studying weak memory systems can be
done in a scalable manner thanks to a framework relying on specifically
designed overlapping distributed data structures that enable fragmentation and
replication of the data across many machines as well as parallelism in
computations. This scheme has been implemented for Apache Spark but is
certainly not system specific. Indeed we prove it is also adapted to leveraging
high bandwidth fragmented memory blocks on GPUs
A Necessary and Sufficient Condition for the Existence of Potential Functions for Heterogeneous Routing Games
We study a heterogeneous routing game in which vehicles might belong to more
than one type. The type determines the cost of traveling along an edge as a
function of the flow of various types of vehicles over that edge. We relax the
assumptions needed for the existence of a Nash equilibrium in this
heterogeneous routing game. We extend the available results to present
necessary and sufficient conditions for the existence of a potential function.
We characterize a set of tolls that guarantee the existence of a potential
function when only two types of users are participating in the game. We present
an upper bound for the price of anarchy (i.e., the worst-case ratio of the
social cost calculated for a Nash equilibrium over the social cost for a
socially optimal flow) for the case in which only two types of players are
participating in a game with affine edge cost functions. A heterogeneous
routing game with vehicle platooning incentives is used as an example
throughout the article to clarify the concepts and to validate the results.Comment: Improved Literature Review; Updated Introductio
Reinforcement Learning versus PDE Backstepping and PI Control for Congested Freeway Traffic
We develop reinforcement learning (RL) boundary controllers to mitigate
stop-and-go traffic congestion on a freeway segment. The traffic dynamics of
the freeway segment are governed by a macroscopic Aw-Rascle-Zhang (ARZ) model,
consisting of quasi-linear partial differential equations (PDEs)
for traffic density and velocity. Boundary stabilization of the linearized ARZ
PDE model has been solved by PDE backstepping, guaranteeing spatial norm
regulation of the traffic state to uniform density and velocity and ensuring
that traffic oscillations are suppressed. Collocated Proportional (P) and
Proportional-Integral (PI) controllers also provide stability guarantees under
certain restricted conditions, and are always applicable as model-free control
options through gain tuning by trail and error, or by model-free optimization.
Although these approaches are mathematically elegant, the stabilization result
only holds locally and is usually affected by the change of model parameters.
Therefore, we reformulate the PDE boundary control problem as a RL problem that
pursues stabilization without knowing the system dynamics, simply by observing
the state values. The proximal policy optimization, a neural network-based
policy gradient algorithm, is employed to obtain RL controllers by interacting
with a numerical simulator of the ARZ PDE. Being stabilization-inspired, the RL
state-feedback boundary controllers are compared and evaluated against the
rigorously stabilizing controllers in two cases: (i) in a system with perfect
knowledge of the traffic flow dynamics, and then (ii) in one with only partial
knowledge. We obtain RL controllers that nearly recover the performance of the
backstepping, P, and PI controllers with perfect knowledge and outperform them
in some cases with partial knowledge
Large Scale Estimation in Cyberphysical Systems using Streaming Data: a Case Study with Smartphone Traces
Controlling and analyzing cyberphysical and robotics systems is increasingly
becoming a Big Data challenge. Pushing this data to, and processing in the
cloud is more efficient than on-board processing. However, current cloud-based
solutions are not suitable for the latency requirements of these applications.
We present a new concept, Discretized Streams or D-Streams, that enables
massively scalable computations on streaming data with latencies as short as a
second.
We experiment with an implementation of D-Streams on top of the Spark
computing framework. We demonstrate the usefulness of this concept with a novel
algorithm to estimate vehicular traffic in urban networks. Our online EM
algorithm can estimate traffic on a very large city network (the San Francisco
Bay Area) by processing tens of thousands of observations per second, with a
latency of a few seconds
Flow: A Modular Learning Framework for Autonomy in Traffic
The rapid development of autonomous vehicles (AVs) holds vast potential for
transportation systems through improved safety, efficiency, and access to
mobility. However, due to numerous technical, political, and human factors
challenges, new methodologies are needed to design vehicles and transportation
systems for these positive outcomes. This article tackles technical challenges
arising from the partial adoption of autonomy: partial control, partial
observation, complex multi-vehicle interactions, and the sheer variety of
traffic settings represented by real-world networks. The article presents a
modular learning framework which leverages deep Reinforcement Learning methods
to address complex traffic dynamics. Modules are composed to capture common
traffic phenomena (traffic jams, lane changing, intersections). Learned control
laws are found to exceed human driving performance by at least 40% with only
5-10% adoption of AVs. In partially-observed single-lane traffic, a small
neural network control law can eliminate stop-and-go traffic -- surpassing all
known model-based controllers, achieving near-optimal performance, and
generalizing to out-of-distribution traffic densities.Comment: 14 pages, 8 figures; new experiments and analysi
On the Approximability of Time Disjoint Walks
We introduce the combinatorial optimization problem Time Disjoint Walks
(TDW), which has applications in collision-free routing of discrete objects
(e.g., autonomous vehicles) over a network. This problem takes as input a
digraph with positive integer arc lengths, and pairs of vertices that
each represent a trip demand from a source to a destination. The goal is to
find a walk and delay for each demand so that no two trips occupy the same
vertex at the same time, and so that a min-max or min-sum objective over the
trip durations is realized.
We focus here on the min-sum variant of Time Disjoint Walks, although most of
our results carry over to the min-max case. We restrict our study to various
subclasses of DAGs, and observe that there is a sharp complexity boundary
between Time Disjoint Walks on oriented stars and on oriented stars with the
central vertex replaced by a path. In particular, we present a poly-time
algorithm for min-sum and min-max TDW on the former, but show that min-sum TDW
on the latter is NP-hard.
Our main hardness result is that for DAGs with max degree ,
min-sum Time Disjoint Walks is APX-hard. We present a natural approximation
algorithm for the same class, and provide a tight analysis. In particular, we
prove that it achieves an approximation ratio of on
bounded-degree DAGs, and on DAGs and bounded-degree digraphs.Comment: 20 pages; extended (full) version; preliminary version appeared in
COCOA 2018; new results in the extended version include those listed in the
second paragraph of the abstrac
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