Alexandre M. Bayen

A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games

We describe and implement an algorithm for computing the set of reachable states of a continuous dynamic game. The algorithm is based on a proof that the reachable set is the zero sublevel set of the viscosity solution of a particular time-dependent Hamilton-Jacobi-Isaacs partial differential equation. While alternative techniques for computing the reachable set have been proposed, the differential game formulation allows treatment of nonlinear systems with inputs and uncertain parameters. Because the time-dependent equations solution is continuous and defined throughout the state space, methods from the level set literature can be used to generate more accurate approximations than are possible for formulations with potentially discontinuous solutions. A numerical implementation of our formulation is described and has been released on the web. Its correctness is verified through a two vehicle, three dimensional collision avoidance example for which an analytic solution is available.

Virtual trip lines for distributed privacy-preserving traffic monitoring

Automotive traffic monitoring using probe vehicles with Global Positioning System receivers promises significant improvements in cost, coverage, and accuracy. Current approaches, however, raise privacy concerns because they require participants to reveal their positions to an external traffic monitoring server. To address this challenge, we propose a system based on virtual trip lines and an associated cloaking technique. Virtual trip lines are geographic markers that indicate where vehicles should provide location updates. These markers can be placed to avoid particularly privacy sensitive locations. They also allow aggregating and cloaking several location updates based on trip line identifiers, without knowing the actual geographic locations of these trip lines. Thus they facilitate the design of a distributed architecture, where no single entity has a complete knowledge of probe identities and fine-grained location information. We have implemented the system with GPS smartphone clients and conducted a controlled experiment with 20 phone-equipped drivers circling a highway segment. Results show that even with this low number of probe vehicles, travel time estimates can be provided with less than 15% error, and applying the cloaking techniques reduces travel time estimation accuracy by less than 5% compared to a standard periodic sampling approach.

Computational techniques for the verification of hybrid systems

Hybrid system theory lies at the intersection of the fields of engineering control theory and computer science verification. It is defined as the modeling, analysis, and control of systems that involve the interaction of both discrete state systems, represented by finite automata, and continuous state dynamics, represented by differential equations. The embedded autopilot of a modern commercial jet is a prime example of a hybrid system: the autopilot modes correspond to the application of different control laws, and the logic of mode switching is determined by the continuous state dynamics of the aircraft, as well as through interaction with the pilot. To understand the behavior of hybrid systems, to simulate, and to control these systems, theoretical advances, analyses, and numerical tools are needed. In this paper, we first present a general model for a hybrid system along with an overview of methods for verifying continuous and hybrid systems. We describe a particular verification technique for hybrid systems, based on two-person zero-sum game theory for automata and continuous dynamical systems. We then outline a numerical implementation of this technique using level set methods, and we demonstrate its use in the design and analysis of aircraft collision avoidance protocols and in verification of autopilot logic.

Understanding Road Usage Patterns in Urban Areas

In this paper, we combine the most complete record of daily mobility, based on large-scale mobile phone data, with detailed Geographic Information System (GIS) data, uncovering previously hidden patterns in urban road usage. We find that the major usage of each road segment can be traced to its own - surprisingly few - driver sources. Based on this finding we propose a network of road usage by defining a bipartite network framework, demonstrating that in contrast to traditional approaches, which define road importance solely by topological measures, the role of a road segment depends on both: its betweeness and its degree in the road usage network. Moreover, our ability to pinpoint the few driver sources contributing to the major traffic flow allows us to create a strategy that achieves a significant reduction of the travel time across the entire road system, compared to a benchmark approach.

Viability Theory: New Directions

Viability theory designs and develops mathematical and algorithmic methods for investigating the adaptation to viability constraints of evolutions governed by complex systems under uncertainty that are found in many domains involving living beings, from biological evolution to economics, from environmental sciences to financial markets, from control theory and robotics to cognitive sciences. It involves interdisciplinary investigations spanning fields that have traditionally developed in isolation. The purpose of this book is to present an initiation to applications of viability theory, explaining and motivating the main concepts and illustrating them with numerous numerical examples taken from various fields.

An ensemble Kalman filtering approach to highway traffic estimation using GPS enabled mobile devices

Traffic state estimation is a challenging problem for the transportation community due to the limited deployment of sensing infrastructure. However, recent trends in the mobile phone industry suggest that GPS equipped devices will become standard in the next few years. Leveraging these GPS equipped devices as traffic sensors will fundamentally change the type and the quality of traffic data collected on large scales in the near future. New traffic models and data assimilation algorithms must be developed to efficiently transform this data into usable traffic information. In this work, we introduce a new partial differential equation (PDE) based on the Lighthill-Whitham-Richards PDE, which serves as a flow model for velocity. We formulate a Godunov discretization scheme to cast the PDE into a Velocity Cell Transmission Model (CTM-v), which is a nonlinear dynamical system with a time varying observation matrix. The Ensemble Kalman Filtering (EnKF) technique is applied to the CTM- v to estimate the velocity field on the highway using data obtained from GPS devices, and the method is illustrated in microsimulation on a fully calibrated model of I880 in California. Experimental validation is performed through the unprecedented 100-vehicle Mobile Century experiment, which used a novel privacy-preserving traffic monitoring system to collect GPS cell phone data specifically for this research.

Cyber Security of Water SCADA Systems—Part I: Analysis and Experimentation of Stealthy Deception Attacks

This brief aims to perform security threat assessment of networked control systems with regulatory and supervisory control layers. We analyze the performance of a proportional-integral controller (regulatory layer) and a model-based diagnostic scheme (supervisory layer) under a class of deception attacks. We adopt a conservative approach by assuming that the attacker has knowledge of: 1) the system dynamics; 2) the parameters of the diagnostic scheme; and 3) the sensor-control signals. The deception attack presented here can enable remote water pilfering from automated canal systems. We also report a field-operational test attack on the Gignac canal system located in Southern France.

Learning the Dynamics of Arterial Traffic From Probe Data Using a Dynamic Bayesian Network

Estimating and predicting traffic conditions in arterial networks using probe data has proven to be a substantial challenge. Sparse probe data represent the vast majority of the data available on arterial roads. This paper proposes a probabilistic modeling framework for estimating and predicting arterial travel-time distributions using sparsely observed probe vehicles. We introduce a model based on hydrodynamic traffic theory to learn the density of vehicles on arterial road segments, illustrating the distribution of delay within a road segment. The characterization of this distribution is essentially to use probe vehicles for traffic estimation: Probe vehicles report their location at random locations, and the travel times between location reports must be properly scaled to match the map discretization. A dynamic Bayesian network represents the spatiotemporal dependence on the network and provides a flexible framework to learn traffic dynamics from historical data and to perform real-time estimation with streaming data. The model is evaluated using data from a fleet of 500 probe vehicles in San Francisco, CA, which send Global Positioning System (GPS) data to our server every minute. The numerical experiments analyze the learning and estimation capabilities on a subnetwork with more than 800 links. The sampling rate of the probe vehicles does not provide detailed information about the location where vehicles encountered delay or the reason for any delay (i.e., signal delay, congestion delay, etc.). The model provides an increase in estimation accuracy of 35% when compared with a baseline approach to process probe-vehicle data.

Estimating arterial traffic conditions using sparse probe data

Estimating and predicting traffic conditions in arterial networks using probe data has proven to be a substantial challenge. In the United States, sparse probe data represents the vast majority of the data available on arterial roads in most major urban environments. This article proposes a probabilistic modeling framework for estimating and predicting arterial travel time distributions using sparsely observed probe vehicles. We evaluate our model using data from a fleet of 500 taxis in San Francisco, CA, which send GPS data to our server every minute. The sampling rate does not provide detailed information about where vehicles encountered delay or the reason for any delay (i.e. signal delay, congestion delay, etc.). Our model provides an increase in estimation accuracy of 35% when compared to a baseline approach for processing probe vehicle data.

Lax–Hopf Based Incorporation of Internal Boundary Conditions Into Hamilton–Jacobi Equation. Part I: Theory

This article proposes a new approach for computing a semi-explicit form of the solution to a class of Hamilton-Jacobi (HJ) partial differential equations (PDEs), using control techniques based on viability theory. We characterize the epigraph of the value function solving the HJ PDE as a capture basin of a target through an auxiliary dynamical system, called ¿characteristic system¿. The properties of capture basins enable us to define components as building blocks of the solution to the HJ PDE in the Barron/Jensen-Frankowska sense. These components can encode initial conditions, boundary conditions, and internal ¿boundary¿ conditions, which are the topic of this article. A generalized Lax-Hopf formula is derived, and enables us to formulate the necessary and sufficient conditions for a mixed initial and boundary conditions problem with multiple internal boundary conditions to be well posed. We illustrate the capabilities of the method with a data assimilation problem for reconstruction of highway traffic flow using Lagrangian measurements generated from Next Generation Simulation (NGSIM) traffic data.