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2018 | Book

Feedback Control Theory for Dynamic Traffic Assignment

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About this book

This book develops a methodology for designing feedback control laws for dynamic traffic assignment (DTA) exploiting the introduction of new sensing and information-dissemination technologies to facilitate the introduction of real-time traffic management in intelligent transportation systems. Three methods of modeling the traffic system are discussed:partial differential equations representing a distributed-parameter setting;continuous-time ordinary differential equations (ODEs) representing a continuous-time lumped-parameter setting; anddiscreet-time ODEs representing a discrete-time lumped-parameter setting.Feedback control formulations for reaching road-user-equilibrium are presented for each setting and advantages and disadvantage of using each are addressed. The closed-loop methods described are proposed expressly to avoid the counter-productive shifting of bottlenecks from one route to another because of driver over-reaction to routing information.

The second edition of Feedback Control Theory for Dynamic Traffic Assignment has been thoroughly updated with completely new chapters:a review of the DTA problem and emphasizing real-time-feedback-based problems;an up-to-date presentation of pertinent traffic-flow theory; anda treatment of the mathematical solution to the traffic dynamics.

Techinques accounting for the importance of entropy are further new inclusions at various points in the text.

Researchers working in traffic control will find the theoretical material presented a sound basis for further research; the continual reference to applications will help professionals working in highway administration and engineering with the increasingly important task of maintaining and smoothing traffic flow; the extensive use of end-of-chapter exercises will help the graduate student and those new to the field to extend their knowledge.

Table of Contents

Frontmatter

Introduction to Feedback Control and Dynamic Traffic Assignment

Frontmatter
Chapter 1. Introduction
Abstract
This chapter provides the motivation behind using feedback control for dynamic traffic diversion and dynamic traffic assignment. It then presents some preliminary ideas about feedback control methodology and describes how traffic control systems are set up for traffic management purposes.
Pushkin Kachroo, Kaan M. A. Özbay
Chapter 2. Traffic Assignment: A Survey of Mathematical Models and Techniques
Abstract
This chapter presents the fundamentals of the theory and techniques of traffic assignment problem. It first presents the steady-state traffic assignment problem formulation which is also called static assignment, followed by Dynamic Traffic Assignment (DTA), where the traffic demand on the network is time varying. The static assignment problem is shown in a mathematical programming setting for two different objectives to be satisfied. The first one where all users experience same travel times in alternate used routes is called user-equilibrium and another setting called system optimum in which the assignment attempts to minimize the total travel time. The alternate formulation uses variational inequality method which is also presented. Dynamic travel routing problem is also reviewed in the variational inequality setting. DTA problem is shown in discrete and continuous time in terms of lumped parameters as well as in a macroscopic setting, where partial differential equations are used for the link traffic dynamics. A Hamilton–Jacobi- based travel time dynamics model is also presented for the links and routes, which is integrated with the macroscopic traffic dynamics. Simulation-based DTA method is also very briefly reviewed. This chapter is taken from the following Springer publication and is reproduced here, with permission and with minor changes: Pushkin Kachroo, and Neveen Shlayan, “Dynamic traffic assignment: A survey of mathematical models and technique,” Advances in Dynamic Network Modeling in Complex Transportation Systems (Editor: Satish V. Ukkusuri and Kaan Özbay) Springer New York, 2013. 1-25.
Pushkin Kachroo, Kaan M. A. Özbay

Traffic Flow Theory and Traffic Assignment Modeling

Frontmatter
Chapter 3. Traffic Flow Theory
Abstract
This chapter presents the basic traffic flow theory which is used in the following chapters for control problem formulations. The theory develops the Lighthill–Whitham–Richards (LWR) model that uses the conservation law for traffic. Additionally, a density-dependent speed formula is used. There are many relationships available for this fundamental diagram, the chapter uses Greenshields’ formula for further analysis. Elementary partial differential equations (PDE) theory is also presented including the method of characteristics needed for the analysis of the traffic model. Shockwaves and weak solutions are discussed followed by a brief discussion of traffic measurements.
Pushkin Kachroo, Kaan M. A. Özbay
Chapter 4. Modeling and Problem Formulation
Abstract
This chapter provides the system dynamics model for control design in continuous and discrete time and space variables for traffic flow and also develops the basic framework for the control design. The problem is formulated as a feedback control design for the traffic as a distributed parameter system, i.e., in terms of Partial Differential Equations (PDE). DTR formulation for two alternate routes and then generalized for n routes is developed. Space discretization of the model results in Ordinary Differential Equation (ODE) representation, and its further time discretization produces difference equations. System dynamic equations for sample problems are developed, and a simple control law is also shown with computer simulation results.
Pushkin Kachroo, Kaan M. A. Özbay

Feedback Control for Dynamic Traffic Routing

Frontmatter
Chapter 5. Dynamic Routing Problem in Distributed Parameter Setting
Abstract
The aim of this chapter is to design control law for the DTR (Dynamic Traffic Routing) problem modeled in the distributed parameter setting. For the control design, the chapter uses the sliding mode control technique for regulating the error. Sliding mode control provides a robust method against bounded uncertainties. The price to pay for that robustness is chattering. The chapter shows methods to deal with chattering reduction in the control implementation. Control design and its software simulation using sliding mode control are presented in this chapter. The chapter also provides a simple study of discretization errors that are obtained in the numerical approximation of the distributed model for software simulation. The chapter shows the development of a simple software simulation code and its simulation to study this problem.
Pushkin Kachroo, Kaan M. A. Özbay
Chapter 6. Dynamic Routing Problem in Distributed Parameter Setting Using Semigroup Theory
Abstract
The aim of this chapter is to present the background information on semigroup theory and evolution equations and motivate their use in the design of controllers, especially feedback controllers, for the dynamic traffic routing and assignment problems. The chapter presents the highway model in the semigroup context and also provides a starting point for traffic routing control design using the semigroup theory. In order to present this framework, this chapter presents the fundamentals of the applicable functional analysis from the ground up. It reviews the topics of topological spaces, vector spaces, metric spaces, normed linear spaces, banach and Hilbert spaces, semigroups and finally how the routing problem can be viewed in this framework as an abstract differential controlled equation.
Pushkin Kachroo, Kaan M. A. Özbay
Chapter 7. Fuzzy Feedback Control for Dynamic Routing Problem
Abstract
The aim of this chapter is to provide a review of fuzzy logic fundamentals for control design and then to show a design of a fuzzy feedback control law for a sample DTR problem. The fundamentals of fuzzy set theory are first presented, and then fuzzy logic is covered in terms of the fuzzy sets. Development of fuzzy control involves the concepts of fuzzification, fuzzy logic based inference engine, and defuzzification. These concepts are explained and finally a fuzzy logic based controller is designed for a routing problem and a computer simulation performed for illustration purposes.
Pushkin Kachroo, Kaan M. A. Özbay
Chapter 8. Feedback Control for Dynamic Traffic Routing in Lumped Parameter Setting
Abstract
The aim of this chapter is to solve the point diversion problem between two nodes using feedback control where the dynamics of the system are written in the lumped parameter form, i.e., in terms of Ordinary Differential Equations (ODE). Chapter presents the system dynamics model for the routing problem as a space discretized version of the LWR traffic model. This model is also a direct consequence of the conservation law in the space discretized setting. The nonlinear control method of feedback linearization is presented and it is shown how that can be utilized to develop feedback traffic routing controllers. Two routes with one section each, followed by multiple sections, as well as multiple routes each with multiple sections are studied for the routing control design, and simulations are performed to show the results of the controller. Sliding mode control is used for the model with uncertainties.
Pushkin Kachroo, Kaan M. A. Özbay
Chapter 9. Feedback Control for Network-Level Dynamic Traffic Routing
Abstract
The aim of this chapter is to develop models for network-level traffic systems and, moreover, to design feedback controllers for network-level traffic problems in user-equilibrium as well as system-optimal settings. In order to do so, the chapter develops the network-level traffic dynamics for the traffic assignment problem and also shows the objective functions for the user-equilibrium and the system-optimal cases. Link-based and route-based models are presented from literature. Dynamic traffic assignment problem is formulated in the structure of an \(H_{\infty }\) control problem, so that the solution techniques available for the controller can be applied to the traffic assignment problem. A sample problem is presented and control design steps for that problem are presented.
Pushkin Kachroo, Kaan M. A. Özbay
Chapter 10. Feedback Routing via Congestion Pricing
Abstract
This chapter addresses a control design for performing dynamic congestion pricing as a method to perform traffic assignment to achieve certain objective. The design uses the methodology of optimal control theory. The formulation allows for modeling tolled and non-tolled lanes or routes. A logit model connects the toll price with the driver choice behavior. A feedback optimal tolling control law is designed based on deriving the corresponding Hamilton–Jacobi–Bellman equation for the model of the system. Simulations are also presented to illustrate the working of the control design. Some of the content of this chapter has been adapted from the following paper: \(\copyright \) 2016 IEEE. Reprinted, with permission, from: Kachroo P, Gupta S, Agarwal S, Özbay K., “Optimal Control for Congestion Pricing: Theory, Simulation, and Evaluation,” IEEE Transactions on Intelligent Transportation Systems. 2017 May; 18(5):1234–40.
Pushkin Kachroo, Kaan M. A. Özbay
Backmatter
Metadata
Title
Feedback Control Theory for Dynamic Traffic Assignment
Authors
Dr. Pushkin Kachroo
Dr. Kaan M.A. Özbay
Copyright Year
2018
Electronic ISBN
978-3-319-69231-9
Print ISBN
978-3-319-69229-6
DOI
https://doi.org/10.1007/978-3-319-69231-9