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Boundary Control Design for Traffic With Nonlinear Dynamics | IEEE Journals & Magazine | IEEE Xplore

Boundary Control Design for Traffic With Nonlinear Dynamics


Abstract:

We address a problem of boundary control for a nonlinear scalar conservation law. Namely, this article is devoted to the boundary control of a Lighthill-Whitham-Richards ...Show More

Abstract:

We address a problem of boundary control for a nonlinear scalar conservation law. Namely, this article is devoted to the boundary control of a Lighthill-Whitham-Richards (LWR) partial differential equation (PDE) with triangular flux function evolving along a single road. The target state is a time- and space-dependent trajectory. The boundary control law is constructed using the analytical solution of the Hamilton–Jacobi (H–J) equation, which is an integral form of the LWR PDE. We design a feedback controller and illustrate its performance on a numerical example using the Godunov scheme.
Published in: IEEE Transactions on Automatic Control ( Volume: 67, Issue: 3, March 2022)
Page(s): 1301 - 1313
Date of Publication: 29 March 2021

ISSN Information:

Funding Agency:

Author image of Liudmila Tumash
Université Grenoble Alpes, CNRS, Inria, Grenoble INP, GIPSA-lab, Grenoble, France
Liudmila Tumash received both the B.Sc. and M.Sc. degrees from the Institute of Theoretical Physics, Technical University of Berlin, Berlin, Germany, specializing on control of chaotic networks.
She is currently a Doctoral Researcher with CNRS, Gipsa-Lab, Grenoble, France. Her research interest includes control of traffic systems in large-scale networks.
Ms. Tumash was the recipient of Physik-Studienpreis for the excellent ...Show More
Liudmila Tumash received both the B.Sc. and M.Sc. degrees from the Institute of Theoretical Physics, Technical University of Berlin, Berlin, Germany, specializing on control of chaotic networks.
She is currently a Doctoral Researcher with CNRS, Gipsa-Lab, Grenoble, France. Her research interest includes control of traffic systems in large-scale networks.
Ms. Tumash was the recipient of Physik-Studienpreis for the excellent ...View more
Author image of Carlos Canudas-de-Wit
Université Grenoble Alpes, CNRS, Inria, Grenoble INP, GIPSA-lab, Grenoble, France
Carlos Canudas-de-Wit (Fellow, IEEE) was born in Villahermosa, Mexico, in 1958. He received the B.S. degree in electronics and communications from the Monterrey Institute of Technology and Higher Education, Monterrey, Mexico, in 1980, and the M.S. and Ph.D. degrees in automatic control from the Grenoble Institute of Technology, Grenoble, France, in 1984 and 1987, respectively.
He is currently a Directeur de recherche (Seni...Show More
Carlos Canudas-de-Wit (Fellow, IEEE) was born in Villahermosa, Mexico, in 1958. He received the B.S. degree in electronics and communications from the Monterrey Institute of Technology and Higher Education, Monterrey, Mexico, in 1980, and the M.S. and Ph.D. degrees in automatic control from the Grenoble Institute of Technology, Grenoble, France, in 1984 and 1987, respectively.
He is currently a Directeur de recherche (Seni...View more
Author image of Maria Laura Delle Monache
Université Grenoble Alpes, Inria, CNRS, Grenoble INP, GIPSA-lab, Grenoble, France
Maria Laura Delle Monache (Member, IEEE) received the B.S. degree in industrial engineering from the University of L’Aquila, L’Aquila, Italy, the M.S. degree in mathematical engineering from the University of L’Aquila, and the University of Hamburg, Hamburg, Germany, and the Ph.D. degree in applied mathematics from the University of Nice-Sophia Antipolis, Nice, France, in 2009, 2011, and 2014, respectively.
Prior to Inria,...Show More
Maria Laura Delle Monache (Member, IEEE) received the B.S. degree in industrial engineering from the University of L’Aquila, L’Aquila, Italy, the M.S. degree in mathematical engineering from the University of L’Aquila, and the University of Hamburg, Hamburg, Germany, and the Ph.D. degree in applied mathematics from the University of Nice-Sophia Antipolis, Nice, France, in 2009, 2011, and 2014, respectively.
Prior to Inria,...View more

I. Introduction

Conservation laws are a very important topic to study nowadays, since a significant part of physical systems is based on the principle of the conservation of some quantity, for example, mass or energy. Urban and highway traffic behavior is modeled via conservation laws with the principle of preserving the number of cars. The first work describing the behavior of traffic on a macroscopic scale was proposed by Lighthill, Whitham [2], and, independently, Richards [3] in the 1950s Lighthill-Whitham-Richards (LWR) model. The LWR model is a first-order model based on the conservation of mass equation. Despite the appearance of more sophisticated first order [14]–[17] or even higher order models capable of covering more realistic traffic behavior [13], [18], the LWR model remains the most used one to study traffic behavior due to its simplicity. Mathematically, the LWR model is a hyperbolic scalar conservation law with a nonlinear concave flux function, which represents an empirical relation between flow and density, see [25] for a detailed review.

Author image of Liudmila Tumash
Université Grenoble Alpes, CNRS, Inria, Grenoble INP, GIPSA-lab, Grenoble, France
Liudmila Tumash received both the B.Sc. and M.Sc. degrees from the Institute of Theoretical Physics, Technical University of Berlin, Berlin, Germany, specializing on control of chaotic networks.
She is currently a Doctoral Researcher with CNRS, Gipsa-Lab, Grenoble, France. Her research interest includes control of traffic systems in large-scale networks.
Ms. Tumash was the recipient of Physik-Studienpreis for the excellent master’s degree.
Liudmila Tumash received both the B.Sc. and M.Sc. degrees from the Institute of Theoretical Physics, Technical University of Berlin, Berlin, Germany, specializing on control of chaotic networks.
She is currently a Doctoral Researcher with CNRS, Gipsa-Lab, Grenoble, France. Her research interest includes control of traffic systems in large-scale networks.
Ms. Tumash was the recipient of Physik-Studienpreis for the excellent master’s degree.View more
Author image of Carlos Canudas-de-Wit
Université Grenoble Alpes, CNRS, Inria, Grenoble INP, GIPSA-lab, Grenoble, France
Carlos Canudas-de-Wit (Fellow, IEEE) was born in Villahermosa, Mexico, in 1958. He received the B.S. degree in electronics and communications from the Monterrey Institute of Technology and Higher Education, Monterrey, Mexico, in 1980, and the M.S. and Ph.D. degrees in automatic control from the Grenoble Institute of Technology, Grenoble, France, in 1984 and 1987, respectively.
He is currently a Directeur de recherche (Senior Researcher) with CNRS, Grenoble, France, where he is the Leader with the NeCS Team, a joint team of GIPSA-Lab (CNRS) and INRIA, on networked controlled systems.
Dr. Canudas-de-Wit is an IFAC Fellow. He was an Associate Editor of the IEEE Transactions on Automatic Control, Automatica, and IEEE Transactions on Control Systems Technology. He is an Associate Editor of the Asian Journal of Control, and the IEEE Transactions on Control of Network Systems. He served as the President of the European Control Association from 2013 to 2015, and a member of the IEEE Board of Governors of the Control System Society from 2011 to 2014. He holds the ERC Advanced Grant Scale-FreeBack from 2016 to 2021.
Carlos Canudas-de-Wit (Fellow, IEEE) was born in Villahermosa, Mexico, in 1958. He received the B.S. degree in electronics and communications from the Monterrey Institute of Technology and Higher Education, Monterrey, Mexico, in 1980, and the M.S. and Ph.D. degrees in automatic control from the Grenoble Institute of Technology, Grenoble, France, in 1984 and 1987, respectively.
He is currently a Directeur de recherche (Senior Researcher) with CNRS, Grenoble, France, where he is the Leader with the NeCS Team, a joint team of GIPSA-Lab (CNRS) and INRIA, on networked controlled systems.
Dr. Canudas-de-Wit is an IFAC Fellow. He was an Associate Editor of the IEEE Transactions on Automatic Control, Automatica, and IEEE Transactions on Control Systems Technology. He is an Associate Editor of the Asian Journal of Control, and the IEEE Transactions on Control of Network Systems. He served as the President of the European Control Association from 2013 to 2015, and a member of the IEEE Board of Governors of the Control System Society from 2011 to 2014. He holds the ERC Advanced Grant Scale-FreeBack from 2016 to 2021.View more
Author image of Maria Laura Delle Monache
Université Grenoble Alpes, Inria, CNRS, Grenoble INP, GIPSA-lab, Grenoble, France
Maria Laura Delle Monache (Member, IEEE) received the B.S. degree in industrial engineering from the University of L’Aquila, L’Aquila, Italy, the M.S. degree in mathematical engineering from the University of L’Aquila, and the University of Hamburg, Hamburg, Germany, and the Ph.D. degree in applied mathematics from the University of Nice-Sophia Antipolis, Nice, France, in 2009, 2011, and 2014, respectively.
Prior to Inria, she was a Postdoctoral Researcher with Rutgers University Camden, Camden, NJ, USA. She is a Research Scientist with the Networked Controlled Systems Team, Inria, and with the Department of Control, GIPSA-Lab, Grenoble, France. Her research interests includes mathematical and engineering aspects of traffic flow and mathematical modeling and control of traffic flow applications.
Maria Laura Delle Monache (Member, IEEE) received the B.S. degree in industrial engineering from the University of L’Aquila, L’Aquila, Italy, the M.S. degree in mathematical engineering from the University of L’Aquila, and the University of Hamburg, Hamburg, Germany, and the Ph.D. degree in applied mathematics from the University of Nice-Sophia Antipolis, Nice, France, in 2009, 2011, and 2014, respectively.
Prior to Inria, she was a Postdoctoral Researcher with Rutgers University Camden, Camden, NJ, USA. She is a Research Scientist with the Networked Controlled Systems Team, Inria, and with the Department of Control, GIPSA-Lab, Grenoble, France. Her research interests includes mathematical and engineering aspects of traffic flow and mathematical modeling and control of traffic flow applications.View more

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