. . Highway Traffic Models, Estimation, and Control Summer School HYCON2 on Complex Systems Trento, Italy C ARLOS C ANUDAS - DE -W IT , Director of research at the CNRS Control System Department, GIPSA-Lab, NeCS INRIA/CNRS Team, France carlos.canudas-de-wit@gipsa-lab.grenoble-inp.fr http://necs.inrialpes.fr/ June 19, 2011 C ARLOS C ANUDAS - DE -W IT , Director of research at the CNRS Control System Department, GIPSA-Lab, NeCS INRIA/CNRS Team, Highway Traffic Models, Estimation, and Control Summer School HYCON2 on Complex June 19, 2011 1 / 50
Models 1 Conservation laws Fundamentals: Observability Fundamentals: Controllability Fundamentals for discretization Inputs/output ramps Variable speed Control 2 Problem formulation Variable-length nonlinear model Actuator operation Best effort control Dynamic constrained Best effort control Traffic state observation 3 Model parametrization Observer architecture Case study: one cell density estimation Density estimation for a highway segment Simulation results C ARLOS C ANUDAS - DE -W IT , Director of research at the CNRS Control System Department, GIPSA-Lab, NeCS INRIA/CNRS Team, Highway Traffic Models, Estimation, and Control Summer School HYCON2 on Complex June 19, 2011 2 / 50
Conservation law in traffic flows N = Vehicles/section [Veh] ϕ ( x , t ) - Flow [Veh/hr] ρ ( x , t ) - Density [Veh/Km] Conservation Law: d d t N = − , ϕ in ϕ out ���� ���� ���� inflows outflows vehicle rate in the cell with N being the number of vehicle in the section [ 0 , L ] � L N = 0 ρ ( x , t ) d x C ARLOS C ANUDAS - DE -W IT , Director of research at the CNRS Control System Department, GIPSA-Lab, NeCS INRIA/CNRS Team, Highway Traffic Models, Estimation, and Control Summer School HYCON2 on Complex June 19, 2011 3 / 50
Conservation law in traffic flows–cont. Fundamental diagram: ϕ = Φ( ρ ) . ϕ in ( x , t ) = Φ( ρ ( 0 , t )) ϕ out ( x , t ) = Φ( ρ ( L , t )) � L d 0 ρ ( x , t ) d x = Φ( ρ ( 0 , t )) − Φ( ρ ( L , t )) d t Assuming ρ and Φ be derivable (in some sense), then � L � L � L d 0 ρ ( x , t ) d x = 0 ∂ t ρ ( x , t ) d x , Φ( ρ ( 0 , t )) − Φ( ρ ( L , t )) = 0 ∂ x Φ( ρ ) d x d t Conservation law turn out into a hyperbolic PDE: ∂ t ρ + ∂ x Φ( ρ ) = ∂ t ρ +Φ � ( ρ ) ∂ x ρ = 0 , ρ I = ρ ( x , 0 ) . C ARLOS C ANUDAS - DE -W IT , Director of research at the CNRS Control System Department, GIPSA-Lab, NeCS INRIA/CNRS Team, Highway Traffic Models, Estimation, and Control Summer School HYCON2 on Complex June 19, 2011 4 / 50
Qualitative Stable/unstable behaviors Stable area for positive slopes, i.e. Φ � ( ρ ) > 0. The section is under φ = Φ( ρ ) [veh/h/lane] v f free flowing conditions and Critical maximum allowed speed is C o n g reached e s Free t e d Unstable area for negative slopes, Jammed i.e. Φ � ( ρ ) < 0. The section is Stopped ρ congested, Jammed or eventually meta-stable meta-stable [veh/km/lane] unstable stopped. stable stable Critical stable The road operates closed to its critical density where Figure from [7] Φ � ( ρ ) ≈ 0. C ARLOS C ANUDAS - DE -W IT , Director of research at the CNRS Control System Department, GIPSA-Lab, NeCS INRIA/CNRS Team, Highway Traffic Models, Estimation, and Control Summer School HYCON2 on Complex June 19, 2011 5 / 50
Quantitative behaviors: the Riemann problem with constant advection velocities ( v , − w ) Solutions for: ∂ t ρ +Φ � ( ρ ) ∂ x ρ = 0, under: � ρ L for x ≤ x 0 ρ ( x , 0 ) = for x > x 0 ρ R Stable area has positive slope, Unstable area has negative slope, i.e. Φ � ( ρ ) = v > 0 i.e. Φ � ( ρ ) = − w < 0 ∂ t ρ + v ∂ x ρ = 0 , ρ I = ρ ( x , 0 ) ∂ t ρ − w ∂ x ρ = 0 , ρ I = ρ ( x , 0 ) ρ ( x , t ) = ρ ( x − vt , 0 ) ρ ( x , t ) = ρ ( x + wt , 0 ) C ARLOS C ANUDAS - DE -W IT , Director of research at the CNRS Control System Department, GIPSA-Lab, NeCS INRIA/CNRS Team, Highway Traffic Models, Estimation, and Control Summer School HYCON2 on Complex June 19, 2011 6 / 50
Solution representation: method of the characteristics Characteristics are curves (invariants) in the ( x − t ) -space along which the density is constant The characteristic curves of the LWR equation are of the form � x = t / v + x 0 1 if free-flow x = Φ � ( ρ I ( 0 )) t + x 0 = ⇒ ���� x = − t / w + x 0 if congested CTM-model C ARLOS C ANUDAS - DE -W IT , Director of research at the CNRS Control System Department, GIPSA-Lab, NeCS INRIA/CNRS Team, Highway Traffic Models, Estimation, and Control Summer School HYCON2 on Complex June 19, 2011 7 / 50
Observability (fundamentals) Downstream observability in free flow mode via downstream measurements y d Upstream observability in congested mode via upstream measurements y u C ARLOS C ANUDAS - DE -W IT , Director of research at the CNRS Control System Department, GIPSA-Lab, NeCS INRIA/CNRS Team, Highway Traffic Models, Estimation, and Control Summer School HYCON2 on Complex June 19, 2011 8 / 50
Flux Controllability (fundamentals) Downstream controllability in free flow mode via upstream boundary control u u Upstream controllability in congested mode via downstream boundary control u d C ARLOS C ANUDAS - DE -W IT , Director of research at the CNRS Control System Department, GIPSA-Lab, NeCS INRIA/CNRS Team, Highway Traffic Models, Estimation, and Control Summer School HYCON2 on Complex June 19, 2011 9 / 50
Numerical discretization: Godunov scheme for LWR links The section is divided in a finite number of cells ρ − , ρ + Each cell is assumed to have constant density Principle To solve a succession of local Riemann problems, ρ Local Riemann Problem x x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 or, equivalently, finding the numerical flux Φ( ρ − , ρ + ) , at the cell interface. Φ( ρ − , ρ + ) = Φ( ρ ∗ ) where ρ ∗ = value of the Riemann problem solution. C ARLOS C ANUDAS - DE -W IT , Director of research at the CNRS Control System Department, GIPSA-Lab, NeCS INRIA/CNRS Team, Highway Traffic Models, Estimation, and Control Summer School HYCON2 on Complex June 19, 2011 10 / 50
Godunov explicit solution(LeVeque, 1992)–cont. Explicit Solution : Φ( ρ − , ρ + ) = Φ( ρ ∗ ) , F= Free, C= Congested. F: Φ � ( ρ − ) ≥ 0 C: Φ � ( ρ − ) < 0 Left/Right � ρ − if Φ( ρ + ) − Φ( ρ − ) > 0 F: Φ � ( ρ − ) ≥ 0 ρ ∗ = ρ − ρ ∗ = ρ + else C: Φ � ( ρ − ) < 0 ρ ∗ = argmax Φ( · ) ρ ∗ = ρ + FF: All characteristics moves forward. Interface flow is determined by its left condition. CC: All characteristics moves backward. Interface flow is determined by its right condition. FC: shock occurs. Sign of the chock speed defines ρ ∗ . CF: rarefaction wave (sonic point). Maximum flow applies. C ARLOS C ANUDAS - DE -W IT , Director of research at the CNRS Control System Department, GIPSA-Lab, NeCS INRIA/CNRS Team, Highway Traffic Models, Estimation, and Control Summer School HYCON2 on Complex June 19, 2011 11 / 50
Godunov discrete-time dynamic equation � − � ρ i ( k + 1 ) = ρ i ( k )+ ∆ t → Φ i ( k ) − ← − Φ i + 1 ( k ) , ∆ x i In-flow: − → Φ i ( k ) = Φ( ρ i − 1 ( k ) , ρ i ( k )) , Out-flow: ← − Φ i + 1 ( k ) = Φ( ρ i ( k ) , ρ i + 1 ( k )) Interface flows a = Φ( ρ i ( k )) − Φ( ρ i − 1 ( k )) , b = Φ( ρ i + 1 ( k )) − Φ( ρ i ( k )) Interface FF CC FC CF − → Φ( ρ i − 1 ( k )) a > 0 Φ i ( k ) = Φ( ρ i − 1 ( k )) Φ( ρ i ( k )) Φ max Φ( ρ i ( k )) else ← − Φ( ρ i ( k )) b > 0 Φ i + 1 ( k ) = Φ( ρ i ( k )) Φ( ρ i + 1 ( k )) Φ max Φ( ρ i + 1 ( k )) else C ARLOS C ANUDAS - DE -W IT , Director of research at the CNRS Control System Department, GIPSA-Lab, NeCS INRIA/CNRS Team, Highway Traffic Models, Estimation, and Control Summer School HYCON2 on Complex June 19, 2011 12 / 50
Recommend
More recommend
