Highway Traffic Models, Estimation, and Control Summer School HYCON2 - - PowerPoint PPT Presentation

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Highway Traffic Models, Estimation, and Control Summer School HYCON2 on Complex Systems Trento, Italy

CARLOS CANUDAS-DE-WIT, 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

CARLOS CANUDAS-DE-WIT, 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

1

Models Conservation laws Fundamentals: Observability Fundamentals: Controllability Fundamentals for discretization Inputs/output ramps

2

Variable speed Control Problem formulation Variable-length nonlinear model Actuator operation Best effort control Dynamic constrained Best effort control

3

Traffic state observation Model parametrization Observer architecture Case study: one cell density estimation Density estimation for a highway segment Simulation results

CARLOS CANUDAS-DE-WIT, 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 dt N

• vehicle rate in the cell

= ϕin

• inflows

− ϕout

• utflows

, with N being the number of vehicle in the section [0,L] N =

L

0 ρ(x,t)dx

CARLOS CANUDAS-DE-WIT, 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.

ϕin(x,t) = Φ(ρ(0,t)) ϕout(x,t) = Φ(ρ(L,t)) d dt

L

0 ρ(x,t)dx = Φ(ρ(0,t))−Φ(ρ(L,t))

Assuming ρ and Φ be derivable (in some sense), then Fundamental diagram: ϕ = Φ(ρ). d dt

L

0 ρ(x,t)dx =

L

0 ∂tρ(x,t)dx,

Φ(ρ(0,t))−Φ(ρ(L,t)) =

L

0 ∂xΦ(ρ)dx

Conservation law turn out into a hyperbolic PDE: ∂tρ +∂xΦ(ρ) = ∂tρ +Φ(ρ)∂xρ = 0, ρI = ρ(x,0) .

CARLOS CANUDAS-DE-WIT, 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

Critical C

• n

g e s t e d Jammed Stopped vf φ = Φ(ρ) ρ [veh/h/lane] stable meta-stable unstable stable meta-stable [veh/km/lane] Free

Figure from [7]

Stable area for positive slopes, i.e. Φ(ρ) > 0. The section is under free flowing conditions and maximum allowed speed is reached Unstable area for negative slopes, i.e. Φ(ρ) < 0. The section is congested, Jammed or eventually stopped. Critical stable The road operates closed to its critical density where Φ(ρ) ≈ 0.

CARLOS CANUDAS-DE-WIT, 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: ∂tρ +Φ(ρ)∂xρ = 0, under: ρ(x,0) = ρL for x ≤ x0 ρR for x > x0 Stable area has positive slope, i.e. Φ(ρ) = v > 0 ∂tρ +v∂xρ = 0, ρI = ρ(x,0) ρ(x,t) = ρ(x −vt,0) Unstable area has negative slope, i.e. Φ(ρ) = −w < 0 ∂tρ −w∂xρ = 0, ρI = ρ(x,0) ρ(x,t) = ρ(x +wt,0)

CARLOS CANUDAS-DE-WIT, 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 = 1 Φ(ρI(0))t +x0 = ⇒

• CTM-model

x = t/v +x0 if free-flow x = −t/w +x0 if congested

CARLOS CANUDAS-DE-WIT, 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 yd Upstream observability in congested mode via upstream measurements yu

  

 

    

CARLOS CANUDAS-DE-WIT, 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 uu Upstream controllability in congested mode via downstream boundary control ud

  

 

 



CARLOS CANUDAS-DE-WIT, 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,

x ρ x8 x7 x6 x5 x4 x3 x2 x1 x0 Local Riemann Problem

 

• r, 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.

CARLOS CANUDAS-DE-WIT, 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.

Left/Right F: Φ(ρ−) ≥ 0 C: Φ(ρ−) < 0 F: Φ(ρ−) ≥ 0 ρ∗ = ρ− ρ∗ = ρ− ifΦ(ρ+)−Φ(ρ−) > 0 ρ+ else C: Φ(ρ−) < 0 ρ∗ = argmaxΦ(·) 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.

CARLOS CANUDAS-DE-WIT, 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

ρi(k +1) = ρi(k)+ ∆t ∆xi − → Φ i(k)− ← − Φ i+1(k)

• ,

  

In-flow:− → Φ i(k) = Φ(ρi−1(k),ρi(k)), Out-flow: ← − Φ i+1(k) = Φ(ρi(k),ρi+1(k))

Interface flows a = Φ(ρi(k))−Φ(ρi−1(k)), b = Φ(ρi+1(k))−Φ(ρi(k))

Interface FF CC FC CF − → Φ i(k) = Φ(ρi−1(k)) Φ(ρi(k)) Φ(ρi−1(k)) a > 0 Φ(ρi(k)) else Φmax ← − Φ i+1(k) = Φ(ρi(k)) Φ(ρi+1(k)) Φ(ρi(k)) b > 0 Φ(ρi+1(k)) else Φmax

CARLOS CANUDAS-DE-WIT, 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

Cell Transmission Model: Demand/Supply formulation



 





 

   





   







  

Fundamental diagram Deman/Suply functions

CTM model is build with a piecewise linear function Φ

ρi(k +1) = ρi(k)+ ∆t ∆xi

• ϕi(k)−ϕi+1(k)
• ϕi

= min{Di−1,Si} with Di−1 = min{vi−1ρi−1,ϕm,i−1},Si = min{ϕm,i,wi(ρm,i −ρi)}.

CARLOS CANUDAS-DE-WIT, 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 13 / 50

Cell Model with inputs/output ramps

  

 

   

s = outflows, r = inflows, β = split ratio

ρi(k +1) = ρi(k)+ ∆t ∆xi

• ϕ+

i (k)−ϕ− i (k)

• ϕ+

i

= 1 1−β min{(1−β)Di−1,Si} ϕ−

i

= min{Di +r,Si+1}−r with Di−1 = min{vi−1ρi−1,ϕm,i−1},Si = min{ϕm,i,wi(ρm,i −ρi)}.

CARLOS CANUDAS-DE-WIT, 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 14 / 50

The CTM model can be captured by a finite set of linear systems with a state-dependent switching function s(k). x(k +1) = As(k)x(k)+Bs(k)u(k)+ ˜ Bs(k)ρm y(k) = Cs(k)x(k)+ Cs(k)ρm s(k) = f(x(k),u(k)) where x ∈ RN = density vectors u ∈ Rm= includes boundary and ramp-input fluxes ρm = (ρm1,1,...,ρmN,N) {As(k),Bs(k), Bs(k),Cs(k), C} set of matrices indexed by s(k) s(k) switching variable.

CARLOS CANUDAS-DE-WIT, 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 15 / 50

Model complexity in a homogenous cell–unconstrained

     

         

Example with 3 cells. Unconstrained: all combinations are considered 2N −1, subsystems s ∈ {1,2....8} FFF

• FFC

FCF CFF FCC CFC CCF CCC

• CARLOS CANUDAS-DE-WIT, 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 16 / 50

Model complexity in a homogenous cell–constrained

 

   

 

            

Example with 3 cells. Constrained: unfeasible physical combination are pruned. N +1, subsystems s ∈ {1,2,3,4}, FFF FFC FCC CCC s(k +1) = s(k)+    −1 1

CARLOS CANUDAS-DE-WIT, 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 17 / 50

Control strategies for traffic control

Possible control strategies:

ramp metering regulation [4, 5, 8, 9, 10] variable speed limit control [6, 14]. combination of the both control strategies: [1, 15, 17]. This talk: Formulate and solve the problem of controlling the congestion front in a single link road section using variable speed limitation control.

Variable speed limit control–strategy

a variable-length two-cell lumped model, composed of one congested cell and another in free flow. design a simple “best-effort” controller that regulates (at its best) the congestion front to some pre-specified value. The control is designed under constraints concerning magnitude step changes and dwell-time.

CARLOS CANUDAS-DE-WIT, 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 18 / 50

Front congestion control problem via VSL control

The front congestion control problem consists at regulating the front congestion to some pre-specified value in order to avoid that the congestion overspread upstream blocking other exit ramps (producing even largest congestion conditions), or to reach critical safety sections (i.e. tunnels, intersections, etc.).

CARLOS CANUDAS-DE-WIT, 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 19 / 50

Two-cell variable-length model

 

˙ Nf = min{ϕin,Sf}−min{Df,Sc} (1) ˙ Nc = min{Df,Sc}−min{Dc,ϕout} (2) Demand and supply functions for the free flow and the congested cell are Df = min{vf ρf,ϕm(vf)}, Dc = min{vcρc,ϕm(vc)}, Sf = min{ϕm(vf ),wf(ρm −ρf)}, Sc = min{ϕm(vc),wc(ρm −ρc)}

CARLOS CANUDAS-DE-WIT, 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 20 / 50

Assumptions

Assumptions: The downstream cell is congested while the upstream cell is free. 0 ≤ ρf ≤ ρ∗, ρ∗ < ρc ≤ ρm From the these equations, and assuming that ϕin ≤ ϕm, ϕout ≤ ϕm, then min{ϕin,Sf} = ϕin min{Dc,ϕout} = ϕout

The model (1)-(2) simplifies to:

˙ Nf = ϕin −min{Df,Sc} ˙ Nc = min{Df,Sc}−ϕout

CARLOS CANUDAS-DE-WIT, 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 21 / 50

Variation law for the congestion line front [?]

˙ l = c

• ϕ−

c −ϕ+ c

• = c (Df −Sc)

(3) where c ≈ 1/ρm [m/Vehicle] is a constant describing the mean spatial

• ccupance per vehicle on the section.

 

• Remark. The equation (3) can be seen as a simplification of the

Rankine-Hugoniot condition (speed of the shock wave), ˙ s = Φ(ρ+)−Φ(ρ−) ρ+ −ρ− = 1 ρ+ −ρ−

ρ+

ρ− Φ(ψ)dψ

CARLOS CANUDAS-DE-WIT, 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 22 / 50

Non-linear model

Noticing that, ˙ Nc = l ˙ ρc + ˙ lρc, ˙ Nf = (L−l) ˙ ρf − ˙ lρf

The complete model writes as:

˙ ρf = 1 L−l

• ϕin −min{Df,Sc}+ ˙

lρf

• (4)

˙ ρc = 1 l

• min{Df,Sc}−ϕout − ˙

lρc

• (5)

˙ l = c (Df −Sc) (6) The model implicitly assume a separation between the free and congested cell, i.e. ρf ∈ [0,ρ∗], ρc ∈ [ρ∗,ρm].

CARLOS CANUDAS-DE-WIT, 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 23 / 50

Actuator operation: speed limit control

 



   





Figure: Variation of the fundamental diagram as a function of the speed limit v; the maximum capacity ϕm = ϕ(vmax) is reach for the maximum available velocity vmax, at the critical density ρ∗(vmax). The capacity of the section will decrease to ϕm = ϕ(vmin) when the speed limit is set to its minimum vmin. Nevertheless the critical density is increased substantially ρ∗(vmin)

CARLOS CANUDAS-DE-WIT, 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 24 / 50

Homogenous sections–hypothesis

 



   





H1) The whole section has the same fundamental diagram. That is; v = vc = vf, and w = wc = wf, where v = v(t) is time-varying but w is assumed to be constant. H2) The critical density ρ∗(v), and its associated road maximum capacity ϕm(v), are both functions of v, ρ∗ = ρ∗(v) = wρm v +w , ϕm(v) = vρ∗ = v wρm v +w but the maximum density ρm is independent of v.

CARLOS CANUDAS-DE-WIT, 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 25 / 50

Model as a function of the control variable v

Velocity dependent variable-length model

˙ ρf = 1 L−l (ϕin −min{Df(v),Sc(v)}+ρf ·c (Df(v)−Sc(v))) ˙ ρc = 1 l (min{Df(v),Sc(v)}−ϕout −ρc ·c (Df(v)−Sc(v))) ˙ l = c (Df(v)−Sc(v)) where: Df(v) = min{vρf ,ϕm(v)} Sc(v) = min{ϕm(v),wc(ρm −ρc)} The model implicitly induce a separation between the free and congested cell, i.e. ρf ∈ [0,ρ∗(v)], ρc ∈ [ρ∗(v),ρm].

CARLOS CANUDAS-DE-WIT, 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 26 / 50

Best effort control



 





  

 

(a) Sc − k

c˜

l ≥ supv maxρ{ϕ(ρ)}, (b) infv minρ{ϕ(ρ)} < Sc − k

c˜

l < supv maxρ{ϕ(ρ)}, (c) Sc − k

c˜

l ≤ infv minρ{ϕ(ρ)}, Let ˜ l = l −lr, then d˜ l dt = Df(v)−Sc = −k c ˜ l = 0 Let U = {vmax,vmin}, then the best-effort constrained control is defined as v∗ = min

v∈U |Df(v)−Sc + k

c ˜ l| This problem can be solved graphically, and leads to the solution v∗ = Satvmax

vmin

1 ρf

• −wρc +wρm + k

c ˜ l

• CARLOS CANUDAS-DE-WIT, 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 27 / 50

Best-effort control under dynamic constraints

Operational constraints: v should respect a prescribed Dwell-time Ts Rate of change of v should be limited by steps changes of a maximum ∆v (Km/h). This implies the rate of variation of dl

dt be approximated as:

l(k +1) = l(k)+cTs (Df(v(k))−Sf (k)) the variation of the control v should be constraint to change as, ∇v(k) = v(k +1)−v(k)∈ V where V is the finite 3-valued set defined as V = {−∆v,0,∆v}

CARLOS CANUDAS-DE-WIT, 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 28 / 50

Best-effort control under dynamic constraints–cont.

The constrained best-effort law can be derived by observing, with γ(v(k),k) = cTs (Df(v(k))−Sf (k)), that ∆˜ l2(k) = ˜ l2(k +1)−˜ l2(k) = 2˜ lγ(v(k),k)+γ2(v(k),k) In this context, the best effort control is v∗ = min

v∈U,∇v∈V

• 2˜

lγ(v(k),k)+γ2(v(k),k)

• A reformulation of this minimization problem as γ-size insensitive

v∗ = min

v∈U,∇v∈V

• sign(˜

l)+sign(γ(v(k),k))

• make the best possible choice for v(k) so as sign of γ be, when

possible, opposed to the one of ˜ l), i.e: v(k +1) = Satvmax

vmin

• v(k)− ∆v

2

• sign(γ(v(k),k))+sign(˜

l)

• (7)

CARLOS CANUDAS-DE-WIT, 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 29 / 50

Control rationality

v(k + 1) = Satvmax

vmin

• v(k)− ∆v

2

• sign(γ(v(k),k))+ sign(˜

l)

• (8)

Control rationality Cases sign(γ(v(k),k)) sign(˜ l(k)) ∇v(k) a) 1

• 1

b) 1 1

• ∆v

c)

• 1

1 d)

• 1
• 1

∆v Table: Set of best possible solutions of the control law (8) (a,b) the font congestion is increasing (c,d) the font congestion is decreasing

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Implementation issues

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 50 60 70 80 90 100 110 120 130 Time evolution of the speed limits v [Km/h] Time [h] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3 3.5 4 Time evolution of the front congestion front congestion l [Km] Time [h]

Without speed regulation With Speed Regulation

Two possible ways to implement the law:

1

by explicit computation of sign(γ(v(k),k)), or by

2

time derivative approximation of the front state l, i.e. v(k + 1) = Satvmax

vmin

• v(k)− ∆v

2

• sign(l(k + 1)− l(k))+ sign(˜

l(k))

• CARLOS CANUDAS-DE-WIT, 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 31 / 50

Conclusions

1

A new variable-length two-cell model has been proposed:

minimize the number of states, avoid many switching and reduce complexity of the model lower dimensional nonlinear system which solutions are continuous.

2

‘best-effort” control strategy using variable speed limits:

best effort control is here linked to the physical variable speed limit constraints which limits its size and as well as its rate variation. This results in simple control in closed-form that can be implemented by using only information about the front congestion location.

3

Further evaluation is needed. Depending of the considered scenario, this will have an important impact on the traveling time and drivers comfort. Acknowledgements HYCON2 (Highly-complex and networked control systems) EU-project from the ICT-FP7

• D. Jacquet from Karrus-ITS for discussions.

CARLOS CANUDAS-DE-WIT, 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 32 / 50

Network structure and problem formulation

Figure: Highway structure and sensors placement.

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Godunov scheme

ρi(k +1) = ρi(k)+ T li

• ϕi(k)−ϕi+1(k)
• (9)

ϕi = min{Di−1,Si} (10) with Di−1 = min{vi−1ρi−1,ϕm,i−1}, Si = min{ϕm,i,wi(ρm,i −ρi)}

Figure: The switching mode model based on cell transition model.

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Constrained switching mode model

s(k) Cell 1 Cell 2 . . . Cell N C C . . . C 1 F C . . . C 2 F F . . . C . . . . . . . . . . . . . . . N F F . . . F      ρ(k + 1) = As(k)ρ(k)+ Bϕ(k)+ Bm,s(k)ρm s(k + 1) = s(k)+ f(ρ(k),ϕ(k)) y(k) = h(ρ(k)) (11) where ϕ = (ϕu,ϕd) f(ρ(k),u(k)) =    −1 if C −(ρ(k),s(k)) if C 0(ρ(k),s(k),ϕ(k)) 1 if C +(ρ(k),s(k)) (12)

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Matrix structure

Γi :=       1− T

l1 v1

...

T l2 v1

1− T

l2 v2

... . . . . . . . . . . . . . . . ...

T li vi−1

1       ∈ Ri2, ∆i :=         1− T

li+1 wi+1 T li+1 wi+2

... 1− T

li+2 wi+2 T li+2 wi+3

... . . . . . . . . . . . . . . . ...

T lN−1wN

... 1− T

lN wN

        ∈ R(N−i)2 (13)

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matrices structure–cont.

                 AN = ΓN, A0 =

• 1

T

l1 w2,0 N−2

• 0N−1

∆1

• Ai =

  Γi

• 0i−1,N−i

(T

li wi+1,0 N−i−1)

• 0N−i,i

∆i  ,1 ≤ i ≤ N −1            Bm,N = 0N,N, Bm,0 = IN −A0 Bm,i =   0i−1,i 0i−1,N−i

i

(− T

li wi+1,0 N−i−1)

0N−i,i IN−i −∆i   B =     

T l1

. . . . . . − T

lN

    

CARLOS CANUDAS-DE-WIT, 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 37 / 50

Observer architecture

    

ϕd(k) ϕu(k) u(k) y(k) ˆ ρ(k) ˆ s(k)

Figure: The observer architecture (with ˆ ϕ = u

Observer equations

ˆ ρ(k +1) = Aˆ

s(k)ˆ

ρ(k)+B ˆ ϕ(k)+Bm,ˆ

s(k)ρm +Lˆ s(k)(y(k)− ˆ

y(k)) ˆ s(k +1) = ˆ s(k)+f(ˆ ρ(k), ˆ ϕ(k)) ˆ y(k) = h(ˆ ρ(k)) where ˆ ϕ will be defined according to different possible situations.

CARLOS CANUDAS-DE-WIT, 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 38 / 50

• ne cell density estimation: F/F situation

FF= Backward observable ρ(k +1) = ρ(k)+ T l

• ϕu(k)−vρ(k)
• ˆ

ϕ(k) = (ϕu,ϕd) = (ϕu(k),v ˆ ρ(k)), ˆ ρ(0) = 0 ˆ ρ(k +1) = ˆ ρ(k)+ T l

• ϕu(k)−v ˆ

ρ(k)

• +(ρ(k)− ˆ

ρ(k)) Denoting by ˜ ρ := ρ − ˆ ρ one obtains ˜ ρ(k +1) = ˜ ρ(k)

• 1− T

l v −

• where is chosen such that 1− T

l v − ∈ (0,1)

|˜ ρ(k)| <

• 1− T

l v −

• k

|˜ ρ(0)|

CARLOS CANUDAS-DE-WIT, 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 39 / 50

• ne cell density estimation: C/C situation

CC= Forward observable ρ(k +1) = ρ(k)+ T l

• w(ρm −ρ(k))−ϕd(k)
• ˆ

ϕ(k) = (ϕu,ϕd) = (w(ρm − ˆ ρ(k)),ϕd(k)), ˆ ρ(0) = ρm ˆ ρ(k +1) = ˆ ρ(k)+ T l

• w(ρm − ˆ

ρ(k))−ϕd(k)

• +(ρ(k)− ˆ

ρ(k)) Denoting by ˜ ρ := ρ − ˆ ρ one obtains ˜ ρ(k +1) = ˜ ρ(k)

• 1− T

l w −

• where is chosen such that 1− T

l w − ∈ (0,1)

|˜ ρ(k)| <

• 1− T

l w −

• k

|˜ ρ(0)|

CARLOS CANUDAS-DE-WIT, 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 40 / 50

• ne cell density estimation: F/C situation

FC= unobservable ρ(k +1) = ρ(k)+ T l

• ϕu(k)−ϕd(k)
• In this case ˆ

ϕ(k) = (ϕu,ϕd), there is no output y usable for observation. ˆ ρ(k +1) = ˆ ρ(k)+ T l

• ϕu(k)−ϕd(k)
• which leads to

|˜ ρ(k +1)| = |˜ ρ(k)|

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The observability for different SMM modes [?]

Upstream Cells Downstream Cells Observable with Free-flow Free-flow Downstream measurement Congested Congested Upstream measurement Congested Free-flow

• Up. and Down. measurement

Free-flow Congested Unobservable

Table: Observability for different SMM modes

CARLOS CANUDAS-DE-WIT, 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 42 / 50

General case: F/F situation

ρ(k +1) = ANρ(k)+Bϕ(k)+Bm,Nρm (14) ˆ ϕ(k) = (ϕu,vN ˆ ρN) and ˆ ρ(0) = (0,0,...,0) ˜ ρ(k +1) =       1− T

l1 v1

...

T l2 v1

1− T

l2 v2

... . . . . . . . . . . . . . . . ...

T lN vN−1

1− T

lN vN

      ˜ ρ(k) E1 ˜ ρ(k) So ||˜ ρ(k +1)||2 < maxi=1,n

• 1− T

li vi

• ||˜

ρ(k)||2,∀k Closed-loop error dynamics: ˜ ρ(k +1) = (E1 −LF ·CF)˜ ρ(k) where LF = LNvN := (F,1,F,2,...,F,N) and CF = (0,...,0,1).

CARLOS CANUDAS-DE-WIT, 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 43 / 50

General case: C/C situation

ρ(k +1) = A0ρ(k)+Bϕ(k)+Bm,0ρm (15) ˆ ϕ(k) = (w1(ρm,1 − ˆ ρ1),ϕd) and ˆ ρ(0) = ρm. ˜ ρ(k +1) =         1− T

l1 w1 T l1 w2

... 1− T

l2 w2 T l2 w3

... . . . . . . . . . . . . . . . ...

T lN−1wN

... 1− T

lN wN

        ˜ ρ(k) E2˜ ρ(k) So ||˜ ρ(k +1)||2 < maxi=1,n

• 1− T

li wi

• ||˜

ρ(k)||2,∀k Closed-loop error dynamics: ˜ ρ(k +1) = (E2 −LC ·CC) ˜ ρ(k) where LC = −L0w1 := (C,1,C,2,...,C,N) and CC = (1,0,...,0).

CARLOS CANUDAS-DE-WIT, 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 44 / 50

General case: F/C situation

• ρ(k + 1) = Aiρ(k)+ Bϕ(k)+ Bm,iρm

ˆ ρ(k + 1) = Ai+j ˆ ρ(k)+ B ˆ ϕ(k)+ Bm,i+jρm With ϕ(k) = ˆ ϕ(k), Ki(k) = T

li

• viρi(k)− wi+1(ρm,i+1 − ρi+1(k))
• ˜

ρ(k + 1) = Aiρ(k)− Ai+j ˆ ρ(k)+ Bm,iρm − Bm,i+jρm = Ai+j ˜ ρ(k)+

j

∑

=1

(Ai+−1 − Ai+)ρ(k)+

j

∑

=1

(Bm,i+−1 − Bm,i+)ρm = Ai+j ˜ ρ(k)+              0i−2 Ki−1(k) Ki(k)− Ki−1(k) Ki+1(k)− Ki(k) . . . Ki+j(k)− Ki+j−1(k) −Ki+j(k) 0N−i−j             

CARLOS CANUDAS-DE-WIT, 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 45 / 50

General case: FC situation–cont.

From previous equation we have that Sum(˜ ρ(k +1)) = Sum(Ai+j ˜ ρ(k)) On the other hand we have that Sum(Aix) = Sum(Γi(x1,...,xi))+ T l wi+1xi+1 +Sum(∆i(xi+1,...,xN) = Sum((x1,...,xi))+Sum((xi+1,...,xN)) = Sum(x), ∀x ∈ Rn, ∀i ∈ {1,...,N} Therefore we have error norm conservation, i.e. Sum(˜ ρ(k +1)) = Sum(˜ ρ(k))

CARLOS CANUDAS-DE-WIT, 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 46 / 50

Convergence properties in the general case

Convergence properties

The error estimation remains bounded, i.e. |˜ ρ(k)| ≤ |˜ ρ(0)|, ∀k ≥ 0 |˜ ρ(k)| is a strictly non-increasing function The error converge to zero if there is persistently enough

• bservable situations, i.e.

lim

k→∞|˜

ρ(k)| = 0, if lim

k→∞ N(k)

∑

i=1

τi = ∞ τi is the time window duration in any of the observable modes.

CARLOS CANUDAS-DE-WIT, 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 47 / 50

Microscopic simulation versus macroscopic simulation

Highway segment with 5 identical cells.

          

Figure: The density evolution given by the macroscopic simulation is smoother but it accurately reproduces the behavior of the measured density.

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Estimation I

                     

Figure: When none of the cell is congested the estimated density approach the real (simulated) one. During the transition period when some cells are congested and some other are free the estimation errors propagate and they are vanishing when all the cells are congested.

CARLOS CANUDAS-DE-WIT, 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 49 / 50

Estimation II

                     

Figure: During the transition period when some cells are congested and some other are free the total estimation error propagates but the distribution

• changes. When all the cells are congested the estimation error disappears.

CARLOS CANUDAS-DE-WIT, 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 50 / 50

L.D. Baskar, B. De Schutter, and H. Hellendoorn, ”Dynamic speed limits and on-ramp metering for IVHS using model predictive control,” Proceedings of the 11th International IEEE Conference on Intelligent Transportation Systems (ITSC 2008), Beijing, China, pp. 821-826, Oct. 2008.

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