optimal perimeter control synthesis for two urban regions...

Macroscopic modeling and control Optimal perimeter control Numerical verifications Conclusions

Optimal Perimeter Control Synthesis forTwo Urban Regions with Boundary Queue

Dynamics

Jack Haddad

Technion Sustainable Mobility and Robust Transportation (T-SMART) LabFaculty of Civil and Environmental Engineering

Technion - Israel Institute of Technologywebpage: haddad.net.technion.ac.il

Nov 11, 2014

Tsmart Technion Sustainable Mobility andRobust Transportation Laboratory

Tsmart Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 1 / 30


Classification of Traffic Flow Models (aggregation level)

6.2 Model Classification 57

ρ (x,t)

v (t)α

xα yα

n=0

MacroscopicModel

MicroscopicModel

CellularAutomaton (CA)

Pedestrian Model (v (t), v (t))

n=1

Fig. 6.2 Comparison of various model categories (with respect to the way they represent reality)including typical model equations

Macroscopic models describe traffic flow analogously to liquids or gases in motion.Hence they are sometimes called hydrodynamic models. The dynamical variables arelocally aggregated quantities such as the traffic density ρ(x, t), flow Q(x, t), meanspeed V (x, t), or the speed variance σ 2

V (x, t). Because the aggregation is local, thesequantities generally vary across space and time, i.e., they correspond to dynamicfields. Thus, macroscopic models are able to describe collective phenomena suchas the evolution of congested regions or the propagation velocity of traffic waves.Furthermore, macroscopic model are useful,

• if effects that are difficult to describe macroscopically need not to be considered(e.g., lane changes, several driver-vehicle types),

• if one is interested in macroscopic quantities, only,• if the computation time of the simulation is critical, e.g., in real-time applications

(due to increasing computing power, this aspect is becoming less important), or• if the available input data come from heterogeneous sources and/or are inconsis-

tent, so data fusion is necessary.

Multiple real-time speed and the capability to incorporate heterogeneous datasources are particularly important for traffic state estimations and predictions. In thisprocess, the future traffic state is predicted over a time horizon τ and the predictionsare updated over smaller time intervals ∆t . The predictions are processed such thatthey can be distributed via traffic message channel, variable-message signs, or serveas input for connected navigation devices.1

Microscopic models including car-following models and most cellular automatadescribe individual “driver-vehicle particles” α, which collectively form the traffic

1 Traffic flow modeling and transportation planning are intertwined in these applications: Trafficflow models provide the basis for the route choice.

6.2 Model Classification 57

ρ (x,t)

v (t)α

xα yα

n=0

MacroscopicModel

MicroscopicModel

CellularAutomaton (CA)

Pedestrian Model (v (t), v (t))

n=1

Fig. 6.2 Comparison of various model categories (with respect to the way they represent reality)including typical model equations

Macroscopic models describe traffic flow analogously to liquids or gases in motion.Hence they are sometimes called hydrodynamic models. The dynamical variables arelocally aggregated quantities such as the traffic density ρ(x, t), flow Q(x, t), meanspeed V (x, t), or the speed variance σ 2

V (x, t). Because the aggregation is local, thesequantities generally vary across space and time, i.e., they correspond to dynamicfields. Thus, macroscopic models are able to describe collective phenomena suchas the evolution of congested regions or the propagation velocity of traffic waves.Furthermore, macroscopic model are useful,

• if effects that are difficult to describe macroscopically need not to be considered(e.g., lane changes, several driver-vehicle types),

• if one is interested in macroscopic quantities, only,• if the computation time of the simulation is critical, e.g., in real-time applications

(due to increasing computing power, this aspect is becoming less important), or• if the available input data come from heterogeneous sources and/or are inconsis-

tent, so data fusion is necessary.

Multiple real-time speed and the capability to incorporate heterogeneous datasources are particularly important for traffic state estimations and predictions. In thisprocess, the future traffic state is predicted over a time horizon τ and the predictionsare updated over smaller time intervals ∆t . The predictions are processed such thatthey can be distributed via traffic message channel, variable-message signs, or serveas input for connected navigation devices.1

Microscopic models including car-following models and most cellular automatadescribe individual “driver-vehicle particles” α, which collectively form the traffic

1 Traffic flow modeling and transportation planning are intertwined in these applications: Trafficflow models provide the basis for the route choice.

Macroscopic network model

u21(t)

q21(t)

q11(t)

q12(t) u12(t)

12

Aggregated network-level approach to large-scale urban modeling



Fundamental diagram for a link i

Three traffic regimes:

• undersaturated,

• saturated,

• oversaturated (if flow is restricted). Link i

[veh]

[veh/sec]

Undersaturated

Saturated

Oversaturated

Accumulation

Trip completionrate

[veh/km]Density

0

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ax {q

i}

Detector #: 10-003D Detector #: T07-005D

Single Detectors

occupancy

flow



Fundamental diagram for a link i

Three traffic regimes:

• undersaturated,

• saturated,

• oversaturated (if flow is restricted). Link i

[veh]

[veh/sec]

Undersaturated

Saturated

Oversaturated

Accumulation

Trip completionrate

[veh/km]Density 0

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Single Detectors

occupancy

flow



Macroscopic Fundamental Diagram (MFD) for an urban regionMFD links space-mean flow, density, and speed of a large urban area.

[veh]

[veh/sec]

Accumulation

Trip completionrate

∑i Trip completion rate for link i

MFDG(n)

nnjamncr




[veh]

[veh/sec]

Accumulation

Trip completionrate


MFDG(n)

nnjamncr

Average network flow and trip completion rate

• Average network flow F

F =

∑i fi · li∑

i li

where: fi (veh/s) - flow in link i, li (m) - length of link i.

• Trip completion rate/average network flow ≈ constant

G/F ≈ constant.




[veh]

[veh/sec]

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Trip completionrate


MFDG(n)

nnjamncr

San Francisco (simulation)

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VKT

Vehicle Accumulation

Geroliminis and Daganzo (2007) – Tr. Res. BoardYokohama (experiment)

0

15

30

45

0 20 40 60 80o u (%)

qu (v

hs/5

min

)

A1B1C1D1A2B2C2D2Av

erage flo

w

Average occupancy

Geroliminis and Daganzo (2008) – Tr. Res. Part B

Other recent MFD studies (UC Berkeley, EPFL, TU Delft, Northwestern, INRETS,TUC, Penn State, UCL, Technion, etc.)



Perimeter Traffic Flow Control for an Urban Region

u21(t)

q21(t)

q11(t)

q12(t) u12(t)

12

G1(n

1(t))

(veh

/s)

Tripco

mpletionflow

n1,jamn∗1

Accumulation, n1(t) (veh)

Literature survey: perimeter control for a single MFD system

• Daganzo (2007): the optimal control policy was presented for a single MFD system

(bang-bang control). Explicit proof in Haddad (2014) based on Modified

Krotov-Bellman sufficient conditions of optimality.

• Keyvan-Ekbatani et al. (2012): a classical feedback control approach.

• Haddad and Shraiber (2014): “Robust perimeter control design for an urban region,”

Transportation Research Part B, 68, pp. 315–332, 2014.



Effect of Perimeter Control

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Video Video(Videos provided by LUTS Laboratory, EPFL, Switzerland)



Well-defined MFD?Mazloumian, Geroliminis, and Helbing (2010): an urban region with small variance

of link densities has well-defined MFD.

homogeneous distribution of congestion

uneven distribution of congestion



Perimeter Traffic Flow Control for Two Regions

Region 1Region 2



Optimal perimeter control for two urban regions with Macroscopic Fundamental Diagrams

Region 1Region 2

u12(t)

q12(t)

q22(t)

q11(t)

R1

q21(t) u21(t)

R2

tkc−1 tkc+Np−1tkcTime

tkc−1 tkc+Np−1tkcTime

Horizon 1

Horizon 2

Prediction horizon

MPC for perimeter control

IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, 3

by the controllers such that only a ratio transfers at timet. Theperimeter controllersu12(t) andu21(t), where0 ≤ u12(t) ≤ 1and 0 ≤ u21(t) ≤ 1, control are the ratio of the transferflow that transfers fromR1 to R2 and R2 to R1 at time t,respectively.

The criterion is to maximize the output of the trafficnetwork, i.e. the number of vehicles that complete their tripsand reach their destinations. Therefore, the two-region MFDscontrol problem with four state variables is formulated asfollows (similarly to [28]):

J = maxu12(t),u21(t)

∫ tf

t0

[M11(t) +M22(t)

]dt (1)

subject to

dn11(t)

dt= q11(t) + u21(t) ·M21(t)−M11(t) (2)

dn12(t)

dt= q12(t)− u12(t) ·M12(t) (3)

dn21(t)

dt= q21(t)− u21(t) ·M21(t) (4)

dn22(t)

dt= q22(t) + u12(t) ·M12(t)−M22(t) (5)

0 ≤ n11(t) + n12(t) (6)

0 ≤ n21(t) + n22(t) (7)

n11(t) + n12(t) ≤ n1,jam (8)

n21(t) + n22(t) ≤ n2,jam (9)

umin ≤ u12(t) ≤ umax (10)

umin ≤ u21(t) ≤ umax (11)

n11(t0) = n11,0 ; n12(t0) = n12,0

n21(t0) = n21,0 ; n22(t0) = n22,0(12)

wheretf [sec] is the final time,nij,0, i, j = 1, 2 are the initialaccumulations att0, n1,jam and n2,jam [veh] are the accu-mulations at the jammed density inR1 andR2, respectively,umin and umax are the lower and upper bounds foru12(t),u21(t), respectively. Recall thatMij = (nij/ni) · Gi(ni(t)),i, j = 1, 2. The equations (2)–(5) are the conservation of massequations fornij(t), while the equations (6), (7) and (8), (9)are the lower and upper bound constraints on accumulationsin R1, R2, respectively.

III. M ODEL PREDICTIVE CONTROL FOR TWO-REGION

MFDS PROBLEM

The two-region MFDs problem (1)–(12) aims to find theperimeter controllers, i.e. ratios of transfer flows ofR1 andR2, that maximize the number of vehicles completing theirtrips (reach their destinations). This problem is an optimalcontrol problem with a nonlinear objective function (1) anddynamic equations (2)–(5), inequality state constraints (6)–(9), and control constraints (10)–(11). Moreover, errors areexpected in the modeling due to the scatter in the MFDs,mainly in the congested regime. Therefore, the optimal controlproblem is solved by applying the model predictive control(MPC) approach which has the ability to handle the stateand control constraints, and the errors in the MFDs modeling.Furthermore, the MPC is a real-time implementable solutionthat can be utilized for real-time urban traffic applications.

The MPC is a form of rolling horizon control in whichthe current control variables are obtained by solving a finite

horizon open-loop optimal control problem at each time stepwith a feedback current state from the plant as the initialstate of the model, see Fig. 2. The open-loop optimizationproblem yields a sequence of optimal control variables afterseveral iterations of solving nonlinear programming, and thefirst control in this sequence is applied to the plant, then theprocedure is carried out again.

This scheme of feedback control, i.e. the feedback loop ofstates from the plant to the model as initial states for theoptimization, can handle the errors between the predictionmodel and the plant.

G2(n2)G1(n1)

dn(t)dt = f (n(t), u(k), q(t), ε(k))

ε(kc)

G2(n2)G1(n1)

dn(t)dt = f (n(t), u(k), q(t))

Two-region MFDs prediction model

Maximizing the number of trips ended

u∗(kc) n(tkc )

Two-region MFDs plant

MPC controllerkc = kc + 1

n(tkc−1) = n(tkc−1)tk−1 ≤ t ≤ tk , k = kc , · · · , kc + Np − 1

tkc−1 ≤ t ≤ tkc

u(kc )u(kc + 1)

u(kc + Np − 1)

...

(Open-loop optimization problem)

q(t)

q(t)

Fig. 2. Model predictive control scheme for two-region MFDs system.

A. Two-region MFDs prediction model and optimization prob-lem

The MPC controller obtains the optimal control sequencefor the current horizon by solving an optimization problemformulated with prediction model, see bottom of Fig. 2.

The prediction model used in the MPC scheme is formulatedwith equations (2)–(5). The dynamic equations predict theevolution of accumulations for the two regions with MFDsgiven the initial accumulations and future values of perimetercontrollers and demand.

In this paper, we follow the direct methods to solve theoptimization problem (other solution methods include dynamicprogramming and indirect methods). The direct methods aremost commonly used methods due to their applicability androbustness, where their basic principle is “first discretizeand then optimize”. These methods can handle inequalityconstraints and use the state-of-the-art methods for nonlinearproblem solvers.

The open-loop optimal control problem is solved using thedirect sequential method, also referred to as single-shootingor control vector parameterization (CVP) in the literature, e.g.

Main contributions

• Formulation the perimeter control problem of two urban regions by the MacroscopicFundamental Diagrams.

• Solving the control problem by Model Predictive Control.Tsmart

Optimal Perimeter Control Synthesis for Two Urban Regions with Boundary Queue Dynamics 9 / 30


Example: different levels of demand

100% demand

5 10 15 20 25 30−5

0

5

10

15

20

25

Np

Impr

ovem

ent [

%]

Nc=2

Nc=3

Nc=4

Nc=5

Fig. 3. Tuning parametersNp andNc for MPC controller.

the prediction horizonNp, however for Np ≥ 20 onlyminor improvement is achieved. The MPC controller is lesssensitive to the control horizonNc, whereNc ≥ 2 yieldssimilar results for trip completion. Therefore, the parametersare set asNp = 20 andNc = 2 for all subsequent case studyexamples. Note that for small prediction horizonNp < 6 theMPC controller does not perform well compared with thegreedy control.

IV. CASE STUDY EXAMPLES

In this section, results of several case study examples arepresented to explore the features of the MPC controller. Forall examples 1, 2, and 3, both regionsR1 andR2 have thesame shape MFD, the selected MPC parameters areNp = 20andNc = 2, the time duration of the control time step is setto 60 [sec], the lower boundsu12,min = u21,min = 0.1, andthe upper boundsu12,max = u21,max = 0.9.

In Example 1, both regionsR1 and R2 are initiallycongested, i.e. the initial accumulationsn1(t0) and n2(t0)are in the decreasing part of the MFD. The time varyingdemand shown in Fig. 4(d) are simulating a morning peakhour with high demandq12(t) for trips fromR1 to R2, e.g.from periphery to city center.

The evolutions of accumulations over timen11(t), n12(t),n21(t), n22(t), 0 ≤ t ≤ 3600, corresponding to the MPCcontroller are presented in Fig. 4(a), while the evolutionspresented in Fig. 4(b) corresponding to the greedy controller.

Note that at the beginning of the control process theMPC and greedy controllers decreases the total accumulationin R1, n1(t), and keeps theR2 total accumulationn2(t)unchanged. Afterwards, the MPC controller tries to decreasen2(t) by changingu21(t) from 0.1 to 0.55 to let morevehicles enter to theR1. In contrast the greedy controllerbrings the two accumulations be equal att = 600 [sec],i.e. n1(590) = n2(590) = 4125 [veh], and from that time,600 ≤ t ≤ 3600, the chattering behavior occurs in theaccumulations corresponding to the greedy control as a resultof umin andumax control switchings; note the saw lines ofaccumulations aftert = 600 [sec].

The cumulative trip completion corresponding to MPC andgreedy controllers are shown in Fig. 4(c), while the controlsequencesu12(t) and u21(t) are shown in Fig. 4(e). Thethird polynomial MFDsG1(n1) andG2(n2) are coincided asshown in Fig. 4(f), while the circle points are the calculatedG1 and G2 according to the accumulations, see (25) and

(26)). In Fig. 4(f) it is assumed that there are no errors inboth MFD whereα1 = α2 = 0, see (23) and (24).

The effect of different levels of error in the MFD isinvestigated with the help of Example 1. The MPC controlperformances for small (α1 = α2 = 0.2) and large (α1 =α2 = 1) errors in the MFD, see (23) and (24), are shownin Fig. 5(a) and Fig. 5(b), respectively. Comparison betweenthe three levels of error, i.e. without errors in Fig. 4(f), smalland large errors in Fig. 5, shows that the controlleru21(t)becomes less smoother when the errors in the MFD of theplant (reality) increase.

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n22

n1

n2

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n2

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eh/s

ec]

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q12

q21

q22

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u [−

]

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u21MPC

u12GC

u21GC

0 5000 100000

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4

6

8

Accumulation [veh](f)

G(n

) [v

eh/s

ec]

MFD1

MFD2

Fig. 4. Example 1: regionsR1 andR2 are initially congested.

The effect of different demand on the MPC controllerare examined by Examples 2 and 3. These examples havethe same initial accumulations of Example 1, however, thedemandsq11(t), q12(t), q21(t), andq22(t) for Example 2 andExample 3 are proportionally decreased by16% and32% ,respectively, compared with the demand for Example 1 inFig. 4(d).

Comparing with the results of Example 1, the MPC andgreedy controllers bring both regions to the uncongested partof the two MFDs in examples 2 and 3, while in Example 1the demand is high such that at the end of the control processboth regions are congested (even with greedy controllerregions move forward to face gridlock).

The difference between the trip completion correspondingto MPC and greedy control without errors, with small, andlarge errors in the plant MFDs are summarized in Table I.

84% demand 68% demand

IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, 7

TABLE IITHE TRIP COMPLETION CORRESPONDING TOMPC AND GREEDY CONTROLLERS, AND THE TOTAL DELAY DIFFERENCE, ·103 .

Example 1without errors (α1 = α2 = 0) small errors (α1 = α2 = 0.2) large errors (α1 = α2 = 1)

MPC [veh] GC [veh] MPC-GC [veh · sec] MPC GC MPC-GC MPC GC MPC-GCwithout noises 23.55 17.07 7791.6 (%22.5) 23.63 17.11 7883.5 (%22.7) 24.04 17.21 8370.0 (%24.1)

low noises 23.37 16.78 7864.3 (%22.9) 23.48 16.82 7997.3 (%23.3) 23.93 17.00 8379.5 (%24.3)high noises 23.37 16.78 7864.3 (%22.9) 23.48 16.82 7997.3 (%23.3) 23.93 17.00 8379.5 (%24.3)










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]

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u12GC

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5

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8

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G(n

) [v

eh/s

ec]

MFD1

MFD2

(a) Example 1: small errors in MFDEample

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]

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u21MPC

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u21GC

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2

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5

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7

8

Accumulation [veh]

G(n

) [v

eh/s

ec]

MFD1

MFD2

(b) Example 1: large errors in MFD

Fig. 6. Example 1: small and large errors in MFDs.

Example 1 with high unbiased noise in demand (σij = 0.5,i, j = 1, 2, see (27)) is illustrated in Fig. 9. The overallresults of MPC remain similar, however, the correspondingapplied MPC shows more fluctuations than base example 1.In Fig. 10, a biased error in demand which occurs at timeinstant 1200 [sec] for duration of 600 [sec] is added to thebase setup of example 1, see Fig. 10(b). The MPC controllerresults to a similar performance than before whereas greedycontroller makes both regions to gridlock (it can be inferredfrom almost horizontal ending part of greedy controller tripcompletion profile, see Fig. 10(c)). Note that the MPC profilein Fig. 10(d) is identical to Fig. 5(e) for times before1200 [sec]

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1

1.5

2

2.5x 104

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ulat

ive

trip

com

plet

ion

[veh

]

MPCGC

0 1000 2000 3000Time [sec]

(b)

MPCGC

0 1000 2000 3000Time [sec]

(c)

MPCGC

Fig. 7. The cumulative trip completion for (a) example 2, (b) example 3,and (c) example 4.

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n

11n

12n

21n

22n

1n

2

0 1000 2000 3000Time [sec]

(b)

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0.8

1

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u [−

]

u12MPC

u21MPC

u12GC

u21GC

MPC GC

Fig. 8. Example 4: regionsR1 andR2 are initially uncongested and finallycongested.

and after that with decreasingu12(t) from umax, the MPC canhandle the unbiased sudden augmentation in demand whichhas a great impact onn2(t) accumulation, see Fig. 10(a).

The differences between the trip completion correspondingto MPC and greedy control without errors, with small (α1 =α2 = 0.2), and large errors (α1 = α2 = 1) in the plant MFDsand without noise, low (σij = 0.25, i, j = 1, 2) and high

• the differences between the total delays are related to the congestion level,• the greedy controller performs similar to the MPC controller in uncongested situation.

Journal paper

• N. Geroliminis, J. Haddad, and M. Ramezani, “Optimal perimeter control for two urban regions with Macroscopic FundamentalDiagrams: A model predictive approach,” IEEE Transactions on Intelligent Transportation Systems, vol. 14, no. 1, pp. 348-359,2013.Tsmart



Stability Analysis of Perimeter Controln1(t)[veh]

n2(t)[veh]n2,cr

n1,cr

n1,jam

n2,jam

MFD for region 2

MFD

forregion

1

G1(n

1)

G2(n2)[veh/sec]

[veh

/sec]

γ2

γ1

I II

IVIII

saddle point unstable node

saddle pointstable node

Main contributions in stability analysis

• analysis of the dynamic equations.

• stability characterization algorithm.

• a state-feedback control strategy.

0 50 100 150 200 250 300 350 400 4500

20

40

60

80

100

120

140

160

180

200

n2(t) [veh]

n 1(t)

[veh

]

Figure 7: Numerical example 1 demonstrates case a: the trajectories are in green and the red curve is the boundary.

4. if n2,B > µ2, then calculate trajectory from pointB to the unstable equilibrium point (n2,eq, n1,eq)IV in reversedirection according to (A.3), (A.4), and (A.5) in Appendix A.1, with initial state pointB andt = 0→ ∞. If thetrajectory B-(n2,eq, n1,eq)IV does not enter the state region III, i.e. does not intersect the linen2(t) = µ2, then it iscase a, otherwise it is case b:

• case a: draw a horizontal line stars from the unstable equilibrium point (n2,eq, n1,eq)IV moves through thesaddle point (n2,eq, n1,eq)III , and ends atn2(t) = 0. The line is horizontal according to the correspondingeigenvector of the negative eigenvalue for the saddle equilibrium point in state region III.

• case b: calculate trajectories from pointsB to C andC to D in reverse way, according to Appendix A.2.

5. if n2,B = µ2 then it is case c. Calculate trajectory from pointsB to C andC to D in reverse way according toAppendix A.3.

Note that the RA boundary curve is combined from several trajectories some of them are calculated numerically, whileother trajectories are calculated analytically, see Appendix A.1, Appendix A.2, and Appendix A.3.

The region of attraction boundaries for cases a, b, and c are demonstrated by examples 1, 2, and 3, as shown inFig. 7, 8, and 9, respectively, where the red curve is the RA boundary. The input data for example 1 are given inSection 2.2. The input data for example 2 are as follows: the traffic flow demand rates areq1 = 0.194 [veh/sec],q2 =

0.319 [veh/sec], the perimeter controlu(t) = umax = 0.8, the MFD parameters are:γ1 = 0.5 [veh/sec],µ1 = 50 [veh],w1 = 200 [veh],γ2 = 0.583 [veh/sec],µ2 = 150 [veh],w2 = 450 [veh]. The input data for example 3 are as follows:the traffic flow demand rates areq1 = 0.194 [veh/sec],q2 = 0.278 [veh/sec], the perimeter controlu(t) = umax = 1,the MFD parameters are:γ1 = 0.5 [veh/sec],µ1 = 50 [veh], w1 = 200 [veh],γ2 = 0.5 [veh/sec],µ2 = 150 [veh],w2 = 450 [veh].

Until this section, the RA boundaries for all numerical examples 1, 2, and 3 were calculated for constant controlu(t) = umax (trajectories drawn by green color). Clearly, different RA boundaries and trajectories are obtained byapplyingu(t) = umin, e.g. the phase portraits with the RA for example 1 corresponding to u(t) = umin = 0.45 andu(t) = umax are shown in Fig. 10, where trajectories are drawn by blue color and the boundary by cyan color forumin.

3.2. Stability characterization

In the previous section, an algorithm is proposed to computetheRAu boundary for a constant controlu. In thissection, the algorithm for computing the RA is used to characterize the stable and unstable regions.

Recall that stable region is defined as the set of all points that have (at least) one trajectory approaches a stableequilibrium point corresponding to controlu(t). If u(t) is assumed to be constant for the whole control period, then

8

(a) Numerical example 6: RAs surface boundaries surfaces in three-state two-region system,RAumax and RAumin surface boundaries are drawn in red and cyan, respectively,and stable andunstable trajectories correspond toumax are drawn in green.

050

100150

200 0

50

100

0

200

400

n12

(t) [veh]n

11 (t) [veh]

n 2 (t)

[veh

]

(b) RAumax surface boundary forq11(t) = 2/3 · q1(t).

050

100150

200 0

50

1000

200

400

n12

(t) [veh]n

11(t) [veh]

n 2(t)

[veh

]

(c) RAumax surface boundaryq11(t) = 1/3 · q1(t).

Figure 10: RAs surface boundaries.

18

Journal paper

• J. Haddad and N. Geroliminis, “On the stability of perimeter traffic control in two-region urban cities,” Transportation ResearchPart B, 46, pp. 1159–1176, 2012.



Cooperative Control for Mixed Urban-Freeway NetworksUrban network

Region 1Region 2

two-region MFDsdynamics

+

Freeway

Region 1Region 2

Region (1)

Region (2) Freeway (3)

Freeway (3)

asymmetric celltransmission model

(ACTM)

⇒

MUF netwok

Region 1Region 2

Region (1)

Region (2) Freeway (3

)

MUF dynamics

Different levels of coordination

• C-MPC: Centralized MPC (network delay).

• CD-MPC: Cooperative Decentralized MPC (network delay).

• D-MPC: Decentralized MPC (freeway and urban delays).

• ALINEA: ALINEA control for the freeway and umax for urban network.

Journal paper

• J. Haddad, M. Ramezani, and N. Geroliminis, “Cooperative Traffic Control of Mixed Urban and Freeway Networks,”Transportation Research Part B, 54, pp. 17-36, 2013.



Main contributions in this talk

• modeling and integrating the boundary queue dynamics,

• perimeter control policy taking into account the maximum andminimum boundary queue constraints.



Two-region MFD system

Two Urban Regions with Boundary Queue DynamicsTraffic terminology:

• demands: q11(t), q12(t), q21(t), q22(t) (veh/s)

• accumulations: n1(t), n2(t), n12(t), n21(t) (veh)

• exit flows of MFDs: G1

(n1(t)

), G2

(n2(t)

)(veh/s)

• perimeter control inputs: u1(t) and u2(t) (-)0 ≤ u1(t) , u2(t) ≤ 1

• perimeter saturation flow: d (veh/s)

u1(t)q12(t)

q22(t)

q11(t)

q21(t)u2(t)

Region 2

Region 1




Two Urban Regions with Boundary Queue Dynamics

u1(t)q12(t)

q22(t)

q11(t)

q21(t)u2(t)

Region 2

Region 1

Traffic terminology:




(n1(t)

), G2

(n2(t)

)(veh/s)






Two Urban Regions with Boundary Queue Dynamics

u1(t)q12(t)

q22(t)

q11(t)

q21(t)u2(t)

Region 2

Region 1

Traffic terminology:




(n1(t)

), G2

(n2(t)

)(veh/s)



Dynamic equations

dn1(t)

dt= q11(t) + q12(t) + u2(t) · d−G1(n1) ,

dn2(t)

dt= q21(t) + q22(t) + u1(t) · d−G2(n2) ,

dn12(t)

dt= q12(t)− u1(t) · d ,

dn21(t)

dt= q21(t)− u2(t) · d .




Optimal perimeter control problem definition

Given:

• time varying demands: q11(t), q12(t), q21(t), q22(t),

• the initial accumulation: n1(0), n2(0), n12(0), n21(0),

• the MFDs: G1(n1), G2(n2),

• accumulation (state) constraints:0 ≤ n12(t) ≤ n12 ,

0 ≤ n21(t) ≤ n21 ,

• control constraints:0 ≤ u1(t) ,0 ≤ u2(t) ,

u1 ≤ u1(t) ≤ u1 ,u2 ≤ u2(t) ≤ u2 ,u1(t) + u2(t) ≤ 1

Manipulate u1(t) and u2(t) to maximize:

J =

∫ tf

t0

(G1(n1) +G2(n2))dt .




Brief description of Pontryagin’s Maximum Principle

Classical optimal control problem (OCP)

∫ T

0f0(x, u)dt→ min (1)

dx(t)

dt= f(x, u) (2)

x(0) = x0, x(T ) = xT (3)

umin ≤ u(t) ≤ umax (4)

where:control variables u(t) ∈ Rm, statevariables x(t) ∈ Rn, f(x, u) ∈ Rn, andm ≤ n.

According to PMP:

H = pT · f(x, u)− f0(x, u) (5)

dp

dt= −∂H

∂x

T

= −∂f∂x

T

p+∂f0

∂x

T

(6)

Hamiltonian = H,costate variables p(t) ∈ Rn.If ∃(x∗, u∗) → ∃ p∗ such that:

(a) H(x∗, u∗, p∗) ≥ H(x∗, u, p∗),

(b) x∗, p∗ satisfy (2) and (6),

(c) u∗ satisfies (4),

(d) the end conditions in (3) must hold.

Advertisement: new course

Optimal Control: Theory and Transportation Applications (undergraduateand graduate levels, civil and environmental engineering faculty).



Optimal control solution synthesis

Optimal control solution via PMPThe augmented Hamiltonian function, H, is formed as

H =pn1(t) ·[q11(t) + q12(t) + u2(t) · d−G1(n1)

]

+pn2(t) ·[q21(t) + q22(t) + u1(t) · d−G2(n2)

]

+pn12(t) ·[q12(t)− u1(t) · d

]+ pn21(t) ·

[q21(t)− u2(t) · d

]+G1(n1) +G2(n2)

−λ12 ·[n12(t)− n12

]− λ21 ·

[n21(t)− n21

]+ λ12 · n12(t) + λ21 · n21(t)

−λ112 ·[q12(t)− u1(t) · d

]− λ121 ·

[q21(t)− u2(t) · d

]− λ112 ·

[− q12(t) + u1(t) · d

]

−λ121[− q21(t) + u2(t) · d

],

where pn1(t), pn2(t), pn12(t), pn21(t) satisfy

dpn1(t)

dt= − ∂H

∂pn1= (pn1(t)− 1) · ∂G1(n1)

∂n1

dpn2(t)

dt= − ∂H

∂pn2= (pn2(t)− 1) · ∂G2(n2)

∂n2

dpn12(t)

dt= − ∂H

∂pn12= λ12 − λ12 ,

dpn21(t)

dt= − ∂H

∂pn21= λ21 − λ21 .




Switching function S(t)

• maxu1(t), u2(t)H subject to the control constraints → simple LP problem.

• If both coefficients for u1(t), u2(t) in the Hamiltonian are positive ⇒u1(t) + u2(t) = 1.

Switching function S(t) (coefficient of u2(t))

S(t) = pn1(t)− pn21(t)− pn2(t) + pn12(t)− λ112 + λ121 + λ112 − λ121 .

• The optimal control solution obtained by maxu1(t),u2(t)H is

u∗2(t) = u2 , u∗1(t) = 1− u2 ∀ S(t) > 0,

u∗2(t) = 1− u1 , u∗1(t) = u1 ∀ S(t) < 0,

singular control ∀ S(t) = 0.




Optimal control cases

• Case 1: n1(t) < n∗1 and n2(t) < n∗

2

∂G1(n1)

∂n1>0 ,

∂G2(n2)

∂n2>0 .

• Case 2: n1(t) > n∗1 and n2(t) > n∗

2

∂G1(n1)

∂n1<0 ,

∂G2(n2)

∂n2<0 .

• Case 3.a: n1(t) < n∗1 and n2(t) > n∗

2

∂G1(n1)

∂n1>0 ,

∂G2(n2)

∂n2<0 .

• Case 3.b: n1(t) > n∗1 and n2(t) < n∗

2

∂G1(n1)

∂n1<0 ,

∂G2(n2)

∂n2>0 .

n1(t)(veh)

n2(t)(veh)n∗2

n∗ 1

n1,jam

n2,jam

MFD for region 2

MFD

forregion

1

G1(n

1)

G2(n2)(veh/s)

(veh

/s)

Case 3.b Case 2

Case 3.aCase 1

• subcases: (state) constrained trajectories.Tsmart




Unbounded trajectories• all Lagrange multipliers are equal to zero.

• one can choose pn12(t) = pn21(t) = 0, for t0 ≤ t ≤ tf .

• one can choose pn1(t) > 0, pn2(t) > 0 and pn1(t) > pn2(t) ⇒ S(t) > 0.


S(t) = pn1(t)−pn21(t)− pn2(t)+pn12(t)− λ112 + λ121 + λ112 − λ121 > 0 .

• the optimal solution is u∗2(t) = u2 , u∗1(t) = 1− u2 .

• Define Pn1(t) = pn1(t)− 1 and Pn2(t) = pn2(t)− 1.











recall that ...

dpn1(t)

dt= − ∂H

∂pn1= (pn1(t)− 1) · ∂G1(n1)

∂n1= Pn1(t) ·

∂G1(n1)

∂n1,

dpn2(t)

dt= − ∂H

∂pn2= (pn2(t)− 1) · ∂G2(n2)

∂n2= Pn2(t) ·

∂G2(n2)

∂n2,











Therefore,

dS

dt= Pn1(t) ·

∂G1(n1)

∂n1− Pn2(t) ·

∂G2(n2)

∂n2

= Pn1(t) ·[∂G1(n1)

∂n1− ∂G2(n2)

∂n2

]+ S(t) · ∂G2(n2)

∂n2.




Therefore,

S(t) = pn1(t)− pn2(t) > 0 ,

dS

dt= Pn1(t) ·

[∂G1(n1)

∂n1− ∂G2(n2)

∂n2

]+ S(t) · ∂G2(n2)

∂n2< 0.

• choosing the initial values of the costate variables Pn1(t) and Pn2(t) to makedS/dt < 0.

• let us consider: ∂G1(n1)/∂n1 > ∂G2(n2)/∂n2 .

• S(t) and Pn1(t), Pn2(t) will decrease.

• Singular solution: dS(t)/dt = 0 holds if S(t) = 0 and

∂G1(n1)

∂n1=∂G2(n2)

∂n2.




Singular solution

• taking full time derivatives of ∂G1(n1)/∂n1 and ∂G2(n2)/∂n2, one gets

d

dt

(∂G1(n1)

∂n1

)=∂2G1(n1)

∂n12·[q11(t) + q12(t) + u2(t) · d−G1(n1)

]=

d

dt

(∂G2(n2)

∂n2

)=∂2G2(n2)

∂n22·[q21(t) + q22(t) + (1− u2(t)) · d−G2(n2)

].

Denoting

a =∂2G1(n1)

∂n12 , b = ∂2G2(n2)

∂n22 ,

c = q11(t) + q12(t)−G1(n1) , e = q21(t) + q22(t)−G2(n2).

The singular control inputs

u∗2(t) = [b · e+ b · d− a · c]/[(a+ b) · d] ,u∗1(t) = 1− u∗2(t) .




• E.g. if the MFD shapes are approximated by second order polynomial functions,then the obtained singular curve n1(t) = θ(n2(t)) is linear:

u∗1(t) = 1− u2, u∗2(t) = u2, ∀n1(t) < θ(n2(t)),

u∗1(t) = u1, u∗2(t) = 1− u1, ∀n1(t) > θ(n2(t)),

singular control ∀n1(t) = θ(n2(t)).

n2(t)

n1(t)

∂G1(n1)∂n1

= ∂G2(n2)∂n2

∂G1(n1)∂n1

< ∂G2(n2)∂n2

∂G1(n1)∂n1

> ∂G2(n2)∂n2

Singular control

u∗1(t) = 1− u2, u∗2(t) = u2

u∗1(t) = u1, u∗2(t) = 1− u1




Kelley condition

Kelley condition: Second order necessary condition of optimality for the singular arc

(−1)q ∂

∂u

(d2q

dt2q∂H

∂u

)≤ 0 ,

where q is a so-called degree of singularity.

∂H

∂u2=S(t) = pn1(t)− pn2(t) = Pn1(t)− Pn2(t) ,

d2q

dt2q∂H

∂u2=d2S

dt2= Pn1(t) ·

{∂2G1(n1)

∂n12·[q11(t) + q12(t)−G1(n1) + d · u2(t)

]

− ∂2G2(n2)

∂n22·[q21(t) + q22(t)−G2(n2) + (1− u2(t)) · d

]},

(−1)q ∂

∂u2

(d2q

dt2q∂H

∂u2

)= −Pn1(t) · (a+ b) · d ≤ 0.

a < 0, b < 0, and Pn1(t) is also negative ⇒ Kelley condition is satisfied.




Switching from unbounded to (state-)constrainedtrajectories

• unbounded trajectories might switch to constrained trajectories, before enteringto the singular arc, if the upper or lower state constraint becomes active.

E.g. : the upper bound n12(t) = n12


S(t) = pn1(t)−pn21(t)− pn2(t) + pn12(t)− λ112+λ121 + λ112 − λ121 .

• λ112(t) will become positive such that the switching function S(t) = 0.

• dS/dt will be held equal to zero by applying corresponding values of λ112(t), and

the upper boundary singular control will be applied such that n12(t) = n12 issatisfied.

• The upper boundary singular control inputs are calculated from dn12/dt = 0,i.e. u∗1(t) = q12(t)/d, u∗2(t) = 1− u∗1(t).

Switching to a lower state constraint can be analyzed in a similar way.



Numerical verificationsUnbounded trajectories (with singular solution)

0 100 200 3001000

2000

3000

Time [s]

Accum

ula

tion [veh]

n1

n2

n12

n21

1500 2000 2500

2000

2200

2400

n1 [veh]

n2 [veh]

Optimal trajectory in (n1,n

2)−plane.

0 100 200 3000

0.2

0.4

0.6

0.8

1

Time [s]

u [−

]

u1

u2

0 5000 100000

2

4

6

8

Accumulation [veh]

G(n

) [v

eh/s

]

MFD1

MFD2



Switching to the upper state constraint n12 = n12

0 200 400 600

1000

2000

3000

Time [s]

Accum

ula

tion [veh]

n1

n2

n12

n21

800 1000 1200 1400

1800

2000

2200

2400

2600

n1 [veh]

n2 [veh]

Optimal trajectory in (n1,n

2)−plane.

0 200 400 6000

0.2

0.4

0.6

0.8

1

Time [s]

u [−

]

u1

u2

0 5000 100000

2

4

6

8

Accumulation [veh]

G(n

) [v

eh/s

]

MFD1

MFD2



Conclusions

Conclusions

• the optimal perimeter control synthesis has been presented for different cases ofinitial accumulation conditions.

• the optimal control law is presented in analytical feedback form, as a function ofcurrent regional accumulations n1(t) and n2(t).

Future research

• perimeter adaptive control based on MFD model with time delays (travel times).

• an application of these strategies in the field and/or in a micro-simulationenvironment.



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