Constrained optimal steady-state control for isolated traffic intersections

Jack HADDAD,David MAHALEL,Ilya IOSLOVICH,Per-Olof GUTMAN

Technion Sustainable Mobility and Robust Transportation(T-SMART)Laboratory,

Technion-Israel Institute of Technology,Haifa 32000,Israel

Constrained optimal steady-state control for isolated traffic intersections

Jack HADDAD?,David MAHALEL,Ilya IOSLOVICH,Per-Olof GUTMAN

Technion Sustainable Mobility and Robust Transportation(T-SMART)Laboratory,

Technion-Israel Institute of Technology,Haifa 32000,Israel

The steady-state or cyclic control problem for a simplified isolated traffic intersection is considered.The optimization problem for the green-red switching sequence is formulated with the help of a discrete-event max-plus model.Two steady-state control problems are formulated:optimal steady-state with green duration constraints,and optimal steady-state control with lost time.In the case when the criterion is a strictly increasing,linear function of the queue lengths,the steady-state control problems can be solved analytically.The structure of constrained optimal steady-state traffic control is revealed,and the effect of the lost time on the optimal solution is illustrated.

Steady-state control;Discrete-event models;Isolated traffic intersections

1 Introduction

Control of an isolated traffic intersection has been proposed and applied by[1–11].The aim of these different models,methods,and strategies is to find the optimal green-red switching sequences in order to minimize delays or to maximize the intersection capacity.

Under the assumptions of a continuous differential equation model for isolated intersection with only two movements,constant total throughput,constant cycle length,constant saturation flows,time varying arrival rates,and over saturation(meaning that the initial queue lengths are bigger than zero),a necessary condition for not increasing queue lengths simultaneously was found in[1].

In[12],the optimal switching sequence for an isolated traffic intersection is computed for the steady-state problem with constant cycle length.When the criterion J is a strictly increasing linear function of the queue lengths,the steady-state problem is a linear programming problem that can also be solved analytically.The green duration constraints,i.e.,the maximum and minimum green durations,are not imposed in the formulation of the steady-state problem in[12].Furthermore,the total lost time,i.e.,time duration which is not effectively used by any movement,is neglected.Based on these assumptions,a necessary condition for the steady-state control with constant cycle length is derived.The N-stage control problem is formulated in[13].It was shown that the N-stage control problem can be solved by linear programming if the criterion J is linear and strictly increasing.Furthermore,the N-stage control can be used to bring initial oversaturated non-optimal queue lengths to optimum,i.e.,to the optimal steady state queues.

In this paper,under the same assumption as in[12]of a discrete-event model with constant arrival and departure rates,and with the optimization criterion being a strictly increasing linear function of the queue lengths,two problems are formulated:the optimal steady-state or cyclic problem with green duration constraints,and the optimal steady-state or cyclic problem with lost time.The green duration and lost time constraints have a practical importance related to traffic safety.The two problems are shown to be solvable by linear programming,and the optimal solution has a simple form that can also be found analytically.Clearly,it would easily be possible to formulate a combined problem with green duration constraints and lost time.The reason for splitting the problem is purely didactic:it becomes possible to explain the solutions graphically.

2 Problem definition

Let us consider a simplified isolated vehicular traffic intersection with two one-way movements(m1and m2),defined as the sets of vehicles having reached but not passed the intersection.Each movement is governed by a traffic signal that each can be either green or red.Since the two movements cannot occupy the intersection area simultaneously,the traffic signals will be opposite,i.e.,when movement m1has green light,movement m2sees red light,and vice versa.Each movement will encounter intertwined green and red periods.Without loss of generality the amber period is not explicitly considered,and a cycle is defined as a pair of one green and one red period,whose durations may be time-varying.The queue length qi(t)veh for movement miat time t,is defined as the number of vehicles belonging to miwhich is behind the stop line,i.e.,the queue does not include the vehicles that are inside the intersection or have passed it.Let ai(t)veh/s,and di(t)veh/s be the arrival and departure rates for queue i,respectively.The following assumptions are made:A1)the arrival rates are known,non-negative constants for each green or red period,and the departure rates are known,nonnegative constants within each green or red period,A2)di(t)＞ai(t),and di(t)=0,i=1,2 for green and red periods,respectively,A3)the queue lengths(veh)are approximated by non-negative real numbers,and A4)for every k=0,1,2,...,the kth cycle length,C(k)s,is limited downwards,C(k)≥Cmin＞0.

For the isolated traffic intersection with constant traffic arrival and departure rates, and constant cycle length,we intend to determine the steady-state cyclic traffic signal control solution that minimizes a given queue length dependent criterion,under green duration constraints,and lost time constraint,respectively.

3 Discrete-event models for isolated traffic intersections

In this paper,we consider the isolated traffic intersection as a switching system,as was done in[12,14–17],and the optimization of the traffic signal switching sequences will be performed with the same discrete-event max-plus model as in the mentioned papers.Next,the discrete-event max-plus model is briefly described.For a full description,see[12].

By definition,in a cycle,each movement(m1or m2)has only one green period.In the steady-state control problem with constant cycle length,it is assumed that all cycles and their flow rates are identical,and hence only one cycle needs to be considered,and the cycle index k may be dropped.The decision variables are then the green duration for m1and m2denoted by g1s and g2s,respectively,and the cycle duration is C=g1+g2.Without loss of generality, let the start time of the steady state cycle be 0 which also coincides with the start of the green light for movement m1.There are two switching times for the constant cycle: τ1s and τ2s,where τ1is the end of the green light for movement m1and the start of the green light for movement m2,while τ2is the end of the green light for movement m2which also coincides with the end of the steady-state cycle.Hence,C=τ2,g1= τ1,and g2= τ2?τ1.We now consider the following problem:for one cycle and starting time 0,compute an optimal switching time sequence τ1,τ2that minimizes a given performance criterion J1Let the queue length vector q be defined as[q1(τ1)q1(τ2)q2(τ1)q2(τ2)]T.The criterion function J is said to be strictly increasing,if,for all queue length vectors?q,?q with?q≤?q(elementwise)and?qi＜?qifor at least one index i,we have J(?q)＜J(?q)..There are a variety of criteria that can be chosen,e.g.,average queue length,maximal queue length,and delay over all queues,[15].The queue length for movement i at the start of the cycle will be equal to the queue length at the start of the next cycle,i.e.,q1(0)=q1(τ2),and q2(0)=q2(τ2).

Recalling Assumption A4,it holds that C=g1+g2≥Cmin＞0.For simplicity,we assume that the arrival rates are constant over the whole cycle,i.e.,a1(t)=a1,a2(t)=a2for 0≤ t≤ τ2.We also write d1(t)=d1for 0 ≤ t＜ τ1and d2(t)=d2for τ1≤ t＜ τ2.Note that these assumptions are reasonable for real world applications.Arrival demand profiles can be accurately predicted for traffic signal design,especially in recurrent congestion,based on historical data.Constant arrival rates for the rush hour can be calculated from the predicted profiles.While the departure rates can be assumed constant within each green period over all cycles,since vehicles discharge at their saturation flow during green light when queue exists at the intersection approach.

The cyclic discrete-event max-plus(CDMP)problem[12]is now defined as follows:

4 Steady-state cyclic control with green duration constraints

4.1 A linear programming formulation

The minimum and maximum green durations constraints are

where i=1,2,gi,mins and gi,maxs are the minimum and maximum green durations for movement i,respectively.The minimum and maximum green durations are given a priori by the traffic controller.The constraints(9)are imposed on the CDMP problem(1)–(8)which leads to the steady-state with green-duration constraints(SSGC)problem(1)–(9).Note that for scalars a,b,c ∈ R we have that a=max(b,c)implies a≥b and a≥c.In a similar way,the SSGC problem can be rewritten in such a way that the max equations are ‘relaxed’to linear inequality equations.However,first,the cyclic queue lengths equations(7)and(8)are substituted into(2)and(4),respectively

The max equations(10)and(5)can then be relaxed into linear inequality equations as follows:

This leads to the ‘relaxed’steady-state with greenduration constraints(R-SSGC)problem:

Proposition 1If the criterion J is a strictly increasing function of the queue lengths,then any optimal solution to the R-SSGC problem has the following properties:i)it holds that?q1(τ1)=0 and?q2(τ2)=0,and ii)it is also an optimal solution to the SSGC problem(1)–(9).

ProofIn[12],Proposition 1 is proven without(20)and(21).The same proof holds here.

In[12,equation(36)],the necessary and sufficient condition for steady-state control was derived for the CDMP problem,notably for the case when the arrival rate could change at the switching times.One can rewrite the derivation based on the R-SSGC problem without the constraints(20)and(21).The necessary and sufficient condition for a steady-state solution for constant arrival rates over the whole cycle is

4.2 Analytic solution for linear criterion

In this section,it is shown that if the criterionJis a strictly increasinglinearfunction of the queue lengths,then the R-SSGC problem can be solved analytically.

The ‘zero-queue-length period’(ZQLP)was defined in[12]as the time period(larger than zero)for which the queue length is equal to zero,refer to in[12,Fig.3].Given the assumptions,a movement can encounter at most one ZQLP per cycle,and it may happen only before the end of the green light,i.e.,between 0 and τ1for movementm1,and between τ1and τ2for movementm2.Let us denote the start of the ZQLP for movementsm1andm2byrespectively.Then,the ZQLP for movementm1starts at timeand ends at time τ1,and the ZQLP for movementm2starts at timeand ends at time τ2.

Now,we focus on a criterionJwhich is a strictly increasinglinearfunction of the queue lengths.LetJbe the weighted sum of the queue lengths,

wherew1,w2＞0.Let us assume that the minimum cycle lengthCminis feasible for the R-SSGC problem,i.e.,(12)–(21).The necessary and sufficient conditions for feasibility of the minimum cycle lengthCminin the R-SSGC problem are formulated in Proposition 2 below.Under this assumption,the optimal cycle length is equal toCminfor the R-SSGC problem with a criterionJthat is a strictly increasing linear function of the queue lengths,according to Proposition 2 in[12]which also holds here.Furthermore,according to Proposition 1,at the optimum it holds thatq1(τ1)=q2(τ2)=0.Hence,at the optimum,the criterionJis a function only of the maximum queue lengths,i.e.,

This leads to the following linear programming(LP)problem whenJis given by(24),

Remark 1The feasible minimum and maximum green durations constraints form2,replacing(21),are implied by(28),(29),and(30).

Remark 2It is assumed that the minimum cycle lengthCminis a feasible solution for steady-state control with green duration constraints(R-SSGC problem),i.e.,(12)–(21).Therefore,according to Proposition 2 in[12]which also holds here,the optimal cycle length is equal to the minimum cycle length for the R-SSGC problem with a criterionJthat is a strictly increasing linear function of the queue lengths,i.e.,the inequality in(19)is re-written as equality in(28).

The feasible set in the(g1,g2)–plane of the LP-problem(25)–(32)is illustrated in Fig.1.In[12],the solution given for the unconstrained problem(12)–(19)was found,i.e.,without the minimum and maximum green duration constraints(20)and(21).When the necessary and sufficient condition(22)is satisfied strictly,i.e.,a1/(d1?a1)＜(d2?a2)/a2,the solution to the problem depends on the slope of the linear is oclines ofJin the(g1,g2)-plane.For the unconstrained problem as shown in Fig.1(a),it holds that ifw2a2＜w1a1the optimal solution is found in PointA,whereby the movementm2will not have a ZQLP as shown in Fig.2,PointA.Whenw2a2＞w1a1the optimal solution is found in PointB,and the movementm1will not have a ZQLP as shown in Fig.2,PointB.PointsAandBare given by

whereg1,Ls,g1,Hs,g2,Ls,andg2,Hs are defined as follows:

Ifw2a2=w1a1every point on the straight line betweenAandB,i.e.,the convex combination of α(g1,g2)A+(1?α)(g1,g2)B,0≤α≤1,is optimal.The inner points will have two ZQLPs,one ZQLP for each movement.In the case when the necessary and sufficient condition(22)is satisfied with equality,i.e.,a1/(d1?a1)=(d2?a2)/a2,the two pointsAandBare equal.In this case the optimal solution will not have any movement with ZQLP.

Next,the optimal steady-state solution for the constrained problem R-SSGC(25)–(32)is introduced.Sinceg1+g2=Cmin,see(28),we can focus on only one variable,arbitrarily chosen asg1.When the green duration constraints(29)and(30)are imposed on the problem,the optimal solution is found in a point on the straight line betweenAandB.There are four different cases according to the relative values of

It follows from the unconstrained problem,i.e.,(26),(27),and(28),that

Hence,it is clear from Fig.1,and proven in Proposition 2 below,that feasible values forg1exist only ifg1,min≤g1,H,andg1,max≥g1,L.As shown in Fig.1,the green duration constraints do not affect the optimal solution in Case I,which is equal to the solution to the unconstrained problem.However,the maximum green duration constraint form1(30)changes PointAtoA′in Case II,and in Case III the minimum green duration constraint form1(29)changes PointBtoB′,while in Case IV both maximum and minimum green durations constraints change PointsAandBtoA′andB′,respectively.PointsA′andB′are given by

PointsA′andB′are inner points on the straight line betweenAandBin Fig.1,i.e.,g1,L＜ (g1)A′＜g1,H,g1,L＜ (g1)B′ ＜g1,H,and therefore,each movementm1,m2will have a ZQLP in the steady-state cycle.

Fig.1 Different cases with the imposed minimum and maximum green durations constraints. The equations of the straight lines connecting the origin and B,and the origin and A,are given by(26)and(27)with equality,respectively.

Let us call the optimal queue lengthsthe optimal steady-state queue lengthsq1,ssveh andq2,ssveh,respectively,which are equal to

Fig.2 Steady-state queue length profiles with green duration constraints.

Proposition 2The feasible solutiong1+g2=Cminof the R-SSGC problem,i.e.,(25)–(32),exists,if and only if the following inequalities hold:(22)and

ProofLet g1,Cmins and g2,Cmins be the green durations for m1and m2,respectively,of the minimum cycle length,i.e.,g1,Cmin+g2,Cmin=Cmin.From the minimum and maximum green durations constraints,i.e.,(9),it holds that:

Combining(45)and(46),we obtain

Recall that by definition g1,Cmin+g2,Cmin=Cmin.Hence,necessary conditions for feasibility of the minimum cycle length Cminin the steady-state control with green duration constraints are(41)and(42).Moreover,it follows from the unconstrained problem with optimal cycle length equalling minimum cycle length,i.e.,(26),(27),and(28),that

The green durations g1and g2for the constrained problem must satisfy(48),(49)and the minimum and maximum green durations constraints,i.e.,(9).Hence,a feasible set of g1,g2,that satisfies g1+g2=Cmin,exists only if(43)and(44)hold,i=1,2.Hence,equations(22),(41)–(44)are necessary conditions for feasible minimum cycle length in the steady-state control with green duration constraints.

Moreover,in the following it is proven that combining together these necessary equations we get the sufficient condition for feasibility of the minimum cycle length Cmin.We have to prove that if equations(22),(41)–(44)hold,then for g1,g2such that g1+g2=Cminthere is a feasible solution g1,g2,and Cminfor the R-SSGC problem,i.e.,(26)–(32)are satisfied.

If(22)holds,then there is a feasible solution g1,g2that satisfies(26)and(27),since(22)is a sufficient condition for the unconstrained steady-state control.From g1+g2=Cminand(22),equations(48)and(49)hold.Combining(48),(43)with i=1,and(44)with i=1,it is clear that a feasible g1exists such that(29)and(31)are satisfied,and combining(49),(43)with i=2,and(44)with i=2,it is clear that a feasible g1exists such that(30)and(32)are satisfied.Clearly,if(41)and(42)hold,g1+g2=Cminis feasible.Hence,combining together equations(22),(41)–(44)we get the sufficient condition for feasibility of the minimum cycle length Cmin,and for the optimal cycle length being equal to Cminfor the steady state control problem with green duration constraints,i.e.,(25)–(30).

Corollary 1For the R-SSGC problem(25)–(32)with a criterion J that is a strictly increasing linear function of the queue lengths,if at least one of the necessary conditions,i.e.,(22),(41)–(44),does not hold and if an optimal solution exists,then the optimal cycle length is larger than the minimum cycle length.

ProofThe proof follows from Proposition 2 and(19).

Remark 3Maximum queue lengths constraints,i.e.,q1(τ2)≤ q1,maxand q2(τ1)≤ q2,max,where q1,maxveh and q2,maxveh are the upper bounds on the queue length for m1and m2,respectively,can be transformed to green duration constraints,see[18].

4.3 Numerical examples for steady-state control with green duration constraints

Four numerical examples are presented demonstrating the optimal solutions in Points A,B,A′,and B′,defined in Fig.1 and(33),(34),(38),and(39).The criterion J is the weighted sum of the queue lengths(25)with w1=w2=1,and the minimum cycle length is Cmin=50s.The arrival and departure rate data(a1,a2,d1,d2)for examples 1,2 and 3,4 are(0.2,0.15,0.55,0.3)and(0.15,0.2,0.45,0.4),respectively.The minimum and maximum green durations(g1,min,g1,max)for examples 1,2,3,and 4 are(14.5,25),(14.5,22.5),(14.5,25),and(20,22.5),respectively.The queue length profiles in Point A,B,A′,and B′are shown in Fig.2.

5 Steady-state cyclic control with lost time

In this section,the optimal steady-state control problem with lost time is formulated based on the CDMP problem(1)–(8).The lost time L s(with L ≥ 0)is defined as the total lost time per cycle and it is determined from the equationwhere C s is the cycle length,and gis is the green duration for movement i.

In the previous sections,the lost time was neglected,and it was assumed that the cycle length is equal to the sum of the green durations,i.e.,C=g1+g2.Furthermore,the departure rate of a given movement(satura-tion flow)was assumed to be constant over its green light period.Here,the lost time is modelled by modifying the departure rate.A ‘zero’rate period,[0,Li]is included before the constant departure rate for movementi,such that the sum of the zero rate periods for the two movements is equal to the lost time,i.e.,The departure rate for movementiis defined asdi(t)=0 if 0≤t＜Li,anddi(t)=diifLi≤t≤gi.

When lost time is imposed on a movement at the start of its green light period,it affects the queue length profile.During the lost time,the departure rate is zero.Therefore,the lost time period is equivalent to a red period and queue lengths are increasing.Taking into account this,the CDMP problem is relaxed by modifying equations(12)and(14),respectively,to

This leads to the ‘relaxed’steady-state with lost time(R-SSLT)problem:

For the R-SSLT problem with a criterionJthat is a strictly increasing linear function of the queue lengths,the optimal cycle length is not necessarily equal to the minimum cycle lengthCmin,but still,at the optimum,q1(τ1)=q2(τ2)=0 holds,according to Proposition 1.This leads to the following linear programming(LP)problem whenJis given by(24),

Remark 4The optimal green duration for each movement must be larger that its lost time for the problem(53)–(56),i.e.,the solution must satisfy the following constraints:L1≤g1andL2≤g2.

In the(g1,g2)-plane,PointCis the intersection between the extremes of the two linear constraints(54)and(55),see Fig.3.It depends on the arrival and departure rates,and the lost time for the two movements.PointCis given by

The feasible(g1,g2)set is defined by(54)–(56).As shown in Fig.3,two possible feasible set shapes exist depending on the minimum cycle length constraint(56),and the position of PointC.Ifg1,C+g2,C＜Cmin(Case A),the optimal cycle length will be equal to the minimum cycle length, and the optimal solution is found in PointDorEdepending on the slope of the linear is oclines ofJin the(g1,g2)-plane.An analogy can be made between PointsDandEand PointsAandB,respectively,in the sense of ZQLP.PointsDandEare given by

Note that PointsDandEare equal to PointsAandBwhenL1=L2=0.Ifg1,C+g2,C≥Cmin(Case B),then the optimal solution is found in PointC.In this point,neither of the two movements will have a ZQLP,and the optimal cycle solution,C=g1,C+g2,C,will be larger than the minimum cycle length.The steady-state queue lengthsq1,ssandq2,ssare given by(40),whereg?2is equal tog2of PointC,D,orE.

Fig.3 Analytic solution for steady-state control with lost time.

Proposition 3For the R-SSLT problem,i.e.,(13),(15),(17)–(19),(50)–(52),with a criterionJthat is a strictly increasing linear function of the queue lengths,the optimal cycle length may be larger than the minimum cycle length.

Two numerical examples demonstrating the optimal solutions in PointsC,D,andEare presented.The criterionJis the weighted sum of the queue lengths(53)withw1=w2=1.The input data(a1,a2,d1,d2,L1,L2,Cmin)for examples 5 and 6 are(0.1,0.2,0.5,0.5,8,8,30)and(0.1,0.1,0.4,0.5,6,6,50),respectively.Sincew2a2=w1a1is also satisfied in example 6,all points on the straight line betweenD,(g1,g2)D=(34,16),andE,(g1,g2)E=(18.5,31.5),are optimal.The numerical results are shown in Figs.4–6.

Fig.4 Optimal solutions in Point C for example 5.

Fig.6 Optimal solutions in Point E for example 6.

6 Conclusions

Two steady-state control problems were formulated for a simplified isolated traffic intersection.Maximum and minimum green durations constraints were imposed in the first problem,while the modeling of lost time was inserted in the second problem.A discreteevent max-plus model was used to optimize the greenred switching sequence for the steady-state problems with constant cycle length.The solutions can be computed analytically when the criterionJis a strictly increasing linear function of the queue lengths.Necessary and sufficient conditions for the existence of constrained steady-state control with constant cycle length have been derived. Since arrival and departure rates, and queue lengths are readily measurable,and LP-problems easily solvable on line,the presented results may form a basis of approximately optimal traffic light control for isolated intersections.

[1]D.C.Gazis,R.B.Potts.The oversaturated intersection.Proceedings of the International Symposium on the Theory of Traffic Flow.London:Elsevier,1963:221–237.

[2]F.V.Webster.Traffic Signal Settings.London:Great Britain Road Research Laboratory,1958.

[3]G.Improta,G.Cantarella.Control systems design for an individual signalised junction.Transportation Research–Part B,1984,18B(2):147–167.

[4]H.R.Kashani,G.N.Saridis.Intelligent control for urban traffic systems.Automatica,1983,19(2):191–197.

[5]I.Talmor,D.Mahalel.Signal design for an isolated intersection during congestion.Journal of the Operational Research Society,2007,58(4):454–466.

[6]D.C.Gazis.Optimum control of a system of oversaturated intersections.Operations Research,1964,12(6):815–831.

[7]T.H.Chang,J.T.Lin.Optimal signal timing for an oversaturated intersection.Transportation Research–Part B,2000,34(6):471–491.

[8]A.Di Febbraro,N.Sacco.On modelling urban transportation networks via hybrid Petri nets.Control Engineering Practice,2004,12(10):1225–1239.

[9]M.Dotoli,M.Pia Fanti.An urban traffic network model via coloured timed Petri nets.Control Engineering Practice,2006,14(10):1213–1229.

[10]J.Lei,U.Ozguner.Decentralized hybrid intersection control.Proceedings of the 40th IEEE Conference on Decision and Control.Orlando:IEEE,2001:1237–1242.

[11]I.A.Villalobos,A.S.Poznyak,A.M.Tamayo.Urban traffic control problem:a game theory approach.Proceedings of the 47th IEEE Conference on Decision and Control.Cancun,Mexico:IEEE,2008:2168–2172.

[12]J.Haddad,B.De Schutter,D.Mahalel,et al.Optimal steadystate control for isolated traffic intersections.IEEE Transactions on Automatic Control,2010,55(11):2612–2617.

[13]J.Haddad,P.O.Gutman,I.Ioslovich,etal.Discrete dynamic optimization of N-stages control for isolated signalized intersections.Control Engineering Practice,2013,21(11):1553–1563.

[14]B.DeSchutter.Optimal control of a class of linear hybrid systems with saturation.SIAM Journal on Control and Optimization,2000,39(3):835–851.

[15]B.De Schutter.Optimizing acyclic traffic signal switching sequences through an extended linear complementarity problem formulation.European Journal of Operational Research,2002,139(2):400–415.

[16]B.De Schutter,B.De Moor.Optimal traffic light control for a single intersection.European Journal of Control,1998,4(3):260–276.

[17]J.Haddad,B.De Schutter,D.Mahalel,et al.Steady-state and N-stages control for isolated controlled intersections.Proceedings of the American Control Conference.St.Louis:IEEE,2009:2843–2848.

[18]J.Haddad.Optimal Control for Isolated Signalised Intersections.Ph.D.thesis.Haifa,Israel:Technion -Israel Institute of Technology,2010.

[19]I.Ioslovich,J.Haddad,P.O.Gutman,et al.Optimal traffic control synthesis for an isolated intersection.Control Engineering Practice,2011,19(8):900–911.

8 November 2012;revised 24 September 2013;accepted 25 September 2013

DOI10.1007/s11768-014-2247-7

?Corresponding author.

E-mail:jh@technion.ac.il.Tel.:+972-4-77887-1742.

This work was supported by the Technion-Israel Institute of Technology.

Jack HADDADreceived his B.S.(Cum Laude),M.S.(Cum Laude),and Ph.D.degrees,in Transportation Engineering from the Technion-Israel Institute of Technology,Haifa,Israel,in 2003,in 2006,and in 2010,respectively.He is an assistant professor and head of the Technion Sustainable Mobility and Robust Transportation(TSMART)Laboratory,Technion-Israel Institute of Technology,Haifa,Israel.In 2010–2013,he served as a postdoctoral researcher at the Urban Transport Systems Laboratory(LUTS),EPFL,Switzerland.His main research interests include management and control of transport networks,traffic light control,optimal control theory,model predictive control,and hybrid systems.In 2008,he was a visiting researcher at the Delft Center for Systems and Control(DCSC),Delft University of Technology,the Netherlands.He received the‘Office of the Executive Vice President for Research at Technion’grant(2008)for his Ph.D.research,the ‘Lenget Award for Excellence in Teaching’(2007),and the ‘Student Travel Grant’from Israeli Association for Automatic Control(IAAC)(2010).E-mail:jh@technion.ac.il.

David MAHALELis an associate professor in Transportation Engineering.He received his B.A.degree in Statistics and Economics from the Hebrew University,M.S.degree in Operation Research,and Ph.D.in Transportation Engineering(1979)from the Technion.He was a postdoctoral fellow in Imperial College in London(1979–1980).He held research and teaching position in University of Massachusetts in Lowell(1988–1989)and was an honorary visiting academic,Middlesex University,London,2000.He is the head of Transport Today and Tomorrow—an NGO for promoting sustainable transport in Israel.He is also a member of the Board of Or Yarok—an NGO for promoting traffic safety.Mahalel was the head of the Transportation Research Institute(2005–2011)and the head of Ran Naor Safety Research Institute(2009–2011).He was the head of the Transportation Section(1991–1993 and 2002–2004).Currently,Mahalel is the chief scientific advisor to the National Road Safety Authority.His main research areas are traffic management and control,traffic engineering,road safety.E-mail:mahalel@technion.ac.il.

Ilya IOSLOVICHwas born in Moscow,Russia,in 1937.He received his M.S.degree in Mechanics from Moscow State University,Moscow,in 1960,and Ph.D.degree in Physics and Mathematics from Moscow Institute of Physics and Technology(Phys-Tech),Moscow,in 1967.He hold positions of head of lab and head of division in different research institutions in Moscow.Since 1991,he has been with the Faculty of Agricultural Engineering,Technion,Haifa,Israel.Since 2002,he was a full professor in the Faculty of Civil and Environmental Engineering,Technion.After his retirement in 2012,he is scientific consultant at Technion Research and Development Foundation Ltd.His current research interests include optimization of agri-cultural,environmental and transportation systems,space research,optimal control,identification,and modeling.Professor Ioslovich is recipient of two silver medals for industrial achievements from the Soviet All-Union Exhibition in 1976 and 1983,respectively.E-mail:agrilya@technion.ac.il.

Per-Olof GUTMANreceived his M.S.degree in Engineering Physics(1973),Ph.D.in Automatic Control(1982)and the title of Docent in Automatic Control(1988),all from the Lund Institute of Technology,Lund,Sweden.He received his MSE degree from the University of California,Los Angeles in 1977 as a Fulbright grant recipient.

From 1973 to 1975 he taught Mathematics in Tanzania.During 1983–1984,he held a post-doctoral position at the Faculty of Electrical Engineering,Technion-Israel Institute of Technology,Haifa,Israel.During 1984–1990,he was a scientist with the Control Systems Section,El-Op Electro-Optics Industries Ltd.,Rehovot,Israel.From 1989,he is with the Technion-Israel Institute of Technology,holding,since 2007,the position of Professor at the Faculty of Civil and Environmental Engineering.

His research interests include robust and adaptive control,control of complex nonlinear systems, computer aided design, vehicle control,and traffic control.Gutman has(co-)authored 70 papers for reviewed international journals,has contributed to 6 books,and co-authored Qsyn-The Toolbox for Robust Control Systems Design for use with MATLAB.He served as an associate editor of Automatica,and as an EU evaluator of research proposals.E-mail:peo@technion.ac.il.