Modeling for Control of Traffic Signal Systems

Yuji Wakasa, Kenichiro Hanaoka, Tadashi Iwasa and Kanya Tanaka

Abstract— This paper proposes two kinds of modeling meth-ods for real-time and network-wide traffic signal control. Theone is a modeling method based on the conventional store-and-forward modeling approach and the least squares method.This method can give a state-space model from practical data ofsimulation without calculating it from ideal traffic parameters.Therefore, this method is more practical for complex trafficsituations. The other is a method based on system identification,which not only can give a more practical model but also canrelax restriction on controllability of traffic signal systems. Asa result, this method extends the applicable range of trafficsignal control. The simulation examples are provided to showthe effectiveness of the proposed methods.

I. INTRODUCTION

Traffic signal control plays a very important role for roadsafety and smoothness of traffic flow. Especially, in orderto avoid traffic congestion in urban road networks, not onlyoff-line scheduling but also real-time control of traffic signalshas been studied [1], [2], [5], [9].

Recently, Diakaki et al. [2] have proposed a model forsuch network-wide traffic signal control, and have appliedoptimal linear quadratic control to the model. This researchseems very promising for intelligent transportation systems,whereas the model and the control method proposed in[2] have some problems on controllability of the modeland optimality of the method. To resolve these problems,an appropriate model and a robust control technique havebeen proposed in [11], [12]. The modeling method in [11],[12] as well as in [2] is based on the so-called store-and-forward modeling approach and needs some ideal orstatistical traffic parameters such as saturation flows andturning movement rates in order to construct the model.Therefore, this modeling method has disadvantages that itis relatively complicated to construct a model from suchparameters and that it might be too ideal to take accountof real traffic phenomena.

On the other hand, many traffic simulators have beendeveloped to evaluate traffic environments and conditions.Indeed, traffic simulators also need various traffic parameters,but it can sufficiently approximate real traffic phenomena.

In this paper, we propose two modeling methods of trafficsignal systems by using a traffic simulator. First, we proposea method for estimating a coefficient matrix of the state-spacemodel proposed in [11], [12] from traffic signal variations

This work is supported by Grant-in-Aid for Young Scientists (B)15760323 of the Ministry of Education, Culture, Sports, Science andTechnology in Japan

All of the authors are with Department of Electrical and Elec-tronic Engineering, Faculty of Engineering, Yamaguchi University, 2-16-1 Tokiwadai, Ube 755-8611, JAPAN. Corresponding author’s email:wakasa@eee.yamaguchi-u.ac.jp

and the resulting traffic volumes. This method is basedon the least squares method. The obtained model is moresuitable for traffic signal control by optimal and robustcontrol methods.

However, when this model and the conventional modelare used to control traffic signal systems, these models arerestrictive from the viewpoint of controllability of the modelsas pointed out in [12]. To cope with the restriction, wenext propose a modeling method by system identificationmethods. By this modeling, we can get controllable modelsfor the most traffic signal systems and can apply variouscontrol methods.

This paper is organized as follows. In Section II, wedescribe the conventional modeling method of traffic signalsystems. Section III proposes a modeling method by theleast squares method and describes a state-feedback controlmethod. In Section IV, we present a modeling method bysystem identification methods and an output feedback controlmethod for the obtained model. In Section V, simulationexamples are given to show the effectiveness of the proposedmethod, which is followed by the conclusion.

II. CONVENTIONAL MODELING

In this section, we describe the conventional modelingmethod proposed in [11], [12].

Traffic signals are controlled mainly by three parameterscalled cycle time, split and offset [2], [4], [8]. Roughlyspeaking, the cycle time is the time required for one completesequence of signal indications, the split is a ratio of the timeof each traffic phase to the cycle time, and the offset is thetime difference between the start of the green indication atone intersection and that at another intersection. In general,it is difficult to simultaneously optimize these control pa-rameters, and therefore, many methods for optimizing eachcontrol parameter have been studied [4], [8]. In practice,a common cycle time for coordinated signal control isoften adopted. Also, it is known that offset control is notso effective for congested or near congested cases [6]. Inthis paper, therefore, we concentrate on control of the splitparameter as well as in [2].

Now we describe a model for traffic signal systems.Consider an urban road network composed by the sets oflinks and intersections denoting L = {L1, L2, . . . , LnL

} andJ = {J1, J2, . . . , JnJ}, respectively. For each intersectionJj ∈ J , Ij and Oj denote the sets of incoming and outgoinglinks, respectively.

We here make the following assumptions:

• Cycle times T at all intersections are equal and fixed;

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

WC2.4

0-7803-9354-6/05/$20.00 ©2005 IEEE 1594

• Lost time (i.e., time of yellow and all red signal) of eachintersection is zero for simplicity;

• The turning movement rates ti,o from Li ∈ Ij to Lo ∈Oj at intersection Jj are assumed to be known andfixed;

• All roads cross at right angles (i.e., are along x and ydirections).

����

����

Jm Jn

� �

�

�

�

�

� �qi ri

�

wi

li

Li

Fig. 1. A road link along x direction.

x

y

phase 1 phase 2

green

red green

red

phase 1 phase 2

cycle time

road alongx-direction

road alongy-direction

Fig. 2. An example of signal cycle.

Consider a single link Li connecting two intersectionsJm, Jn as shown in Fig. 1. This link Li is a road link alongx direction. Also, as a simple example, these intersectionsare assumed to have two traffic phases as shown in Fig. 2. Atintersection Jj , let gj ∈ [0, 1] be a split (i.e., a ratio of greentime to the cycle time) for the road along x direction. Then,the split for the road along y direction at the intersection is1− gj . Note that the split needs to be about in between 0.2and 0.8 in practice.

According to the store-and-forward modeling approach[2], [3], the dynamics of link Li is expressed by

li(k + 1) = li(k) + qi(k) − ri(k) + wi(k), (1)

where li is the number of vehicles within link Li; qi andri are the inflow and outflow, respectively, of link Li overthe period [kT, (k + 1)T ] with the cycle time T ; and wi isa disturbance within link Li such as demand and exit flows.Moreover we make the following assumptions:

• The inflow to link Li is given by qi(k) =∑Lj∈Im

tj,irj(k);• li is sufficiently large, and therefore, the outflow ri of

link Li is given by ri = pign where pi is the saturationflow that is the product of the saturation flow rate andthe cycle time T ;

• When wi(k) = 0 for Li ∈ L, nominal splits gNm and gN

n

that lead to a steady-state system with an equilibriumpoint lNi for Li are available.

Note that the dynamics of links along y direction can bemodeled in the almost same way as the links along xdirection as mentioned above.

To construct a controllable system, we must carefully de-termine state variables from the links. To this end, we shouldnot choose the origin and destination links as state variablesbecause these links are connected with single intersectionsand thus are incomplete. Now let L(⊂ L) denote a set oflinks chosen as state variables.

Applying the above modeling way to all links Li ∈ L andarranging li(k), gj(k) and wi(k) as the vectors l(k) ∈ �nL ,g(k) ∈ �ng and w(k) ∈ �nL , respectively, we obtain thesystem

l(k + 1) = l(k) + Bg(k) + w(k) + c, (2)

where c ∈ �nL is the vector independent of l, g and w,and B ∈ �nL×ng and c are the matrix and the vector,respectively, linearly including the saturation flow parametersand turning movement rate parameters. From the assumptionof the existence of an equilibrium point, the equilibriumequation is given by

lN = lN + BgN + c, (3)

where lN and gN are the vectors of lNi and gNi for all links

Li ∈ L.Subtracting (3) from (2) and arranging differences l(k)−

lN and g(k) − gN as the state vector x ∈ �nL and thecontrol input vector u ∈ �ng , respectively, we obtain a lineardynamical model

x(k + 1) = x(k) + Bu(k) + w(k). (4)

The matrix B in system (4) includes all information ontraffic situations such as the saturation flow parameters theturning movement rates. System (4) is constructed based onan ideal traffic situation. In real-life traffic systems, however,various factors as well as the saturation flow parameters arechangeable and some of them might not be ignorable fortraffic signal control.

A. Controllability of the Traffic Signal Model

When we design a control system based on its model,controllability and observability of the model are importantproperties to be checked. In our case, x is assumed to bemeasurable, and hence, system (4) is observable. However,the controllability of system (4) is not always satisfied.Related to this, the following proposition holds for system(4).

1595

Proposition 1: System (4) is controllable if and only ifrankB = nL.

We see from Proposition 1 that nL ≤ ng is a necessarycondition for the controllability of system (4).

Next, we consider a rectangle grid network. Although thisis a special case, we can get some useful information onstructural design of traffic signal systems. Fig. 3 shows anm × n two-ways rectangle grid network. For this traffic

m

n

Fig. 3. A grid network.

network, the following theorem holds [12].Theorem 1: Suppose that the traffic network in Fig. 3 is

modeled by the method stated in this section and that allintersections have the same number of phases. Then, if theobtained model is controllable, the number of phases np

satisfies

np ≥ 5 − 2(m + n)mn

. (5)Proof The traffic network has mn intersections and 4mn−2(m + n) links. The number of control inputs ng is (np −1)mn and that of state variables nL is 4mn − 2(m + n).From nL ≤ ng , we obtain (5).

From Theorem 1, we see that the minimum np is between2 and 5 in the traffic network in Fig. 3. Table 1 showsthe minimum np for various m,n. This relation betweenthe controllability of the model and the number of phasesis helpful as a guide of structural design of traffic signalsystems.

TABLE I

MINIMUM np FOR m, n

m\n 1 2 3 4 5 6

1 - 2 3 3 3 32 2 3 4 4 4 43 3 4 4 4 4 44 3 4 4 4 5 55 3 4 4 5 5 56 3 4 4 5 5 5

III. MODELING BY THE LEAST SQUARESMETHOD

In the previous section, we have derived the state-spacemodel (4) from ideal or statistical information on saturationflow parameters and turning movement rates. As stated inthe previous section, the matrix B in system (4) includes

all information on these traffic situations. This model is rea-sonable for an ideal case. However, real-life traffic situationis complicated, and hence, the matrix B of a real-life casemight be different from that of the ideal case.

From this viewpoint, we propose direct estimation of thematrix B in the state-space model by using a traffic simulatorin this section. Since our goal is to control traffic volumesby split parameters, we give random split parameters g(k)and store the resulting traffic volumes l(k) from the trafficsimulator which can approximate real traffic situations. Thenwe obtain data u(k) and x(k), k = 1, . . . , N , and considerthe following least squares problem:

minB

N−1∑k=1

‖x(k + 1) − x(k) − Bu(k)‖2,

where ‖ ·‖ is the Euclidean norm. To solve this problem, wedefine the matrices

Ξ = [x(2) − x(1) · · · x(N) − x(N − 1)] ,U = [u(1) · · · u(N − 1)] .

By using these matrices, we can express the above leastsquares problem by

minB

‖Ξ − BU‖2F ,

where ‖ · ‖F is the Frobenius norm:

‖M‖F :=

√√√√m∑

i=1

n∑j=1

M2i,j for M ∈ �m×n.

Since

‖Ξ − BU‖2F

= Tr[(Ξ − BU)(Ξ − BU)T ]= Tr[ΞΞT − ΞUT BT − BUΞT + BUUT BT ],

the function ‖Ξ − BU‖2F is a convex quadratic function of

B. Therefore, from the normal equation

∂‖Ξ − BU‖2F

∂B= −2ΞUT + 2BUUT

= 0,

the optimal solution to the least squares problem is given by

B = ΞUT (UUT )−1,

if UUT is nonsingular. The nonsingularity of UUT is usuallyachieved by generating the split parameters g(k) randomly.

For the model (4) with the estimated matrix B, we canapply the state-feedback H∞ control as shown in [11]. Anaugmented plant for the H∞ control is as follows:

x(k + 1) = x(k) + w(k) + Bu(k)

z(k) =[

Q1/2

0

]x(k) +

[0

R1/2

]u(k)

where z is the control output, and Q and R are symmetricpositive definite matrices. By applying this plant to the state-feedback H∞ control, we can obtain a state-feedback gain

1596

K that minimizes the H∞ norm of the transfer function Gzw

from w to z

sup‖w‖2 �=0

‖z‖2

‖w‖2,

where ‖ · ‖2 is the l2 norm defined by ‖v(k)‖22 :=∑∞

k=0 v(k)T v(k).We see that ‖z(k)‖2

2 =∑∞

k=0 z(k)T z(k) gives a linearquadratic (LQ) performance index since

z(k)T z(k) = x(k)T Qx(k) + u(k)T Ru(k).

Therefore, we can minimize the worst-case LQ performanceagainst the disturbance by the above control system design.

IV. MODELING BY SYSTEM IDENTIFICATION

As pointed out in Subsection II. A, traffic signal systemsthat we can control based on the model (4) are restricteddue to the controllability. This restriction stems from thatthe matrix A in the state-space model is identity. In practice,however, the matrix A is not always identity because thestore-and-forward modeling and other ideal assumptionsmight be disturbed. In order to relax the restriction of trafficsignal systems, we can use a modeling method by systemidentification. In this case, the approximated matrix A isnot usually identity, and therefore, we can remove the aboverestriction in the most cases.

We use a traffic simulator to obtain traffic volumes l(k) forrandomly generated split parameters g(k) as in the previoussection. We now define y(k) := l(k)− lN instead of x(k) =l(k) − lN . For the input and output sequences u(k), y(k),k = 1, . . . , N , we apply a system identification method toobtain a state-space model.

Various system identification methods have been proposedso far, and are available by means of software such asMATLAB System Identification Toolbox [7]. By such a tool,we can generally obtain a state-space model as follows:

x(k + 1) = Ax(k) + Bu(k)y(k) = Cx(k) + Du(k). (6)

When we design a control system based on the model (6), wecan only use the output y(k) but cannot obtain x(k) directly.In this case, therefore, we apply an output-feedback controlmethod for the model (6). Note that the feedthrough term isusually zero, namely, D = 0 because traffic volumes changeafter split parameters change.

As in the previous section, we consider an augmented plantfor the model (6) as follows:

x(k + 1) = Ax(k) + w(k) + Bu(k)

z(k) =[

Q1/2C0

]x(k) +

[0

R1/2

]u(k)

y(k) = Cx(k), (7)

where the disturbance w is added to the state equation. TheH∞ norm to be minimized is

sup‖w‖2 �=0

‖z‖2

‖w‖2,

and an LQ performance index is dealt with as in the previoussection since

z(k)T z(k) = x(k)T CT QCx(k) + u(k)T Ru(k)= y(k)T Qy(k) + u(k)T Ru(k).

Applying the output-feedback H∞ control in order tominimize the H∞ norm of the plant, we obtain a controllerof the form:

x(k + 1) = Ax(k) + By(k)u(k) = Cx(k) + Dy(k).

The output-feedback control needs more on-line computa-tional complexity than the state-feedback control. However,it should be noted that we can deal with more various trafficsignal systems by this modeling and control approach.

V. SIMULATION EXAMPLES

In this section, we provide simple simulation examplesto show the effectiveness of the proposed modeling andcontrol methods. We use the traffic simulator AIMSUN andGETRAM Extensions 1 where the GETRAM Extensionsmodule is an application programming interface and is usedfor evaluating external signal control laws.

A. Example for Modeling Based on the Least SquaresMethod

����

����

J1 J2

��

L5

L6

��

L12

L11

�

�L7

L8 �

�L9

L10

�

�L4

L3 �

�L14

L13

��

L1

L2

Fig. 4. Traffic network with two intersections.

Fig. 4 shows a simple traffic network composed by thesets of two intersections and 14 links denoting J = {J1, J2}and L = {L1, L2, . . . , L14}, respectively. The incoming linksand the outgoing links are as follows:

I1 = {L2, L3, L5, L7}, I2 = {L1, L9, L11, L13}O1 = {L1, L4, L6, L8}, O2 = {L2, L10, L12, L14}.

All links have two lanes. We must to set state vari-ables so that the control system is controllable. Here wechoose the numbers of vehicles in links L1, L2 as thestate variables, i.e., L = {L1, L2}. Suppose that the suffi-cient vehicles (4000 [veh/h]) are generated in source linksL3, L5, L7, L9, L11, L13 and that the turning movement ratesare 0.5 for through movements and 0.25 for right and leftturns. The cycle time is set to be 60[sec] and the lengthof links L1 and L2 is set to be 230[m] in the traffic

1Microscopic traffic simulation systems developed by TSS.

1597

simulator AIMSUN. Under the above situations, the matrixB is estimated according to the modeling method presentedin Section III as follows:

Bprop =[

1.9290 −40.9158−24.5631 3.2832

].

On the other hand, the matrix B is calculated by theconventional method as follows:

Bconv =[

0 −60−60 0

],

where all saturation flow rates are assumed to be3600[veh/h]. We see that there is a large difference betweenideal and practical models.

In the above traffic situations, the links L1 and L2 arecongested for more than 50 vehicles, and near congestedfor between 30 and 50 vehicles. Based on the obtainedmodels with Bprop and Bconv, we compute state-feedbackcontrol laws and implement them to AIMSUN/GETRAMExtensions. We set the control parameters as Q = R = I ,lNi = 20 and gN

i = 0.5 for i = 1, 2. Figs. 5–7 show thenumbers of vehicles by the fixed splits (g1 = g2 = 0.5),the conventional modeling and control and the proposedmodeling and control, respectively. We see that the linksL1 and L2 get congested for the fixed splits while theconventional and proposed methods can avoid congestion.In order to compare performance of the conventional andproposed methods, we show histograms in the near congestedcondition where there are more than 30 vehicles within L1

or L2 in Figs. 8 and 9. We see from these figures that theproposed method is better than the conventional one.

0 20 40 60 80

10

20

30

40

50

60

70

80

Cycle

Num

ber

of v

ehic

les

Fig. 5. Number of vehicles by the fixed splits; l1 (solid) and l2 (dashed).

B. Example for Modeling Based on the System IdentificationMethod

We consider a control problem of a traffic system shownin Fig. 10. The length of Li, i = 1, . . . , 4 is 350[m] andthe other traffic conditions are the same as the previoussubsection. If the numbers of vehicles in 4 links Li, i =1, . . . , 4 are controlled by changing the split parameters in3 junctions with two phases, nL > ng holds from nL = 4and ng = 3. Namely, the conventional modeling method and

0 20 40 60 80

10

20

30

40

50

60

70

80

Cycle

Num

ber

of v

ehic

les

Fig. 6. Numbers of vehicles by the conventional method; l1 (solid) andl2 (dashed).

0 20 40 60 80

10

20

30

40

50

60

70

80

Cycle

Num

ber

of v

ehic

les

Fig. 7. Numbers of vehicles by the proposed method; l1 (solid) and l2(dashed).

the modeling method based on the least squares method arenot applicable because these models become uncontrollable.For this reason, we apply the modeling method by systemidentification.

According to the modeling and control method presentedin Section IV, we obtain a 4th order model and a controllerwith the same order. The control parameters are set as Q =diag(1, 1, 1, 10), R = 0.001I , lNi = 30 for i = 1, . . . , 4and gN

i = 0.5 for i = 1, 2, 3. Figs. 10 and 11 show thenumbers of vehicles by the fixed splits (g1 = g2 = g3 =0.5) and the proposed method, respectively. We see that theproposed method controls the numbers of vehicle within thelinks, while the fixed splits lead to congestion.

VI. CONCLUSION

In this paper, we have proposed two modeling methodsof traffic signal systems by using a traffic simulator. Also,we have presented the control methods corresponding tothese modeling methods and have shown the effectivenessof the proposed modeling and control methods by simpletraffic simulation examples. Although we have shown justsimulation results at this stage, the proposed approachesseem to be effective for field implementation, which needsfurther research.

1598

����

����

����

J1 J2 J3

��

L7

L8

�

�L9

L10 �

�L11

L12

�

�L6

L5 �

�L20

L19

��

L1

L3

�

�L13

L14

�

�L18

L17

��

L16

L15

��

L2

L4

Fig. 10. Traffic network with three intersections.

30 31 32 33 34 35 36 37 38 390

1

2

3

4

5

Number of vehicles

Cyc

le

Fig. 8. Histogram of the number of vehicles by the conventional method.

30 31 32 33 34 35 36 37 38 390

1

2

3

4

5

Number of vehicles

Cyc

le

Fig. 9. Histogram of the number of vehicles by the proposed method.

REFERENCES

[1] B. De Schutter, “Optimal Traffic Light Control for a Single Intersec-tion,” Proc. American Control Conference, pp. 2195–2199, 1999.

[2] C. Diakaki, M. Papageorgiou and K. Aboudolas, “A multivariableregulator approach to traffic-responsive network-wide signal control,”Control Engineering Practice, vol. 10, pp. 183–195, 2002.

[3] D. C. Gazis, “Network modeling and control: store-and-forward ap-proach,” Concise Encyclopedia of Traffic & Transportation System,Pergamon Press, pp. 473-478, 1991.

[4] P. B. Hunt, D. L. Robertson, R. D. Bretherton and M. C. Royle,“The SCOOT on-line traffic signal optimisation technique,” TrafficEngineering and Control, vol. 23, pp. 190–199, 1982.

[5] K. Iwaoka, T. Otokita and S. Niikura, “Optimization strategy on urbansignal control”, Proc. the 8th World Congress on ITS ’01, 2001.

[6] S. Kawakami and H. Matsui, Traffic Engineering, Morikita, 1987 (inJapanese).

0 20 40 60 80 100 120

10

20

30

40

50

60

70

80

90

100

110

CycleN

umbe

r of

veh

icle

s

Fig. 11. Numbers of vehicles by the fixed splits; l1 (solid), l2 (dashed),l3 (dash-dotted) and l4 (dotted).

0 20 40 60 80 100 120

10

20

30

40

50

60

70

80

90

100

110

Cycle

Num

ber

of v

ehic

les

Fig. 12. Numbers of vehicles by the proposed method; l1 (solid), l2(dashed), l3 (dash-dotted) and l4 (dotted).

[7] L. Ljung, System Identification Toolbox, For Use with MATLAB, TheMathWorks, Inc., 2001.

[8] Manual on Traffic Signal Control, Japan Society of Traffic Engineers,1992 (in Japanese).

[9] T. Oda and J. Leo, “Optimization of coordinated traffic signal timingsin urban road network,” Trans. ISCIE, vol. 11, no. 7, pp. 364–374,1998 (in Japanese).

[10] H. Shimizu, “Congestion control system of arterials,” Journal of theSociety of Instrument and Control Engineers, vol. 41, no. 3, pp. 199–204, 2002 (in Japanese).

[11] Y. Wakasa, K. Iwaoka and K. Tanaka, “Modelling and robust controlof traffic signal systems,” Proc. European Control Conference 2003(CD-ROM), 2003.

[12] Y. Wakasa, K. Hanaoka, K. Iwaoka and K. Tanaka, “Controllabilityof network-wide traffic signal systems and state-feedback controllertuning,” Proc. 11th World Congress on ITS Nagoya, Aichi 2004, Japan(CD-ROM), 2004.

1599