transfer function representation approach of the lwr...

Transfer Function Representation Approach of the LWR Macroscopic Traffic Flow Model Based on the Green Function

EMIL NIKOLOV Department Automation

Technical University of Sofia 8, Kliment Ohridski St.

Sofia-1000 BULGARIA

Abstract - In the present article we offer a method for the highway traffic flow modeling, based on the Green function as a transformation tool. This approach is suitable for distributed parameter systems in order to re-formulate their partial differential equation description as sets of transfer functions. The model used in our work is the macroscopic continuous Lighthill-Whitham-Richards representation. The physical phenomena char-acteristic for the vehicular flow on a highway section are analyzed over some important length. This implies the description of the traffic as a distributed parameter plant. We present the Green function transformation’s gen-eral case and after we apply it to the considered traffic flow model in order to obtain its transfer function form. The sets of expressions obtained can be specified using concrete fundamental diagrams and data, and also can be implemented in the design of control strategies. Key-Words: - macroscopic traffic flow model; Green function transformations; representation transfer function form 1. Introduction. The application of any control strategy requires the appropriated and suitable model of the plant which behavior is to be analyzed and governed. The mathematical representation of the processes and phenomena tends to minimize the gap between model and reality according to the limitations and criteria involved. The traffic flow as a control plant, is char-acterized by high complexity the variables’ interac-tions as well as by a large variety of environmental conditions. Modeling in that domain has been devel-oped in several axes depending on the desired appli-cation and functionality. The most used and know classification of traffic flow models is composed ac-cordingly to the level of detail. Our work here con-siders one of the macroscopic models, based on the hydrodynamic analogy of vehicular and fluid streams. This point of view over the highway traffic flow delivers a mathematical description in terms of partial differential equations (PDEs), because repre-sents it as a distributed parameter plant. In such a system the notion of transfer function is meaningful after some transformations that are needed to substitute the PDEs. The space distribution of the highway section we consider is one-dimen-sional, as we are not interested in the individual vehi-cles, nor in their type, dimensions, lane changes. The

distributed nature of traffic is shown mostly through the presence in the final mathematical form of an im-portant time delay, proportional to the section’s length. 2. Traffic flow modeling. The variety of approaches for traffic flow modeling are developed according to several of its characteristics and serve different purposes. Traffic flow models can be classified according to the level of detail as micro-, meso- and macroscopic. In terms of behavior and con-trol variables, we distinguish stochastic and determi-nistic representations composed by random or average values, respectively. Traffic flow can be treated as con-tinuous or discrete medium and plant. Modeling in-volves an analytical description of it or a simulation program, delivering its states. It this work we adopt a macroscopic representation for the traffic, which in-volves its global variables, namely density, speed and flow, without considering the individual behavior of the vehicles. The model chosen for the actual work is proposed first by Lighthill and Whitham [15] while independently developed by Richards [19]. It is based on the hydro-dynamic analogy of traffic to river streams and de-scribes the traffic flow as a one-dimensional com-pressible fluid. The Lighthill-Whitham-Richards (LWR) representation has some strong disadvantages, criticized and overcome by numerous authors like

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ISBN: 978-960-6766-60-2 53 ISSN: 1790-5117

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Payne [17] and Daganzo [5]. The higher order mod-els they proposed describe more precisely the nature and complexity of the traffic flow phenomena ob-served in practice. Nevertheless, we have chosen the LWR, neglecting its shortcomings, basically for its simplicity due to the low order in the mathematical form. We also con-sider that it has been validated through several re-searches and implementations. Among them we can quote the acceptable interpretation of the LWR model characteristics [16], and the further improvements [4,14]. As we assume the macroscopic description, the fundamental variables are supposed continuous and the physical dimensions of vehicles - negligible. Thus, the amount of cars is conserved and another re-lation defines the correspondence of the speed or the flow and the density. This leads to the conservation (or continuity) equation [9,20]. The relation between speed v and density ρ delivers the correspondent flow-rate (1). The LWR model is defined by the conservation equation (2.a) and its initial (2.b) and boundary conditions (2.c). This ex-pression states that the number of vehicles is constant in each point of a considered highway section. Re-solving (2) requires one more equation or hypothesis. In the case of first order models, like the LWR, the afore mentioned speed-density relation is taken (3). As a consequence, we obtain the expression (4) de-fining the fundamental diagram of traffic flow (Fig.1).

q

( ) ( ) ( t,xvt,xt,xq )ρ= (1)

( ) ( )0

t

t,x

x

t,xq=

∂

∂+

∂

∂ ρ (2.a)

( ) ( )( ) ( )(

0

entree0

0

tt,Lx0tt,0

,t,00,xq

≥≤≤=

=ρρ )ρ

(2.b)

( ) ( )( t,xFt,xv )ρ= (3)

( ) ( ) ( )( t,xFt,xt,xq )ρρ= (4) In general, in fundamental diagrams, two particular regions are observed. The fluid and the congested states of traffic are separated by the critical point of density crρ and maximal flow-rate , called also capacity. The other important parameter is the maxi-mal density

maxq

maxρ corresponding to the completely blocked state of the highway.

Several equations have been developed to describe the fundamental diagram. They involve the parameters mentioned above, as well as the free-flow speed , reflecting the equilibrium traffic state.

fv

q

ρ

cρmaxρ

maxq

fluid traffic congested traffic

Iv

q

ρ

cρmaxρ

maxq

fluid traffic congested traffic

Iv

Fig. 1. General form for the fundamental diagram of traffic flow The dependence that connects speed and density has been derived, based on the observations made by Chandler [3] (5a), Greenberg [11] (5b), Drew [6] (5c), Greenshields [12] (5d) and Edie [7] (5e).

( )( )

( )⎟⎟⎠

⎞⎜⎜⎝

⎛−=

max

max t,x1

t,xt,xv

ρ

ρ

ρ

ρ (5.a)

( )( )

⎟⎟⎠

⎞⎜⎜⎝

⎛=

max

f

t,xnlvt,xv

ρ

ρ (5.b)

( )( )

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

5,0

max

f

t,x1vt,xv

ρ

ρ (5.c)

( )( )

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

max

f

t,x1vt,xv

ρ

ρ (5.d)

( ) ( )( )maxt,xf evt,xv ρρ−= (5.e)

As relation (4) is introduced in the conservation equa-tion (2.a), we can form (6), which is a function of the partial derivative ρ∂ . After some transformations, we obtain (7) where

( )( ) ( ) ρρρ dFdt,xF =′ holds. The derivative (8) according to x , of the relation (1), in common with (3), leads also to the conservation equation (9) and the same result (7). The derivative according to , of the relation (2), in common with (3), leads to the form (10), used in the conservation equation. Several transformations deliver an equation depending on the partial derivatives

t

q∂ and imply (11).

( ) ( )( )( ) ( )0

t

t,x

x

t,xFt,x=

∂

∂+

∂

∂ ρρρ (6)

( )( ) ( ) ( )( )( )( ) ( )

0t

t,x

x

t,xt,xFt,xt,xF =

∂

∂+

∂

∂′+

ρρρρρ (7)


ISBN: 978-960-6766-60-2 54 ISSN: 1790-5117

( ) ( )( ) ( )

( )t

t,xvt,xt,xv

x

t,x

x

t,xq

∂

∂+

∂

∂=

∂

∂ρ

ρ (8)

( ) ( )( ) ( )

( )0

x

t,xvt,xt,xv

x

t,x

t

t,x=

∂

∂+

∂

∂+

∂

∂ρ

ρρ (9)

( ) ( )( )( ) ( ) (( )( t,xFt,xt,xF

x

t,x

x

t,xqρρρ ) )

ρ′+

∂

∂=

∂

∂ (10)

( )( )( ) ( ) ( )( )( )

( )0

t

t,xq.

t,xFt,xt,xF

1

x

t,xq=

∂

∂

′++

∂

∂

ρρρ (11)

3. Representation of distributed pa-rameter systems using the Green- func-tion Transfer functions in the multidimensional case [18], where not only time, but also space dependencies of the variables are observed, allow considering in an autonomous way the impact on the system output of excitation functions, initial and boundary conditions. The basic advantages of the transfer function ap-proach in system modeling and control stem from their high precision and easy implementation in cal-culations and simulations, as well as from their im-proved stability. The LWR model, used in this work, reveals through its partial differential equation form the distributed character of the traffic as a control plant. Distributed systems can be transformed from their PDE description to transfer functions by several ap-proaches. We can enumerate the spectral methods (Gottlieb and Orszag, 1977), the finite element and

finite volumes (Strang and Fix, 1973), the separation of variables, the Fourier transform the method of characteristics (Woodward and Colella, 1984), etc. For the purposes of our work, we have chosen a transfor-mation technique applying the Green function (Ar-fken, 1985, Ford and Diethelm, 2001), which is a mathematical modeling tool. In general, a signal ( )t,xw in a control system is called distributed when being a function of both time and space. Consequently, a system element (block) (Fig.2) can receive as inputs and deliver as outputs dis-tributed signals of arbitrary dimensions. Input and out-put variables are measured for different and distant points in space, which involves the symbolic descrip-tion (13) of a distributed parameter element. This ex-pression can be rewritten as (12), which states that an integral operator matches to every input ( )t,xw 1 ,

( )011 tt,Dx ≥∈ an output of the form ( )t,xQ 2 ( )022 tt,Dx ≥∈ ,

( ) ( ) ( )τξτξ ,w,t,,xGt,xQ 22 = (13)

( ) ( ) ( )∫ ∫=

t

t D

22

0 2

dd,wt,,xGt,xQ τξτξξ (12)

Element( )t,xw 1 ( )t,xQ 2

Element( )t,xw 1 ( )t,xQ 2

Fig. 2. Control system element with distributed parameter sig-

l

Accordingly, a distributed parameter element is de-scribed by the kernel of the integral operator

(( )τξ ,t,,xG has two time ( )00 tt,t ≥≥τ

)τξ ,t,,xG , called Green function, impulse re-sponse function, source or influence function [2]. Analytical deriving of the Green function, used for solving PDEs and FDEs, requires information on at least one of the boundary or initial conditions. The function

and two space ( )12 D,Dx ∈∈ ξ arguments. The Green function of a distributed parameter element gives, in the moment and for the point 0tt ≥

22 Dx ∈ , its reaction to an unit impulse input in τ=t and ξ=1x . The output of this element represents the


ISBN: 978-960-6766-60-2 55 ISSN: 1790-5117

solution of a mathematical boundary value problem, integro-differential equation, etc. The symbolic relation between input and output is given by an operator (14). The function is

the univocal solution of the equation (14), which in-volves the existence of an inverse operator, such that (15) holds. Consequently, the impulse response func-tion ( t,xw 1 ) ( )τξ ,t,,xG 2 fulfils the equation (16) or (17).

( )( ) ( ) ( 02221122 tt,Dx,Dx,t,xwt,xQ,t,xl > )∈∈= (14)

( ) ( )( ) ( ) ( )t,xw,,t,xGt,xw,t,xlt,xQ 12121

2 τξ== − (15)

( )( ) ( ) ( )τδξδτξ −−= tx,,t,xG,t,xl 122 (16)

( ) ( ) ( )( )τδξδτξ −−= − tx,t,xl,t,,xG 121

2 (17)

Let be the function that defines the mathe-matical problem to be solved. The space coordinate x takes values in the open domain with boundary

, as the time coordinate does along the half-axe .

( t,xu )

DD∂

0tt ≥The considered PDE problem represents processes evolving into the region and having their begin-ning since . Let (18) be the mathematical prob-lem’s equation, with the corresponding initial (19) and boundary (20) conditions where are linear operators and , ,

D0t

N,,l Γ( )t,xf ( )t,xg ( )xu 0

are preliminary known or defined functions.

( )( ) ( ) ( 0tt,Dx,t,xft,xul >∈= )

( )( ) ( ) ( )tt,Dx,xut,xuN =∈=

(18)

00 (19)

( )( ) ( ) ( )0tt,Dx,t,xgt,xu >∈=Γ (20)

( )( ) ( ) ( 0tt,Dx,t,xt,xul >∈= )ϖ (21)

( )( ) ( 0tt,Dx,0t,xu >∂∈=Γ )

)

(22)

( )( ) ( 0tt,Dx,0t,xuN =∈= (23)

Solving the problem means to find the function ( )t,xu , that fulfils the system of equations

(18) ÷ (20). Every mathematical problem, stated in this way, exists a standardization function ( )t,xϖ , line-arly dependent on ( )( )t,xf ( ,xg, and )t xu 0 . This standardization function defines the equivalent problem in its standard form (21) (23). ÷It is possible to derive also the problem with zero ini-tial and uniform boundary conditions. For ( ) 0t,xg = and ( ) 0x0 =u , we have the prob-lem directly in its standard form which involves

( ) ( )t,xft,x =ϖ . The Green function ( )τξ ,t,,xG is a sufficient characteristic of the standard problem, as it fulfils the system of equations (21) ÷ (23), for D∈ξ ≥, 0tτ and

( ) ( ) ( )τδξδ −−= txt,xf , ( )0tt,Dx ≥∈

It equals that the Green function fulfils (24) ÷ (26). It is obvious that ( ) 0,t,,xG =τξ for 0t<τ , because the reaction of a system to an excitation cannot appear before that excitation itself.

( )( ) ( ) ( ) ( )012 tt,Dx,tx,,t,xGl >∈−−= τδξδτξ (24)

( )( ) ( )0tt,Dx,0,t,,xG >∂∈=τξΓ (25)

( )( ) ( )0tt,Dx,0,t,,xGN =∈=τξ (26)

( ) ( ) ( )∫ ∫=

t

t D0

dd,t,,xGt,xu τξτξϖξ (27)

( ) ( ) ( ωσξξ jp,tdt,,xGep,,xW

0

tp +== ∫∞

− ) (28)


ISBN: 978-960-6766-60-2 56 ISSN: 1790-5117

Given the Green ( )τξ ,t,,xG and the standardiza-tion functions of ( t,x )ϖ , the problem (18) ÷ (20) solution for arbitrary , , ( )t,xf ( )t,xg ( )xu 0 is obtained conform to the expression (27). The trans-fer function ( p,,xW )ξ of the considered prob-lem (18) (20), respective to is given by the integral (28). This is the Laplace transform, ac-cording to time, of the Green function. The fre-quency characteristic

÷ ( t,xu )

)( ωξ j,,xW , corresponding to the defined transfer function, is in fact the fre-quency characteristic of the considered problem (18) (20), or the temporal Fourier transform of the Green function.

÷

As our further efforts will point to the design and analysis of control strategies for the traffic flow regu-lation, we need to study a system composed by a plant and a controller in the context of the assumed transformation method by the Green function. Let us consider the feedback connection of a distributed pa-rameter plant to a localized parameter controller (Fig.3), which reflects the most common situation in practice. In this case, the system’s state is measured for one or several points in the definition domain of the output.

( )pW controller

( )p,,xW plant ξ

( )p,xu ( )p,xy

( )bx −δ ( )a−ξδ

( )p,,xW ξ

( )pW controller

( )p,,xW plant ξ

( )p,xu ( )p,xy

( )bx −δ ( )a−ξδ

( )p,,xW ξ

The control signal delivered is applied as well in one or several points of the input definition region. The transfer element

( ) ( axp,W −= )δξξ

is used to obtain the input in a given point ax = , as the other transfer block

( ) ( xbp,xW x −= )δ

is needed to deliver, for a given point , the er-ror signal to the plant.

bx =

4. Transformation of the LWR traffic flow model by the green function. We have applied the aforementioned transformation method, based on the Green function, to the concrete model of traffic flow, chosen for the purposes of our work. With that approach, we have led the PDE repre-sentation of the LWR governing equation to a transfer function form, according to the general statements of the Green function operability. The particular limita-tions that originate from the nature of the traffic mac-roscopic phenomena were considered. As well, the specific requirements of flow control were taken into account in order to achieve a useful and easy to imple-ment mathematical and functional form for the LWR model. Our reasoning started from the two forms derived for the conservation equation, depending on the partial de-rivatives ρ∂ (7) and q∂ (11), respectively. The use of the Green function for these expressions delivers two general forms for the flow model transfer function. When choosing one fundamental diagram and the corresponding relationship for the speed-density dependence (5a-e), one can specify the result-ing particular traffic representation.

0x = Lx =[ ]L0,x ÷∈ξ

outputinput

( )p,,xW ξ

bx = ax =

0x = Lx =[ ]L0,x ÷∈ξ

outputinput

( )p,,xW ξ

bx = ax =

Fig. 4. A highway section as a distributed parameter plant. If we consider a highway section of a certain length (Fig.4) as a distributed parameter plant, the respective transfer function defines the reaction (output) in a point to the (input) signal in another point b up-stream. This reaction depends on the space variables

L

a

ξ and x , which show the current position along the highway section. The Green function, as well as the corresponding standardization and transfer functions, are derived for a multitude of typical cases for the PDE and their form can be found in the mathematical literature (Boutkovskii A.G., 1979). Among the avail-able variants, we have chosen the one that fits best the LWR model considered. After some transformations, the traffic flow description (7) takes the form (29), which is a particular case of a typical PDE.

Fig. 3. Connection of a controller to a distributed parameter plant.

( ) ( )( ) ( )

( ) ( ) ( ) ( )⎪⎩

⎪⎨

⎧

≠≠≥≥==

=+∂

∂+

∂

∂

0b,0a,0t,0x,tgt,0u,xu0,xu

t,xft,xucx

t,xub

t

t,xua

0

(29)

( ) ( )( )

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−−−=

−−

ξδξξξ

xb

ate

b

1x1t,,xG

xb

c

(30.a)


ISBN: 978-960-6766-60-2 57 ISSN: 1790-5117

( ) ( ) ( ) ( ) ( ) ( )tgxlbtxuat,xft,x 0 −++= δδϖ (30.b)

( ) ( )( )

( ) ( )( )( )(⎪⎩

⎪⎨⎧

⎩⎨⎧

>−≤−

=−−=−

+−

0x,10x,0

x1;eb

1x1p,,xW

xb

cpa

ξξ

ξξξξ

) (30.c)

( ) ( )

( )( )( ) ( ) ( )( )( )

( )( ) ( ) ( )( )( )

( )( )( ) ( ) ( )( )( )t,xFt,xt,xF

xpt,xFt,xt,xF

x

et,xFt,xt,xF

ex1p,,xW ρρρ

ξρρρ

ξ

ρρρξξ ′+

−−′+

−−

′+−= (30.d)

( )( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )⎪⎩

⎪⎨

⎧

≠≠≥≥==

=+∂

∂+

∂

∂

0b,0a,0t,0x,tgt,0u,xu0,xu

t,xft,xuxcx

t,xub

t

t,xuxa

0

(31)

( ) ( )( )

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

∫−= ∫

− xydb

c

ydb

yate

b

1x1t,,xG

x

ξ

δξξ ξ (32.a)

( ) ( ) ( ) ( ) ( ) ( ) ( )tgxbtxuxat,xft,x 0 δδϖ ++= (32.b)

( ) ( )( ) ( )( )

( ) ( )( )( )( )⎪

⎪

⎩

⎪⎪

⎨

⎧

⎩⎨⎧

>−≤−

=−

∫−=

+−

0x,10x,0

x1

eb

1x1p,,xW

ydycpyab

1 x

ξξ

ξ

ξξ ξ

(32.c)

( ) ( ) ( )( ) ( ) ( )( )( )

( ) ( )( )( )( )⎪

⎪

⎩

⎪⎪

⎨

⎧

⎩⎨⎧

>−≤−

=−

∫−=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

′+−

0x,10x,0

x1

ex1p,,xWyd1

t,xFt,xt,xF

p

q

x

ξξ

ξ

ξξ ξρρρ

(32.d)

Thus, we have obtained the Green, the standardiza-tion and the transfer functions for that specific situa-tion in the form (30.a,b,c). The expression (30.c) describes a distributed plant with parameters depending on the Laplace operator p , as well as on the spatial variables ξ and x . This

transfer function is rewritten in terms of traffic flow, with the appropriated parameters of the LWR model, as (30.d). We have proceeded by analogy concerning the de-scription (11) of traffic flow and have obtained the relation (31), as well as the corresponding Green, standardization and transfer functions (32.a,b,c). Analyzing the results obtained so far, we have ob-served that for a fixed value of x , the other space variable ξ can take its values in the interval from 0 to x . The time-delay in both representations (30.a) and (32.a) is directly proportional to the distance be-tween the two points and b discussed, as ex-pected, it depends on the highway section’s length. Thus, the transfer functions (30.d) or (32.d), for den-sity or flow, respectively, reflects the relation for the

time instant between the variable’s values in point

a

t x and in another point ξ upstream ( ξ≥x ). It is easily seen that for ξ=x , both transfer functions are equal to unity, as it involves the same values of the variable considered. As our goal is to represent the LWR traffic flow model in a form, suitable for further control implementations, we consider the fundamental diagram (Fig.1) to find the optimal traffic conditions. A highway is best used, in terms of density, for values around the critical, and in terms of flow - for these, in-ferior to the maximum. A control law could be pro-posed for both cases. The task of regulation consists in the capability to guarantee the desired output of the considered highway section in a point x , treated as fi-nal, based on the data for the variable involved, avail-able in another point ξ upstream, viewed as initial. We assume that a control strategy developed according to the speed should maintain its value as closest as possible to the critical, in a small region around it. If the flow-rate is to be regulated, its maximum value should not be exceeded. In terms of performance and


ISBN: 978-960-6766-60-2 58 ISSN: 1790-5117

shape of the transitory processes in the corresponding step responses for the two cases cited, different crite-ria are respected. For the first, an oscillatory damping with an off-set is acceptable, whereas in the second only critical damping shape for the response fits. 5. Representation of the solution. The structural model of the highway traffic flow one uniform highway segment of fixed length is shown on Fig.5. The solutions (9) (32), reached by means of transformation of the LWR traffic flow model by the Green function, give the basis by means of con-

crete information to simulating characteristics of the process with the distributed parameters.

÷

Characteristics (when input is the density of a traffic flow inputρ , and the output is outputtρ ) of the highway traffic flow are shown on Fig.6 for

véh/h8800 q, véh/km 16 ρ1,0 ρ

véh/km80 ρ5,0 ρ véh/km , 160 ρρ

km/h , 220 v

max

max rampecr

max cr

max j

f

=

== = =

==

=

_

UniformHighwaySegmentof FixedLength

flow-rate

density

speed

density

flow-rate

speed

number of gateways

recommanded speednumber and capacity

of the ramps

outp

uts

inpu

ts

disturbancesatmospheric and highway conditions, accidents,

psychological and personal factors,etc…

inputρ

outoputρUniformHighwaySegmentof FixedLength

flow-rate

density

speed

density

flow-rate

speed

number of gateways

recommanded speednumber and capacity

of the ramps

outp

uts

inpu

ts

disturbancesatmospheric and highway conditions, accidents,

psychological and personal factors,etc…

inputρ

outoputρ

Fig. 5. Structural model of the highway traffic flow.

0 50 100 150 200 250 3000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Time (sec)

Am

plitu

de (-

)

Step Response Procede

( )ρ,L,th

( )ρτ ,LF,T,k =

0 50 100 150 200 250 300

0

0.005

0.01

0.015

0.02

0.025

0.03

Time (sec)

Am

plitu

de (

-)

Impulse Response Procede

( )ρτ ,LF,T,k =

( )ρ,L,ti

-1000 -900 -800 -700 -600 -500 -400 -300 -200 -100 0-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

20lo

g10[

G(jw

)] (d

b)

Phase(deg)

Nichols plot Procede

( )ρτ ,LF,T,k =

( )ρω ,L,jW

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Real (jw)

Imag

(jw

)

Nyquist plot Procede

( )ρτ ,LF,T,k =

( )ρω ,L,jW

0 50 100 150 200 250 3000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Time (sec)

Am

plitu

de (-

)


( )ρ,L,th

( )ρτ ,LF,T,k =

0 50 100 150 200 250 3000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Time (sec)

Am

plitu

de (-

)


( )ρ,L,th

( )ρτ ,LF,T,k =

0 50 100 150 200 250 300

0

0.005

0.01

0.015

0.02

0.025

0.03

Time (sec)

Am

plitu

de (

-)


( )ρτ ,LF,T,k =

( )ρ,L,ti

0 50 100 150 200 250 300

0

0.005

0.01

0.015

0.02

0.025

0.03

Time (sec)

Am

plitu

de (

-)


( )ρτ ,LF,T,k =

( )ρ,L,ti

-1000 -900 -800 -700 -600 -500 -400 -300 -200 -100 0-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

20lo

g10[

G(jw

)] (d

b)

Phase(deg)


( )ρτ ,LF,T,k =

( )ρω ,L,jW

-1000 -900 -800 -700 -600 -500 -400 -300 -200 -100 0-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

20lo

g10[

G(jw

)] (d

b)

Phase(deg)


( )ρτ ,LF,T,k =

( )ρω ,L,jW

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Real (jw)

Imag

(jw

)


( )ρτ ,LF,T,k =

( )ρω ,L,jW

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Real (jw)

Imag

(jw

)


( )ρτ ,LF,T,k =

( )ρω ,L,jW

Fig. 6. Characteristics of the highway traffic flow.


ISBN: 978-960-6766-60-2 59 ISSN: 1790-5117

6. Conclusion and further work. In this work we have proposed an approach for the traffic flow mathematical representation. The main idea consists in the application of a mathematical method or tool for that transformation. The final goal is to obtain a transfer function form for the highway traffic flow. The model we have chosen is the macro-scopic continuous hydrodynamic representation, pro-posed by Lighthill-Whitham-Richards. It is a first-order equation, that neglects some appropriate traffic phenomena, but even though we have selected it for it is simple and easy to implement. The LWR model of the highway vehicular flow represents a distrib-uted parameter unity and consist of partial differen-tial equations. Since, it requires a reformulation in terms of sets of transfer functions to become suitable for the implementation in control strategies based on this particular plant description. Among the existing methods for solving PDEs, we have preferred an ap-proach based on the Green function as a transforma-tion tool. Starting from the general form of a problem statement and solution, we have derived the sets of transfer functions corresponding to the two formula-tions for the conservation equation in the LWR flow model. For a highway section of a given length, we have analyzed the possibilities for regulation in terms of the variables concerned, the step response and the performance of the traffic system. For both cases we obtain sets of transfer functions, strongly dependent on the time-delay, that is to say, on the section’s length involved. Depending on whether the density or the flow-rate is controlled, different criteria for the damping of the time characteristic are imposed, namely limited off-set or non-oscillatory step re-sponse, respectively. Our hope is that the results we have obtained could be useful in further traffic flow modeling and control. The Green function approach could be applied to other higher order macroscopic and by consequence, more realistic models, like these of Payne, Helbing, etc. It can be specified for differ-ent concrete cases from practice with the correspond-ing data from measures, in order to deliver accuracy in traffic flow representation. As our future work will point essentially on the design of regulation strate-gies, we consider this first tentative very important for the conception of control systems with this par-ticular representation of traffic in the structure. Among our goals are the possibilities for application of our research results in real traffic conditions as to estimate its efficacy in terms of travel time, pollution, risk and costs reduction, which are the main indexes for traffic performance.

References. [1] Arfken G., 1985. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 332-335. [2] Boutkovskii A.G., 1979. Characteristics of distributed parameter systems, Moskva, Nauka, G.R.F-M.L, 1979

[3] Chandler F. E., R. Herman, and E. W. Montroll, 1958. Traffic Dynamics: Studies in Car Following, Operations Re-search, 6, p. 165-184. [4] Daganzo C.F., 1995. A finite difference approximation of the kinematic wave model of traffic flow, Transportation Re-search Part B, vol. 29B, no. 4, p. 261-276. [5] Daganzo C.F., 1995. Requiem for second-order fluid approximations of traffic flow, Transportation Research Part B, vol. 29B, no. 4, p. 277–286. [6] Drew D. R., 1965. Deterministic Aspects of Freeway Op-erations and Control. Highway Research Record, 99, p. 48-58. [7] Edie L. C., 1961. Car Following and Steady-State The-ory for Non-Congested Traffic. Operations Research, 9, p. 66-76. [8] Ford N. J. and K. Diethelm, 2001. Numerical solution of linear and non-linear fractional differential equations involv-ing fractional derivatives of several orders, Numerical Analy-sis Report 379, Manchester Centre for Computational Mathe-matics, Manchester. [9] Gartner N. H., C. J. Messer, and A. Rathi, 1998. Traffic Flow Theory: A State of the Art Report. U.S. Project of De-partment of Transportation Federal Highway Administration, Transportation Research Board, Oak Ridge National Labora-tory. [10] Gottlieb D. and S.A. Orszag, 1977. Numerical Analysis of Spectral Methods: Theory and Applications (Philadelphia: S.I.A.M.). [11] Greenberg H., 1959. An Analysis of Traffic Flow. Opera-tions Research, Vol 7, p. 78-85. [12] Greenshields B. D., 1935. A Study of Traffic Capacity. Highway Research Board Proceedings 14, p. 448-477. [13] Helbing and al., 2001. MASTER: macroscopic traffic simulation based on a gas-kinetic, non-local traffic model, Transportation Research Part B 35, p. 183-211. [14] Lesort J.B. and J.P. Lebacque, 1996. The Godunov scheme and what is means for first order traffic flow models,” in Proceedings of the 13th International Symposium on Trans-portation and Traffic Theory, Lyon, France, p. 647–677. [15] Lighthill M. J. and J. B. Whitham, 1955. On kinematic waves. I: Flow movement in long rivers. II: A Theory of traf-fic flow on long crowded roads. Proceedings of the Royal So-ciety London series A, 229, p.281–345. [16] Papageorgiou M., 1998. Some remarks on macroscopic traffic flow modelling, TransportationResearch Part A, vol. 32, no. 5, p. 323–329. [17] Payne H.J., 1971. Models of freeway traffic and control, Simulation Council Proceedings, no. 1, p. 51–61. [18] Rabenstein R. and L. Trautmann, 1999. Solution Of Vec-tor Partial Differential Equations By Transfer Function Mod-els. New York: McGraw-Hill. IEEE Int. Symposium on Cir-cuits and Systems (ISCAS), Orlando, USA. [19] Richards P. I., 1956. Shock waves on highways. Opera-tions Research 4, p.42-51. [20] Shvetsov V. I., 2003. Mathematical Modeling of Traffic Flows, Institute for Systems Analysis, Russian Academy of Sciences, Moscow, Russia. [21] Strang G. and G. Fix, 1973. An Analysis of the Finite Element Method (Englewood Cliffs, NJ: Prentice-Hall). [22] Woodward P. and P. Colella, 1984. Journal of Computa-tional Physics, vol. 54, p. 115–173.G. O. Young, “Synthetic structure of industrial plastics (Book style with paper title and editor),”in Plastics, 2nd ed. vol. 3, J. Peters, Ed. New York: McGraw-Hill, 1964, pp. 15–64. [23] W.-K. Chen, Linear Networks and Systems (Book style), Belmont, CA: Wadsworth, 1993, pp. 123


ISBN: 978-960-6766-60-2 60 ISSN: 1790-5117

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