Research ArticleAn Adaptive Fuzzy-Logic Traffic Control System inConditions of Saturated Transport Stream

N R Yusupbekov A R Marakhimov H Z Igamberdiev and Sh X Umarov

Department of Information Technology Tashkent State Technical University 100095 Tashkent Uzbekistan

Correspondence should be addressed to N R Yusupbekov dodabekmailru

Received 26 December 2015 Accepted 16 May 2016

Academic Editor Oleg H Huseynov

Copyright copy 2016 N R Yusupbekov et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper considers the problem of building adaptive fuzzy-logic traffic control systems (AFLTCS) to deal with informationfuzziness and uncertainty in case of heavy traffic streams Methods of formal description of traffic control on the crossroads basedon fuzzy sets and fuzzy logic are proposed This paper also provides efficient algorithms for implementing AFLTCS and developsthe appropriate simulation models to test the efficiency of suggested approach

1 Introduction

One of the most important and promising tasks of transportsystems in modern cities is to satisfy the needs of bothstate and citizens in efficient transport services Prior studiesinvestigated methods of improving operational efficiencyof transport management systems in different countries [12] The results demonstrate that the most promising direc-tion is the development and implementation of intelligenttransport systems (ITS) Intelligent transportation system(ITS) is an integration of advanced information and com-munication technologies computer-aided control and trafficmanagement transport infrastructure vehicles and users toimprove safety and efficiency of road traffic [3]

Today the scope of the ITS implementation ranges fromsolving problems of traffic-light management and road safetyto increasing the efficiency of the existing transport systemCapabilities of the ITS are not limited to improving thesafety and efficiency of the transporting processes It buildsinformation retrieval systems focused on the identificationof vehicles the analysis of traffic situations and detecting vio-lations Furthermore ITS allows searching violators stolenvehicles and suspected criminals Such operations are basedon remote video monitoring collection and intellectualprocessing of large amounts of data as well as automaticgeneration of analytical reports The reports include bothsimple and integrated decision-making mechanisms

The review of the extant ITS projects showed that thelatest advances in information and communications tech-nologies control systems theory data mining and analy-sis techniques are not widely implemented in road trafficmanagement The reason is the lack of extensive researchand efficient methods of synthesis of multilevel intelligentsystems in conditions of fuzziness of parameters of the roadtraffic There are a number of studies on automating thedevelopment of control systems and their components Theyare focused on developing and improving the systems ofcoordinated traffic management on highways and adaptivemanagement that addresses problems of vehicle throughput[2ndash4] Furthermore there are a number of studies thatfocus on optimizing the traffic-light management [3ndash8] Asignificant number of works are devoted to the developmentof technical methods for measuring the parameters of roadtraffic and their processing [4ndash9] However these works arehighly scattered and focus on solving local problems of auto-mated control of road traffic They do not take into accountthat the improvement of road traffic in a particular area maylead to deterioration of the traffic situation on the other siteAnother factor that is not considered by previous studiesis that synthesized parameters of traffic-light managementsystems are optimal only under certain conditionsmdashwhen theparameters of road traffic do not change over time (or changein short intervals) In most cases the synthesis of control

Hindawi Publishing Corporatione Scientific World JournalVolume 2016 Article ID 6719459 9 pageshttpdxdoiorg10115520166719459

2 The Scientific World Journal

systems of road traffic does not consider unpredictable trafficsituations fuzziness of the parameters of the road traffic

Therefore developing conceptual framework and effec-tivemethods of structural and parametric synthesis of controlsystems of road traffic and development of information tech-nology support systems on their basis is important

2 Statement of a Problem

Below we consider a class of traffic control systems withimprecise information about the object in which the quali-tative characteristics of the road traffic are dominant

In general for synthesis of the traffic control systems theobject of control can be formally represented as follows

119872 = ⟨Ω119866119883 119884 119880 119879 120588 120574 120577⟩ (1)

where Ω = Ω1 Ω

2 Ω

119899times119898 is the range of conditions

(eg object output) 119866 = ⟨119881 (119863119882)⟩ is a model of thecrossroad represented by the graph 119866 in the space R3 119881 =

V119894 are the vertices (nodes) of the graph corresponding

to the nodes of possible branches of the road traffic 119863 sube

119881 times 119881 is the plurality of arcs of the graph (road sectionconnecting nodes of possible branches of road traffic) withcorrespondingweight coefficients as119882 = ⟨Λ 119875 119885⟩ (intensityof road traffic density of road traffic and average speed onthis particular road section) 119883 = 119883

1 119883

2 119883

119899 is the

plurality of characteristics and determinants describing thestate of the control object Ω and taking their values each inhis own set of values 119883

119894 119884 = 119884

1 119884

2 119884

119898 is the range

of output values (observed processes parameters estimatesetc)119880 = 119880

1 119880

2 119880

119903 is the range of controls (decisions)

119879 = 1199051 1199052 119905

119897 is the time (discrete or continuous)

120588 119883 times 119880 times 119879 rarr Ω is the description of the dynamics ofthe state of the object the reaction of the dynamic systemin a particular state to control actions 120574 Ω times 119879 rarr

119884 is the output describing the observation process of thecontrol object (obtaining estimates opinions etc) 120577 are someexternal uncontrollable factors conditions and others thathave an impact on the dynamics of the control object

The analysis of transport flows on the crossroads ascontrolled objects demonstrated that their mathematicalmodel should be able to deal with information fuzziness anddescribe random values and processes invariantly to theirdistribution Mathematical models should also provide mathformalism to express expert knowledge rich empiricism andheuristics

According to aforementioned requirements a general-ized dynamic model of the object controls (OC) can bedefined as the following linear equationwith fuzzy state space[9 10]

119909 = 119860 otimes 119909 oplus 119861 otimes 119906 120583

119878(119878) 119878 sube Ω

119910 = 119862 otimes 119909

(2)

with fuzzy initial conditions

1199091(0) = 119863

1

1199092(0) = 119863

2

119909119899(0) = 119863

119899

(3)

where otimes oplus are the fuzzy operations of multiplication andaddition 119906 is a control signal (scalar) that accepts discretenumerical value 119909 = 119909

1 119909

2 119909

119899 is the state space vector

119910 = 1199101 119910

2 119910

119898 is the output variables vector 120583

119878(119878) is

themembership function (MF) of the state space of the objectcontrols 119878 = 119904

1 1199042 119904

119873

119860 =

[[[[

[

1198601

1sdot sdot sdot 119860

1

119899

119860119899

1sdot sdot sdot 119860

119899

119899

]]]]

]

119861 =

[[[[

[

1198611

119861119899

]]]]

]

119862 =

[[[[[

[

1198621

1sdot sdot sdot 119862

1

119899

1198621

1sdot sdot sdot 119862

1

119899

]]]]]

]

(4)

are the matrix of the fuzzy coefficients of the model where

1198601

1=

1198601

1

1205831198601

1

(1198601

1)

119860119899

119899=

119860119899

119899

120583119860119899

119899

(119860119899119899)

1198611

= 1198611

1205831198611 (1198611)

119861119899

= 119861119899

120583119861119899 (119861

119899)

1198621

1=

1198621

1

1205831198621

1

(1198621

1)

1198621

119899=

1198621

119899

120583119862119899

119899

(1198621119899)

(5)

The Scientific World Journal 3

Some 119894th state space vector as a time function 119905 can bedescribed as fuzzy relation [1 3]

119909119894(119905) = 119905

119909119894

120583119909119894

(119905 119909119894) 119894 = 1 2 119899 (6)

At a fixed time this variable can be expressed as a fuzzyset

119909119894=

119909119894

120583119909119894

(119909119894) (7)

A similar description is 119895th output variable

119910119895(119905) =

119905

119910119895

120583119910119895

(119905 119910119895)

119895 = 1 2 119898

119910119895=

119910119895

120583119910119895

(119910119895)

(8)

Initial conditions and the number of variables of the statevector also can be described as the following fuzzy sets

119863119894=

119909119894

120583119863119894

(119909119894)

119878 = 119878

120583119878(119878)

(9)

Suppose that the membership functions of the inputand output linguistic variables are defined as the followinganalytical functions

120583119883119894

(119909119894) = 120593 (119909 119886

119883119894

1198871119883119894

1198872119883119894

1205731119883119894

1205732119883119894

)

= ((1198871119883119894

(119886119883119894

minus 119909))1205731119883119894

sign (1198871119883119894

(119886119883119894

minus 119909)) + 1

2

+ (1198872119883119894

(119909 minus 119886119883119894

))1205732119883119894

sign (1198872119883119894

(119909 minus 119886119883119894

)) + 1

2

+ 1)

minus1

(10)

120583119884119895

(119910119895) = 120593 (119910 119886

119884119895

1198871119884119895

1198872119884119895

1205731119884119895

1205732119884119895

)

= ((1198871119884119895

(119886119884119895

minus 119910))

1205731119884119895

sign (1198871119884119895

(119886119884119895

minus 119910)) + 1

2

+ (1198872119884119895

(119910 minus 119886119884119895

))

1205732119884119895

sign (1198872119884119895

(119910 minus 119886119884119895

)) + 1

2

+ 1)

minus1

(11)

In (10) and (11) the coefficients 119886119883119894

119886119884119895

1198871119883119894

1198871119884119895

1198872119883119894

1198872119884119895

1205731119883119894

1205731119884119895

1205732119883119894

and 1205732119884119895

are mood width and slope ofthe membership functions of the input and output linguisticvariables These coefficients allow forming high variety ofshapers ofmembership functionsThey can also act as indica-tors of information fuzziness for a formalmodel of the controlobjects which are provided as state space equation (2)

We assume that the indicators of quality of the controlsystem (eg time of transient process overshoot and bugtracking) are given as the following utility functions gener-ated by experts

119876 = 1199021

120583119876(1199021)

1199022

120583119876(1199022)

119902119896

120583119876(119902119896)

120583119876(119902119896) = 120593 (119902

119896 119886119876119896

1198871119876119896

1198872119876119896

1205731119876119896

1205732119876119896

)

(12)

under certain limitations for variablesrsquo state space and controlsignal

1198921(119909 119906 120574 119905) lt 119909

1max

1198922(119909 119906 120574 119905) gt 119909

2min

1198922119899minus1

(119909 119906 120574 119905) lt 119909119899max

1198922119899(119909 119906 120574 119905) gt 119909

119899min

119892119898minus1

(119909 119906 120574 119905) lt 119906max

119892119898(119909 119906 120574 119905) gt 119906min

(13)

where 120593 is membership function of the quality index of thecontrol system provided as (6) Then the problem of thesynthesis of control system for control objects provided asin a fuzzy state equation (2) with the quality evaluationindicator (12) and the system of constraints (13) can beformulated as described below

First it is necessary to synthesize control systems withembedded adaptive adjustment parameters of the controllerfor object controls (2) All signals should be limited in (13)and the transient processes in the system have to satisfy thepredetermined quality parameters (12)

For a given statement of the problem gradient speedsearch algorithm in the parametric formwith a proportional-integral controller and circuit bootstrapping can be selected[2 3] Among available methods of adaptive control withrobust characteristics the gradient speed search algorithm isthe least complex It also fits the constraints on the controlsignal and its velocity Then the control signal is generatedbased on the fuzzy set values of state space parametersaccording to the following modified control signal

119906 (119905 + 1) = 119896119906(119905) sdot 119906

119898(119905) +

119899

sum

119894=1

119896sum

119909119894

(119905) sdot 119909sum

119894(119905) (14)

4 The Scientific World Journal

119896sum

1199091

[119905 + 1] = 119896sum

1199091

[119905] (1 minus 1205961205743)

+ 120596 (1205745minus 120574

4) 120575

lowast[119905] 119909

sum

1[119905]

minus 1205961205745120575lowast[119905 + 1] 119909

sum

1[119905 + 1]

119896sum

1199092

[119905 + 1] = 119896sum

1199092

[119905] (1 minus 1205961205743)

+ 120596 (1205745minus 120574

4) 120575

lowast[119905] 119909

sum

2[119905]

minus 1205961205745120575lowast[119905 + 1] 119909

sum

2[119905 + 1]

119896sum

119909119899

[119905 + 1] = 119896sum

119909119899

[119905] (1 minus 1205961205743)

+ 120596 (1205745minus 120574

4) 120575

lowast[119905] 119909

sum

119899[119905]

minus 1205961205745120575lowast[119905 + 1] 119909

sum

119899[119905 + 1]

119896119906 [119905 + 1] = 119896

119906 [119905] (1 minus 1205961205741) + 120596 (1205746 minus 1205742) 120575lowast[119905] 119906119898 [119905]

minus 1205961205746120575lowast[119905 + 1] 119906119898 [119905 + 1]

(15)

where 119905 = 119898120596 120596 gt 0 is discretization step 120574 = 1205741 1205742 1205743

1205744 1205745 1205746 are the parameters of the adaptive controller 119890sum

119894=

int119909119894

(119909119894minus119909

119894119898)120583119890119894

(119890119894)119889119909

119894is a signal mismatch between the actual

parameters of the state vector and desirable model parame-ters 120583

119890119894

(119890119894) = 120593(119890

119894 119886119890119894

1198871119890119894

1198872119890119894

]1119890119894

]2119890119894

) is the membershipfunction of the signal mismatch provided as (10) 119886

119890119894

= 119886119909119894

minus

119909119894119898 ]

1119890119894

= ]1119909119894

]2119890119894

= ]2119909119894

1198871119890119894

= 1198871119909119894

1198872119890119894

= 1198872119909119894

119909sum119894

=

int119909119894

119909119894119889119909

119894are the integrated parameters of the state vector

120575lowast[119905] = sum

119899

119894=1ℎ119894sdot 119890sum

119894 ℎ

119894are the coefficients obtained by Lya-

punovrsquos decision equation and the matrix equation withreference model 119861M

The synthesis of the control law based on fuzzy modelcan increase the robustness of the gradient speed searchalgorithm it can also maintain limitations of the phase tra-jectories in certain areas under uncompensated informationfuzziness [11 12]

Quality indicators of the control systems are based notonly on the best possible dynamic characteristics of objectcontrolling 119886

119909119894

(119905) 119886119910119894

(119905) but also on the parameters of itsfuzziness 119886

119883119894

119886119884119895

1198871119883119894

1198871119884119895

1198872119883119894

1198872119884119895

1205731119883119894

1205731119884119895

1205732119883119894

and1205732119884119895

provided as (10) and (11)Reducing the information fuzziness and consequently

improving the quality control can be achieved by optimizingthe parameters of the membership functions of the input andoutput linguistic variables and fuzzy-logic controllers This isthe core idea of the algorithm for traffic control systems incrossroads presented in this study

3 The Concept of the ProblemSynthesis Adaptive Fuzzy-Logic TrafficControl Systems

Let us consider the problem of synthesis of fuzzy controlsystem of road traffic under conditions of intense traffic inmore detail

We suppose that there is a crossroad regulated with anequal number of intersecting lanes and well-known trafficintensity of 120582

1 120582

2 120582

3 120582

4 The directions of traffic 1 and 3 are

perpendicular to the directions 2 and 4We introduce the following terms Δ119879119906

1is the duration of

the green signal of the traffic light to the first direction of thetraffic 120582

1 120582

3 Δ119879119906

2is the duration of the green signal of the

traffic light to the opposite direction of the traffic 1205822 120582

4

Taking into consideration the principle of progressivecorrection the control object can be defined as (2) Thecontrol process assumes that there is a target goal andthe control system functions to achieve it The quality offunctioning (criterion of efficiency) of the control system (12)assumes a degree of adaptability to fulfill its taskmdashensuring asafe traffic with a minimum delay Therefore the problem ofoptimal traffic control at crossroads can be defined as follows

It is necessary to identify control signals Δ1198791199061and Δ119879

119906

2

(the duration of the green signal of the traffic light for eachlane) so that the actual state of the traffic 119910(119905) was as closeas possible to the desired state when the control signalshave certain limitations and the system exposed uncontrolledexternal and parametric impacts 119910lowast(119905)

That is it is necessary to find such parameters of thetraffic control system that the total delay at the crossroads isminimal [12 13]

119869 (119905) = int

119905

1199050

[119910lowast(120591 119909 (120591) 119906 (120591)) minus 119910 (120591 119909 (120591) 119906 (120591))] 119889120591

997888rarr min(16)

or

sum

119894

sum

119895

[11989013

119894119891lowast(119906

13

119894 11989013

119894) + 119897

13

119894119891lowast

119901119899(119906

13

119894 11989713

119894)

+ 11989024

119894119891lowast(119906

24

119894 11989024

119894) + 119897

24

119894119891lowast

119901119899(119906

24

119894 11989724

119894)] 997888rarr min

(17)

under the constraints

11989013

119894= 119902

leaving(13)119894

minus 119902arriv(13)119894

le 0

11989024

119894= 119902

leaving(24)119894

minus 119902arriv(24)119894

le 0

11990613

min le 11990613

119894le 119906

13

max

11990624

min le 11990624

119894le 119906

24

max

0 le 11989713

119894le 119871

13

max

0 le 11989724

119894le 119871

24

max

(18)

where 119891lowast(11990613119894 120582

13

119894) and 119891lowast

119901119899(11990613

119894 11989713

119894) are the delay evaluation

function and penalty function for the length of queue infront of the stop line of the traffic light in directions 1 and3 respectively 119891lowast(11990624

119894 120582

24

119894) and 119891

lowast

119901119899(11990624

119894 11989724

119894) are the delay

evaluation function and penalty function for the length ofqueue in front of the stop line of the traffic light in directions 2and 4 respectively 119906

119894is the duration of green signal of traffic

light in the i-stage 119902left(13)119894

119902arriv(13)119894

119902left(24)119894

and 119902arriv(24)119894

are

The Scientific World Journal 5

Fuzzy-logic controller(FLC)

The control object(transport flows)

Optimization module ofthe parameters FLC

Quality estimationmodule of the functioningof the system

Ylowast(t)

Ylowast(t)

E(t) U(t)

J(t)

Y(t)

sum

120583jT119890(x alowastj b

lowastj c

lowastj )

Figure 1 A traffic control system at the crossroads with an adaptive fuzzy-logic controller

the number of vehicles entered and left the control zone atthe crossroad in the 119894th phase respectively 119906min and 119906max arethe minimum and maximum duration of the green signal ofthe traffic light respectively 11989713

119894and 11989724

119894are the length of the

queue in front of the stop line at directions 1 and 3 and 2 and4 in the 119894th stage respectively 119871max is the maximum allowedlength of the queue in front of the stop line

In general the process control system at the T-shapedcrossroads can be rearranged as a feedback control sys-tem with adaptive fuzzy-logic controller (Figure 1) In the

proposed scheme the input vector 119864lowast = (119890lowast

1 119890lowast

2) is converted

into fuzzy-logic controller (FLC) through fuzzification blockNext fuzzy inference is performed based on rule base whichresults in a fuzzy output variable 119906lowast The output values fromthe FLC are then transferred from the fuzzy region 119906

lowast toaccurate region 119906 through defuzzification block [14 15]

In general rule bases or knowledge bases contain descrip-tions of the linguistic variables of the FLC that are predefinedfor each of the input and output variables Therefore weintroduce the following linguistic variables

1198901= (ldquoDifference in the number of vehicles that entered and left the control zonerdquo 119885

1198901

1198641)

1198902= (ldquoQueue lengthsrdquo 119885

1198902

1198642)

119906 = (ldquoControlrdquo 119885119906 119880)

(19)

where 119885119890119894

= 1198851

119890119894

1198852

119890119894

119885119896

119890119894

119894 = 1 2 and 119885119906= 119885

1

119906 119885

2

119906

119885119896

119906 are the term-sets of linguistic variables 119890

1 119890

2 and

119906 with corresponding membership functions (MF) 119885119897119890119894

=

120583119897

119890119894

(119890119894) 119885119897

119906= 120583

119897(119906) 119897 = 1 119896 given by universal sets 119864

119894=

[119864119894min 119864119894max] and 119880 = [119880min 119880max]

We assume that seven terms 119885119909= ldquoNBrdquo ldquoNMrdquo ldquoNSrdquo

ldquoZErdquo ldquoPSrdquo ldquoPMrdquo ldquoPBrdquo correspond to each of the input andoutput linguistic variables 119885

119909= 119879

1198901

1198791198902

119879119906 with trian-

gular membership functions provided as (10) As a result thefollowing linguistic variables are derived from the fuzzifica-tion processes

1198901= (ldquoDifference of the number of vehicles that entered and left the control zonerdquo 119885

1198901

1198641)

= [

120583NB1198901

(1198901)

NB1198901

120583NM1198901

(1198901)

NM1198901

120583NS1198901

(1198901)

NS1198901

120583ZE1198901

(1198901)

ZE1198901

120583PS1198901

(1198901)

PS1198901

120583PM1198901

(1198901)

PM1198901

120583PB1198901

(1198901)

PB1198901

]

1198902= (ldquoQueue lengthsrdquo 119885

1198902

1198642) = [

120583NB1198902

(1198902)

NB1198902

120583NM1198902

(1198902)

NM1198902

120583NS1198902

(1198902)

NS1198902

120583ZE1198902

(1198902)

ZE1198902

120583PS1198902

(1198902)

PS1198902

120583PM1198902

(1198902)

PM1198902

120583PB1198902

(1198902)

PB1198902

]

119906 = (ldquoControlrdquo 119885119906 119880) = [

120583NB119906(119906)

NB119906

120583NM119906(119906)

NM119906

120583NS119906(119906)

NS119906

120583ZE119906(119906)

ZE119906

120583PS119906(119906)

PS119906

120583PM119906(119906)

PM119906

120583PB119906(119906)

PB119906

]

(20)

6 The Scientific World Journal

Next we develop a rule base for the FLC inferences asfollows

IF (119885119895

1198901

times 119885119895

1198902

)

ELSE 119885119895

119906 119895 = 1 7

(21)

where (1198851198951198901

times119885119895

1198902

) is the Cartesian product of the fuzzy sets 1198641

and 1198642 with the following membership function

120583(119885119895

1198901times119885119895

1198902)(1198901 1198902) = 120583

119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902) (22)

The corresponding output fuzzy set is defined by fuzzyrelation 119877119895 = (119885

119895

1198901

times 119885119895

1198902

) times 119885119895

119906 119895 = 1 7 with the following

membership function

120583119877119895 ((119890

1 1198902) 119906

lowast) = (120583

119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902))

and 120583119885119895

119906

(119906lowast)

(23)

The set of all rules 119877 = ⋃7

119895=1119877119895 with fuzzy membership

function

120583119877((119890

1 1198902) 119906

lowast)

=

7

⋁

119895=1

[(120583119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902)) and 120583

119885119895

119906

(119906lowast)]

(24)

which defines the rule base of the FLC and produces analgorithm of fuzzy control system

As a result for given values of the input linguistic var-iables 119885

119895

1198901

and 119885119895

1198902

the fuzzy output value of the logicalcontroller119885119895

119906can be determined based on the following com-

posite rule [12]

119861119895= (119885

119895

1198901

times 119885119895

1198902

) sdot 119877 (25)

with a degree of membership

120583119885119895

119906

(119906lowast) = ⋁

1198901isin1198641 1198902isin1198642

[(120583119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902))

and 120583119877(1198901 1198902 119906lowast)]

(26)

When the linguistic input variables 1198901and 119890

2correspond

to fuzzy sets11988510158401198901

and11988510158401198902

fuzzy set1198851015840119906of the linguistic variable

of the control signal 119906lowast is determined as follows

1205831198791015840

119906

1015840 (119906lowast) = max

11989011198902

[

119899

prod

119894=1

1205831198791015840

119890119894

(119890119894)]

sdot [

119898

min119895=1

[

119899

prod

119894=1

120583119879119895

119890119894

(119890119894)] sdot 120583

119879119895

119906

(119906lowast)]

(27)

In order to obtain a true output value following fuzzycomputations it is necessary to perform defuzzificationmdashthetransformation of fuzzy linguistic variable value (qualitative)

119906lowast into discrete numerical value of 119906 For this purpose center

of gravity method is applied [12]

119906 =

sum7

119899=1119906lowast

119899120583119885119906

(119906lowast

119899)

sum7

119899=1120583119885119906

(119906lowast119899)

(28)

Membership function of fuzzy values 1198851015840119906can be repre-

sented as

1205831198851015840

119906

1015840 (119906) =

119899

prod

119894=1

120583119885119895

119890119894

(119890119894) 119906 = 120593

119895

0 119906 = 120593119895

(29)

where 120593119895 is the discrete numerical value of the output signalThen the final output value of the FLC at the defuzzifica-

tion stage can be computed as follows

119906 =

sum119898

119895=1120593119895[prod

119899

119894=1120583119885119895

119890119894

(119890119894)]

sum119898

119895=1prod119899

119894=1120583119885119895

119890119894

(119890119894)

or 119906 (119890 120593) =119898

sum

119895=1

120593119895120595119895(119890)

(30)

where

120595119895(119890) =

prod119899

119894=1120583119879119895

119890119894

(119890119894)

sum119898

119895=1prod119899

119894=1120583119879119895

119890119894

(119890119894) (31)

If we consider that the basic equation of the road trafficcontroller can be written as [12]

119906 (119905) = 1199060+ 119870(119890 (119905) +

1

119879119906

int

119905

0

119890 (120591) 119889120591 + 119879119889

119889119890 (119905)

119889119905) (32)

then the rule of the fuzzy traffic control system can berepresented as a logic controller with the varying coefficientas in (14)

119906 (119905) = 1199060+ 119870

lowast(120572 120573) ∘ 120595

lowast

1199060

(119890)lowast (33)

where 119870lowast(120572 120573) is the variable part of the gain that depends

on the current dynamic parameters of the road traffic [16]This allows implementing an adaptive traffic control systemas a fuzzy system consisting of two interconnected modulesldquomodule of fuzzy controller of the switching phase trafficlightsrdquo and ldquomodule of fuzzy adaptive correction of theparameters of the main fuzzy controller of the switchingphase traffic lightsrdquo 120572 and 120574

Furthermore we introduce the following linguistic vari-ables to assess the quality of the control system and adaptivefuzzy correction of the parameters of the fuzzy-logic con-troller119889119890

2

= (ldquoThe Derivative Queue Lengths of Vehiclesrdquo 1198851198891198902

1198641198891198902

)

119876 = (ldquoThe Quality Control Systemrdquo 119885119876 119864

119876)

(34)

The Scientific World Journal 7

120583intQ119896

and 1205833intQ119896

997888rarr (alowastX119894 alowastY119894 b

lowast1X119894

blowast2X119894 blowast1Y119894 b

lowast2Y119894 120573lowast1X119894

120573lowast2X119894 120573lowast1XY119894 120573

lowast2Y119894

)1205833Q119896

120583Q119896

120583Q119896

Qk

Figure 2 Graphic illustration of a choice of optimum adjusting parameters

where 1198851198891198902119894

= 119885(1)

1198891198902

119885(2)

1198891198902

119885(119899)

1198891198902

119885119876119894

= 119885(1)

119876 119890119894

119885(2)

119876

119885(119896)

119876 t are the term-sets of linguistic variables 119889119890

2and119876with

corresponding membership functions

1198851198891198902

=

119889119890(119894)

2

1205831198851198891198902

(119889119890(119894)

2)

119894 = 1 119899

119885119876=

119902(119895)

120583119876(119902(119895))

119895 = 1 119896

(35)

given by universal sets 1198641198891198902

= [1198641198891198902min 119864119889119890

2max] and 119864

119876=

[119864119876min 119864119876max]Then according to (12) and (25) the choice of the optimal

adjustment parameters of the FLC can be represented as thefollowing fuzzy relations

119877119895

119876lowast = (119885

119895

1198902

times 119885119895

1198902

) times 119885119895

119876 119895 = 1119873

119876 (36)

where

119885119876=

119876119879119875

1

120583119879119875

1198761

(119876119879119875

1)

120583119879119875

119876119896

(119876119879119875

119896) = 120593 (119876

119879119875

119896 119886lowast

119876119896

119887lowast

1119876119896

119887lowast

2119876119896

120573lowast

1119876119896

120573lowast

2119876119896

)

120593 (119876119879119875

119896 119886lowast

119876119896

119887lowast

1119876119896

119887lowast

2119876119896

120573lowast

1119876119896

120573lowast

2119876119896

)

= ((119887lowast

1119876119896

(119886lowast

119876119896

minus 119909))

120573lowast

1119876119896

sdot

sign (119887lowast1119876119896

(119886lowast

119876119896

minus 119909)) + 1

2

+ (119887lowast

2119876119896

(119909 minus 119886lowast

119876119896

))

120573lowast

2119876119896

sign (119887lowast2119876119896

(119909 minus 119886lowast

119876119896

)) + 1

2

+ 1)

minus1

119876 = 1199021

120583119876(1199021)

1199022

120583119876(1199022)

119902119896

120583119876(119902119896)

120583119876(119902119896) = 120593 (119902

119896 119886119876119896

1198871119876119896

1198872119876119896

1205731119876119896

1205732119876119896

)

(37)

This fuzzy relationship served as the basis for developingalgorithms of adaptive parameter settings of the fuzzy-logiccontroller (Figure 2)

4 Simulation Modeling and Analysis ofFuzzy Traffic Control System

We develop a simulation model of the adaptive fuzzy controlsystem of road traffic on Matlab software package Thesimulation model is used to test the efficiency of the fuzzycontrol system through series of computational experiments

As an object of simulation we chose one of the heavy-loaded and problematic crossroads in Tashkent city Uzbek-istan The simulation model considered the road trafficintensity observed at the crossroad during the peak-hoursfrom 0700 to 1000 onMay 5 2015The experiment consistedof two phases At each phase of the simulation variouspatterns of traffic-light management were used In the firstphase the traffic lights with fixed cycles which are used inreal life are simulated In the second phase traffic lightscontrolled with adaptive fuzzy system were simulated Forthis purpose the rules of fuzzy control system with a modulefor adaptive adjustment of parameters were synthesized usingthe proposed method

The simulation results are presented below Figure 3shows the results of simulation of the road intersection thatis managed by traffic lights with fixed cycles Figure 4 showsthe results of simulation of the crossroad that is managed byadaptive fuzzy controller

8 The Scientific World Journal

The length of the queue of vehicles in directions 1ndash3

The length of the queue of vehicles in directions 2ndash4

710

720

910

900

820

800

810

920

700

840

850

750

740

730

930

940

950

830

1000

05101520253035404550

2ndash4The intensity of the traffic flow in directions

The intensity of the traffic flow in directions 1ndash3

Figure 3 The results of the simulation modeling of the trafficcontrol system with optimum fixed cycle of traffic lights

1ndash3The length of the queue of vehicles in directions

The length of the queue of vehicles in directions 2ndash4

710

720

910

900

820

800

810

920

700

840

850

750

740

730

930

940

950

830

1000

2ndash4The intensity of the traffic flow in directions

The intensity of the traffic flow in directions 1ndash3

05101520253035404550

Figure 4 The results of the simulation modeling of the adaptivetraffic control system

Simulation results show that the synthesized fuzzy controlsystem successfully solved the problem of queues at trafficlights and performed more effective than the current trafficcontrol system with fixed-cycle traffic lights Particularly thelength of queue for the synthesized fuzzy control system isless than 26 times for traffic control system with fixed-cycletraffic lights Also delay time for synthesized fuzzy controlsystem is 27 less than traffic control system with fixed-cycletraffic lights

Based on the results of computational experiments wecan conclude that the synthesized adaptive fuzzy controlsystem of road traffic in conditions of intensive traffic isrobust and allows controlling road traffic within a wide rangeof parameters

5 Conclusion

In this study we formally described the problems of synthesisof adaptive fuzzy traffic control systems in conditions ofintense road traffic We developed efficient algorithms of thesynthesis of the adaptive fuzzy-logic traffic control systemsWe also developed a simulationmodel of adaptive fuzzy-logiccontrol system for managing road traffic and conducted aseries of computational experiments

The results showed that the synthesized fuzzy-logic traf-fic control system based on the proposed method allowscontrolling crossroads more effectively compared to trafficcontrol system with fixed-cycle traffic lights We conclude

that adaptive fuzzy-logic controller can successfully functionin the condition of information fuzziness It can also success-fully function when exposed to uncontrolled external andparametric impacts The implementation of the solution inheavy-loaded crossroads allows reducing the length of queuesat the traffic lights up to 26 times and the delay time up to27

Competing Interests

The authors declare that they have no competing interests

References

[1] F Sattar F Karray M Kamel L Nassar and K GolestanldquoRecent advances on context-awareness and datainformationfusion in ITSrdquo International Journal of Intelligent TransportationSystems Research vol 14 no 1 pp 1ndash19 2016

[2] J-D Schmocker and M G H Bell ldquoTraffic control currentsystems and future vision of citiesrdquo International Journal ofIntelligent Transportation Systems Research vol 8 no 1 pp 56ndash65 2010

[3] S Sundaram H N Koutsopoulos M Ben-Akiva C Anto-niou and R Balakrishna ldquoSimulation-based dynamic trafficassignment for short-term planning applicationsrdquo SimulationModelling Practice andTheory vol 19 no 1 pp 450ndash462 2011

[4] Dunn Engineering Associates Traffic Control Systems Hand-book Dunn Engineering Associates Westhampton Beach NYUSA 2005

[5] C P Pappis and E H Mamdani ldquoFuzzy logic controller fora traffic junctionrdquo IEEE Transactions on Systems Man andCybernetics vol 7 no 10 pp 707ndash717 1977

[6] Y S Murat and E Gedizlioglu ldquoA fuzzy logic multi-phasedsignal control model for isolated junctionsrdquo TransportationResearch Part C Emerging Technologies vol 13 no 1 pp 19ndash362005

[7] B Yulianto ldquoApplication of fuzzy logic to traffic signal controlundermixed traffic conditionsrdquoTraffic Engineering andControlvol 44 no 9 pp 332ndash336 2003

[8] B Madhavan Nair and J Cai ldquoA fuzzy logic controller forisolated signalized intersection with traffic abnormality consid-eredrdquo in Proceedings of the IEEE Intelligent Vehicles Symposium(IV rsquo07) pp 1229ndash1233 June 2007

[9] V T Zoriktuev S G Ts and J F Mesyagutov ldquoMethodologyrepresented and derivation of knowledge in control systems ofthemechatronicmachine systemsrdquo STIN Systems vol 11 pp 13ndash18 2007

[10] R A Aliev and R R AlievTheTheory of Intelligent Systems andIts Application Chashiogli Baku Azerbaijan 2001

[11] R A Aliev E G Zakharova and S V Ulyanov Fuzzy Modelsof Dynamic Systems Results of Science and Technology SerTechnology Cybernetics 1990

[12] N R Yusupbekov and A R Marakhimov ldquoSynthesis of theintelligent traffic control systems in conditions saturated trans-port streamrdquo International Journal of International Journal ofChemical Technology Control and Management Jointly with theJournal of Korea Multimedia Society vol 3-4 pp 12ndash18 2015

[13] Y Ge ldquoA two-stage fuzzy logic control method of traffic signalbased on traffic urgency degreerdquo Modelling and Simulation inEngineering vol 2014 Article ID 694185 6 pages 2014

The Scientific World Journal 9

[14] I M Makarov and V M Lokhin Eds Intelligent AutomaticControl Systems Fizmatlit Moscow Russia 2001

[15] A Pegat Fuzzy Modeling and Control Knowledge Laboratory2009

[16] RMartinez-Soto O Castillo L T Aguilar and PMelin ldquoFuzzylogic controllers optimization using genetic algorithms andparticle swarm optimizationrdquo in Advances in Soft ComputingG Sidorov A H Aguirre and C A R Garcıa Eds vol 6438of Lecture Notes in Computer Science pp 475ndash486 SpringerBerlin Germany 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 The Scientific World Journal

systems of road traffic does not consider unpredictable trafficsituations fuzziness of the parameters of the road traffic

Therefore developing conceptual framework and effec-tivemethods of structural and parametric synthesis of controlsystems of road traffic and development of information tech-nology support systems on their basis is important

2 Statement of a Problem

Below we consider a class of traffic control systems withimprecise information about the object in which the quali-tative characteristics of the road traffic are dominant

In general for synthesis of the traffic control systems theobject of control can be formally represented as follows

119872 = ⟨Ω119866119883 119884 119880 119879 120588 120574 120577⟩ (1)

where Ω = Ω1 Ω

2 Ω

119899times119898 is the range of conditions

(eg object output) 119866 = ⟨119881 (119863119882)⟩ is a model of thecrossroad represented by the graph 119866 in the space R3 119881 =

V119894 are the vertices (nodes) of the graph corresponding

to the nodes of possible branches of the road traffic 119863 sube

119881 times 119881 is the plurality of arcs of the graph (road sectionconnecting nodes of possible branches of road traffic) withcorrespondingweight coefficients as119882 = ⟨Λ 119875 119885⟩ (intensityof road traffic density of road traffic and average speed onthis particular road section) 119883 = 119883

1 119883

2 119883

119899 is the

plurality of characteristics and determinants describing thestate of the control object Ω and taking their values each inhis own set of values 119883

119894 119884 = 119884

1 119884

2 119884

119898 is the range

of output values (observed processes parameters estimatesetc)119880 = 119880

1 119880

2 119880

119903 is the range of controls (decisions)

119879 = 1199051 1199052 119905

119897 is the time (discrete or continuous)

120588 119883 times 119880 times 119879 rarr Ω is the description of the dynamics ofthe state of the object the reaction of the dynamic systemin a particular state to control actions 120574 Ω times 119879 rarr

119884 is the output describing the observation process of thecontrol object (obtaining estimates opinions etc) 120577 are someexternal uncontrollable factors conditions and others thathave an impact on the dynamics of the control object

The analysis of transport flows on the crossroads ascontrolled objects demonstrated that their mathematicalmodel should be able to deal with information fuzziness anddescribe random values and processes invariantly to theirdistribution Mathematical models should also provide mathformalism to express expert knowledge rich empiricism andheuristics

According to aforementioned requirements a general-ized dynamic model of the object controls (OC) can bedefined as the following linear equationwith fuzzy state space[9 10]

119909 = 119860 otimes 119909 oplus 119861 otimes 119906 120583

119878(119878) 119878 sube Ω

119910 = 119862 otimes 119909

(2)

with fuzzy initial conditions

1199091(0) = 119863

1

1199092(0) = 119863

2

119909119899(0) = 119863

119899

(3)

where otimes oplus are the fuzzy operations of multiplication andaddition 119906 is a control signal (scalar) that accepts discretenumerical value 119909 = 119909

1 119909

2 119909

119899 is the state space vector

119910 = 1199101 119910

2 119910

119898 is the output variables vector 120583

119878(119878) is

themembership function (MF) of the state space of the objectcontrols 119878 = 119904

1 1199042 119904

119873

119860 =

[[[[

[

1198601

1sdot sdot sdot 119860

1

119899

119860119899

1sdot sdot sdot 119860

119899

119899

]]]]

]

119861 =

[[[[

[

1198611

119861119899

]]]]

]

119862 =

[[[[[

[

1198621

1sdot sdot sdot 119862

1

119899

1198621

1sdot sdot sdot 119862

1

119899

]]]]]

]

(4)

are the matrix of the fuzzy coefficients of the model where

1198601

1=

1198601

1

1205831198601

1

(1198601

1)

119860119899

119899=

119860119899

119899

120583119860119899

119899

(119860119899119899)

1198611

= 1198611

1205831198611 (1198611)

119861119899

= 119861119899

120583119861119899 (119861

119899)

1198621

1=

1198621

1

1205831198621

1

(1198621

1)

1198621

119899=

1198621

119899

120583119862119899

119899

(1198621119899)

(5)

The Scientific World Journal 3

Some 119894th state space vector as a time function 119905 can bedescribed as fuzzy relation [1 3]

119909119894(119905) = 119905

119909119894

120583119909119894

(119905 119909119894) 119894 = 1 2 119899 (6)

At a fixed time this variable can be expressed as a fuzzyset

119909119894=

119909119894

120583119909119894

(119909119894) (7)

A similar description is 119895th output variable

119910119895(119905) =

119905

119910119895

120583119910119895

(119905 119910119895)

119895 = 1 2 119898

119910119895=

119910119895

120583119910119895

(119910119895)

(8)

Initial conditions and the number of variables of the statevector also can be described as the following fuzzy sets

119863119894=

119909119894

120583119863119894

(119909119894)

119878 = 119878

120583119878(119878)

(9)

Suppose that the membership functions of the inputand output linguistic variables are defined as the followinganalytical functions

120583119883119894

(119909119894) = 120593 (119909 119886

119883119894

1198871119883119894

1198872119883119894

1205731119883119894

1205732119883119894

)

= ((1198871119883119894

(119886119883119894

minus 119909))1205731119883119894

sign (1198871119883119894

(119886119883119894

minus 119909)) + 1

2

+ (1198872119883119894

(119909 minus 119886119883119894

))1205732119883119894

sign (1198872119883119894

(119909 minus 119886119883119894

)) + 1

2

+ 1)

minus1

(10)

120583119884119895

(119910119895) = 120593 (119910 119886

119884119895

1198871119884119895

1198872119884119895

1205731119884119895

1205732119884119895

)

= ((1198871119884119895

(119886119884119895

minus 119910))

1205731119884119895

sign (1198871119884119895

(119886119884119895

minus 119910)) + 1

2

+ (1198872119884119895

(119910 minus 119886119884119895

))

1205732119884119895

sign (1198872119884119895

(119910 minus 119886119884119895

)) + 1

2

+ 1)

minus1

(11)

In (10) and (11) the coefficients 119886119883119894

119886119884119895

1198871119883119894

1198871119884119895

1198872119883119894

1198872119884119895

1205731119883119894

1205731119884119895

1205732119883119894

and 1205732119884119895

are mood width and slope ofthe membership functions of the input and output linguisticvariables These coefficients allow forming high variety ofshapers ofmembership functionsThey can also act as indica-tors of information fuzziness for a formalmodel of the controlobjects which are provided as state space equation (2)

We assume that the indicators of quality of the controlsystem (eg time of transient process overshoot and bugtracking) are given as the following utility functions gener-ated by experts

119876 = 1199021

120583119876(1199021)

1199022

120583119876(1199022)

119902119896

120583119876(119902119896)

120583119876(119902119896) = 120593 (119902

119896 119886119876119896

1198871119876119896

1198872119876119896

1205731119876119896

1205732119876119896

)

(12)

under certain limitations for variablesrsquo state space and controlsignal

1198921(119909 119906 120574 119905) lt 119909

1max

1198922(119909 119906 120574 119905) gt 119909

2min

1198922119899minus1

(119909 119906 120574 119905) lt 119909119899max

1198922119899(119909 119906 120574 119905) gt 119909

119899min

119892119898minus1

(119909 119906 120574 119905) lt 119906max

119892119898(119909 119906 120574 119905) gt 119906min

(13)

where 120593 is membership function of the quality index of thecontrol system provided as (6) Then the problem of thesynthesis of control system for control objects provided asin a fuzzy state equation (2) with the quality evaluationindicator (12) and the system of constraints (13) can beformulated as described below

First it is necessary to synthesize control systems withembedded adaptive adjustment parameters of the controllerfor object controls (2) All signals should be limited in (13)and the transient processes in the system have to satisfy thepredetermined quality parameters (12)

For a given statement of the problem gradient speedsearch algorithm in the parametric formwith a proportional-integral controller and circuit bootstrapping can be selected[2 3] Among available methods of adaptive control withrobust characteristics the gradient speed search algorithm isthe least complex It also fits the constraints on the controlsignal and its velocity Then the control signal is generatedbased on the fuzzy set values of state space parametersaccording to the following modified control signal

119906 (119905 + 1) = 119896119906(119905) sdot 119906

119898(119905) +

119899

sum

119894=1

119896sum

119909119894

(119905) sdot 119909sum

119894(119905) (14)

4 The Scientific World Journal

119896sum

1199091

[119905 + 1] = 119896sum

1199091

[119905] (1 minus 1205961205743)

+ 120596 (1205745minus 120574

4) 120575

lowast[119905] 119909

sum

1[119905]

minus 1205961205745120575lowast[119905 + 1] 119909

sum

1[119905 + 1]

119896sum

1199092

[119905 + 1] = 119896sum

1199092

[119905] (1 minus 1205961205743)

+ 120596 (1205745minus 120574

4) 120575

lowast[119905] 119909

sum

2[119905]

minus 1205961205745120575lowast[119905 + 1] 119909

sum

2[119905 + 1]

119896sum

119909119899

[119905 + 1] = 119896sum

119909119899

[119905] (1 minus 1205961205743)

+ 120596 (1205745minus 120574

4) 120575

lowast[119905] 119909

sum

119899[119905]

minus 1205961205745120575lowast[119905 + 1] 119909

sum

119899[119905 + 1]

119896119906 [119905 + 1] = 119896

119906 [119905] (1 minus 1205961205741) + 120596 (1205746 minus 1205742) 120575lowast[119905] 119906119898 [119905]

minus 1205961205746120575lowast[119905 + 1] 119906119898 [119905 + 1]

(15)

where 119905 = 119898120596 120596 gt 0 is discretization step 120574 = 1205741 1205742 1205743

1205744 1205745 1205746 are the parameters of the adaptive controller 119890sum

119894=

int119909119894

(119909119894minus119909

119894119898)120583119890119894

(119890119894)119889119909

119894is a signal mismatch between the actual

parameters of the state vector and desirable model parame-ters 120583

119890119894

(119890119894) = 120593(119890

119894 119886119890119894

1198871119890119894

1198872119890119894

]1119890119894

]2119890119894

) is the membershipfunction of the signal mismatch provided as (10) 119886

119890119894

= 119886119909119894

minus

119909119894119898 ]

1119890119894

= ]1119909119894

]2119890119894

= ]2119909119894

1198871119890119894

= 1198871119909119894

1198872119890119894

= 1198872119909119894

119909sum119894

=

int119909119894

119909119894119889119909

119894are the integrated parameters of the state vector

120575lowast[119905] = sum

119899

119894=1ℎ119894sdot 119890sum

119894 ℎ

119894are the coefficients obtained by Lya-

punovrsquos decision equation and the matrix equation withreference model 119861M

The synthesis of the control law based on fuzzy modelcan increase the robustness of the gradient speed searchalgorithm it can also maintain limitations of the phase tra-jectories in certain areas under uncompensated informationfuzziness [11 12]

Quality indicators of the control systems are based notonly on the best possible dynamic characteristics of objectcontrolling 119886

119909119894

(119905) 119886119910119894

(119905) but also on the parameters of itsfuzziness 119886

119883119894

119886119884119895

1198871119883119894

1198871119884119895

1198872119883119894

1198872119884119895

1205731119883119894

1205731119884119895

1205732119883119894

and1205732119884119895

provided as (10) and (11)Reducing the information fuzziness and consequently

improving the quality control can be achieved by optimizingthe parameters of the membership functions of the input andoutput linguistic variables and fuzzy-logic controllers This isthe core idea of the algorithm for traffic control systems incrossroads presented in this study

3 The Concept of the ProblemSynthesis Adaptive Fuzzy-Logic TrafficControl Systems

Let us consider the problem of synthesis of fuzzy controlsystem of road traffic under conditions of intense traffic inmore detail

We suppose that there is a crossroad regulated with anequal number of intersecting lanes and well-known trafficintensity of 120582

1 120582

2 120582

3 120582

4 The directions of traffic 1 and 3 are

perpendicular to the directions 2 and 4We introduce the following terms Δ119879119906

1is the duration of

the green signal of the traffic light to the first direction of thetraffic 120582

1 120582

3 Δ119879119906

2is the duration of the green signal of the

traffic light to the opposite direction of the traffic 1205822 120582

4

Taking into consideration the principle of progressivecorrection the control object can be defined as (2) Thecontrol process assumes that there is a target goal andthe control system functions to achieve it The quality offunctioning (criterion of efficiency) of the control system (12)assumes a degree of adaptability to fulfill its taskmdashensuring asafe traffic with a minimum delay Therefore the problem ofoptimal traffic control at crossroads can be defined as follows

It is necessary to identify control signals Δ1198791199061and Δ119879

119906

2

(the duration of the green signal of the traffic light for eachlane) so that the actual state of the traffic 119910(119905) was as closeas possible to the desired state when the control signalshave certain limitations and the system exposed uncontrolledexternal and parametric impacts 119910lowast(119905)

That is it is necessary to find such parameters of thetraffic control system that the total delay at the crossroads isminimal [12 13]

119869 (119905) = int

119905

1199050

[119910lowast(120591 119909 (120591) 119906 (120591)) minus 119910 (120591 119909 (120591) 119906 (120591))] 119889120591

997888rarr min(16)

or

sum

119894

sum

119895

[11989013

119894119891lowast(119906

13

119894 11989013

119894) + 119897

13

119894119891lowast

119901119899(119906

13

119894 11989713

119894)

+ 11989024

119894119891lowast(119906

24

119894 11989024

119894) + 119897

24

119894119891lowast

119901119899(119906

24

119894 11989724

119894)] 997888rarr min

(17)

under the constraints

11989013

119894= 119902

leaving(13)119894

minus 119902arriv(13)119894

le 0

11989024

119894= 119902

leaving(24)119894

minus 119902arriv(24)119894

le 0

11990613

min le 11990613

119894le 119906

13

max

11990624

min le 11990624

119894le 119906

24

max

0 le 11989713

119894le 119871

13

max

0 le 11989724

119894le 119871

24

max

(18)

where 119891lowast(11990613119894 120582

13

119894) and 119891lowast

119901119899(11990613

119894 11989713

119894) are the delay evaluation

function and penalty function for the length of queue infront of the stop line of the traffic light in directions 1 and3 respectively 119891lowast(11990624

119894 120582

24

119894) and 119891

lowast

119901119899(11990624

119894 11989724

119894) are the delay

evaluation function and penalty function for the length ofqueue in front of the stop line of the traffic light in directions 2and 4 respectively 119906

119894is the duration of green signal of traffic

light in the i-stage 119902left(13)119894

119902arriv(13)119894

119902left(24)119894

and 119902arriv(24)119894

are

The Scientific World Journal 5

Fuzzy-logic controller(FLC)

The control object(transport flows)

Optimization module ofthe parameters FLC

Quality estimationmodule of the functioningof the system

Ylowast(t)

Ylowast(t)

E(t) U(t)

J(t)

Y(t)

sum

120583jT119890(x alowastj b

lowastj c

lowastj )

Figure 1 A traffic control system at the crossroads with an adaptive fuzzy-logic controller

the number of vehicles entered and left the control zone atthe crossroad in the 119894th phase respectively 119906min and 119906max arethe minimum and maximum duration of the green signal ofthe traffic light respectively 11989713

119894and 11989724

119894are the length of the

queue in front of the stop line at directions 1 and 3 and 2 and4 in the 119894th stage respectively 119871max is the maximum allowedlength of the queue in front of the stop line

In general the process control system at the T-shapedcrossroads can be rearranged as a feedback control sys-tem with adaptive fuzzy-logic controller (Figure 1) In the

proposed scheme the input vector 119864lowast = (119890lowast

1 119890lowast

2) is converted

into fuzzy-logic controller (FLC) through fuzzification blockNext fuzzy inference is performed based on rule base whichresults in a fuzzy output variable 119906lowast The output values fromthe FLC are then transferred from the fuzzy region 119906

lowast toaccurate region 119906 through defuzzification block [14 15]

In general rule bases or knowledge bases contain descrip-tions of the linguistic variables of the FLC that are predefinedfor each of the input and output variables Therefore weintroduce the following linguistic variables

1198901= (ldquoDifference in the number of vehicles that entered and left the control zonerdquo 119885

1198901

1198641)

1198902= (ldquoQueue lengthsrdquo 119885

1198902

1198642)

119906 = (ldquoControlrdquo 119885119906 119880)

(19)

where 119885119890119894

= 1198851

119890119894

1198852

119890119894

119885119896

119890119894

119894 = 1 2 and 119885119906= 119885

1

119906 119885

2

119906

119885119896

119906 are the term-sets of linguistic variables 119890

1 119890

2 and

119906 with corresponding membership functions (MF) 119885119897119890119894

=

120583119897

119890119894

(119890119894) 119885119897

119906= 120583

119897(119906) 119897 = 1 119896 given by universal sets 119864

119894=

[119864119894min 119864119894max] and 119880 = [119880min 119880max]

We assume that seven terms 119885119909= ldquoNBrdquo ldquoNMrdquo ldquoNSrdquo

ldquoZErdquo ldquoPSrdquo ldquoPMrdquo ldquoPBrdquo correspond to each of the input andoutput linguistic variables 119885

119909= 119879

1198901

1198791198902

119879119906 with trian-

gular membership functions provided as (10) As a result thefollowing linguistic variables are derived from the fuzzifica-tion processes

1198901= (ldquoDifference of the number of vehicles that entered and left the control zonerdquo 119885

1198901

1198641)

= [

120583NB1198901

(1198901)

NB1198901

120583NM1198901

(1198901)

NM1198901

120583NS1198901

(1198901)

NS1198901

120583ZE1198901

(1198901)

ZE1198901

120583PS1198901

(1198901)

PS1198901

120583PM1198901

(1198901)

PM1198901

120583PB1198901

(1198901)

PB1198901

]

1198902= (ldquoQueue lengthsrdquo 119885

1198902

1198642) = [

120583NB1198902

(1198902)

NB1198902

120583NM1198902

(1198902)

NM1198902

120583NS1198902

(1198902)

NS1198902

120583ZE1198902

(1198902)

ZE1198902

120583PS1198902

(1198902)

PS1198902

120583PM1198902

(1198902)

PM1198902

120583PB1198902

(1198902)

PB1198902

]

119906 = (ldquoControlrdquo 119885119906 119880) = [

120583NB119906(119906)

NB119906

120583NM119906(119906)

NM119906

120583NS119906(119906)

NS119906

120583ZE119906(119906)

ZE119906

120583PS119906(119906)

PS119906

120583PM119906(119906)

PM119906

120583PB119906(119906)

PB119906

]

(20)

6 The Scientific World Journal

Next we develop a rule base for the FLC inferences asfollows

IF (119885119895

1198901

times 119885119895

1198902

)

ELSE 119885119895

119906 119895 = 1 7

(21)

where (1198851198951198901

times119885119895

1198902

) is the Cartesian product of the fuzzy sets 1198641

and 1198642 with the following membership function

120583(119885119895

1198901times119885119895

1198902)(1198901 1198902) = 120583

119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902) (22)

The corresponding output fuzzy set is defined by fuzzyrelation 119877119895 = (119885

119895

1198901

times 119885119895

1198902

) times 119885119895

119906 119895 = 1 7 with the following

membership function

120583119877119895 ((119890

1 1198902) 119906

lowast) = (120583

119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902))

and 120583119885119895

119906

(119906lowast)

(23)

The set of all rules 119877 = ⋃7

119895=1119877119895 with fuzzy membership

function

120583119877((119890

1 1198902) 119906

lowast)

=

7

⋁

119895=1

[(120583119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902)) and 120583

119885119895

119906

(119906lowast)]

(24)

which defines the rule base of the FLC and produces analgorithm of fuzzy control system

As a result for given values of the input linguistic var-iables 119885

119895

1198901

and 119885119895

1198902

the fuzzy output value of the logicalcontroller119885119895

119906can be determined based on the following com-

posite rule [12]

119861119895= (119885

119895

1198901

times 119885119895

1198902

) sdot 119877 (25)

with a degree of membership

120583119885119895

119906

(119906lowast) = ⋁

1198901isin1198641 1198902isin1198642

[(120583119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902))

and 120583119877(1198901 1198902 119906lowast)]

(26)

When the linguistic input variables 1198901and 119890

2correspond

to fuzzy sets11988510158401198901

and11988510158401198902

fuzzy set1198851015840119906of the linguistic variable

of the control signal 119906lowast is determined as follows

1205831198791015840

119906

1015840 (119906lowast) = max

11989011198902

[

119899

prod

119894=1

1205831198791015840

119890119894

(119890119894)]

sdot [

119898

min119895=1

[

119899

prod

119894=1

120583119879119895

119890119894

(119890119894)] sdot 120583

119879119895

119906

(119906lowast)]

(27)

In order to obtain a true output value following fuzzycomputations it is necessary to perform defuzzificationmdashthetransformation of fuzzy linguistic variable value (qualitative)

119906lowast into discrete numerical value of 119906 For this purpose center

of gravity method is applied [12]

119906 =

sum7

119899=1119906lowast

119899120583119885119906

(119906lowast

119899)

sum7

119899=1120583119885119906

(119906lowast119899)

(28)

Membership function of fuzzy values 1198851015840119906can be repre-

sented as

1205831198851015840

119906

1015840 (119906) =

119899

prod

119894=1

120583119885119895

119890119894

(119890119894) 119906 = 120593

119895

0 119906 = 120593119895

(29)

where 120593119895 is the discrete numerical value of the output signalThen the final output value of the FLC at the defuzzifica-

tion stage can be computed as follows

119906 =

sum119898

119895=1120593119895[prod

119899

119894=1120583119885119895

119890119894

(119890119894)]

sum119898

119895=1prod119899

119894=1120583119885119895

119890119894

(119890119894)

or 119906 (119890 120593) =119898

sum

119895=1

120593119895120595119895(119890)

(30)

where

120595119895(119890) =

prod119899

119894=1120583119879119895

119890119894

(119890119894)

sum119898

119895=1prod119899

119894=1120583119879119895

119890119894

(119890119894) (31)

If we consider that the basic equation of the road trafficcontroller can be written as [12]

119906 (119905) = 1199060+ 119870(119890 (119905) +

1

119879119906

int

119905

0

119890 (120591) 119889120591 + 119879119889

119889119890 (119905)

119889119905) (32)

then the rule of the fuzzy traffic control system can berepresented as a logic controller with the varying coefficientas in (14)

119906 (119905) = 1199060+ 119870

lowast(120572 120573) ∘ 120595

lowast

1199060

(119890)lowast (33)

where 119870lowast(120572 120573) is the variable part of the gain that depends

on the current dynamic parameters of the road traffic [16]This allows implementing an adaptive traffic control systemas a fuzzy system consisting of two interconnected modulesldquomodule of fuzzy controller of the switching phase trafficlightsrdquo and ldquomodule of fuzzy adaptive correction of theparameters of the main fuzzy controller of the switchingphase traffic lightsrdquo 120572 and 120574

Furthermore we introduce the following linguistic vari-ables to assess the quality of the control system and adaptivefuzzy correction of the parameters of the fuzzy-logic con-troller119889119890

2

= (ldquoThe Derivative Queue Lengths of Vehiclesrdquo 1198851198891198902

1198641198891198902

)

119876 = (ldquoThe Quality Control Systemrdquo 119885119876 119864

119876)

(34)

The Scientific World Journal 7

120583intQ119896

and 1205833intQ119896

997888rarr (alowastX119894 alowastY119894 b

lowast1X119894

blowast2X119894 blowast1Y119894 b

lowast2Y119894 120573lowast1X119894

120573lowast2X119894 120573lowast1XY119894 120573

lowast2Y119894

)1205833Q119896

120583Q119896

120583Q119896

Qk

Figure 2 Graphic illustration of a choice of optimum adjusting parameters

where 1198851198891198902119894

= 119885(1)

1198891198902

119885(2)

1198891198902

119885(119899)

1198891198902

119885119876119894

= 119885(1)

119876 119890119894

119885(2)

119876

119885(119896)

119876 t are the term-sets of linguistic variables 119889119890

2and119876with

corresponding membership functions

1198851198891198902

=

119889119890(119894)

2

1205831198851198891198902

(119889119890(119894)

2)

119894 = 1 119899

119885119876=

119902(119895)

120583119876(119902(119895))

119895 = 1 119896

(35)

given by universal sets 1198641198891198902

= [1198641198891198902min 119864119889119890

2max] and 119864

119876=

[119864119876min 119864119876max]Then according to (12) and (25) the choice of the optimal

adjustment parameters of the FLC can be represented as thefollowing fuzzy relations

119877119895

119876lowast = (119885

119895

1198902

times 119885119895

1198902

) times 119885119895

119876 119895 = 1119873

119876 (36)

where

119885119876=

119876119879119875

1

120583119879119875

1198761

(119876119879119875

1)

120583119879119875

119876119896

(119876119879119875

119896) = 120593 (119876

119879119875

119896 119886lowast

119876119896

119887lowast

1119876119896

119887lowast

2119876119896

120573lowast

1119876119896

120573lowast

2119876119896

)

120593 (119876119879119875

119896 119886lowast

119876119896

119887lowast

1119876119896

119887lowast

2119876119896

120573lowast

1119876119896

120573lowast

2119876119896

)

= ((119887lowast

1119876119896

(119886lowast

119876119896

minus 119909))

120573lowast

1119876119896

sdot

sign (119887lowast1119876119896

(119886lowast

119876119896

minus 119909)) + 1

2

+ (119887lowast

2119876119896

(119909 minus 119886lowast

119876119896

))

120573lowast

2119876119896

sign (119887lowast2119876119896

(119909 minus 119886lowast

119876119896

)) + 1

2

+ 1)

minus1

119876 = 1199021

120583119876(1199021)

1199022

120583119876(1199022)

119902119896

120583119876(119902119896)

120583119876(119902119896) = 120593 (119902

119896 119886119876119896

1198871119876119896

1198872119876119896

1205731119876119896

1205732119876119896

)

(37)

This fuzzy relationship served as the basis for developingalgorithms of adaptive parameter settings of the fuzzy-logiccontroller (Figure 2)

4 Simulation Modeling and Analysis ofFuzzy Traffic Control System

We develop a simulation model of the adaptive fuzzy controlsystem of road traffic on Matlab software package Thesimulation model is used to test the efficiency of the fuzzycontrol system through series of computational experiments

As an object of simulation we chose one of the heavy-loaded and problematic crossroads in Tashkent city Uzbek-istan The simulation model considered the road trafficintensity observed at the crossroad during the peak-hoursfrom 0700 to 1000 onMay 5 2015The experiment consistedof two phases At each phase of the simulation variouspatterns of traffic-light management were used In the firstphase the traffic lights with fixed cycles which are used inreal life are simulated In the second phase traffic lightscontrolled with adaptive fuzzy system were simulated Forthis purpose the rules of fuzzy control system with a modulefor adaptive adjustment of parameters were synthesized usingthe proposed method

The simulation results are presented below Figure 3shows the results of simulation of the road intersection thatis managed by traffic lights with fixed cycles Figure 4 showsthe results of simulation of the crossroad that is managed byadaptive fuzzy controller

8 The Scientific World Journal

The length of the queue of vehicles in directions 1ndash3

The length of the queue of vehicles in directions 2ndash4

710

720

910

900

820

800

810

920

700

840

850

750

740

730

930

940

950

830

1000

05101520253035404550

2ndash4The intensity of the traffic flow in directions

The intensity of the traffic flow in directions 1ndash3

Figure 3 The results of the simulation modeling of the trafficcontrol system with optimum fixed cycle of traffic lights

1ndash3The length of the queue of vehicles in directions

The length of the queue of vehicles in directions 2ndash4

710

720

910

900

820

800

810

920

700

840

850

750

740

730

930

940

950

830

1000

2ndash4The intensity of the traffic flow in directions

The intensity of the traffic flow in directions 1ndash3

05101520253035404550

Figure 4 The results of the simulation modeling of the adaptivetraffic control system

Simulation results show that the synthesized fuzzy controlsystem successfully solved the problem of queues at trafficlights and performed more effective than the current trafficcontrol system with fixed-cycle traffic lights Particularly thelength of queue for the synthesized fuzzy control system isless than 26 times for traffic control system with fixed-cycletraffic lights Also delay time for synthesized fuzzy controlsystem is 27 less than traffic control system with fixed-cycletraffic lights

Based on the results of computational experiments wecan conclude that the synthesized adaptive fuzzy controlsystem of road traffic in conditions of intensive traffic isrobust and allows controlling road traffic within a wide rangeof parameters

5 Conclusion

In this study we formally described the problems of synthesisof adaptive fuzzy traffic control systems in conditions ofintense road traffic We developed efficient algorithms of thesynthesis of the adaptive fuzzy-logic traffic control systemsWe also developed a simulationmodel of adaptive fuzzy-logiccontrol system for managing road traffic and conducted aseries of computational experiments

The results showed that the synthesized fuzzy-logic traf-fic control system based on the proposed method allowscontrolling crossroads more effectively compared to trafficcontrol system with fixed-cycle traffic lights We conclude

that adaptive fuzzy-logic controller can successfully functionin the condition of information fuzziness It can also success-fully function when exposed to uncontrolled external andparametric impacts The implementation of the solution inheavy-loaded crossroads allows reducing the length of queuesat the traffic lights up to 26 times and the delay time up to27

Competing Interests

The authors declare that they have no competing interests

References

[1] F Sattar F Karray M Kamel L Nassar and K GolestanldquoRecent advances on context-awareness and datainformationfusion in ITSrdquo International Journal of Intelligent TransportationSystems Research vol 14 no 1 pp 1ndash19 2016

[2] J-D Schmocker and M G H Bell ldquoTraffic control currentsystems and future vision of citiesrdquo International Journal ofIntelligent Transportation Systems Research vol 8 no 1 pp 56ndash65 2010

[3] S Sundaram H N Koutsopoulos M Ben-Akiva C Anto-niou and R Balakrishna ldquoSimulation-based dynamic trafficassignment for short-term planning applicationsrdquo SimulationModelling Practice andTheory vol 19 no 1 pp 450ndash462 2011

[4] Dunn Engineering Associates Traffic Control Systems Hand-book Dunn Engineering Associates Westhampton Beach NYUSA 2005

[5] C P Pappis and E H Mamdani ldquoFuzzy logic controller fora traffic junctionrdquo IEEE Transactions on Systems Man andCybernetics vol 7 no 10 pp 707ndash717 1977

[6] Y S Murat and E Gedizlioglu ldquoA fuzzy logic multi-phasedsignal control model for isolated junctionsrdquo TransportationResearch Part C Emerging Technologies vol 13 no 1 pp 19ndash362005

[7] B Yulianto ldquoApplication of fuzzy logic to traffic signal controlundermixed traffic conditionsrdquoTraffic Engineering andControlvol 44 no 9 pp 332ndash336 2003

[8] B Madhavan Nair and J Cai ldquoA fuzzy logic controller forisolated signalized intersection with traffic abnormality consid-eredrdquo in Proceedings of the IEEE Intelligent Vehicles Symposium(IV rsquo07) pp 1229ndash1233 June 2007

[9] V T Zoriktuev S G Ts and J F Mesyagutov ldquoMethodologyrepresented and derivation of knowledge in control systems ofthemechatronicmachine systemsrdquo STIN Systems vol 11 pp 13ndash18 2007

[10] R A Aliev and R R AlievTheTheory of Intelligent Systems andIts Application Chashiogli Baku Azerbaijan 2001

[11] R A Aliev E G Zakharova and S V Ulyanov Fuzzy Modelsof Dynamic Systems Results of Science and Technology SerTechnology Cybernetics 1990

[12] N R Yusupbekov and A R Marakhimov ldquoSynthesis of theintelligent traffic control systems in conditions saturated trans-port streamrdquo International Journal of International Journal ofChemical Technology Control and Management Jointly with theJournal of Korea Multimedia Society vol 3-4 pp 12ndash18 2015

[13] Y Ge ldquoA two-stage fuzzy logic control method of traffic signalbased on traffic urgency degreerdquo Modelling and Simulation inEngineering vol 2014 Article ID 694185 6 pages 2014

The Scientific World Journal 9

[14] I M Makarov and V M Lokhin Eds Intelligent AutomaticControl Systems Fizmatlit Moscow Russia 2001

[15] A Pegat Fuzzy Modeling and Control Knowledge Laboratory2009

[16] RMartinez-Soto O Castillo L T Aguilar and PMelin ldquoFuzzylogic controllers optimization using genetic algorithms andparticle swarm optimizationrdquo in Advances in Soft ComputingG Sidorov A H Aguirre and C A R Garcıa Eds vol 6438of Lecture Notes in Computer Science pp 475ndash486 SpringerBerlin Germany 2010

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Stochastic AnalysisInternational Journal of

The Scientific World Journal 3

Some 119894th state space vector as a time function 119905 can bedescribed as fuzzy relation [1 3]

119909119894(119905) = 119905

119909119894

120583119909119894

(119905 119909119894) 119894 = 1 2 119899 (6)

At a fixed time this variable can be expressed as a fuzzyset

119909119894=

119909119894

120583119909119894

(119909119894) (7)

A similar description is 119895th output variable

119910119895(119905) =

119905

119910119895

120583119910119895

(119905 119910119895)

119895 = 1 2 119898

119910119895=

119910119895

120583119910119895

(119910119895)

(8)

Initial conditions and the number of variables of the statevector also can be described as the following fuzzy sets

119863119894=

119909119894

120583119863119894

(119909119894)

119878 = 119878

120583119878(119878)

(9)

Suppose that the membership functions of the inputand output linguistic variables are defined as the followinganalytical functions

120583119883119894

(119909119894) = 120593 (119909 119886

119883119894

1198871119883119894

1198872119883119894

1205731119883119894

1205732119883119894

)

= ((1198871119883119894

(119886119883119894

minus 119909))1205731119883119894

sign (1198871119883119894

(119886119883119894

minus 119909)) + 1

2

+ (1198872119883119894

(119909 minus 119886119883119894

))1205732119883119894

sign (1198872119883119894

(119909 minus 119886119883119894

)) + 1

2

+ 1)

minus1

(10)

120583119884119895

(119910119895) = 120593 (119910 119886

119884119895

1198871119884119895

1198872119884119895

1205731119884119895

1205732119884119895

)

= ((1198871119884119895

(119886119884119895

minus 119910))

1205731119884119895

sign (1198871119884119895

(119886119884119895

minus 119910)) + 1

2

+ (1198872119884119895

(119910 minus 119886119884119895

))

1205732119884119895

sign (1198872119884119895

(119910 minus 119886119884119895

)) + 1

2

+ 1)

minus1

(11)

In (10) and (11) the coefficients 119886119883119894

119886119884119895

1198871119883119894

1198871119884119895

1198872119883119894

1198872119884119895

1205731119883119894

1205731119884119895

1205732119883119894

and 1205732119884119895

are mood width and slope ofthe membership functions of the input and output linguisticvariables These coefficients allow forming high variety ofshapers ofmembership functionsThey can also act as indica-tors of information fuzziness for a formalmodel of the controlobjects which are provided as state space equation (2)

We assume that the indicators of quality of the controlsystem (eg time of transient process overshoot and bugtracking) are given as the following utility functions gener-ated by experts

119876 = 1199021

120583119876(1199021)

1199022

120583119876(1199022)

119902119896

120583119876(119902119896)

120583119876(119902119896) = 120593 (119902

119896 119886119876119896

1198871119876119896

1198872119876119896

1205731119876119896

1205732119876119896

)

(12)

under certain limitations for variablesrsquo state space and controlsignal

1198921(119909 119906 120574 119905) lt 119909

1max

1198922(119909 119906 120574 119905) gt 119909

2min

1198922119899minus1

(119909 119906 120574 119905) lt 119909119899max

1198922119899(119909 119906 120574 119905) gt 119909

119899min

119892119898minus1

(119909 119906 120574 119905) lt 119906max

119892119898(119909 119906 120574 119905) gt 119906min

(13)

where 120593 is membership function of the quality index of thecontrol system provided as (6) Then the problem of thesynthesis of control system for control objects provided asin a fuzzy state equation (2) with the quality evaluationindicator (12) and the system of constraints (13) can beformulated as described below

First it is necessary to synthesize control systems withembedded adaptive adjustment parameters of the controllerfor object controls (2) All signals should be limited in (13)and the transient processes in the system have to satisfy thepredetermined quality parameters (12)

For a given statement of the problem gradient speedsearch algorithm in the parametric formwith a proportional-integral controller and circuit bootstrapping can be selected[2 3] Among available methods of adaptive control withrobust characteristics the gradient speed search algorithm isthe least complex It also fits the constraints on the controlsignal and its velocity Then the control signal is generatedbased on the fuzzy set values of state space parametersaccording to the following modified control signal

119906 (119905 + 1) = 119896119906(119905) sdot 119906

119898(119905) +

119899

sum

119894=1

119896sum

119909119894

(119905) sdot 119909sum

119894(119905) (14)

4 The Scientific World Journal

119896sum

1199091

[119905 + 1] = 119896sum

1199091

[119905] (1 minus 1205961205743)

+ 120596 (1205745minus 120574

4) 120575

lowast[119905] 119909

sum

1[119905]

minus 1205961205745120575lowast[119905 + 1] 119909

sum

1[119905 + 1]

119896sum

1199092

[119905 + 1] = 119896sum

1199092

[119905] (1 minus 1205961205743)

+ 120596 (1205745minus 120574

4) 120575

lowast[119905] 119909

sum

2[119905]

minus 1205961205745120575lowast[119905 + 1] 119909

sum

2[119905 + 1]

119896sum

119909119899

[119905 + 1] = 119896sum

119909119899

[119905] (1 minus 1205961205743)

+ 120596 (1205745minus 120574

4) 120575

lowast[119905] 119909

sum

119899[119905]

minus 1205961205745120575lowast[119905 + 1] 119909

sum

119899[119905 + 1]

119896119906 [119905 + 1] = 119896

119906 [119905] (1 minus 1205961205741) + 120596 (1205746 minus 1205742) 120575lowast[119905] 119906119898 [119905]

minus 1205961205746120575lowast[119905 + 1] 119906119898 [119905 + 1]

(15)

where 119905 = 119898120596 120596 gt 0 is discretization step 120574 = 1205741 1205742 1205743

1205744 1205745 1205746 are the parameters of the adaptive controller 119890sum

119894=

int119909119894

(119909119894minus119909

119894119898)120583119890119894

(119890119894)119889119909

119894is a signal mismatch between the actual

parameters of the state vector and desirable model parame-ters 120583

119890119894

(119890119894) = 120593(119890

119894 119886119890119894

1198871119890119894

1198872119890119894

]1119890119894

]2119890119894

) is the membershipfunction of the signal mismatch provided as (10) 119886

119890119894

= 119886119909119894

minus

119909119894119898 ]

1119890119894

= ]1119909119894

]2119890119894

= ]2119909119894

1198871119890119894

= 1198871119909119894

1198872119890119894

= 1198872119909119894

119909sum119894

=

int119909119894

119909119894119889119909

119894are the integrated parameters of the state vector

120575lowast[119905] = sum

119899

119894=1ℎ119894sdot 119890sum

119894 ℎ

119894are the coefficients obtained by Lya-

punovrsquos decision equation and the matrix equation withreference model 119861M

The synthesis of the control law based on fuzzy modelcan increase the robustness of the gradient speed searchalgorithm it can also maintain limitations of the phase tra-jectories in certain areas under uncompensated informationfuzziness [11 12]

Quality indicators of the control systems are based notonly on the best possible dynamic characteristics of objectcontrolling 119886

119909119894

(119905) 119886119910119894

(119905) but also on the parameters of itsfuzziness 119886

119883119894

119886119884119895

1198871119883119894

1198871119884119895

1198872119883119894

1198872119884119895

1205731119883119894

1205731119884119895

1205732119883119894

and1205732119884119895

provided as (10) and (11)Reducing the information fuzziness and consequently

improving the quality control can be achieved by optimizingthe parameters of the membership functions of the input andoutput linguistic variables and fuzzy-logic controllers This isthe core idea of the algorithm for traffic control systems incrossroads presented in this study

3 The Concept of the ProblemSynthesis Adaptive Fuzzy-Logic TrafficControl Systems

Let us consider the problem of synthesis of fuzzy controlsystem of road traffic under conditions of intense traffic inmore detail

We suppose that there is a crossroad regulated with anequal number of intersecting lanes and well-known trafficintensity of 120582

1 120582

2 120582

3 120582

4 The directions of traffic 1 and 3 are

perpendicular to the directions 2 and 4We introduce the following terms Δ119879119906

1is the duration of

the green signal of the traffic light to the first direction of thetraffic 120582

1 120582

3 Δ119879119906

2is the duration of the green signal of the

traffic light to the opposite direction of the traffic 1205822 120582

4

Taking into consideration the principle of progressivecorrection the control object can be defined as (2) Thecontrol process assumes that there is a target goal andthe control system functions to achieve it The quality offunctioning (criterion of efficiency) of the control system (12)assumes a degree of adaptability to fulfill its taskmdashensuring asafe traffic with a minimum delay Therefore the problem ofoptimal traffic control at crossroads can be defined as follows

It is necessary to identify control signals Δ1198791199061and Δ119879

119906

2

(the duration of the green signal of the traffic light for eachlane) so that the actual state of the traffic 119910(119905) was as closeas possible to the desired state when the control signalshave certain limitations and the system exposed uncontrolledexternal and parametric impacts 119910lowast(119905)

That is it is necessary to find such parameters of thetraffic control system that the total delay at the crossroads isminimal [12 13]

119869 (119905) = int

119905

1199050

[119910lowast(120591 119909 (120591) 119906 (120591)) minus 119910 (120591 119909 (120591) 119906 (120591))] 119889120591

997888rarr min(16)

or

sum

119894

sum

119895

[11989013

119894119891lowast(119906

13

119894 11989013

119894) + 119897

13

119894119891lowast

119901119899(119906

13

119894 11989713

119894)

+ 11989024

119894119891lowast(119906

24

119894 11989024

119894) + 119897

24

119894119891lowast

119901119899(119906

24

119894 11989724

119894)] 997888rarr min

(17)

under the constraints

11989013

119894= 119902

leaving(13)119894

minus 119902arriv(13)119894

le 0

11989024

119894= 119902

leaving(24)119894

minus 119902arriv(24)119894

le 0

11990613

min le 11990613

119894le 119906

13

max

11990624

min le 11990624

119894le 119906

24

max

0 le 11989713

119894le 119871

13

max

0 le 11989724

119894le 119871

24

max

(18)

where 119891lowast(11990613119894 120582

13

119894) and 119891lowast

119901119899(11990613

119894 11989713

119894) are the delay evaluation

function and penalty function for the length of queue infront of the stop line of the traffic light in directions 1 and3 respectively 119891lowast(11990624

119894 120582

24

119894) and 119891

lowast

119901119899(11990624

119894 11989724

119894) are the delay

evaluation function and penalty function for the length ofqueue in front of the stop line of the traffic light in directions 2and 4 respectively 119906

119894is the duration of green signal of traffic

light in the i-stage 119902left(13)119894

119902arriv(13)119894

119902left(24)119894

and 119902arriv(24)119894

are

The Scientific World Journal 5

Fuzzy-logic controller(FLC)

The control object(transport flows)

Optimization module ofthe parameters FLC

Quality estimationmodule of the functioningof the system

Ylowast(t)

Ylowast(t)

E(t) U(t)

J(t)

Y(t)

sum

120583jT119890(x alowastj b

lowastj c

lowastj )

Figure 1 A traffic control system at the crossroads with an adaptive fuzzy-logic controller

the number of vehicles entered and left the control zone atthe crossroad in the 119894th phase respectively 119906min and 119906max arethe minimum and maximum duration of the green signal ofthe traffic light respectively 11989713

119894and 11989724

119894are the length of the

queue in front of the stop line at directions 1 and 3 and 2 and4 in the 119894th stage respectively 119871max is the maximum allowedlength of the queue in front of the stop line

In general the process control system at the T-shapedcrossroads can be rearranged as a feedback control sys-tem with adaptive fuzzy-logic controller (Figure 1) In the

proposed scheme the input vector 119864lowast = (119890lowast

1 119890lowast

2) is converted

into fuzzy-logic controller (FLC) through fuzzification blockNext fuzzy inference is performed based on rule base whichresults in a fuzzy output variable 119906lowast The output values fromthe FLC are then transferred from the fuzzy region 119906

lowast toaccurate region 119906 through defuzzification block [14 15]

In general rule bases or knowledge bases contain descrip-tions of the linguistic variables of the FLC that are predefinedfor each of the input and output variables Therefore weintroduce the following linguistic variables

1198901= (ldquoDifference in the number of vehicles that entered and left the control zonerdquo 119885

1198901

1198641)

1198902= (ldquoQueue lengthsrdquo 119885

1198902

1198642)

119906 = (ldquoControlrdquo 119885119906 119880)

(19)

where 119885119890119894

= 1198851

119890119894

1198852

119890119894

119885119896

119890119894

119894 = 1 2 and 119885119906= 119885

1

119906 119885

2

119906

119885119896

119906 are the term-sets of linguistic variables 119890

1 119890

2 and

119906 with corresponding membership functions (MF) 119885119897119890119894

=

120583119897

119890119894

(119890119894) 119885119897

119906= 120583

119897(119906) 119897 = 1 119896 given by universal sets 119864

119894=

[119864119894min 119864119894max] and 119880 = [119880min 119880max]

We assume that seven terms 119885119909= ldquoNBrdquo ldquoNMrdquo ldquoNSrdquo

ldquoZErdquo ldquoPSrdquo ldquoPMrdquo ldquoPBrdquo correspond to each of the input andoutput linguistic variables 119885

119909= 119879

1198901

1198791198902

119879119906 with trian-

gular membership functions provided as (10) As a result thefollowing linguistic variables are derived from the fuzzifica-tion processes

1198901= (ldquoDifference of the number of vehicles that entered and left the control zonerdquo 119885

1198901

1198641)

= [

120583NB1198901

(1198901)

NB1198901

120583NM1198901

(1198901)

NM1198901

120583NS1198901

(1198901)

NS1198901

120583ZE1198901

(1198901)

ZE1198901

120583PS1198901

(1198901)

PS1198901

120583PM1198901

(1198901)

PM1198901

120583PB1198901

(1198901)

PB1198901

]

1198902= (ldquoQueue lengthsrdquo 119885

1198902

1198642) = [

120583NB1198902

(1198902)

NB1198902

120583NM1198902

(1198902)

NM1198902

120583NS1198902

(1198902)

NS1198902

120583ZE1198902

(1198902)

ZE1198902

120583PS1198902

(1198902)

PS1198902

120583PM1198902

(1198902)

PM1198902

120583PB1198902

(1198902)

PB1198902

]

119906 = (ldquoControlrdquo 119885119906 119880) = [

120583NB119906(119906)

NB119906

120583NM119906(119906)

NM119906

120583NS119906(119906)

NS119906

120583ZE119906(119906)

ZE119906

120583PS119906(119906)

PS119906

120583PM119906(119906)

PM119906

120583PB119906(119906)

PB119906

]

(20)

6 The Scientific World Journal

Next we develop a rule base for the FLC inferences asfollows

IF (119885119895

1198901

times 119885119895

1198902

)

ELSE 119885119895

119906 119895 = 1 7

(21)

where (1198851198951198901

times119885119895

1198902

) is the Cartesian product of the fuzzy sets 1198641

and 1198642 with the following membership function

120583(119885119895

1198901times119885119895

1198902)(1198901 1198902) = 120583

119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902) (22)

The corresponding output fuzzy set is defined by fuzzyrelation 119877119895 = (119885

119895

1198901

times 119885119895

1198902

) times 119885119895

119906 119895 = 1 7 with the following

membership function

120583119877119895 ((119890

1 1198902) 119906

lowast) = (120583

119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902))

and 120583119885119895

119906

(119906lowast)

(23)

The set of all rules 119877 = ⋃7

119895=1119877119895 with fuzzy membership

function

120583119877((119890

1 1198902) 119906

lowast)

=

7

⋁

119895=1

[(120583119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902)) and 120583

119885119895

119906

(119906lowast)]

(24)

which defines the rule base of the FLC and produces analgorithm of fuzzy control system

As a result for given values of the input linguistic var-iables 119885

119895

1198901

and 119885119895

1198902

the fuzzy output value of the logicalcontroller119885119895

119906can be determined based on the following com-

posite rule [12]

119861119895= (119885

119895

1198901

times 119885119895

1198902

) sdot 119877 (25)

with a degree of membership

120583119885119895

119906

(119906lowast) = ⋁

1198901isin1198641 1198902isin1198642

[(120583119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902))

and 120583119877(1198901 1198902 119906lowast)]

(26)

When the linguistic input variables 1198901and 119890

2correspond

to fuzzy sets11988510158401198901

and11988510158401198902

fuzzy set1198851015840119906of the linguistic variable

of the control signal 119906lowast is determined as follows

1205831198791015840

119906

1015840 (119906lowast) = max

11989011198902

[

119899

prod

119894=1

1205831198791015840

119890119894

(119890119894)]

sdot [

119898

min119895=1

[

119899

prod

119894=1

120583119879119895

119890119894

(119890119894)] sdot 120583

119879119895

119906

(119906lowast)]

(27)

In order to obtain a true output value following fuzzycomputations it is necessary to perform defuzzificationmdashthetransformation of fuzzy linguistic variable value (qualitative)

119906lowast into discrete numerical value of 119906 For this purpose center

of gravity method is applied [12]

119906 =

sum7

119899=1119906lowast

119899120583119885119906

(119906lowast

119899)

sum7

119899=1120583119885119906

(119906lowast119899)

(28)

Membership function of fuzzy values 1198851015840119906can be repre-

sented as

1205831198851015840

119906

1015840 (119906) =

119899

prod

119894=1

120583119885119895

119890119894

(119890119894) 119906 = 120593

119895

0 119906 = 120593119895

(29)

where 120593119895 is the discrete numerical value of the output signalThen the final output value of the FLC at the defuzzifica-

tion stage can be computed as follows

119906 =

sum119898

119895=1120593119895[prod

119899

119894=1120583119885119895

119890119894

(119890119894)]

sum119898

119895=1prod119899

119894=1120583119885119895

119890119894

(119890119894)

or 119906 (119890 120593) =119898

sum

119895=1

120593119895120595119895(119890)

(30)

where

120595119895(119890) =

prod119899

119894=1120583119879119895

119890119894

(119890119894)

sum119898

119895=1prod119899

119894=1120583119879119895

119890119894

(119890119894) (31)

If we consider that the basic equation of the road trafficcontroller can be written as [12]

119906 (119905) = 1199060+ 119870(119890 (119905) +

1

119879119906

int

119905

0

119890 (120591) 119889120591 + 119879119889

119889119890 (119905)

119889119905) (32)

then the rule of the fuzzy traffic control system can berepresented as a logic controller with the varying coefficientas in (14)

119906 (119905) = 1199060+ 119870

lowast(120572 120573) ∘ 120595

lowast

1199060

(119890)lowast (33)

where 119870lowast(120572 120573) is the variable part of the gain that depends

on the current dynamic parameters of the road traffic [16]This allows implementing an adaptive traffic control systemas a fuzzy system consisting of two interconnected modulesldquomodule of fuzzy controller of the switching phase trafficlightsrdquo and ldquomodule of fuzzy adaptive correction of theparameters of the main fuzzy controller of the switchingphase traffic lightsrdquo 120572 and 120574

Furthermore we introduce the following linguistic vari-ables to assess the quality of the control system and adaptivefuzzy correction of the parameters of the fuzzy-logic con-troller119889119890

2

= (ldquoThe Derivative Queue Lengths of Vehiclesrdquo 1198851198891198902

1198641198891198902

)

119876 = (ldquoThe Quality Control Systemrdquo 119885119876 119864

119876)

(34)

The Scientific World Journal 7

120583intQ119896

and 1205833intQ119896

997888rarr (alowastX119894 alowastY119894 b

lowast1X119894

blowast2X119894 blowast1Y119894 b

lowast2Y119894 120573lowast1X119894

120573lowast2X119894 120573lowast1XY119894 120573

lowast2Y119894

)1205833Q119896

120583Q119896

120583Q119896

Qk

Figure 2 Graphic illustration of a choice of optimum adjusting parameters

where 1198851198891198902119894

= 119885(1)

1198891198902

119885(2)

1198891198902

119885(119899)

1198891198902

119885119876119894

= 119885(1)

119876 119890119894

119885(2)

119876

119885(119896)

119876 t are the term-sets of linguistic variables 119889119890

2and119876with

corresponding membership functions

1198851198891198902

=

119889119890(119894)

2

1205831198851198891198902

(119889119890(119894)

2)

119894 = 1 119899

119885119876=

119902(119895)

120583119876(119902(119895))

119895 = 1 119896

(35)

given by universal sets 1198641198891198902

= [1198641198891198902min 119864119889119890

2max] and 119864

119876=

[119864119876min 119864119876max]Then according to (12) and (25) the choice of the optimal

adjustment parameters of the FLC can be represented as thefollowing fuzzy relations

119877119895

119876lowast = (119885

119895

1198902

times 119885119895

1198902

) times 119885119895

119876 119895 = 1119873

119876 (36)

where

119885119876=

119876119879119875

1

120583119879119875

1198761

(119876119879119875

1)

120583119879119875

119876119896

(119876119879119875

119896) = 120593 (119876

119879119875

119896 119886lowast

119876119896

119887lowast

1119876119896

119887lowast

2119876119896

120573lowast

1119876119896

120573lowast

2119876119896

)

120593 (119876119879119875

119896 119886lowast

119876119896

119887lowast

1119876119896

119887lowast

2119876119896

120573lowast

1119876119896

120573lowast

2119876119896

)

= ((119887lowast

1119876119896

(119886lowast

119876119896

minus 119909))

120573lowast

1119876119896

sdot

sign (119887lowast1119876119896

(119886lowast

119876119896

minus 119909)) + 1

2

+ (119887lowast

2119876119896

(119909 minus 119886lowast

119876119896

))

120573lowast

2119876119896

sign (119887lowast2119876119896

(119909 minus 119886lowast

119876119896

)) + 1

2

+ 1)

minus1

119876 = 1199021

120583119876(1199021)

1199022

120583119876(1199022)

119902119896

120583119876(119902119896)

120583119876(119902119896) = 120593 (119902

119896 119886119876119896

1198871119876119896

1198872119876119896

1205731119876119896

1205732119876119896

)

(37)

This fuzzy relationship served as the basis for developingalgorithms of adaptive parameter settings of the fuzzy-logiccontroller (Figure 2)

4 Simulation Modeling and Analysis ofFuzzy Traffic Control System

We develop a simulation model of the adaptive fuzzy controlsystem of road traffic on Matlab software package Thesimulation model is used to test the efficiency of the fuzzycontrol system through series of computational experiments

As an object of simulation we chose one of the heavy-loaded and problematic crossroads in Tashkent city Uzbek-istan The simulation model considered the road trafficintensity observed at the crossroad during the peak-hoursfrom 0700 to 1000 onMay 5 2015The experiment consistedof two phases At each phase of the simulation variouspatterns of traffic-light management were used In the firstphase the traffic lights with fixed cycles which are used inreal life are simulated In the second phase traffic lightscontrolled with adaptive fuzzy system were simulated Forthis purpose the rules of fuzzy control system with a modulefor adaptive adjustment of parameters were synthesized usingthe proposed method

The simulation results are presented below Figure 3shows the results of simulation of the road intersection thatis managed by traffic lights with fixed cycles Figure 4 showsthe results of simulation of the crossroad that is managed byadaptive fuzzy controller

8 The Scientific World Journal

The length of the queue of vehicles in directions 1ndash3

The length of the queue of vehicles in directions 2ndash4

710

720

910

900

820

800

810

920

700

840

850

750

740

730

930

940

950

830

1000

05101520253035404550

2ndash4The intensity of the traffic flow in directions

The intensity of the traffic flow in directions 1ndash3

Figure 3 The results of the simulation modeling of the trafficcontrol system with optimum fixed cycle of traffic lights

1ndash3The length of the queue of vehicles in directions

The length of the queue of vehicles in directions 2ndash4

710

720

910

900

820

800

810

920

700

840

850

750

740

730

930

940

950

830

1000

2ndash4The intensity of the traffic flow in directions

The intensity of the traffic flow in directions 1ndash3

05101520253035404550

Figure 4 The results of the simulation modeling of the adaptivetraffic control system

Simulation results show that the synthesized fuzzy controlsystem successfully solved the problem of queues at trafficlights and performed more effective than the current trafficcontrol system with fixed-cycle traffic lights Particularly thelength of queue for the synthesized fuzzy control system isless than 26 times for traffic control system with fixed-cycletraffic lights Also delay time for synthesized fuzzy controlsystem is 27 less than traffic control system with fixed-cycletraffic lights

Based on the results of computational experiments wecan conclude that the synthesized adaptive fuzzy controlsystem of road traffic in conditions of intensive traffic isrobust and allows controlling road traffic within a wide rangeof parameters

5 Conclusion

In this study we formally described the problems of synthesisof adaptive fuzzy traffic control systems in conditions ofintense road traffic We developed efficient algorithms of thesynthesis of the adaptive fuzzy-logic traffic control systemsWe also developed a simulationmodel of adaptive fuzzy-logiccontrol system for managing road traffic and conducted aseries of computational experiments

The results showed that the synthesized fuzzy-logic traf-fic control system based on the proposed method allowscontrolling crossroads more effectively compared to trafficcontrol system with fixed-cycle traffic lights We conclude

that adaptive fuzzy-logic controller can successfully functionin the condition of information fuzziness It can also success-fully function when exposed to uncontrolled external andparametric impacts The implementation of the solution inheavy-loaded crossroads allows reducing the length of queuesat the traffic lights up to 26 times and the delay time up to27

Competing Interests

The authors declare that they have no competing interests

References

[1] F Sattar F Karray M Kamel L Nassar and K GolestanldquoRecent advances on context-awareness and datainformationfusion in ITSrdquo International Journal of Intelligent TransportationSystems Research vol 14 no 1 pp 1ndash19 2016

[2] J-D Schmocker and M G H Bell ldquoTraffic control currentsystems and future vision of citiesrdquo International Journal ofIntelligent Transportation Systems Research vol 8 no 1 pp 56ndash65 2010

[3] S Sundaram H N Koutsopoulos M Ben-Akiva C Anto-niou and R Balakrishna ldquoSimulation-based dynamic trafficassignment for short-term planning applicationsrdquo SimulationModelling Practice andTheory vol 19 no 1 pp 450ndash462 2011

[4] Dunn Engineering Associates Traffic Control Systems Hand-book Dunn Engineering Associates Westhampton Beach NYUSA 2005

[5] C P Pappis and E H Mamdani ldquoFuzzy logic controller fora traffic junctionrdquo IEEE Transactions on Systems Man andCybernetics vol 7 no 10 pp 707ndash717 1977

[6] Y S Murat and E Gedizlioglu ldquoA fuzzy logic multi-phasedsignal control model for isolated junctionsrdquo TransportationResearch Part C Emerging Technologies vol 13 no 1 pp 19ndash362005

[7] B Yulianto ldquoApplication of fuzzy logic to traffic signal controlundermixed traffic conditionsrdquoTraffic Engineering andControlvol 44 no 9 pp 332ndash336 2003

[8] B Madhavan Nair and J Cai ldquoA fuzzy logic controller forisolated signalized intersection with traffic abnormality consid-eredrdquo in Proceedings of the IEEE Intelligent Vehicles Symposium(IV rsquo07) pp 1229ndash1233 June 2007

[9] V T Zoriktuev S G Ts and J F Mesyagutov ldquoMethodologyrepresented and derivation of knowledge in control systems ofthemechatronicmachine systemsrdquo STIN Systems vol 11 pp 13ndash18 2007

[10] R A Aliev and R R AlievTheTheory of Intelligent Systems andIts Application Chashiogli Baku Azerbaijan 2001

[11] R A Aliev E G Zakharova and S V Ulyanov Fuzzy Modelsof Dynamic Systems Results of Science and Technology SerTechnology Cybernetics 1990

[12] N R Yusupbekov and A R Marakhimov ldquoSynthesis of theintelligent traffic control systems in conditions saturated trans-port streamrdquo International Journal of International Journal ofChemical Technology Control and Management Jointly with theJournal of Korea Multimedia Society vol 3-4 pp 12ndash18 2015

[13] Y Ge ldquoA two-stage fuzzy logic control method of traffic signalbased on traffic urgency degreerdquo Modelling and Simulation inEngineering vol 2014 Article ID 694185 6 pages 2014

The Scientific World Journal 9

[14] I M Makarov and V M Lokhin Eds Intelligent AutomaticControl Systems Fizmatlit Moscow Russia 2001

[15] A Pegat Fuzzy Modeling and Control Knowledge Laboratory2009

[16] RMartinez-Soto O Castillo L T Aguilar and PMelin ldquoFuzzylogic controllers optimization using genetic algorithms andparticle swarm optimizationrdquo in Advances in Soft ComputingG Sidorov A H Aguirre and C A R Garcıa Eds vol 6438of Lecture Notes in Computer Science pp 475ndash486 SpringerBerlin Germany 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 The Scientific World Journal

119896sum

1199091

[119905 + 1] = 119896sum

1199091

[119905] (1 minus 1205961205743)

+ 120596 (1205745minus 120574

4) 120575

lowast[119905] 119909

sum

1[119905]

minus 1205961205745120575lowast[119905 + 1] 119909

sum

1[119905 + 1]

119896sum

1199092

[119905 + 1] = 119896sum

1199092

[119905] (1 minus 1205961205743)

+ 120596 (1205745minus 120574

4) 120575

lowast[119905] 119909

sum

2[119905]

minus 1205961205745120575lowast[119905 + 1] 119909

sum

2[119905 + 1]

119896sum

119909119899

[119905 + 1] = 119896sum

119909119899

[119905] (1 minus 1205961205743)

+ 120596 (1205745minus 120574

4) 120575

lowast[119905] 119909

sum

119899[119905]

minus 1205961205745120575lowast[119905 + 1] 119909

sum

119899[119905 + 1]

119896119906 [119905 + 1] = 119896

119906 [119905] (1 minus 1205961205741) + 120596 (1205746 minus 1205742) 120575lowast[119905] 119906119898 [119905]

minus 1205961205746120575lowast[119905 + 1] 119906119898 [119905 + 1]

(15)

where 119905 = 119898120596 120596 gt 0 is discretization step 120574 = 1205741 1205742 1205743

1205744 1205745 1205746 are the parameters of the adaptive controller 119890sum

119894=

int119909119894

(119909119894minus119909

119894119898)120583119890119894

(119890119894)119889119909

119894is a signal mismatch between the actual

parameters of the state vector and desirable model parame-ters 120583

119890119894

(119890119894) = 120593(119890

119894 119886119890119894

1198871119890119894

1198872119890119894

]1119890119894

]2119890119894

) is the membershipfunction of the signal mismatch provided as (10) 119886

119890119894

= 119886119909119894

minus

119909119894119898 ]

1119890119894

= ]1119909119894

]2119890119894

= ]2119909119894

1198871119890119894

= 1198871119909119894

1198872119890119894

= 1198872119909119894

119909sum119894

=

int119909119894

119909119894119889119909

119894are the integrated parameters of the state vector

120575lowast[119905] = sum

119899

119894=1ℎ119894sdot 119890sum

119894 ℎ

119894are the coefficients obtained by Lya-

punovrsquos decision equation and the matrix equation withreference model 119861M

The synthesis of the control law based on fuzzy modelcan increase the robustness of the gradient speed searchalgorithm it can also maintain limitations of the phase tra-jectories in certain areas under uncompensated informationfuzziness [11 12]

Quality indicators of the control systems are based notonly on the best possible dynamic characteristics of objectcontrolling 119886

119909119894

(119905) 119886119910119894

(119905) but also on the parameters of itsfuzziness 119886

119883119894

119886119884119895

1198871119883119894

1198871119884119895

1198872119883119894

1198872119884119895

1205731119883119894

1205731119884119895

1205732119883119894

and1205732119884119895

provided as (10) and (11)Reducing the information fuzziness and consequently

improving the quality control can be achieved by optimizingthe parameters of the membership functions of the input andoutput linguistic variables and fuzzy-logic controllers This isthe core idea of the algorithm for traffic control systems incrossroads presented in this study

3 The Concept of the ProblemSynthesis Adaptive Fuzzy-Logic TrafficControl Systems

Let us consider the problem of synthesis of fuzzy controlsystem of road traffic under conditions of intense traffic inmore detail

We suppose that there is a crossroad regulated with anequal number of intersecting lanes and well-known trafficintensity of 120582

1 120582

2 120582

3 120582

4 The directions of traffic 1 and 3 are

perpendicular to the directions 2 and 4We introduce the following terms Δ119879119906

1is the duration of

the green signal of the traffic light to the first direction of thetraffic 120582

1 120582

3 Δ119879119906

2is the duration of the green signal of the

traffic light to the opposite direction of the traffic 1205822 120582

4

Taking into consideration the principle of progressivecorrection the control object can be defined as (2) Thecontrol process assumes that there is a target goal andthe control system functions to achieve it The quality offunctioning (criterion of efficiency) of the control system (12)assumes a degree of adaptability to fulfill its taskmdashensuring asafe traffic with a minimum delay Therefore the problem ofoptimal traffic control at crossroads can be defined as follows

It is necessary to identify control signals Δ1198791199061and Δ119879

119906

2

(the duration of the green signal of the traffic light for eachlane) so that the actual state of the traffic 119910(119905) was as closeas possible to the desired state when the control signalshave certain limitations and the system exposed uncontrolledexternal and parametric impacts 119910lowast(119905)

That is it is necessary to find such parameters of thetraffic control system that the total delay at the crossroads isminimal [12 13]

119869 (119905) = int

119905

1199050

[119910lowast(120591 119909 (120591) 119906 (120591)) minus 119910 (120591 119909 (120591) 119906 (120591))] 119889120591

997888rarr min(16)

or

sum

119894

sum

119895

[11989013

119894119891lowast(119906

13

119894 11989013

119894) + 119897

13

119894119891lowast

119901119899(119906

13

119894 11989713

119894)

+ 11989024

119894119891lowast(119906

24

119894 11989024

119894) + 119897

24

119894119891lowast

119901119899(119906

24

119894 11989724

119894)] 997888rarr min

(17)

under the constraints

11989013

119894= 119902

leaving(13)119894

minus 119902arriv(13)119894

le 0

11989024

119894= 119902

leaving(24)119894

minus 119902arriv(24)119894

le 0

11990613

min le 11990613

119894le 119906

13

max

11990624

min le 11990624

119894le 119906

24

max

0 le 11989713

119894le 119871

13

max

0 le 11989724

119894le 119871

24

max

(18)

where 119891lowast(11990613119894 120582

13

119894) and 119891lowast

119901119899(11990613

119894 11989713

119894) are the delay evaluation

function and penalty function for the length of queue infront of the stop line of the traffic light in directions 1 and3 respectively 119891lowast(11990624

119894 120582

24

119894) and 119891

lowast

119901119899(11990624

119894 11989724

119894) are the delay

evaluation function and penalty function for the length ofqueue in front of the stop line of the traffic light in directions 2and 4 respectively 119906

119894is the duration of green signal of traffic

light in the i-stage 119902left(13)119894

119902arriv(13)119894

119902left(24)119894

and 119902arriv(24)119894

are

The Scientific World Journal 5

Fuzzy-logic controller(FLC)

The control object(transport flows)

Optimization module ofthe parameters FLC

Quality estimationmodule of the functioningof the system

Ylowast(t)

Ylowast(t)

E(t) U(t)

J(t)

Y(t)

sum

120583jT119890(x alowastj b

lowastj c

lowastj )

Figure 1 A traffic control system at the crossroads with an adaptive fuzzy-logic controller

the number of vehicles entered and left the control zone atthe crossroad in the 119894th phase respectively 119906min and 119906max arethe minimum and maximum duration of the green signal ofthe traffic light respectively 11989713

119894and 11989724

119894are the length of the

queue in front of the stop line at directions 1 and 3 and 2 and4 in the 119894th stage respectively 119871max is the maximum allowedlength of the queue in front of the stop line

In general the process control system at the T-shapedcrossroads can be rearranged as a feedback control sys-tem with adaptive fuzzy-logic controller (Figure 1) In the

proposed scheme the input vector 119864lowast = (119890lowast

1 119890lowast

2) is converted

into fuzzy-logic controller (FLC) through fuzzification blockNext fuzzy inference is performed based on rule base whichresults in a fuzzy output variable 119906lowast The output values fromthe FLC are then transferred from the fuzzy region 119906

lowast toaccurate region 119906 through defuzzification block [14 15]

In general rule bases or knowledge bases contain descrip-tions of the linguistic variables of the FLC that are predefinedfor each of the input and output variables Therefore weintroduce the following linguistic variables

1198901= (ldquoDifference in the number of vehicles that entered and left the control zonerdquo 119885

1198901

1198641)

1198902= (ldquoQueue lengthsrdquo 119885

1198902

1198642)

119906 = (ldquoControlrdquo 119885119906 119880)

(19)

where 119885119890119894

= 1198851

119890119894

1198852

119890119894

119885119896

119890119894

119894 = 1 2 and 119885119906= 119885

1

119906 119885

2

119906

119885119896

119906 are the term-sets of linguistic variables 119890

1 119890

2 and

119906 with corresponding membership functions (MF) 119885119897119890119894

=

120583119897

119890119894

(119890119894) 119885119897

119906= 120583

119897(119906) 119897 = 1 119896 given by universal sets 119864

119894=

[119864119894min 119864119894max] and 119880 = [119880min 119880max]

We assume that seven terms 119885119909= ldquoNBrdquo ldquoNMrdquo ldquoNSrdquo

ldquoZErdquo ldquoPSrdquo ldquoPMrdquo ldquoPBrdquo correspond to each of the input andoutput linguistic variables 119885

119909= 119879

1198901

1198791198902

119879119906 with trian-

gular membership functions provided as (10) As a result thefollowing linguistic variables are derived from the fuzzifica-tion processes

1198901= (ldquoDifference of the number of vehicles that entered and left the control zonerdquo 119885

1198901

1198641)

= [

120583NB1198901

(1198901)

NB1198901

120583NM1198901

(1198901)

NM1198901

120583NS1198901

(1198901)

NS1198901

120583ZE1198901

(1198901)

ZE1198901

120583PS1198901

(1198901)

PS1198901

120583PM1198901

(1198901)

PM1198901

120583PB1198901

(1198901)

PB1198901

]

1198902= (ldquoQueue lengthsrdquo 119885

1198902

1198642) = [

120583NB1198902

(1198902)

NB1198902

120583NM1198902

(1198902)

NM1198902

120583NS1198902

(1198902)

NS1198902

120583ZE1198902

(1198902)

ZE1198902

120583PS1198902

(1198902)

PS1198902

120583PM1198902

(1198902)

PM1198902

120583PB1198902

(1198902)

PB1198902

]

119906 = (ldquoControlrdquo 119885119906 119880) = [

120583NB119906(119906)

NB119906

120583NM119906(119906)

NM119906

120583NS119906(119906)

NS119906

120583ZE119906(119906)

ZE119906

120583PS119906(119906)

PS119906

120583PM119906(119906)

PM119906

120583PB119906(119906)

PB119906

]

(20)

6 The Scientific World Journal

Next we develop a rule base for the FLC inferences asfollows

IF (119885119895

1198901

times 119885119895

1198902

)

ELSE 119885119895

119906 119895 = 1 7

(21)

where (1198851198951198901

times119885119895

1198902

) is the Cartesian product of the fuzzy sets 1198641

and 1198642 with the following membership function

120583(119885119895

1198901times119885119895

1198902)(1198901 1198902) = 120583

119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902) (22)

The corresponding output fuzzy set is defined by fuzzyrelation 119877119895 = (119885

119895

1198901

times 119885119895

1198902

) times 119885119895

119906 119895 = 1 7 with the following

membership function

120583119877119895 ((119890

1 1198902) 119906

lowast) = (120583

119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902))

and 120583119885119895

119906

(119906lowast)

(23)

The set of all rules 119877 = ⋃7

119895=1119877119895 with fuzzy membership

function

120583119877((119890

1 1198902) 119906

lowast)

=

7

⋁

119895=1

[(120583119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902)) and 120583

119885119895

119906

(119906lowast)]

(24)

which defines the rule base of the FLC and produces analgorithm of fuzzy control system

As a result for given values of the input linguistic var-iables 119885

119895

1198901

and 119885119895

1198902

the fuzzy output value of the logicalcontroller119885119895

119906can be determined based on the following com-

posite rule [12]

119861119895= (119885

119895

1198901

times 119885119895

1198902

) sdot 119877 (25)

with a degree of membership

120583119885119895

119906

(119906lowast) = ⋁

1198901isin1198641 1198902isin1198642

[(120583119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902))

and 120583119877(1198901 1198902 119906lowast)]

(26)

When the linguistic input variables 1198901and 119890

2correspond

to fuzzy sets11988510158401198901

and11988510158401198902

fuzzy set1198851015840119906of the linguistic variable

of the control signal 119906lowast is determined as follows

1205831198791015840

119906

1015840 (119906lowast) = max

11989011198902

[

119899

prod

119894=1

1205831198791015840

119890119894

(119890119894)]

sdot [

119898

min119895=1

[

119899

prod

119894=1

120583119879119895

119890119894

(119890119894)] sdot 120583

119879119895

119906

(119906lowast)]

(27)

In order to obtain a true output value following fuzzycomputations it is necessary to perform defuzzificationmdashthetransformation of fuzzy linguistic variable value (qualitative)

119906lowast into discrete numerical value of 119906 For this purpose center

of gravity method is applied [12]

119906 =

sum7

119899=1119906lowast

119899120583119885119906

(119906lowast

119899)

sum7

119899=1120583119885119906

(119906lowast119899)

(28)

Membership function of fuzzy values 1198851015840119906can be repre-

sented as

1205831198851015840

119906

1015840 (119906) =

119899

prod

119894=1

120583119885119895

119890119894

(119890119894) 119906 = 120593

119895

0 119906 = 120593119895

(29)

where 120593119895 is the discrete numerical value of the output signalThen the final output value of the FLC at the defuzzifica-

tion stage can be computed as follows

119906 =

sum119898

119895=1120593119895[prod

119899

119894=1120583119885119895

119890119894

(119890119894)]

sum119898

119895=1prod119899

119894=1120583119885119895

119890119894

(119890119894)

or 119906 (119890 120593) =119898

sum

119895=1

120593119895120595119895(119890)

(30)

where

120595119895(119890) =

prod119899

119894=1120583119879119895

119890119894

(119890119894)

sum119898

119895=1prod119899

119894=1120583119879119895

119890119894

(119890119894) (31)

If we consider that the basic equation of the road trafficcontroller can be written as [12]

119906 (119905) = 1199060+ 119870(119890 (119905) +

1

119879119906

int

119905

0

119890 (120591) 119889120591 + 119879119889

119889119890 (119905)

119889119905) (32)

then the rule of the fuzzy traffic control system can berepresented as a logic controller with the varying coefficientas in (14)

119906 (119905) = 1199060+ 119870

lowast(120572 120573) ∘ 120595

lowast

1199060

(119890)lowast (33)

where 119870lowast(120572 120573) is the variable part of the gain that depends

on the current dynamic parameters of the road traffic [16]This allows implementing an adaptive traffic control systemas a fuzzy system consisting of two interconnected modulesldquomodule of fuzzy controller of the switching phase trafficlightsrdquo and ldquomodule of fuzzy adaptive correction of theparameters of the main fuzzy controller of the switchingphase traffic lightsrdquo 120572 and 120574

Furthermore we introduce the following linguistic vari-ables to assess the quality of the control system and adaptivefuzzy correction of the parameters of the fuzzy-logic con-troller119889119890

2

= (ldquoThe Derivative Queue Lengths of Vehiclesrdquo 1198851198891198902

1198641198891198902

)

119876 = (ldquoThe Quality Control Systemrdquo 119885119876 119864

119876)

(34)

The Scientific World Journal 7

120583intQ119896

and 1205833intQ119896

997888rarr (alowastX119894 alowastY119894 b

lowast1X119894

blowast2X119894 blowast1Y119894 b

lowast2Y119894 120573lowast1X119894

120573lowast2X119894 120573lowast1XY119894 120573

lowast2Y119894

)1205833Q119896

120583Q119896

120583Q119896

Qk

Figure 2 Graphic illustration of a choice of optimum adjusting parameters

where 1198851198891198902119894

= 119885(1)

1198891198902

119885(2)

1198891198902

119885(119899)

1198891198902

119885119876119894

= 119885(1)

119876 119890119894

119885(2)

119876

119885(119896)

119876 t are the term-sets of linguistic variables 119889119890

2and119876with

corresponding membership functions

1198851198891198902

=

119889119890(119894)

2

1205831198851198891198902

(119889119890(119894)

2)

119894 = 1 119899

119885119876=

119902(119895)

120583119876(119902(119895))

119895 = 1 119896

(35)

given by universal sets 1198641198891198902

= [1198641198891198902min 119864119889119890

2max] and 119864

119876=

[119864119876min 119864119876max]Then according to (12) and (25) the choice of the optimal

adjustment parameters of the FLC can be represented as thefollowing fuzzy relations

119877119895

119876lowast = (119885

119895

1198902

times 119885119895

1198902

) times 119885119895

119876 119895 = 1119873

119876 (36)

where

119885119876=

119876119879119875

1

120583119879119875

1198761

(119876119879119875

1)

120583119879119875

119876119896

(119876119879119875

119896) = 120593 (119876

119879119875

119896 119886lowast

119876119896

119887lowast

1119876119896

119887lowast

2119876119896

120573lowast

1119876119896

120573lowast

2119876119896

)

120593 (119876119879119875

119896 119886lowast

119876119896

119887lowast

1119876119896

119887lowast

2119876119896

120573lowast

1119876119896

120573lowast

2119876119896

)

= ((119887lowast

1119876119896

(119886lowast

119876119896

minus 119909))

120573lowast

1119876119896

sdot

sign (119887lowast1119876119896

(119886lowast

119876119896

minus 119909)) + 1

2

+ (119887lowast

2119876119896

(119909 minus 119886lowast

119876119896

))

120573lowast

2119876119896

sign (119887lowast2119876119896

(119909 minus 119886lowast

119876119896

)) + 1

2

+ 1)

minus1

119876 = 1199021

120583119876(1199021)

1199022

120583119876(1199022)

119902119896

120583119876(119902119896)

120583119876(119902119896) = 120593 (119902

119896 119886119876119896

1198871119876119896

1198872119876119896

1205731119876119896

1205732119876119896

)

(37)

This fuzzy relationship served as the basis for developingalgorithms of adaptive parameter settings of the fuzzy-logiccontroller (Figure 2)

4 Simulation Modeling and Analysis ofFuzzy Traffic Control System

We develop a simulation model of the adaptive fuzzy controlsystem of road traffic on Matlab software package Thesimulation model is used to test the efficiency of the fuzzycontrol system through series of computational experiments

As an object of simulation we chose one of the heavy-loaded and problematic crossroads in Tashkent city Uzbek-istan The simulation model considered the road trafficintensity observed at the crossroad during the peak-hoursfrom 0700 to 1000 onMay 5 2015The experiment consistedof two phases At each phase of the simulation variouspatterns of traffic-light management were used In the firstphase the traffic lights with fixed cycles which are used inreal life are simulated In the second phase traffic lightscontrolled with adaptive fuzzy system were simulated Forthis purpose the rules of fuzzy control system with a modulefor adaptive adjustment of parameters were synthesized usingthe proposed method

The simulation results are presented below Figure 3shows the results of simulation of the road intersection thatis managed by traffic lights with fixed cycles Figure 4 showsthe results of simulation of the crossroad that is managed byadaptive fuzzy controller

8 The Scientific World Journal

The length of the queue of vehicles in directions 1ndash3

The length of the queue of vehicles in directions 2ndash4

710

720

910

900

820

800

810

920

700

840

850

750

740

730

930

940

950

830

1000

05101520253035404550

2ndash4The intensity of the traffic flow in directions

The intensity of the traffic flow in directions 1ndash3

Figure 3 The results of the simulation modeling of the trafficcontrol system with optimum fixed cycle of traffic lights

1ndash3The length of the queue of vehicles in directions

The length of the queue of vehicles in directions 2ndash4

710

720

910

900

820

800

810

920

700

840

850

750

740

730

930

940

950

830

1000

2ndash4The intensity of the traffic flow in directions

The intensity of the traffic flow in directions 1ndash3

05101520253035404550

Figure 4 The results of the simulation modeling of the adaptivetraffic control system

Simulation results show that the synthesized fuzzy controlsystem successfully solved the problem of queues at trafficlights and performed more effective than the current trafficcontrol system with fixed-cycle traffic lights Particularly thelength of queue for the synthesized fuzzy control system isless than 26 times for traffic control system with fixed-cycletraffic lights Also delay time for synthesized fuzzy controlsystem is 27 less than traffic control system with fixed-cycletraffic lights

Based on the results of computational experiments wecan conclude that the synthesized adaptive fuzzy controlsystem of road traffic in conditions of intensive traffic isrobust and allows controlling road traffic within a wide rangeof parameters

5 Conclusion

In this study we formally described the problems of synthesisof adaptive fuzzy traffic control systems in conditions ofintense road traffic We developed efficient algorithms of thesynthesis of the adaptive fuzzy-logic traffic control systemsWe also developed a simulationmodel of adaptive fuzzy-logiccontrol system for managing road traffic and conducted aseries of computational experiments

The results showed that the synthesized fuzzy-logic traf-fic control system based on the proposed method allowscontrolling crossroads more effectively compared to trafficcontrol system with fixed-cycle traffic lights We conclude

that adaptive fuzzy-logic controller can successfully functionin the condition of information fuzziness It can also success-fully function when exposed to uncontrolled external andparametric impacts The implementation of the solution inheavy-loaded crossroads allows reducing the length of queuesat the traffic lights up to 26 times and the delay time up to27

Competing Interests

The authors declare that they have no competing interests

References

[1] F Sattar F Karray M Kamel L Nassar and K GolestanldquoRecent advances on context-awareness and datainformationfusion in ITSrdquo International Journal of Intelligent TransportationSystems Research vol 14 no 1 pp 1ndash19 2016

[2] J-D Schmocker and M G H Bell ldquoTraffic control currentsystems and future vision of citiesrdquo International Journal ofIntelligent Transportation Systems Research vol 8 no 1 pp 56ndash65 2010

[3] S Sundaram H N Koutsopoulos M Ben-Akiva C Anto-niou and R Balakrishna ldquoSimulation-based dynamic trafficassignment for short-term planning applicationsrdquo SimulationModelling Practice andTheory vol 19 no 1 pp 450ndash462 2011

[4] Dunn Engineering Associates Traffic Control Systems Hand-book Dunn Engineering Associates Westhampton Beach NYUSA 2005

[5] C P Pappis and E H Mamdani ldquoFuzzy logic controller fora traffic junctionrdquo IEEE Transactions on Systems Man andCybernetics vol 7 no 10 pp 707ndash717 1977

[6] Y S Murat and E Gedizlioglu ldquoA fuzzy logic multi-phasedsignal control model for isolated junctionsrdquo TransportationResearch Part C Emerging Technologies vol 13 no 1 pp 19ndash362005

[7] B Yulianto ldquoApplication of fuzzy logic to traffic signal controlundermixed traffic conditionsrdquoTraffic Engineering andControlvol 44 no 9 pp 332ndash336 2003

[8] B Madhavan Nair and J Cai ldquoA fuzzy logic controller forisolated signalized intersection with traffic abnormality consid-eredrdquo in Proceedings of the IEEE Intelligent Vehicles Symposium(IV rsquo07) pp 1229ndash1233 June 2007

[9] V T Zoriktuev S G Ts and J F Mesyagutov ldquoMethodologyrepresented and derivation of knowledge in control systems ofthemechatronicmachine systemsrdquo STIN Systems vol 11 pp 13ndash18 2007

[10] R A Aliev and R R AlievTheTheory of Intelligent Systems andIts Application Chashiogli Baku Azerbaijan 2001

[11] R A Aliev E G Zakharova and S V Ulyanov Fuzzy Modelsof Dynamic Systems Results of Science and Technology SerTechnology Cybernetics 1990

[12] N R Yusupbekov and A R Marakhimov ldquoSynthesis of theintelligent traffic control systems in conditions saturated trans-port streamrdquo International Journal of International Journal ofChemical Technology Control and Management Jointly with theJournal of Korea Multimedia Society vol 3-4 pp 12ndash18 2015

[13] Y Ge ldquoA two-stage fuzzy logic control method of traffic signalbased on traffic urgency degreerdquo Modelling and Simulation inEngineering vol 2014 Article ID 694185 6 pages 2014

The Scientific World Journal 9

[14] I M Makarov and V M Lokhin Eds Intelligent AutomaticControl Systems Fizmatlit Moscow Russia 2001

[15] A Pegat Fuzzy Modeling and Control Knowledge Laboratory2009

[16] RMartinez-Soto O Castillo L T Aguilar and PMelin ldquoFuzzylogic controllers optimization using genetic algorithms andparticle swarm optimizationrdquo in Advances in Soft ComputingG Sidorov A H Aguirre and C A R Garcıa Eds vol 6438of Lecture Notes in Computer Science pp 475ndash486 SpringerBerlin Germany 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 5

Fuzzy-logic controller(FLC)

The control object(transport flows)

Optimization module ofthe parameters FLC

Quality estimationmodule of the functioningof the system

Ylowast(t)

Ylowast(t)

E(t) U(t)

J(t)

Y(t)

sum

120583jT119890(x alowastj b

lowastj c

lowastj )

Figure 1 A traffic control system at the crossroads with an adaptive fuzzy-logic controller

the number of vehicles entered and left the control zone atthe crossroad in the 119894th phase respectively 119906min and 119906max arethe minimum and maximum duration of the green signal ofthe traffic light respectively 11989713

119894and 11989724

119894are the length of the

queue in front of the stop line at directions 1 and 3 and 2 and4 in the 119894th stage respectively 119871max is the maximum allowedlength of the queue in front of the stop line

In general the process control system at the T-shapedcrossroads can be rearranged as a feedback control sys-tem with adaptive fuzzy-logic controller (Figure 1) In the

proposed scheme the input vector 119864lowast = (119890lowast

1 119890lowast

2) is converted

into fuzzy-logic controller (FLC) through fuzzification blockNext fuzzy inference is performed based on rule base whichresults in a fuzzy output variable 119906lowast The output values fromthe FLC are then transferred from the fuzzy region 119906

lowast toaccurate region 119906 through defuzzification block [14 15]

In general rule bases or knowledge bases contain descrip-tions of the linguistic variables of the FLC that are predefinedfor each of the input and output variables Therefore weintroduce the following linguistic variables

1198901= (ldquoDifference in the number of vehicles that entered and left the control zonerdquo 119885

1198901

1198641)

1198902= (ldquoQueue lengthsrdquo 119885

1198902

1198642)

119906 = (ldquoControlrdquo 119885119906 119880)

(19)

where 119885119890119894

= 1198851

119890119894

1198852

119890119894

119885119896

119890119894

119894 = 1 2 and 119885119906= 119885

1

119906 119885

2

119906

119885119896

119906 are the term-sets of linguistic variables 119890

1 119890

2 and

119906 with corresponding membership functions (MF) 119885119897119890119894

=

120583119897

119890119894

(119890119894) 119885119897

119906= 120583

119897(119906) 119897 = 1 119896 given by universal sets 119864

119894=

[119864119894min 119864119894max] and 119880 = [119880min 119880max]

We assume that seven terms 119885119909= ldquoNBrdquo ldquoNMrdquo ldquoNSrdquo

ldquoZErdquo ldquoPSrdquo ldquoPMrdquo ldquoPBrdquo correspond to each of the input andoutput linguistic variables 119885

119909= 119879

1198901

1198791198902

119879119906 with trian-

gular membership functions provided as (10) As a result thefollowing linguistic variables are derived from the fuzzifica-tion processes

1198901= (ldquoDifference of the number of vehicles that entered and left the control zonerdquo 119885

1198901

1198641)

= [

120583NB1198901

(1198901)

NB1198901

120583NM1198901

(1198901)

NM1198901

120583NS1198901

(1198901)

NS1198901

120583ZE1198901

(1198901)

ZE1198901

120583PS1198901

(1198901)

PS1198901

120583PM1198901

(1198901)

PM1198901

120583PB1198901

(1198901)

PB1198901

]

1198902= (ldquoQueue lengthsrdquo 119885

1198902

1198642) = [

120583NB1198902

(1198902)

NB1198902

120583NM1198902

(1198902)

NM1198902

120583NS1198902

(1198902)

NS1198902

120583ZE1198902

(1198902)

ZE1198902

120583PS1198902

(1198902)

PS1198902

120583PM1198902

(1198902)

PM1198902

120583PB1198902

(1198902)

PB1198902

]

119906 = (ldquoControlrdquo 119885119906 119880) = [

120583NB119906(119906)

NB119906

120583NM119906(119906)

NM119906

120583NS119906(119906)

NS119906

120583ZE119906(119906)

ZE119906

120583PS119906(119906)

PS119906

120583PM119906(119906)

PM119906

120583PB119906(119906)

PB119906

]

(20)

6 The Scientific World Journal

Next we develop a rule base for the FLC inferences asfollows

IF (119885119895

1198901

times 119885119895

1198902

)

ELSE 119885119895

119906 119895 = 1 7

(21)

where (1198851198951198901

times119885119895

1198902

) is the Cartesian product of the fuzzy sets 1198641

and 1198642 with the following membership function

120583(119885119895

1198901times119885119895

1198902)(1198901 1198902) = 120583

119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902) (22)

The corresponding output fuzzy set is defined by fuzzyrelation 119877119895 = (119885

119895

1198901

times 119885119895

1198902

) times 119885119895

119906 119895 = 1 7 with the following

membership function

120583119877119895 ((119890

1 1198902) 119906

lowast) = (120583

119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902))

and 120583119885119895

119906

(119906lowast)

(23)

The set of all rules 119877 = ⋃7

119895=1119877119895 with fuzzy membership

function

120583119877((119890

1 1198902) 119906

lowast)

=

7

⋁

119895=1

[(120583119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902)) and 120583

119885119895

119906

(119906lowast)]

(24)

which defines the rule base of the FLC and produces analgorithm of fuzzy control system

As a result for given values of the input linguistic var-iables 119885

119895

1198901

and 119885119895

1198902

the fuzzy output value of the logicalcontroller119885119895

119906can be determined based on the following com-

posite rule [12]

119861119895= (119885

119895

1198901

times 119885119895

1198902

) sdot 119877 (25)

with a degree of membership

120583119885119895

119906

(119906lowast) = ⋁

1198901isin1198641 1198902isin1198642

[(120583119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902))

and 120583119877(1198901 1198902 119906lowast)]

(26)

When the linguistic input variables 1198901and 119890

2correspond

to fuzzy sets11988510158401198901

and11988510158401198902

fuzzy set1198851015840119906of the linguistic variable

of the control signal 119906lowast is determined as follows

1205831198791015840

119906

1015840 (119906lowast) = max

11989011198902

[

119899

prod

119894=1

1205831198791015840

119890119894

(119890119894)]

sdot [

119898

min119895=1

[

119899

prod

119894=1

120583119879119895

119890119894

(119890119894)] sdot 120583

119879119895

119906

(119906lowast)]

(27)

In order to obtain a true output value following fuzzycomputations it is necessary to perform defuzzificationmdashthetransformation of fuzzy linguistic variable value (qualitative)

119906lowast into discrete numerical value of 119906 For this purpose center

of gravity method is applied [12]

119906 =

sum7

119899=1119906lowast

119899120583119885119906

(119906lowast

119899)

sum7

119899=1120583119885119906

(119906lowast119899)

(28)

Membership function of fuzzy values 1198851015840119906can be repre-

sented as

1205831198851015840

119906

1015840 (119906) =

119899

prod

119894=1

120583119885119895

119890119894

(119890119894) 119906 = 120593

119895

0 119906 = 120593119895

(29)

where 120593119895 is the discrete numerical value of the output signalThen the final output value of the FLC at the defuzzifica-

tion stage can be computed as follows

119906 =

sum119898

119895=1120593119895[prod

119899

119894=1120583119885119895

119890119894

(119890119894)]

sum119898

119895=1prod119899

119894=1120583119885119895

119890119894

(119890119894)

or 119906 (119890 120593) =119898

sum

119895=1

120593119895120595119895(119890)

(30)

where

120595119895(119890) =

prod119899

119894=1120583119879119895

119890119894

(119890119894)

sum119898

119895=1prod119899

119894=1120583119879119895

119890119894

(119890119894) (31)

If we consider that the basic equation of the road trafficcontroller can be written as [12]

119906 (119905) = 1199060+ 119870(119890 (119905) +

1

119879119906

int

119905

0

119890 (120591) 119889120591 + 119879119889

119889119890 (119905)

119889119905) (32)

then the rule of the fuzzy traffic control system can berepresented as a logic controller with the varying coefficientas in (14)

119906 (119905) = 1199060+ 119870

lowast(120572 120573) ∘ 120595

lowast

1199060

(119890)lowast (33)

where 119870lowast(120572 120573) is the variable part of the gain that depends

on the current dynamic parameters of the road traffic [16]This allows implementing an adaptive traffic control systemas a fuzzy system consisting of two interconnected modulesldquomodule of fuzzy controller of the switching phase trafficlightsrdquo and ldquomodule of fuzzy adaptive correction of theparameters of the main fuzzy controller of the switchingphase traffic lightsrdquo 120572 and 120574

Furthermore we introduce the following linguistic vari-ables to assess the quality of the control system and adaptivefuzzy correction of the parameters of the fuzzy-logic con-troller119889119890

2

= (ldquoThe Derivative Queue Lengths of Vehiclesrdquo 1198851198891198902

1198641198891198902

)

119876 = (ldquoThe Quality Control Systemrdquo 119885119876 119864

119876)

(34)

The Scientific World Journal 7

120583intQ119896

and 1205833intQ119896

997888rarr (alowastX119894 alowastY119894 b

lowast1X119894

blowast2X119894 blowast1Y119894 b

lowast2Y119894 120573lowast1X119894

120573lowast2X119894 120573lowast1XY119894 120573

lowast2Y119894

)1205833Q119896

120583Q119896

120583Q119896

Qk

Figure 2 Graphic illustration of a choice of optimum adjusting parameters

where 1198851198891198902119894

= 119885(1)

1198891198902

119885(2)

1198891198902

119885(119899)

1198891198902

119885119876119894

= 119885(1)

119876 119890119894

119885(2)

119876

119885(119896)

119876 t are the term-sets of linguistic variables 119889119890

2and119876with

corresponding membership functions

1198851198891198902

=

119889119890(119894)

2

1205831198851198891198902

(119889119890(119894)

2)

119894 = 1 119899

119885119876=

119902(119895)

120583119876(119902(119895))

119895 = 1 119896

(35)

given by universal sets 1198641198891198902

= [1198641198891198902min 119864119889119890

2max] and 119864

119876=

[119864119876min 119864119876max]Then according to (12) and (25) the choice of the optimal

adjustment parameters of the FLC can be represented as thefollowing fuzzy relations

119877119895

119876lowast = (119885

119895

1198902

times 119885119895

1198902

) times 119885119895

119876 119895 = 1119873

119876 (36)

where

119885119876=

119876119879119875

1

120583119879119875

1198761

(119876119879119875

1)

120583119879119875

119876119896

(119876119879119875

119896) = 120593 (119876

119879119875

119896 119886lowast

119876119896

119887lowast

1119876119896

119887lowast

2119876119896

120573lowast

1119876119896

120573lowast

2119876119896

)

120593 (119876119879119875

119896 119886lowast

119876119896

119887lowast

1119876119896

119887lowast

2119876119896

120573lowast

1119876119896

120573lowast

2119876119896

)

= ((119887lowast

1119876119896

(119886lowast

119876119896

minus 119909))

120573lowast

1119876119896

sdot

sign (119887lowast1119876119896

(119886lowast

119876119896

minus 119909)) + 1

2

+ (119887lowast

2119876119896

(119909 minus 119886lowast

119876119896

))

120573lowast

2119876119896

sign (119887lowast2119876119896

(119909 minus 119886lowast

119876119896

)) + 1

2

+ 1)

minus1

119876 = 1199021

120583119876(1199021)

1199022

120583119876(1199022)

119902119896

120583119876(119902119896)

120583119876(119902119896) = 120593 (119902

119896 119886119876119896

1198871119876119896

1198872119876119896

1205731119876119896

1205732119876119896

)

(37)

This fuzzy relationship served as the basis for developingalgorithms of adaptive parameter settings of the fuzzy-logiccontroller (Figure 2)

4 Simulation Modeling and Analysis ofFuzzy Traffic Control System

We develop a simulation model of the adaptive fuzzy controlsystem of road traffic on Matlab software package Thesimulation model is used to test the efficiency of the fuzzycontrol system through series of computational experiments

As an object of simulation we chose one of the heavy-loaded and problematic crossroads in Tashkent city Uzbek-istan The simulation model considered the road trafficintensity observed at the crossroad during the peak-hoursfrom 0700 to 1000 onMay 5 2015The experiment consistedof two phases At each phase of the simulation variouspatterns of traffic-light management were used In the firstphase the traffic lights with fixed cycles which are used inreal life are simulated In the second phase traffic lightscontrolled with adaptive fuzzy system were simulated Forthis purpose the rules of fuzzy control system with a modulefor adaptive adjustment of parameters were synthesized usingthe proposed method

The simulation results are presented below Figure 3shows the results of simulation of the road intersection thatis managed by traffic lights with fixed cycles Figure 4 showsthe results of simulation of the crossroad that is managed byadaptive fuzzy controller

8 The Scientific World Journal

The length of the queue of vehicles in directions 1ndash3

The length of the queue of vehicles in directions 2ndash4

710

720

910

900

820

800

810

920

700

840

850

750

740

730

930

940

950

830

1000

05101520253035404550

2ndash4The intensity of the traffic flow in directions

The intensity of the traffic flow in directions 1ndash3

Figure 3 The results of the simulation modeling of the trafficcontrol system with optimum fixed cycle of traffic lights

1ndash3The length of the queue of vehicles in directions

The length of the queue of vehicles in directions 2ndash4

710

720

910

900

820

800

810

920

700

840

850

750

740

730

930

940

950

830

1000

2ndash4The intensity of the traffic flow in directions

The intensity of the traffic flow in directions 1ndash3

05101520253035404550

Figure 4 The results of the simulation modeling of the adaptivetraffic control system

Simulation results show that the synthesized fuzzy controlsystem successfully solved the problem of queues at trafficlights and performed more effective than the current trafficcontrol system with fixed-cycle traffic lights Particularly thelength of queue for the synthesized fuzzy control system isless than 26 times for traffic control system with fixed-cycletraffic lights Also delay time for synthesized fuzzy controlsystem is 27 less than traffic control system with fixed-cycletraffic lights

Based on the results of computational experiments wecan conclude that the synthesized adaptive fuzzy controlsystem of road traffic in conditions of intensive traffic isrobust and allows controlling road traffic within a wide rangeof parameters

5 Conclusion

In this study we formally described the problems of synthesisof adaptive fuzzy traffic control systems in conditions ofintense road traffic We developed efficient algorithms of thesynthesis of the adaptive fuzzy-logic traffic control systemsWe also developed a simulationmodel of adaptive fuzzy-logiccontrol system for managing road traffic and conducted aseries of computational experiments

The results showed that the synthesized fuzzy-logic traf-fic control system based on the proposed method allowscontrolling crossroads more effectively compared to trafficcontrol system with fixed-cycle traffic lights We conclude

that adaptive fuzzy-logic controller can successfully functionin the condition of information fuzziness It can also success-fully function when exposed to uncontrolled external andparametric impacts The implementation of the solution inheavy-loaded crossroads allows reducing the length of queuesat the traffic lights up to 26 times and the delay time up to27

Competing Interests

The authors declare that they have no competing interests

References

[1] F Sattar F Karray M Kamel L Nassar and K GolestanldquoRecent advances on context-awareness and datainformationfusion in ITSrdquo International Journal of Intelligent TransportationSystems Research vol 14 no 1 pp 1ndash19 2016

[2] J-D Schmocker and M G H Bell ldquoTraffic control currentsystems and future vision of citiesrdquo International Journal ofIntelligent Transportation Systems Research vol 8 no 1 pp 56ndash65 2010

[3] S Sundaram H N Koutsopoulos M Ben-Akiva C Anto-niou and R Balakrishna ldquoSimulation-based dynamic trafficassignment for short-term planning applicationsrdquo SimulationModelling Practice andTheory vol 19 no 1 pp 450ndash462 2011

[4] Dunn Engineering Associates Traffic Control Systems Hand-book Dunn Engineering Associates Westhampton Beach NYUSA 2005

[5] C P Pappis and E H Mamdani ldquoFuzzy logic controller fora traffic junctionrdquo IEEE Transactions on Systems Man andCybernetics vol 7 no 10 pp 707ndash717 1977

[6] Y S Murat and E Gedizlioglu ldquoA fuzzy logic multi-phasedsignal control model for isolated junctionsrdquo TransportationResearch Part C Emerging Technologies vol 13 no 1 pp 19ndash362005

[7] B Yulianto ldquoApplication of fuzzy logic to traffic signal controlundermixed traffic conditionsrdquoTraffic Engineering andControlvol 44 no 9 pp 332ndash336 2003

[8] B Madhavan Nair and J Cai ldquoA fuzzy logic controller forisolated signalized intersection with traffic abnormality consid-eredrdquo in Proceedings of the IEEE Intelligent Vehicles Symposium(IV rsquo07) pp 1229ndash1233 June 2007

[9] V T Zoriktuev S G Ts and J F Mesyagutov ldquoMethodologyrepresented and derivation of knowledge in control systems ofthemechatronicmachine systemsrdquo STIN Systems vol 11 pp 13ndash18 2007

[10] R A Aliev and R R AlievTheTheory of Intelligent Systems andIts Application Chashiogli Baku Azerbaijan 2001

[11] R A Aliev E G Zakharova and S V Ulyanov Fuzzy Modelsof Dynamic Systems Results of Science and Technology SerTechnology Cybernetics 1990

[12] N R Yusupbekov and A R Marakhimov ldquoSynthesis of theintelligent traffic control systems in conditions saturated trans-port streamrdquo International Journal of International Journal ofChemical Technology Control and Management Jointly with theJournal of Korea Multimedia Society vol 3-4 pp 12ndash18 2015

[13] Y Ge ldquoA two-stage fuzzy logic control method of traffic signalbased on traffic urgency degreerdquo Modelling and Simulation inEngineering vol 2014 Article ID 694185 6 pages 2014

The Scientific World Journal 9

[14] I M Makarov and V M Lokhin Eds Intelligent AutomaticControl Systems Fizmatlit Moscow Russia 2001

[15] A Pegat Fuzzy Modeling and Control Knowledge Laboratory2009

[16] RMartinez-Soto O Castillo L T Aguilar and PMelin ldquoFuzzylogic controllers optimization using genetic algorithms andparticle swarm optimizationrdquo in Advances in Soft ComputingG Sidorov A H Aguirre and C A R Garcıa Eds vol 6438of Lecture Notes in Computer Science pp 475ndash486 SpringerBerlin Germany 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 The Scientific World Journal

Next we develop a rule base for the FLC inferences asfollows

IF (119885119895

1198901

times 119885119895

1198902

)

ELSE 119885119895

119906 119895 = 1 7

(21)

where (1198851198951198901

times119885119895

1198902

) is the Cartesian product of the fuzzy sets 1198641

and 1198642 with the following membership function

120583(119885119895

1198901times119885119895

1198902)(1198901 1198902) = 120583

119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902) (22)

The corresponding output fuzzy set is defined by fuzzyrelation 119877119895 = (119885

119895

1198901

times 119885119895

1198902

) times 119885119895

119906 119895 = 1 7 with the following

membership function

120583119877119895 ((119890

1 1198902) 119906

lowast) = (120583

119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902))

and 120583119885119895

119906

(119906lowast)

(23)

The set of all rules 119877 = ⋃7

119895=1119877119895 with fuzzy membership

function

120583119877((119890

1 1198902) 119906

lowast)

=

7

⋁

119895=1

[(120583119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902)) and 120583

119885119895

119906

(119906lowast)]

(24)

which defines the rule base of the FLC and produces analgorithm of fuzzy control system

As a result for given values of the input linguistic var-iables 119885

119895

1198901

and 119885119895

1198902

the fuzzy output value of the logicalcontroller119885119895

119906can be determined based on the following com-

posite rule [12]

119861119895= (119885

119895

1198901

times 119885119895

1198902

) sdot 119877 (25)

with a degree of membership

120583119885119895

119906

(119906lowast) = ⋁

1198901isin1198641 1198902isin1198642

[(120583119885119895

1198901

(1198901) and 120583

119885119895

1198902

(1198902))

and 120583119877(1198901 1198902 119906lowast)]

(26)

When the linguistic input variables 1198901and 119890

2correspond

to fuzzy sets11988510158401198901

and11988510158401198902

fuzzy set1198851015840119906of the linguistic variable

of the control signal 119906lowast is determined as follows

1205831198791015840

119906

1015840 (119906lowast) = max

11989011198902

[

119899

prod

119894=1

1205831198791015840

119890119894

(119890119894)]

sdot [

119898

min119895=1

[

119899

prod

119894=1

120583119879119895

119890119894

(119890119894)] sdot 120583

119879119895

119906

(119906lowast)]

(27)

In order to obtain a true output value following fuzzycomputations it is necessary to perform defuzzificationmdashthetransformation of fuzzy linguistic variable value (qualitative)

119906lowast into discrete numerical value of 119906 For this purpose center

of gravity method is applied [12]

119906 =

sum7

119899=1119906lowast

119899120583119885119906

(119906lowast

119899)

sum7

119899=1120583119885119906

(119906lowast119899)

(28)

Membership function of fuzzy values 1198851015840119906can be repre-

sented as

1205831198851015840

119906

1015840 (119906) =

119899

prod

119894=1

120583119885119895

119890119894

(119890119894) 119906 = 120593

119895

0 119906 = 120593119895

(29)

where 120593119895 is the discrete numerical value of the output signalThen the final output value of the FLC at the defuzzifica-

tion stage can be computed as follows

119906 =

sum119898

119895=1120593119895[prod

119899

119894=1120583119885119895

119890119894

(119890119894)]

sum119898

119895=1prod119899

119894=1120583119885119895

119890119894

(119890119894)

or 119906 (119890 120593) =119898

sum

119895=1

120593119895120595119895(119890)

(30)

where

120595119895(119890) =

prod119899

119894=1120583119879119895

119890119894

(119890119894)

sum119898

119895=1prod119899

119894=1120583119879119895

119890119894

(119890119894) (31)

If we consider that the basic equation of the road trafficcontroller can be written as [12]

119906 (119905) = 1199060+ 119870(119890 (119905) +

1

119879119906

int

119905

0

119890 (120591) 119889120591 + 119879119889

119889119890 (119905)

119889119905) (32)

then the rule of the fuzzy traffic control system can berepresented as a logic controller with the varying coefficientas in (14)

119906 (119905) = 1199060+ 119870

lowast(120572 120573) ∘ 120595

lowast

1199060

(119890)lowast (33)

where 119870lowast(120572 120573) is the variable part of the gain that depends

on the current dynamic parameters of the road traffic [16]This allows implementing an adaptive traffic control systemas a fuzzy system consisting of two interconnected modulesldquomodule of fuzzy controller of the switching phase trafficlightsrdquo and ldquomodule of fuzzy adaptive correction of theparameters of the main fuzzy controller of the switchingphase traffic lightsrdquo 120572 and 120574

Furthermore we introduce the following linguistic vari-ables to assess the quality of the control system and adaptivefuzzy correction of the parameters of the fuzzy-logic con-troller119889119890

2

= (ldquoThe Derivative Queue Lengths of Vehiclesrdquo 1198851198891198902

1198641198891198902

)

119876 = (ldquoThe Quality Control Systemrdquo 119885119876 119864

119876)

(34)

The Scientific World Journal 7

120583intQ119896

and 1205833intQ119896

997888rarr (alowastX119894 alowastY119894 b

lowast1X119894

blowast2X119894 blowast1Y119894 b

lowast2Y119894 120573lowast1X119894

120573lowast2X119894 120573lowast1XY119894 120573

lowast2Y119894

)1205833Q119896

120583Q119896

120583Q119896

Qk

Figure 2 Graphic illustration of a choice of optimum adjusting parameters

where 1198851198891198902119894

= 119885(1)

1198891198902

119885(2)

1198891198902

119885(119899)

1198891198902

119885119876119894

= 119885(1)

119876 119890119894

119885(2)

119876

119885(119896)

119876 t are the term-sets of linguistic variables 119889119890

2and119876with

corresponding membership functions

1198851198891198902

=

119889119890(119894)

2

1205831198851198891198902

(119889119890(119894)

2)

119894 = 1 119899

119885119876=

119902(119895)

120583119876(119902(119895))

119895 = 1 119896

(35)

given by universal sets 1198641198891198902

= [1198641198891198902min 119864119889119890

2max] and 119864

119876=

[119864119876min 119864119876max]Then according to (12) and (25) the choice of the optimal

adjustment parameters of the FLC can be represented as thefollowing fuzzy relations

119877119895

119876lowast = (119885

119895

1198902

times 119885119895

1198902

) times 119885119895

119876 119895 = 1119873

119876 (36)

where

119885119876=

119876119879119875

1

120583119879119875

1198761

(119876119879119875

1)

120583119879119875

119876119896

(119876119879119875

119896) = 120593 (119876

119879119875

119896 119886lowast

119876119896

119887lowast

1119876119896

119887lowast

2119876119896

120573lowast

1119876119896

120573lowast

2119876119896

)

120593 (119876119879119875

119896 119886lowast

119876119896

119887lowast

1119876119896

119887lowast

2119876119896

120573lowast

1119876119896

120573lowast

2119876119896

)

= ((119887lowast

1119876119896

(119886lowast

119876119896

minus 119909))

120573lowast

1119876119896

sdot

sign (119887lowast1119876119896

(119886lowast

119876119896

minus 119909)) + 1

2

+ (119887lowast

2119876119896

(119909 minus 119886lowast

119876119896

))

120573lowast

2119876119896

sign (119887lowast2119876119896

(119909 minus 119886lowast

119876119896

)) + 1

2

+ 1)

minus1

119876 = 1199021

120583119876(1199021)

1199022

120583119876(1199022)

119902119896

120583119876(119902119896)

120583119876(119902119896) = 120593 (119902

119896 119886119876119896

1198871119876119896

1198872119876119896

1205731119876119896

1205732119876119896

)

(37)

This fuzzy relationship served as the basis for developingalgorithms of adaptive parameter settings of the fuzzy-logiccontroller (Figure 2)

4 Simulation Modeling and Analysis ofFuzzy Traffic Control System

We develop a simulation model of the adaptive fuzzy controlsystem of road traffic on Matlab software package Thesimulation model is used to test the efficiency of the fuzzycontrol system through series of computational experiments

As an object of simulation we chose one of the heavy-loaded and problematic crossroads in Tashkent city Uzbek-istan The simulation model considered the road trafficintensity observed at the crossroad during the peak-hoursfrom 0700 to 1000 onMay 5 2015The experiment consistedof two phases At each phase of the simulation variouspatterns of traffic-light management were used In the firstphase the traffic lights with fixed cycles which are used inreal life are simulated In the second phase traffic lightscontrolled with adaptive fuzzy system were simulated Forthis purpose the rules of fuzzy control system with a modulefor adaptive adjustment of parameters were synthesized usingthe proposed method

The simulation results are presented below Figure 3shows the results of simulation of the road intersection thatis managed by traffic lights with fixed cycles Figure 4 showsthe results of simulation of the crossroad that is managed byadaptive fuzzy controller

8 The Scientific World Journal

The length of the queue of vehicles in directions 1ndash3

The length of the queue of vehicles in directions 2ndash4

710

720

910

900

820

800

810

920

700

840

850

750

740

730

930

940

950

830

1000

05101520253035404550

2ndash4The intensity of the traffic flow in directions

The intensity of the traffic flow in directions 1ndash3

Figure 3 The results of the simulation modeling of the trafficcontrol system with optimum fixed cycle of traffic lights

1ndash3The length of the queue of vehicles in directions

The length of the queue of vehicles in directions 2ndash4

710

720

910

900

820

800

810

920

700

840

850

750

740

730

930

940

950

830

1000

2ndash4The intensity of the traffic flow in directions

The intensity of the traffic flow in directions 1ndash3

05101520253035404550

Figure 4 The results of the simulation modeling of the adaptivetraffic control system

Simulation results show that the synthesized fuzzy controlsystem successfully solved the problem of queues at trafficlights and performed more effective than the current trafficcontrol system with fixed-cycle traffic lights Particularly thelength of queue for the synthesized fuzzy control system isless than 26 times for traffic control system with fixed-cycletraffic lights Also delay time for synthesized fuzzy controlsystem is 27 less than traffic control system with fixed-cycletraffic lights

Based on the results of computational experiments wecan conclude that the synthesized adaptive fuzzy controlsystem of road traffic in conditions of intensive traffic isrobust and allows controlling road traffic within a wide rangeof parameters

5 Conclusion

In this study we formally described the problems of synthesisof adaptive fuzzy traffic control systems in conditions ofintense road traffic We developed efficient algorithms of thesynthesis of the adaptive fuzzy-logic traffic control systemsWe also developed a simulationmodel of adaptive fuzzy-logiccontrol system for managing road traffic and conducted aseries of computational experiments

The results showed that the synthesized fuzzy-logic traf-fic control system based on the proposed method allowscontrolling crossroads more effectively compared to trafficcontrol system with fixed-cycle traffic lights We conclude

that adaptive fuzzy-logic controller can successfully functionin the condition of information fuzziness It can also success-fully function when exposed to uncontrolled external andparametric impacts The implementation of the solution inheavy-loaded crossroads allows reducing the length of queuesat the traffic lights up to 26 times and the delay time up to27

Competing Interests

The authors declare that they have no competing interests

References

[1] F Sattar F Karray M Kamel L Nassar and K GolestanldquoRecent advances on context-awareness and datainformationfusion in ITSrdquo International Journal of Intelligent TransportationSystems Research vol 14 no 1 pp 1ndash19 2016

[2] J-D Schmocker and M G H Bell ldquoTraffic control currentsystems and future vision of citiesrdquo International Journal ofIntelligent Transportation Systems Research vol 8 no 1 pp 56ndash65 2010

[3] S Sundaram H N Koutsopoulos M Ben-Akiva C Anto-niou and R Balakrishna ldquoSimulation-based dynamic trafficassignment for short-term planning applicationsrdquo SimulationModelling Practice andTheory vol 19 no 1 pp 450ndash462 2011

[4] Dunn Engineering Associates Traffic Control Systems Hand-book Dunn Engineering Associates Westhampton Beach NYUSA 2005

[5] C P Pappis and E H Mamdani ldquoFuzzy logic controller fora traffic junctionrdquo IEEE Transactions on Systems Man andCybernetics vol 7 no 10 pp 707ndash717 1977

[6] Y S Murat and E Gedizlioglu ldquoA fuzzy logic multi-phasedsignal control model for isolated junctionsrdquo TransportationResearch Part C Emerging Technologies vol 13 no 1 pp 19ndash362005

[7] B Yulianto ldquoApplication of fuzzy logic to traffic signal controlundermixed traffic conditionsrdquoTraffic Engineering andControlvol 44 no 9 pp 332ndash336 2003

[8] B Madhavan Nair and J Cai ldquoA fuzzy logic controller forisolated signalized intersection with traffic abnormality consid-eredrdquo in Proceedings of the IEEE Intelligent Vehicles Symposium(IV rsquo07) pp 1229ndash1233 June 2007

[9] V T Zoriktuev S G Ts and J F Mesyagutov ldquoMethodologyrepresented and derivation of knowledge in control systems ofthemechatronicmachine systemsrdquo STIN Systems vol 11 pp 13ndash18 2007

[10] R A Aliev and R R AlievTheTheory of Intelligent Systems andIts Application Chashiogli Baku Azerbaijan 2001

[11] R A Aliev E G Zakharova and S V Ulyanov Fuzzy Modelsof Dynamic Systems Results of Science and Technology SerTechnology Cybernetics 1990

[12] N R Yusupbekov and A R Marakhimov ldquoSynthesis of theintelligent traffic control systems in conditions saturated trans-port streamrdquo International Journal of International Journal ofChemical Technology Control and Management Jointly with theJournal of Korea Multimedia Society vol 3-4 pp 12ndash18 2015

[13] Y Ge ldquoA two-stage fuzzy logic control method of traffic signalbased on traffic urgency degreerdquo Modelling and Simulation inEngineering vol 2014 Article ID 694185 6 pages 2014

The Scientific World Journal 9

[14] I M Makarov and V M Lokhin Eds Intelligent AutomaticControl Systems Fizmatlit Moscow Russia 2001

[15] A Pegat Fuzzy Modeling and Control Knowledge Laboratory2009

[16] RMartinez-Soto O Castillo L T Aguilar and PMelin ldquoFuzzylogic controllers optimization using genetic algorithms andparticle swarm optimizationrdquo in Advances in Soft ComputingG Sidorov A H Aguirre and C A R Garcıa Eds vol 6438of Lecture Notes in Computer Science pp 475ndash486 SpringerBerlin Germany 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 7

120583intQ119896

and 1205833intQ119896

997888rarr (alowastX119894 alowastY119894 b

lowast1X119894

blowast2X119894 blowast1Y119894 b

lowast2Y119894 120573lowast1X119894

120573lowast2X119894 120573lowast1XY119894 120573

lowast2Y119894

)1205833Q119896

120583Q119896

120583Q119896

Qk

Figure 2 Graphic illustration of a choice of optimum adjusting parameters

where 1198851198891198902119894

= 119885(1)

1198891198902

119885(2)

1198891198902

119885(119899)

1198891198902

119885119876119894

= 119885(1)

119876 119890119894

119885(2)

119876

119885(119896)

119876 t are the term-sets of linguistic variables 119889119890

2and119876with

corresponding membership functions

1198851198891198902

=

119889119890(119894)

2

1205831198851198891198902

(119889119890(119894)

2)

119894 = 1 119899

119885119876=

119902(119895)

120583119876(119902(119895))

119895 = 1 119896

(35)

given by universal sets 1198641198891198902

= [1198641198891198902min 119864119889119890

2max] and 119864

119876=

[119864119876min 119864119876max]Then according to (12) and (25) the choice of the optimal

adjustment parameters of the FLC can be represented as thefollowing fuzzy relations

119877119895

119876lowast = (119885

119895

1198902

times 119885119895

1198902

) times 119885119895

119876 119895 = 1119873

119876 (36)

where

119885119876=

119876119879119875

1

120583119879119875

1198761

(119876119879119875

1)

120583119879119875

119876119896

(119876119879119875

119896) = 120593 (119876

119879119875

119896 119886lowast

119876119896

119887lowast

1119876119896

119887lowast

2119876119896

120573lowast

1119876119896

120573lowast

2119876119896

)

120593 (119876119879119875

119896 119886lowast

119876119896

119887lowast

1119876119896

119887lowast

2119876119896

120573lowast

1119876119896

120573lowast

2119876119896

)

= ((119887lowast

1119876119896

(119886lowast

119876119896

minus 119909))

120573lowast

1119876119896

sdot

sign (119887lowast1119876119896

(119886lowast

119876119896

minus 119909)) + 1

2

+ (119887lowast

2119876119896

(119909 minus 119886lowast

119876119896

))

120573lowast

2119876119896

sign (119887lowast2119876119896

(119909 minus 119886lowast

119876119896

)) + 1

2

+ 1)

minus1

119876 = 1199021

120583119876(1199021)

1199022

120583119876(1199022)

119902119896

120583119876(119902119896)

120583119876(119902119896) = 120593 (119902

119896 119886119876119896

1198871119876119896

1198872119876119896

1205731119876119896

1205732119876119896

)

(37)

This fuzzy relationship served as the basis for developingalgorithms of adaptive parameter settings of the fuzzy-logiccontroller (Figure 2)

4 Simulation Modeling and Analysis ofFuzzy Traffic Control System

We develop a simulation model of the adaptive fuzzy controlsystem of road traffic on Matlab software package Thesimulation model is used to test the efficiency of the fuzzycontrol system through series of computational experiments

As an object of simulation we chose one of the heavy-loaded and problematic crossroads in Tashkent city Uzbek-istan The simulation model considered the road trafficintensity observed at the crossroad during the peak-hoursfrom 0700 to 1000 onMay 5 2015The experiment consistedof two phases At each phase of the simulation variouspatterns of traffic-light management were used In the firstphase the traffic lights with fixed cycles which are used inreal life are simulated In the second phase traffic lightscontrolled with adaptive fuzzy system were simulated Forthis purpose the rules of fuzzy control system with a modulefor adaptive adjustment of parameters were synthesized usingthe proposed method

The simulation results are presented below Figure 3shows the results of simulation of the road intersection thatis managed by traffic lights with fixed cycles Figure 4 showsthe results of simulation of the crossroad that is managed byadaptive fuzzy controller

8 The Scientific World Journal

The length of the queue of vehicles in directions 1ndash3

The length of the queue of vehicles in directions 2ndash4

710

720

910

900

820

800

810

920

700

840

850

750

740

730

930

940

950

830

1000

05101520253035404550

2ndash4The intensity of the traffic flow in directions

The intensity of the traffic flow in directions 1ndash3

Figure 3 The results of the simulation modeling of the trafficcontrol system with optimum fixed cycle of traffic lights

1ndash3The length of the queue of vehicles in directions

The length of the queue of vehicles in directions 2ndash4

710

720

910

900

820

800

810

920

700

840

850

750

740

730

930

940

950

830

1000

2ndash4The intensity of the traffic flow in directions

The intensity of the traffic flow in directions 1ndash3

05101520253035404550

Figure 4 The results of the simulation modeling of the adaptivetraffic control system

Simulation results show that the synthesized fuzzy controlsystem successfully solved the problem of queues at trafficlights and performed more effective than the current trafficcontrol system with fixed-cycle traffic lights Particularly thelength of queue for the synthesized fuzzy control system isless than 26 times for traffic control system with fixed-cycletraffic lights Also delay time for synthesized fuzzy controlsystem is 27 less than traffic control system with fixed-cycletraffic lights

Based on the results of computational experiments wecan conclude that the synthesized adaptive fuzzy controlsystem of road traffic in conditions of intensive traffic isrobust and allows controlling road traffic within a wide rangeof parameters

5 Conclusion

In this study we formally described the problems of synthesisof adaptive fuzzy traffic control systems in conditions ofintense road traffic We developed efficient algorithms of thesynthesis of the adaptive fuzzy-logic traffic control systemsWe also developed a simulationmodel of adaptive fuzzy-logiccontrol system for managing road traffic and conducted aseries of computational experiments

The results showed that the synthesized fuzzy-logic traf-fic control system based on the proposed method allowscontrolling crossroads more effectively compared to trafficcontrol system with fixed-cycle traffic lights We conclude

that adaptive fuzzy-logic controller can successfully functionin the condition of information fuzziness It can also success-fully function when exposed to uncontrolled external andparametric impacts The implementation of the solution inheavy-loaded crossroads allows reducing the length of queuesat the traffic lights up to 26 times and the delay time up to27

Competing Interests

The authors declare that they have no competing interests

References

[1] F Sattar F Karray M Kamel L Nassar and K GolestanldquoRecent advances on context-awareness and datainformationfusion in ITSrdquo International Journal of Intelligent TransportationSystems Research vol 14 no 1 pp 1ndash19 2016

[2] J-D Schmocker and M G H Bell ldquoTraffic control currentsystems and future vision of citiesrdquo International Journal ofIntelligent Transportation Systems Research vol 8 no 1 pp 56ndash65 2010

[3] S Sundaram H N Koutsopoulos M Ben-Akiva C Anto-niou and R Balakrishna ldquoSimulation-based dynamic trafficassignment for short-term planning applicationsrdquo SimulationModelling Practice andTheory vol 19 no 1 pp 450ndash462 2011

[4] Dunn Engineering Associates Traffic Control Systems Hand-book Dunn Engineering Associates Westhampton Beach NYUSA 2005

[5] C P Pappis and E H Mamdani ldquoFuzzy logic controller fora traffic junctionrdquo IEEE Transactions on Systems Man andCybernetics vol 7 no 10 pp 707ndash717 1977

[6] Y S Murat and E Gedizlioglu ldquoA fuzzy logic multi-phasedsignal control model for isolated junctionsrdquo TransportationResearch Part C Emerging Technologies vol 13 no 1 pp 19ndash362005

[7] B Yulianto ldquoApplication of fuzzy logic to traffic signal controlundermixed traffic conditionsrdquoTraffic Engineering andControlvol 44 no 9 pp 332ndash336 2003

[8] B Madhavan Nair and J Cai ldquoA fuzzy logic controller forisolated signalized intersection with traffic abnormality consid-eredrdquo in Proceedings of the IEEE Intelligent Vehicles Symposium(IV rsquo07) pp 1229ndash1233 June 2007

[9] V T Zoriktuev S G Ts and J F Mesyagutov ldquoMethodologyrepresented and derivation of knowledge in control systems ofthemechatronicmachine systemsrdquo STIN Systems vol 11 pp 13ndash18 2007

[10] R A Aliev and R R AlievTheTheory of Intelligent Systems andIts Application Chashiogli Baku Azerbaijan 2001

[11] R A Aliev E G Zakharova and S V Ulyanov Fuzzy Modelsof Dynamic Systems Results of Science and Technology SerTechnology Cybernetics 1990

[12] N R Yusupbekov and A R Marakhimov ldquoSynthesis of theintelligent traffic control systems in conditions saturated trans-port streamrdquo International Journal of International Journal ofChemical Technology Control and Management Jointly with theJournal of Korea Multimedia Society vol 3-4 pp 12ndash18 2015

[13] Y Ge ldquoA two-stage fuzzy logic control method of traffic signalbased on traffic urgency degreerdquo Modelling and Simulation inEngineering vol 2014 Article ID 694185 6 pages 2014

The Scientific World Journal 9

[14] I M Makarov and V M Lokhin Eds Intelligent AutomaticControl Systems Fizmatlit Moscow Russia 2001

[15] A Pegat Fuzzy Modeling and Control Knowledge Laboratory2009

[16] RMartinez-Soto O Castillo L T Aguilar and PMelin ldquoFuzzylogic controllers optimization using genetic algorithms andparticle swarm optimizationrdquo in Advances in Soft ComputingG Sidorov A H Aguirre and C A R Garcıa Eds vol 6438of Lecture Notes in Computer Science pp 475ndash486 SpringerBerlin Germany 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 The Scientific World Journal

The length of the queue of vehicles in directions 1ndash3

The length of the queue of vehicles in directions 2ndash4

710

720

910

900

820

800

810

920

700

840

850

750

740

730

930

940

950

830

1000

05101520253035404550

2ndash4The intensity of the traffic flow in directions

The intensity of the traffic flow in directions 1ndash3

Figure 3 The results of the simulation modeling of the trafficcontrol system with optimum fixed cycle of traffic lights

1ndash3The length of the queue of vehicles in directions

The length of the queue of vehicles in directions 2ndash4

710

720

910

900

820

800

810

920

700

840

850

750

740

730

930

940

950

830

1000

2ndash4The intensity of the traffic flow in directions

The intensity of the traffic flow in directions 1ndash3

05101520253035404550

Figure 4 The results of the simulation modeling of the adaptivetraffic control system

Simulation results show that the synthesized fuzzy controlsystem successfully solved the problem of queues at trafficlights and performed more effective than the current trafficcontrol system with fixed-cycle traffic lights Particularly thelength of queue for the synthesized fuzzy control system isless than 26 times for traffic control system with fixed-cycletraffic lights Also delay time for synthesized fuzzy controlsystem is 27 less than traffic control system with fixed-cycletraffic lights

Based on the results of computational experiments wecan conclude that the synthesized adaptive fuzzy controlsystem of road traffic in conditions of intensive traffic isrobust and allows controlling road traffic within a wide rangeof parameters

5 Conclusion

In this study we formally described the problems of synthesisof adaptive fuzzy traffic control systems in conditions ofintense road traffic We developed efficient algorithms of thesynthesis of the adaptive fuzzy-logic traffic control systemsWe also developed a simulationmodel of adaptive fuzzy-logiccontrol system for managing road traffic and conducted aseries of computational experiments

The results showed that the synthesized fuzzy-logic traf-fic control system based on the proposed method allowscontrolling crossroads more effectively compared to trafficcontrol system with fixed-cycle traffic lights We conclude

that adaptive fuzzy-logic controller can successfully functionin the condition of information fuzziness It can also success-fully function when exposed to uncontrolled external andparametric impacts The implementation of the solution inheavy-loaded crossroads allows reducing the length of queuesat the traffic lights up to 26 times and the delay time up to27

Competing Interests

The authors declare that they have no competing interests

References

[1] F Sattar F Karray M Kamel L Nassar and K GolestanldquoRecent advances on context-awareness and datainformationfusion in ITSrdquo International Journal of Intelligent TransportationSystems Research vol 14 no 1 pp 1ndash19 2016

[2] J-D Schmocker and M G H Bell ldquoTraffic control currentsystems and future vision of citiesrdquo International Journal ofIntelligent Transportation Systems Research vol 8 no 1 pp 56ndash65 2010

[3] S Sundaram H N Koutsopoulos M Ben-Akiva C Anto-niou and R Balakrishna ldquoSimulation-based dynamic trafficassignment for short-term planning applicationsrdquo SimulationModelling Practice andTheory vol 19 no 1 pp 450ndash462 2011

[4] Dunn Engineering Associates Traffic Control Systems Hand-book Dunn Engineering Associates Westhampton Beach NYUSA 2005

[5] C P Pappis and E H Mamdani ldquoFuzzy logic controller fora traffic junctionrdquo IEEE Transactions on Systems Man andCybernetics vol 7 no 10 pp 707ndash717 1977

[6] Y S Murat and E Gedizlioglu ldquoA fuzzy logic multi-phasedsignal control model for isolated junctionsrdquo TransportationResearch Part C Emerging Technologies vol 13 no 1 pp 19ndash362005

[7] B Yulianto ldquoApplication of fuzzy logic to traffic signal controlundermixed traffic conditionsrdquoTraffic Engineering andControlvol 44 no 9 pp 332ndash336 2003

[8] B Madhavan Nair and J Cai ldquoA fuzzy logic controller forisolated signalized intersection with traffic abnormality consid-eredrdquo in Proceedings of the IEEE Intelligent Vehicles Symposium(IV rsquo07) pp 1229ndash1233 June 2007

[9] V T Zoriktuev S G Ts and J F Mesyagutov ldquoMethodologyrepresented and derivation of knowledge in control systems ofthemechatronicmachine systemsrdquo STIN Systems vol 11 pp 13ndash18 2007

[10] R A Aliev and R R AlievTheTheory of Intelligent Systems andIts Application Chashiogli Baku Azerbaijan 2001

[11] R A Aliev E G Zakharova and S V Ulyanov Fuzzy Modelsof Dynamic Systems Results of Science and Technology SerTechnology Cybernetics 1990

[12] N R Yusupbekov and A R Marakhimov ldquoSynthesis of theintelligent traffic control systems in conditions saturated trans-port streamrdquo International Journal of International Journal ofChemical Technology Control and Management Jointly with theJournal of Korea Multimedia Society vol 3-4 pp 12ndash18 2015

[13] Y Ge ldquoA two-stage fuzzy logic control method of traffic signalbased on traffic urgency degreerdquo Modelling and Simulation inEngineering vol 2014 Article ID 694185 6 pages 2014

The Scientific World Journal 9

[14] I M Makarov and V M Lokhin Eds Intelligent AutomaticControl Systems Fizmatlit Moscow Russia 2001

[15] A Pegat Fuzzy Modeling and Control Knowledge Laboratory2009

[16] RMartinez-Soto O Castillo L T Aguilar and PMelin ldquoFuzzylogic controllers optimization using genetic algorithms andparticle swarm optimizationrdquo in Advances in Soft ComputingG Sidorov A H Aguirre and C A R Garcıa Eds vol 6438of Lecture Notes in Computer Science pp 475ndash486 SpringerBerlin Germany 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 9

[14] I M Makarov and V M Lokhin Eds Intelligent AutomaticControl Systems Fizmatlit Moscow Russia 2001

[15] A Pegat Fuzzy Modeling and Control Knowledge Laboratory2009

[16] RMartinez-Soto O Castillo L T Aguilar and PMelin ldquoFuzzylogic controllers optimization using genetic algorithms andparticle swarm optimizationrdquo in Advances in Soft ComputingG Sidorov A H Aguirre and C A R Garcıa Eds vol 6438of Lecture Notes in Computer Science pp 475ndash486 SpringerBerlin Germany 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of