A SOFT COMPUTING METHOD FOR EFFICIENT MODELLING OF SMART CITIES NOISE POLLUTION

Transcription

1 Iterdiscipliary Descriptio of Complex Systems 16(3-A), , 2018 A SOFT COMPUTING METHOD FOR EFFICIENT MODELLING OF SMART CITIES NOISE POLLUTION Attila Nemes 1, *, Gyula Mester 1 ad Tibor Mester 2 1 Óbuda Uiversity, Doctoral School o Safety ad Security Scieces 1 Budapest, Hugary 2 Geomat-Algotech 2 Budapest, Hugary DOI: /idecs Regular article Received: 23 rd Jauary Accepted: 31 st August ABSTRACT Noise pollutio is oe of the most relevat problems i urba area. The mai source of oise pollutio is the umber ad type of motor vehicles, but other parameters depedig o street cofiguratio yield to a system hardly to be exactly modelled by classical mathematical methods. Smart cities are expected to dyamically cotrol the urba traffic to reduce ot just traffic jams, but also to esure a comfortable oise level for ihabitats. This article gives a desig method for efficiet geetic fuzzy modellig of traffic geerated smart cities oise pollutio based o fuzzy logic, multi objective geetic algorithm, gradiet descet optimisatio ad sigular value decompositio i the MATLAB eviromet. Geetic algorithms with objectives to miimise the maximum absolute idetificatio error, the root mea square of the idetificatio error, reduce model complexity ad esure maximal umerical robustess are applied to Zadeh type fuzzy partitio membership fuctio parameters prelimiary idetificatio, ad the gradiet descet method is used for their fie-tuig optimizatio, while the fuzzy rule cosequece liear parameters are calculated by sigular value decompositio method to fid the least squares optimal traiig data fittig of the model. The traiig data set is built from measured data, combied with carefully selected simulatio data to esure the completeess of the model ad its umerical robustess. Detailed aalysis of the method ad results by computer simulatio of the idetificatio process show the validity of the proposed method. KEY WORDS oise pollutio, mathematical modellig, fuzzy logic, sigular value decompositio, geetic algorithm CLASSIFICATION JEL: Q53, R41 *Correspodig author, : Mr.Attila.Nemes@gmail.com; ; *

2 A soft computig method for efficiet modellig of smart cities oise pollutio INTRODUCTION Noise pollutio is a ucomfortable soud level obstructig the quality of life, typical to idustrial road side areas with high traffic. Traffic jams ad heavy traffic are highly aoyig ad obstructig ot just because of time loss ad icreased egie exhaust air pollutio effects, but also because of oise pollutio. Itelliget, flexible traffic cotrol is oe of the popular subjects of smart city research. Solutios takig care of the life quality of residet people alog roads do cosider the oise levels. There are may passive costructio desig solutios to reduce the traffic oise startig from buildig protective walls, chagig the road surface or platig appropriate vegetatio betwee the road ad houses [1]. The bottom lie is that the traffic ose primarily depeds o two hard to chage factors the street width ad the house heights defiig the geometry, the air volume which is subject to the third obvious factor, the oise source level itself, which is produced by the umber passig vehicles. So a smart city traffic cotrol system balacig the ose pollutio has to primarily relay o balacig the umber of vehicles passig by each fixed cofiguratio street. The oise level depeds ot oly o the umber of vehicles, but also o their type. There are multiple levels of differetiatios; the most basic is: cars, motorcycles ad heavy vehicles [2]. This paper presets a desig method for efficiet geetic fuzzy modellig of the traffic geerated urba oise pollutio. Fuzzy systems are kow to be traiable for uiversal oliear fuctio approximatios eve for as complex problems as multi-rotor copter flight dyamics modellig [3]. By usig Zadeh-type fuzzy partitios for the atecedet part we ca guaratee the cotiuity of the fuzzy system output for the complete iput space [4]. Geetic algorithms are kow powerful tools for global oliear search, thus suitable for efficiet prelimiary idetificatio of fuzzy membership parameters [5] ad also capable of fuzzy structure optimisatio [6]. We use a alterative multi-objective vector compariso i a domiace based rakig method scheme [7], sice the two essetial quality idicators of ay fuctio approximatio the maximum absolute error ad the root mea square error are idepedet properties usually competig with the model complexity, while all are to be simultaeously miimised. Oce the proximity of the global optimum of membership parameters is foud, we ca rely o fast gradiet descet methods to pipoit the exact optimal values of membership parameters that miimise the approximatio error. Usig a appropriate represetatio of the Takagi-Sugeo-Kag type fuzzy logic system the calculatio of fuzzy rule cosequet liear free parameters becomes a high dimesioal, but still a mere liear equatio solvig problem, which we ca solve by a umerically robust, least square error optimal sigular value decompositio method [8]. Aalysig the sigular values of fuzzy rule parameters we ca also evaluate the quality of the used traiig data relative to the plaed atecedet membership fuctio fuzzy-cluster complexity, more eve we ca use this iformatio to select oly importat samples from the traiig data, thus reducig its size, while maitaiig the umerical quality of the solutio [9]. As the (for us) available measured data is by far isufficiet for creatig a uiversal oise level model [10], we iclude to our traiig set results by other models to obtai a well-defied robust complete fuzzy system of cotiuous output, o worse tha the curret kow models, a also havig the potetial for seamless further quality improvemets by traiig for ay further real life measured data or ay ew model data. NOISE POLLUTION MODELLING Noise pollutio soud pressure systems have very complex structure ad therefore it is very hard to idetify. Previous efforts [1, 2] have bee devoted to determiig liear 303

3 A. Nemes, Gy. Mester ad T. Mester represetatios ad liear combiatios of the system parameter logarithmic values i the followig geeral form: L AeqT = α 0 + α 1 eq + α 2 X, (1) where L AeqT is the equivalet cotiuous soud level i decibel (db), α i deotes idetificatio parameters, eq is the umber of or the logarithm of the umber of equivalet vehicles per hour, X is a fuctio of the observer distace from the oise source or W that is either the width or the logarithm of the width of the road, advaced models also accout for H, the average height or the logarithm of the height of buildigs i the cosidered road sectio, which is resposible for the sheer air volume where the oise is distributed, ad also the refractio surface, further o some models also iclude dampeig ad amplifyig modifiers as the road surface quality, the elevatio of the road, distace from juctios ad other vehicle velocity factors. The equivalet umber of vehicles eq, the most importat factor for the oise level ca be i geeral defied as: eq = c + β 1 mc + β 2 hv, (2) where c is the umber of passig cars, mc of motorcycles, hv of heavy vehicles, ad c i are multiplicatio coefficiets usually β 1 is betwee 2 ad 4, β 2 is betwee 8 ad 16. Such liear models are ofte used because they are easy to implemet, albeit they allow oly a approximate modellig of the real urba traffic oise pollutio. For more accurate results i modellig real oliear systems it is advised to use oliear models [11-13]. NOISE MEASUREMENT The set of measuremets is performed o various street roads of a typical medium-size orther Serbia tow of Subotica. The suitable method for determiig the oise pollutio is the measuremet of the effective equivalet A-weighted eergy-average soud pressure level, with the followig approximatio: T L AeqT = 10log ( 1 T 10L i/10 i=1 ), (3) where T is the observatio time, the umber of soud level samples to be accumulated, L i is the A-weighted soud level i db(a), defied i the soud level meter stadards IEC 60651, IEC 60804, IEC 61672, ANSI S1.4. The umber of cars, motorcycles ad heavy vehicles per miute was visually moitored i perso, ad maually couted. FUNCTION IDENTIFICATION BY ZADEH-TYPE FUZZY PARTITIONS For the oise pollutio model idetificatio method we use fuzzy logic systems (FLSs) with Zadeh-formed membership fuctios (MFs). More exactly we use Takagi-Sugeo-Kag (TSK) type FLSs havig =3 iputs ad 1 output. These FLSs ca be expressed as: M M f(x) = l=1 ω l (x) y l (x) / l=1 ω l (x), (4) where M is the umber of fuzzy rules, x is the vector of iput variables, y l is a scalar fuctio of iput variables, ad y is a liear fuctio of iputs for the first order TSKs. Thus y l is defied by + 1 parameters, respectively. The atecedet, the premise part of a fuzzy rule is defied by MFs as: ω l (x) = i=1 μ Fl (i)(x i ), (5) where μ Fl (i)(x i ) is the membership fuctio of the i th iput variable i the l th rule that defies the liguistic value F l(i). The huma readable, ituitive liguistic form of the l th rule from the previously described first order TSK FLS is [14]: 304

4 A soft computig method for efficiet modellig of smart cities oise pollutio IF (x 1 is F l(1) ) AND (x 2 is F l(2) ) AND (x is F l() ) THEN y l = j=1 c l(j) x j + c l(0). (6) We ca use fuzzy partitios defied by Zadeh-formed MFs. For b parameters b 1 b 2 b 3 b 4 defiig the MFs, which are o-liear secod order polyomial, so called Z-, S-, ad π-fuctios, amed after their shape, defied respectively as: 1 x b 1 mfz(x, b 1, b 2 ) = 1 2((x b 1 )/(b 2 b 1 )) 2 b 1 < x ½(b 2 + b 1 ) 2((b 2 x)/(b 2 b 1 )) 2 ½(b 2 + b 1 ) < x b 2 { 0 x > b 2 mfs(x, b 1, b 2 ) = 1 mfz(x, b 1, b 2 ) mfπ(x, b 1, b 2, b 3, b 4 ) = { mfs(x, b 1, b 2 ) 1 x b 2 b 2 < x b 3 mfz(x, b 3, b 4 ) x > b 3. For setups whe more tha oe value x exists, such that the degree of membership of x is equal to oe, the iterval where the μ k (x, b) = 1 (the iterval [b 2, b 3 ] for mfπ type μ k ) is the so-called plateau of the μ k MF. Whe havig for example three aturally ordered liguistic values l {α, β, γ} (for example α = low, β = medium, γ = large ) there are hard costraits o b i parameters to preserve the atural liguistic orderig: b a1 < b b1 < b c1 b a2 b a3 < b b2 b b3 < b c2 b c3 b a4 < b b4 < b c4. Whe a liguistic variable ca be assiged K differet liguistic values, each described by a MF μ k (x, b) such that for every iput x it holds that K k=1 μ (x, b) = 1, the MFs are said to k form a fuzzy-partitio. By imposig restrictios (8) o all liguistic variables of the FLS, ad assumig the rule base is complete it covers the whole iput domai it follows that the TSK model structure (4) for a multi-iput x vector simplifies to [15]: f(x) = M l=1 ω l (x, b l ) y l (x, c l ) = W(x, b) c, (9) where c is the compoud vector of all the so called liear c l ad b is the compoud vector of o-liear b l parameters (parameters defiig the liear ad o-liear sub-fuctios) of the fuzzy system ad W is the appropriate compoud coefficiet matrix such that equatio (9) holds. This way the resultig formula is simple, still guaratyig uiform iput coverage ad cotiuous system output TSK FLSs of fuzzy partitios ca eve be made periodic [3, 18]. MULTI-OBJECTIVE GENETIC ALGORITHMS A geetic algorithm (GA) is costructed o bases of imitatig atural biological processes ad atural Darwiia evolutio. GAs are widely used as a search ad optimisatio tool [16]. Real-life optimisatio problems ofte have multiple objectives. The compariso of two vectors i this case is ot trivial. Multiobjective optimisatio ca be defied as the problem of fidig a vector of decisio variables which satisfies costraits ad optimises a vector fuctio whose elemets represet the objective fuctios. These fuctios form a mathematical descriptio of performace criteria that are usually i coflict with each other. Hece, the term optimise meas fidig such a solutio that would give the values of all the objective fuctios acceptable to the desiger [17]. Itroduced i [5] ad elaborated i [18] the idea behid the defiitio of a ew vector compariso algorithm, amed quality-domiace is to exted the Pareto-domiace relatio i a way that a domiatio decisio could be also made for vectors, which are ot comparable (7) (8) 305

5 A. Nemes, Gy. Mester ad T. Mester by Pareto domiace, while a huma heuristic would ame a clear preferece. Defiig a domiace relatio < q (r, s) (or briefly g < q s) betwee two vectors of elemets r = (r i ) ad s = (s i ), for i=1.., N +, where each i th elemet type has a well-defied scalar < (less tha) strict partial order biary edorelatio ad also the equivalece relatio = is defied. Defiig a helper fuctio # q< (r, s), which for vectors r ad s defies two values (g r, l r ) = # q< (r, s), where g r, l r N 0 ad g r is equal to the cardiality of set G rs ={ r i s i < r i }, i=1..; ad l r is equal to the cardiality of set L rs ={ r j r j < s j }, j=1... For a miimisatio problem vector r quality-domiates vector s, or briefly: r < q s if g r < l r or i case of g r = l r r qualitydomiates vector s if i (r i s i ) < j (s j r j ), where i is such that r i G rs ad j is such that r j L rs. I [5] a measuremet value for < q (r, s) is defied as d <q (r, s) = { (s j r j ) j l r g r, for g r < l r, where i is such that r i G rs ad j is such that r j L rs. (r i s i ) l i, for g r g r = l r, r We use the domiace measuremet based rakig method [18]. At geeratio t the domiace measuremet based rak of the i th t idividual r i i a GA populatio, which domiates all s t j idividuals i the curret populatio is the i th idividual curret positio, the idividual s rak is defied as: rak i (r i t ) is the sum of the domiated compariso measuremets for every other s j t idividual of geeratio t i correlatio to the i th idividual. rak(r t i ) = j=1 d < (r t i, s t j ), where * ca stad for ay compariso method (the classical Pareto vector-, the ew quality vector-, or the simple weighted sum scalar-compariso) [18]. The key questio for a GA is the codig of possible solutios, which will evolve through several geeratios. Fudametal schemata theory suggests that small alphabets are good, because they maximise the umber of schema available for geetic processig, so biary codig is implemeted [16]. To avoid Hammig cliffs Gray codig is used. Low probability (1 %) mutatio is a importat part of every GA. The chromosomes are simply the cocateated bit strigs of all the parameters with fixed positio for every gee, high probability (0.8) simple two poit crossover will esure low disruptiveess ad high rate of iheritace durig the reproductive phase. Stochastic uiversal samplig havig miimal spread ad zero bias is used for selectio with a rather low (1.2) selectio pressure. Cotiuous exploratio of the search space is achieved together with cosistet covergece by the combiatio of geetic operators i this maer [3-5, 7-9, 18]. SINGULAR VALUE DECOMPOSITION AND GRADIENT DESCENT BASED FUZZY MODELLING We use the miimal uiversal represetatio of Zadeh partitios, by which it ca be easily optimised without ay costraits, the method as itroduced i [3, 4] ad elaborated i [18]. To represet a fuzzy partitio of K membership fuctios we take K pieces of ratioal, positive or zero parameters a κ R 0 +, κ = 1,, K ad form the b k oliear parameters of the Zadeh-type MFs formig fuzzy-partitios as: k (10) b k = j=1 a j / K κ=1 a κ. (11) For cases whe a a κ = 0 we obtai b k = 0, thus the fuzzy partitio structure complexity is reduced by oe MF, thus the complete fuzzy system structure complexity, the umber of rules are sigificatly reduced. I cases whe there remais oly a sigle a κ > 0 the fuzzy system 306

6 A soft computig method for efficiet modellig of smart cities oise pollutio rule degrades to a simple liear equatio. Fially, i the case whe all K pieces of a κ = 0 the fuzzy partitio ad the whole fuzzy cosequet degrades to a costat, ad the fuzzy system becomes idifferet to the reduced iput chael; it becomes idepedet of the correspodig iput variable. These a κ parameters are suitable for efficiet stochastic global search as by GAs, ad ca be freely fie-tued with ay gradiet descet-based method [3-5, 8, 18]; also the Jacobia of the FLS ca be calculated with regards to a κ for advaced oliear least squares data fittig methods. The proposed fuzzy-partitio represetatio has bee used i a multi-objective geetic algorithm [5, 18]. Oe chromosome cosists of the umber of MFs for every iput ad the correspodig parameters as i equatio (11) for every possible MF partitio of each iput. The populatio size is selected to be 50 times the umber of parameters. For parameter codig we use 16 bit biary Grey-coded chromosomes. The objective fuctios we have used are: maxe the maximum absolute error of the idetificatio, MSE the mea squared error of the idetificatio, RuleN the umber of used fuzzy rules for the idetificatio divided by the maximum possible umber of rules for the selected desig, RakW is calculated from the matrix rak of W(x, b) of equatio (9), which is ormed to the theoretically maximal possible rak of the same FLS structure. Based o the trasformatio of the TSK FLS equatio (9) to the f(x) = W(x, b) c format, we split the idetificatio ito two problems: First part is the oliear problem of fidig optimal W(x, b(a κ )) values, which are defied as a fuctio of system iput vector x ad the oliear MF b k parameters of equatio (7). By represetig b k as a fuctio of a κ via equatio (11) ay umerical oliear ucostraied optimisatio ca be applied to a κ parameters. The secod part, whe W(x, b) is fixed (by selected optimal, or cadidates for optimal b parameters) to costat values, of the problem is a simple solutio to a liear equatio system, which is best solved for a give traiig data set of {iput, output} sample vectors of { x traiig, f traiig } time series i a error square optimal maer by the umerically robust, geeral SVD decompositio method as: c = VS 1 U T f traiig for the SVD decompositio of W traiig (x traiig, a κ ) = USV T. I the applied GA chromosomes are evaluated through the followig eight steps. 1.) For each i of iputs: K i, umber of MFs of the i th iput is decoded from the chromosome. 2.) For each K i : correspodig a κ parameters are decoded from the chromosome. 3.) All required parameters b of Zadeh-formed MFs of equatio (7) that form fuzzy partitios are calculated as proposed i (11). 4.) Usig traiig iputs x traiig, all possible traiig atecedets ω l (x traiig ) are formed from are evaluated as i equatio (5) to get W traiig (x traiig, a κ ) of equatio (9). 5.) y l (x traiig ), the correspodig fuzzy rule traiig cosequet parts for all traiig umerical iputs are evaluated by SVD based LS method as: c = VS 1 U T f traiig, (12) by the SVD of the compoud atecedet matrix of USV T = W traiig (x traiig, a κ ) ad f traiig is the measured traiig oise level. 6.) parameters a κ of step 2.) are further optimised by a gradiet decet based method for o more tha i=1 K i steps, where is the dimesio of the iput space ad K i is the umber of used MFs. For every gradiet descet iteratio step 4.) ad 5.) are repeated. 307

7 A. Nemes, Gy. Mester ad T. Mester 7.) The resultat fuzzy system f(x) is evaluated by equatio (9) for the test data set, which is disjuct to the traiig data set separated radomly from available samples before the idetificatio process starts ad its cardiality is 10% of the traiig data set. 8.) For the test data set the maxe, MSE, RuleN/maxRuleN, RakW/maxRakW of the idetificatio is calculated, where maxrulen =, ad maxrakw = (+1) maxrulen. i=1 K i To icrease the efficiecy of the GA after each evaluatio the iitial chromosome defiig a fuzzy system used i step 1.) ad 2.) is updated with its optimised MF parameter values resultig from step 6.). FUZZY SYSTEMS TRAINING DATA SET QUALITY ANALYSIS AND REDUCTION Traiig data set, a prerequisite for system idetificatio is a set of measuremets of system resposes, outputs to be modelled f traiig while the system is beig drive alog a predefied trajectory of iputs x traiig. As this iput traiig path must be sufficietly exitig so that all system output characteristics ca be observed, it is atural that for a good quality idetificatio we must operate with very large iput-output traiig data sets. Evaluatig models, iteratively optimisig parameters alog large traiig sets ad especially performig SVD of large matrices is extremely time cosumig. Thus, it is always a challege for idetificatio tasks to fid sufficietly excitig, but ot a oversized traiig data set. I case of modellig complex systems filterig out uecessary samples, while still leavig all the ecessary data for good quality idetificatio is ot a trivial strait forward process. I case of every idetificatio problem, specific approaches are used whe deemed ecessary. The coditio umber of a matrix i our case the compoud atecedet data set USV T = W traiig which defies a liear system of equatios i our case equatio (9) is called the coditio umber of the equatio; it is the ratio of the largest ad the smallest sigular value i our case for S = [s ij ] it is cod(w traiig ) = max(s ii ) / mi(s ii ). The higher the coditio umber, the more ucertai the solutio is; the more chaotic, the more sesitive the solutio is to small disturbaces of system parameters. The atural goal for a good quality, robust liear system solutio is to have as small a coditio umber as possible. A very well-coditioed liear system of equatios has a coditio umber of three orders of magitude less tha the reciprocal of the umerical precisio of our calculatios. I case of double precisio floatig poit digital computer calculous we ca rely o a umerical precisio of at least ~10-16, which meas that ay coditio umber < will already result i a sufficiet precisio of up to six decimal places, but of course a < 10 3 coditio umber will be preferred. Rak deficiecy of W traiig is obviously out of questio whe the goal is to get a complete fuzzy system with o udefied rules o udefied outputs for ay iput regio. For a idetificatio error least squares (LS) optimal c vector oe ca use the SVD trasformatio property as i equatio (12) c = VS 1 U T f traiig.full for the SVD decompositio of W traiig.full = [W(x traiig.full, b)] = USV T, where f traiig.full is the vector of all available traiig data, for the iput traiig series [x traiig.i ], i = 1,.., N; where N is the umber of all available traiig data iputs. As itroduced i [9] ad elaborated i [18] we apply a greedy selectio algorithm to [x traiig.i ] ad thus to the correspodig f traiig.i traiig data set, such that we ca determie a arbitrary quality per size balaced, but reduced traiig data set x traiig.reduced = [x traiig.j ] ad a correspodig f traiig.reduced for FLS based fuctio idetificatios. The objective of the 308

8 A soft computig method for efficiet modellig of smart cities oise pollutio greedy search is to select iput-output traiig data pairs ([x traiig.j ], f traiig.j ), such that they maximise the coditio umber decrease of the W traiig.reduced = [W(x traiig.j, b)] of the FLS atecedet matrix [18]. Cosiderig the proof of the uiversal fuzzy approximatio theorem, for a geeral case we caot boudlessly geeralise our fuzzy model, we caot reduce the umber of MFs ad by this the required rak of W to a arbitrary low level, sice by this we lose the uiversal fuctio approximatio quality of our FLS. O the other had if we go for a too complex fuzzy structure, however large the measured traiig data set we capture, there is a poit after which there will be o idepedet samples (rows of W will ot be liearly idepedet), the traiig path is called isufficietly excitig, the traiig data is ot fit for defiig a complete TSK FLS based o fuzzy partitio atecedets, the matrix rak of W of equatio (9) will ot be complete, it will be rak(w) < ( + 1) K i, where is the dimesio of the iput space ad K i is the umber of used MFs. Cosequeces of such a idetificatio attempt is popularly called idetificatio overfittig, where the model ca oly precisely match the traiig data, while it performs poorly o disjuct test data sets, which did ot take part i the idetificatio process. The approach take i this paper is to exted the real measured traiig data set of 120 samples (Figure 1.) with a carefully reduced set of 800 simulatio values based o equatios (1) ad (2) of [1, 2]. The same approach ca be take for ay further ew measuremet data or ay further ew reliable models, which we wat to merge i, thus to re-trai our model for a higher precisio or eve to iclude ew iput regios. i=1 a) b) c) Figure 1. Traiig data set measured system iputs of a) average buildig heights (m), b) average Street width (m) ad c) umber of equivalet vehicles per secod. IDENTIFICATION RESULTS The GA process covergece is preseted alog the idetificatio error objective chages through 50 geeratios of evolutio i Figure 2. The selected o-domiated idetificatio solutio has 48 rules defied for [4, 3, 4] MFs correspodig to iputs [ eq, W, H]. The a κ parameters are [291, 25068, 43316; 30436, 29813, 0; 29442, 26577, 56388]. The correspodig b parameters are [0, , , 1; 0, , 1; 0, , , 1]. The fuzzy partitios are graphically preseted i Figure 3. The traiig data had bee exteded to result i system output as preseted o Figure 4., which also presets the overlappig output of a selected o-domiated solutio ad separately the correspodig traiig data idetificatio error. The expected system output, the system idetificatio output, ad the idetificatio error alog radomly selected test samples, which did ot take part i the traiig process, are preseted i Figure

9 A. Nemes, Gy. Mester ad T. Mester Figure 2. GA covergece a) maximal absolute error evolutio, b) mea square error evolutio. Figure 3. Zadeh type fuzzy partitios for the premise part of iputs [ eq, W, H]. Figure 4. Traiig data set outputs a) desired ad idetified output (dba), b) traiig error (dba). 310

10 A soft computig method for efficiet modellig of smart cities oise pollutio Figure 5. System idetificatio test results: expected ad idetified output, dba (left), idetificatio error, dba (right). CONCLUSIONS The preseted idetificatio method is capable of sufficietly precise modellig of real measured systems (which data has oise ad measuremet error) ad approximatig simulated system (o oise, o measuremet error) models to a extreme precisio. The defiitio of fuzzy partitios provides a guaratee that the resultig FLS uiformly covers the complete defied iput space, more eve the FLS structure guaraties its output is cotiuous for the complete iput space. By extedig the measured traiig data set with simulated values, we have esured that our model output is owhere sigificatly worse tha a selected mathematical model, while it precisely approximates the measured data of the real system for all iputs that are i the appropriate rage of the provided measured traiig data set. By extedig the traiig data set ad repeatig the idetificatio process our model is easily fie-tued to match further real data. This robust oise pollutio model ca be used for makig traffic cotrol decisios where the iduced oise levels have to be cofied. The cotrolled street cofiguratio must be defied by the street width ad average buildig height, the cotrol iput is the equivalet car umber ad the system model output is the resultig oise level. REFERENCES [1] Gulliver, J. et al.: Developmet of a ope-source road traffic oise model for exposure assessmet. Evirometal Modellig ad Software 74, , 2015, [2] Vicezo, B.; Crocco, F. ad Mogelli, D.W.E.: A mathematical model for traffic oise predictio i a urba area. I: Mastorakis, N., et al., eds.: Recet Advaces i Mechaics. Proceedigs of the 4th WSEAS Iteratioal Coferece o Urba Plaig ad Trasportatio. WSEAS Press, Corfu, pp , 2011, [3] Nemes, A.: Cotiuous Periodic Fuzzy Logic Systems ad Smooth Trajectory Plaig for Multi-Rotor Dyamic Modelig. Acta Polytechica Hugarica 13(6), , 2016, [4] Nemes, A.: Fuctio Idetificatio by Ucostraied Tuig of Zadeh-type Fuzzy Partitios. 2d Iteratioal Symposium of Hugaria Researchers o Computatioal Itelligece. Budapest, 2001, [5] Nemes, A.: System Idetificatio Based o Multi-Objective Optimisatio ad Ucostraied Tuig of Zadeh-type Fuzzy Partitios. 2d IEEE Iteratioal Symposium o Itelliget Systems ad Iformatics, Subotica, 2003, 311

11 A. Nemes, Gy. Mester ad T. Mester [6] Mester, G.; Pletl, S.; Nemes, A. ad Mester, T.: Structure optimizatio of fuzzy cotrol systems by multi-populatio geetic algorithm. 6th Europea Cogress o Itelliget Techiques ad Soft Computig. EUFIT 98, 1, pp , 1998, [7] Nemes, A.: New Geetic Algorithms for Multi-objective Optimisatio. 1st Iteratioal Symposium of Hugaria Researchers o Computatioal Itelligece, Budapest, 2000, [8] Nemes, A. ad Mester, G.: Ucostraied Evolutioary ad Gradiet Descet-Based Tuig of Fuzzy partitios for UAV Dyamic Modelig. FME Trasactios 45(1), 1-8, 2017, [9] Nemes, A. ad Latos, B.: Traiig Data Reductio for Optimisatio of Fuzzy Logic Systems for Dyamic Modellig of Robot Maipulators by Geetic Algorithms. IEEE Istrumetatio ad Measuremet Techology Coferece, IMTC 2001, Budapest, 2001, [10] Mester, G.; Nemes, A.; Pletl, S. ad Mester, T.: Soft Computimg metoda za optimalo modeliraje buke. 5 th Symposium o Computer Scieces ad Iformatio Techologies, YUINFO '99, Kopaoik, pp.1-5, 1999, [11] Rodic, A.; Mester, G. ad Stojkovic, I.: Qualitative Evaluatio of Flight Cotroller Performaces for Autoomous Quadrotors. I: Pap, E., ed.: Itelliget Systems: Models ad Applicatios. Topics i Itelliget Egieerig ad Iformatics 3. Spriger-Verlag, Berli & Heidelberg, pp , 2013, [12] Mester, G.; Pletl, S.; Pajor, G. ad Rudas, I.: Adaptive Cotrol of Robot Maipulators with Fuzzy Supervisor Usig Geetic Algorithms. I: Kayak, O., ed.: Iteratioal Coferece o Recet Advaces i Mechatroics, ICRAM 95. 2, Istabul, pp , 1995, [13] Mester, G.; Pletl, S.; Pajor, G. ad Jeges, Z.: Flexible Plaetary Gear Drives i Robotics Iteratioal Coferece o Idustrial Electroics, Cotrol, Istrumetatio ad Automatio Robotics, CIM ad Automatio, Emergig Techologies, IEEE IECON '92, 2, pp , Sa Diego, 1992, [14] Wag, L.: Adaptive Fuzzy Systems ad Cotrol: Desig ad Stability Aalysis. Pretice Hall, Eglewood Cliffs, 1994, [15] Helledoor, H. ad Driakov D., eds.: Fuzzy Model Idetificatio, Selected approaches. Spriger-Verlag, Berli & Heidelberg, 1997, [16] Goldberg, D.E.: Geetic Algorithms i Search, Optimizatio ad Machie Learig. Addiso-Wesley Logma Publishig Co. Ic., Bosto, 1989, [17] Osyczka, A.: Multicriteria optimisatio for egieerig desig. I: Gero, J.S., ed.: Desig Optimisatio. Academic Press, pp , 1985, [18] Nemes, A.: Tools for Efficiet Soft Computig Modellig ad Feasible Optimal Cotrol of Complex Dyamic Systems. Ph.D. Thesis. Óbuda Uivesity, Budapest,