DATA DEPENDENT SYSTEMS AS A TOOL FOR MODELLING OF STOCHASTIC PROCESSES IN TRANSPORT AREA

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1 DATA DEPENDENT SYSTEMS AS A TOOL FOR MODELLING OF STOCHASTIC PROCESSES IN TRANSPORT AREA Bohuš Leiner, Lucia Ďuranová 1 Summary: The aim of he paper is shor descripion of possibiliies of daa dependen sysems (DDS) mahemaical apparaus using. Basic form of DDS expression is form a sochasic ime series of process which can be used o descripion and modelling of sochasic process, e.g. operaional process. The paper conains a briefly characerisics of fundamenal erms and equaions of ime series heory mahemaical apparaus, algorihm of a saisically adequae discree model of sochasically dynamic sysem and developed sraegy applicaion on real process. Key words: daa dependen sysem, sochasic operaional process, ime serie. INTRODUCTION One of possible way of complex sysems analysis wihou loss of accuracy and wihou necessiy of complicaed mahemaical apparaus uilisaion is observaion of sysem during is working and uilisaion of produced daa o is analyses. Such analysed sysem we call Daa Dependen Sysems (DDS). I means ha we do no need o know anyhing abou composiion of sysem and all analysis and conclusions are made jus based on observed sysem oupu. Daa dependen sysems are represened by ses of coninuous oupu signals which are discreized wih uniform sampling inerval ges a series of daa (values) which forms a base for descripion and analysis of invesigaed sysem. The main aim is o ge possibiliy of sysem behaviour forecasing and evenually influencing of is behaviour no o ge sysem in any unwaned sae (2). For descripion of DDS, one can use auoregressive models, advanageously Auo- Regressive Moving Average Models (nex as ARMA models). Their advanages are mainly precisely formulaed saisically crieria and relaively simple mahemaical apparaus of saisically regressive analysis and esing of saisical hypohesis. Their disadvanages are complicaions of idenificaion procedures and ime consuming compuer calculaions despie relaively simple mahemaical apparaus. Therefore, hey are suiable jus for off-line dynamic sysems idenificaion and modelling and heir main applicaion forecasing of dynamic sysem behaviour is suiable jus by saionary sysems, which do no change heir parameers wih ime. 1 doc. Ing. Bohuš Leiner, PhD., Ing. Lucia Ďuranová, Univerziy of Zilina, Faculy of Special Engineering, Deparmen of Technical Sciences and Informaics, 1.mája 32, Žilina, SR, Tel.: , Fax: , Bohus.Leiner@fsi.uniza.sk, Lucia.Duranova@fsi.uniza.sk Leiner, Ďuranová: Daa depende sysem as a Tool for Modelling of Sochasic 106

2 1. DATA DEPENDENT SYSTEMS AND ITS THEORETICAL APPARATUS Firs logical echnique for idenificaion is o derive he mahemaical model from sysem behaviour and direcly from naure of is physical characerisics or as he dependence on inpu facors influencing his behaviour. Mahemaical represenaions of such relaions are ofen saed in form of differenial equaions sysems. Soluions of such sysems wheher numerical or exac is very ime-consuming herefore no usable for he real ime conrol. Anoher problem wih his possible way of idenificaion is: some of he influencing facors are no well known and some can no be quanified. Anoher possible echnique for sysem idenificaion is o derive he mahemaical model from inernal inerdependencies of given process only. Clear Auoregressive (AR) model is suiable mahemaical model for discree nonsaionary signal (Fig.1) ha is considered o be visible represenaion of sysem dynamics (2). Dependence of X () values on previous (in ime) values X (-1) is picured in Fig X() [m] , ,5 30 Time [s] Fig.1 - Example of real non-saionary process Source: (2) Linear regression can be clearly seen. This relaion can be described by simple 1s-order Auoregressive model (1) in form (2): X ( ) a1. X ( 1) (1) where is he ime, X( ) is discree signal value in ime, X ( -1) is discree signal value in ime (-1), a 1 is coefficien of AR model and is error resuling from model imperfecions. Leiner, Ďuranová: Daa depende sysem as a Tool for Modelling of Sochasic 107

3 X(-1) Source: (2) Fig.2 - Inerdependence X ()=f {X (-1)} This simples model can be exended o universal n-h order of clear auo-regressive model AR (n) X X() ( ) a1. X ( 1) a2. X ( 2)... an. X ( n) (2) where n is he model order, X (), X (-1), X (-2)..., X (-n) are discree signal values in ime, -1, n, a 1, a 2,..., a n are coefficiens of Auoregressive model and is error resuling from model imperfecions [2,3]. If X () depends no only on preceding values X (-1), X (- 2) X (-n) bu also on he values of preceding errors -1, simple AR models are exended o Auoregressive moving average models (ARMA). The ARMA models can cover more complex characer of process inerdependencies and heir coefficiens relae closely o physical principles of observed process. Universal ARMA model of n-h order in Auoregressive par and (n-1)-h order in moving average par is given by equaion (3). X ( ) a. X ( 1) a. X ( 2)... a. X ( n) 1 b. 1 2 b.... b n1 n1 n. where X(), X(-1), X(-2)... X(-n) are he discree signal value in ime, -1, n, values, -1, n + 1 is error in he ime, -1, -2 -n+1, values a 1, a 2... a n are coefficiens of Auoregressive par of model, b 1, b 2... b n are coefficiens of moving average par of model. Main problem in idenificaion, modelling and simulaion by Auoregressive models is finding coefficiens of AR and MA pars and deerminaion of adequae order of he model. Coefficiens of AR model can be simply found using leas square mehod (LSM). For universal ARMA models non-linear LSM mus be used. Boh mehods use marix calculaions for finding needed coefficiens, which are very ime-consuming and herefore no usable for on-line process, conrol or idenificaion and hey also can no be used for modelling of nonsaionary ime-varying process (3). (3) Leiner, Ďuranová: Daa depende sysem as a Tool for Modelling of Sochasic 108

4 2. ALGORITHM OF ADAPTIVE AR AND ARMA MODELS Because of he above saed reasons, procedure based on heory of adapive and selflearning sysems is used for describing sysem behaviour in real ime. Algorihm for adapive modelling (3) is based on a gradien mehod (seepes descen mehod) and can also be used for non-saionary processes. Model is able o adap iself o he changes in process characer. I is supposed ha n-h order Auoregresive model (2) is a any given ime defined by he vecor of is coefficiens: a k ) [ a ( k), a ( k)... a n ( k)] ( 1 2 T (4) Using he seepes descen mehod, poin of leas squares 2 is searched for. Search begins wih an iniial guess as o where he minimum poin of 2 may be. Minimal sum of squares S is in he poin where S 2 0 a k a k (5) The updaed values of AR model coefficiens are obained from S a( k 1) a( k). a (6) S T where 2. ( k 1) f a X is he gradien direcion and posiive value of in equaion (6) scales he amoun of readjusmen of he model coefficiens in one ime sep. Then, he ieraive correcions of coefficiens are T a( k 1) a( k). e. X ( k 1). (7) In [3] adapive AR models were exended o include also MA par o adapive AuoRegresive models wih moving average. To achieve his, vecor of moving average coefficiens mus be considered T b ( k ) [ b1( k ), b 2( k )... b n( k )] (8) Same procedure as for vecor of AR coefficiens was used o derive formula (10) for ieraive correcions calculaions of MA par coefficiens T b( k 1) b( k).. ( k 1) where is error from he las ieraive sep and (k-1) is vecor of preceding errors ε ( k 1) T k 1, k 2,... k n 1 (10) Anoher problem arises when deciding value of convergence consan. I influences converging speed of algorihm and also is sensiiviy o random or sysemaic changes in process environmen characer. Procedure for calculaing consan, based on experimenal work [2,3] was presened for use in area of adapive conrol: C (11) and C 1 C. k k k, (12) k1 (9) Leiner, Ďuranová: Daa depende sysem as a Tool for Modelling of Sochasic 109

5 where is consan describing sysem memory, i influences model sensiiviy o random process changes, is consan characerising sysem dynamics and is consan for correcion of numeric calculaion errors. Acual values of hese consans can be chosen so he model sensiiviy o sochasic evens and response speed o process characer changes are as required. 3. SOFTWARE SUPPORT FOR STOCHASTIC PROCESSES MODELLING The scalar models of a simple descripion of dynamic sysem can no express saisically adequae descripion of complex sysems. For his reason here was developed an effecive sofware sysem which enables o creae he saisically models of dynamic sochasic sysem by using ARMA or vecor auoregressive models VARMA. The final form of an idenificaion sofware was creaed in such a way ha i enables he use of an idenificaion library and o realise he own idenificaion of sysem parameers. The resul of a proposed applicaion of his mehodology is ArmaGe sofware, his was developed on auhors deparmen. Sofware suppor is fully compaible wih Microsof Windows sysems. I conains users menu, which apar from basic funcions wih file, configuraions, work wih windows and help funcions conains wo submenus submenu of Simulaion and submenu of Idenificaion (Fig.3). Fig. 3 - Main window of creaed applicaion ArmaGe The hear of he program is submenu Idenificaion, by means, which is possible o make selecion of he idenificaion mehod and way of chosen ime series, whereupon i is possible o use eiher adapive algorihm of ime series idenificaion or make idenificaion using non-linear leas squares mehod. Idenificaion by means of higher presened non-linear (respecively for AR models linear) leas square mehod is available in iem Idenificaion and is sub-menu NLINLS. Here are four opions. Firs wo - Model AR afer orders and Model AR- complee calculaion give as resuls of idenificaion AR model, described by (2). Nex iem Model ARMA - afer orders gives coefficiens of beforehand seleced order of ARMA (n, n-1) models deerminaion. I means, ha we i is necessary beforehand o deermine required order (known number of coefficiens) of auoregressive par - a k and Leiner, Ďuranová: Daa depende sysem as a Tool for Modelling of Sochasic 110

6 moving average par - b k which principally deermine number of former values he calculaed value depends on. The iniial guess is of coefficiens of ARMA (n, n-1) model. To coefficiens of moving average par are assigned value of zero and coefficiens of auoregressive par obained by applicaion of linear leas square mehod. Simulaneously he sum of squares of deviaions value expressing deviaion of heoreical model from real model is calculaed. Then he proper ieraive calculaion follows - oupus are he coefficiens of model. The las imporan iem from submenu Idenificaion is - Model ARMA - complee calculaion - which aim is o find an opimal ARMA (n, n-1) model. This model is he bes describe of sochasic sysem, which oupu is a ime serie. Because, in mos cases we don know opimal order of model, beforehand i is necessary o deermine by an ieraive procedure an opimal order of model for descripion of given sysem (Fig.4). An algorihm of opimum auoregressive model deerminaion used in ARMAGET is idenical as algorihm from par 2 of his paper. Fig.4 - Resuls of idenificaion Fig.5 - Resuls of simulaion of ime serie from parameers of ARMA model Iem Simulaion enables adjusmen and conversion of incompaible inpu files of ime series o compaible ones and simulaion (generaion) of ime series basing on given AR or ARMA models order and parameers wih possibiliies of mean and dispersion selecion of simulaed serie (Fig.5). The accuracy and he reliabiliy of he developed mehods and algorihms were verified by use of commercial sofware packages (MS Excel, MATLAB) and will be compared and verified by using of uiliy ARMASA Package [3] ec.). 4. MODELLING OF REAL WORKING CONDITIONS PROCESSES IN TRANSPORT SYSTEMS Because he chosen mahemaical apparaus allows geing of models based of physical principles of researched processes which are developed from recorded courses of examined values i was necessary o obain experimenal daa, which characerise ypical working condiions (1, 4). Therefore a o of experimenal measuremens was done which describe ypical working condiions of rucks (unevenness of road surface, segmenaion of errain, ypical working speed during differen ypes of working modes e..c). Longiudinal and ransverse unevenness had a dominan posiion as a ypical of ranspor machines working condiions generally. Leiner, Ďuranová: Daa depende sysem as a Tool for Modelling of Sochasic 111

7 The experimenal measuremen was realised which consised of following seps: Finding of roads and heir pars represening individual caegories of roadways from poin of view of qualiy of heir coverage, Selecion of acual pars of chosen lengh, Own measuremen using suiable equipmen, Filering of rends and rough unevenness wih suiable sofware package Sagraphics Evaluaion of chosen pars by means of seleced dynamic mehods [1,4] and verificaion of heir classificaion for relevan caegories of qualiy. As a resul of former seps here were obained courses of longiudinal unevenness ogeher from 6 secors (5 qualiaive caegories of road and one secor of errain). Because of limied scope of his paper jus course of 5 h caegory of road and he values of errain are presened on Fig.6. Absolue aliude [m] 130,5 130,0 129,5 129,0 Par of road 5 h caegory (Dolný Hričov) 128, Lengh[m] Absolue aliude [m] , ,5 Par of errain (Terrain near a river Váh) Lengh [m] Fig.6 - Courses of absolue high unevenness of 5 h caegory road and errain The values of evaluaing parameers C and IRI are presened in Tab.1. If was deermined ha from obained parameers of C and IRI chosen secions really are represened chosen qualiaive caegories and here for hey can be judged as represenaive of qualiaive differen surfaces [1]. Leiner, Ďuranová: Daa depende sysem as a Tool for Modelling of Sochasic 112

8 Tab.1 - Calculaed values of C and IRI parameers for chosen secions of road Locaion "C" "IRI" Caegory Žilina 0,85 1,61 I Brodno 2,31 2,62 II Čičmany 3,92 4,28 III Varín 13,85 8,54 IV Dolný Hričov 22,81 14,37 V Povodie Váhu 42,05 22,74 Terrain I was necessary from of poin of view of furher pracical applicaion of experimenally obained models o separae he random par of unevenness from is rend. I was used he sofware pack Sagraphics which allows direcly selecion of rends and seasonally from recorded values. To demonsrae his are on Fig.7 shown courses of sochasic pars of profile unevenness for series from Fig.6. 0,10 Course of high unevennesses (5h caegorie) High of unevennesses [m] 0,05 0, ,05 Value of he parameer C : 22,81 Value of he parameer IRI : 14,34 Caegorie of road : 5-0,10 Lengh [m] 0,6 Course of high unevennesses (Terrain) High of unevennesses [m] 0,3 0, ,3 Value of he parameer C : 42,05 Value of he parameer IRI : 22,74 Caegorie of road : Terrain -0,6 Lengh [m] Fig.7 - Random componens of aliude of unevenness for 5h caegory and errain I was used he ArmaGe sofware o ge real and simulaed daa. Pracically i means opimal auoregressive model and is parameer deerminaion (order and coefficiens of model) for all inpus of discree values of aliude unevenness o seleced secors of roads. Leiner, Ďuranová: Daa depende sysem as a Tool for Modelling of Sochasic 113

9 Finding parameers of adequae models for secor of 5h caegory road and secor of errain are presened in Tab. 2. Ca. 5. Tab.2 - Parameers of adequae ARMA models for secors of 5h caegory and errain Coefficiens of opimal ARMA model a(1) = 0, a(2) = 0, a(3) = 0, a(4) = -0, a(5) = 0, a(6) = -0, b(1) = -0, b(2) = -0, b(3) = 0, b(4) = 0, b(5) = 0, Sum of squares SSC = 0, Terr a(1) = 0, a(2) = 0, a(3) = -0, a(4) = 0, b(1) = 0, b(2) = 0, b(3) = -0, SSC = 2, These models were used for generaion of new, saisically adequae, series of analysing process values. For his purpose sofware ArmaGe conains he iem Simulaion which was successfully applied by generaion of new courses aliude of unevenness of reference secors. For presened pars of 5h caegory and errain here are wo new simulaed courses and iniial experimenal course presens on Fig.8. 0,10 Caegorie 5 High of unevennesses [m] 0,05 0, ,05-0,10 0,60 Experimenal serie Simulaed serie Nr.1 Simulaed serie Nr.2 Lengh [m] Terrain High of unevennesses [m] 0,30 0, ,30-0,60 Experimenal serie Simulaed serie Nr.1 Simulaed serie Nr.2 Lengh [m] Fig.8 - The courses of simulaed series of 5 h caegory road and errain secors Leiner, Ďuranová: Daa depende sysem as a Tool for Modelling of Sochasic 114

10 For compuer simulaion of working condiion characerisics available especially on compuer sress analysis of criically pars of rucks, on influence analyses o seleced working properies of ranspor machines and for auomaic conrol of various load machines oo was used he finding adequae models. During verifying heir physical adequacies were once more used dynamic mehods of evaluaion secors of roads from poin of view longiudinal unevenness. I was found and verified ha deermined mahemaical models very good describe boh analysed characerisics. Deermined differences were in boh of esed facors neglec able and saisically non-significan. CONCLUSIONS I is well known ha working of majoriy of machines is significanly influenced by differen kinds of sochasic loads. There is possible o respec he endency a limiaion of energeically and maerial consumpion o oversize heir dimensions. Bu i is necessary o look for some more ingenious mehods o deal wih hese problems. Some of hem are he ways o conrol (influence) he working of a mechanical sysem in respec o heir proposed behaviour. Bu i needs o follow of he sysem behaviour in he real ime and o make some necessary conrolling inervenions. I was no possible from poin of view of deermined scope of paper o show all of obained oupus from realised calculaions and experimen (mainly made on compuers). Therefore hey are presened jus some of hem in his paper. Based on above-menioned resuls i is possible o sae ha chosen heoreical apparaus and mehodology of auoregressive modelling (namely adapive one). Is a suiable ool for modelling and simulaion of differen working condiion facors. REFERENCES (1) KOVÁČ, M., REMIŠOVÁ, DECKÝ, M., ĎURČANSKÁ, D., ČELKO, J.: Diagnosika paramerov prevádzkovej spôsobilosi vozoviek. 1. vyd., Žilina: Žilinská univerzia, s., ISBN (2) LEITNER, B.: Auoregressive models in modelling and simulaion of ranspor means working condiions. In: Transpor means: he 14h inernaional conference: Kaunas Universiy of Technology, Kaunas, Lihuania. 2010, s , ISSN X. (3) LEITNER, B.: The sofware ool for mechanical srucures dynamic sysems idenificaion.. In: 15h inernaional conference Transpor means 2011: Ocober 20-21, 2011, Kaunas Universiy of Technology, Lihuania, s , ISSN X. (4) ČELKO, J., DECKÝ, M., KOVÁČ, M.: An analysis of vehicle road surface ineracion for classificaion of IRI in frame of Slovak PMS. In: Mainenance and reliabiliy. Polish Mainenance sociey, Nr 1 (41) 2009, sr.15-21, ISSN Leiner, Ďuranová: Daa depende sysem as a Tool for Modelling of Sochasic 115