The 8 h Inernaonal Conference RELIABILITY an STATISTICS n TRANSPORTATION an COMMUNICATION - 008 MODELLING DISTURBANCES IN SYSTEM TRACK RAIL VEHICLE Taeus Csosk, Anrej Necas, Jóef Sokłosa Hgh School of Economcs an Innovaon Melgeska sr. 7/9, 0-09 Lubln, Polan Ph: 8 86776,.csosk@pr.raom.pl Ph: 8 888, a.necas@pollub.pl Ph: 8 8677, j.soklosa@pr.raom.pl All real sysems ork n surbances conons. Dsurbances erm belongs o specal caegory of enry values hch are no knon before an o no make subjec o conrol, means hey are unconrolle enry sgnals. Dsurbances are an mporan elemen of sysems regulaon, because usually lea o unexpece effecs n funconng of seere sysems. An alernave approach has been propose n hs arcle n surbng sgnals escrpon, relae on heory of ave surbances, hch allo o escrbe e surbances scale, occurre n suspensons of acve vehcles. Dsurbances moels base on ne ave nerpreaon can escrbe e class of real, nefne surbances hch occur n suspensons of acve vehcles. Keyors: suspensons, acve vehcles suspensons, avy nerpreaon of surbances, Laplace ransform, ranom surbances. Inroucon The basc ask of suspensons s guaranee expece fluy of vehcles move by solaon boy from heels vbraon cause by roa unevenness. In ha ay boes ece abou comfor of rvng for rver an passengers an also reuce ynamc loas of vehcle ses ncreasng her urably. In seerng heory here s knon a commonly fac ha all real sysems ork n surbances conons. Dsurbances erm belongs o specal caegory of enry values hch are no knon before an o no make subjec o conrol, means hey are unconrolle enry sgnals. Dsurbances are an mporan elemen of sysems regulaon, because usually lea o unexpece effecs n funconng of seere sysems. The ypcal surbances examples hch occur n vehcle rvng are: guss of n an oher aeroynamc forces hch nfluence on vehcle, frcon an neurals n suspenson sysem, unevenness of roa pavemen, movng cenre of gravy an anoher unefne effecs of splacemens n mechancal ses of vehcles. In scence researches, run from he en of 70-es n he las cenury, relae o acve boes n vehcles eermnsc or ranom nerpreaon of surbng sgnals s accepe. If frs of menone approach presens oo smple meanng of surbances naure hen secon complcae oo much her escrpon. Sascs properes of surbances such as average value, varaon, specral ensy an oher suppore on average are long erm values. Meanhle rver rvng vehcle on chosen par of roa by changng n follo he curren behave. Effecve seerng n case of surbances requres curren nformaon abou her funconng. Sasc nformaon suppore on long-erm observaons oesn mee curren nformaon emans an becomes n seerng n real me oally useless. An alernave approach propose n surbng sgnals escrpon, relae on heory of ave surbances, hch allo o escrbe e surbances scale, occurre n suspensons of acve vehcles s presene n he lecure.. Wavy Inerpreaon of Dsurbances n acve vehcles suspensons Dsurbances hch characere avy srucure can be mahemacally escrbe by half-eermnsc analycal epenences n he follong ay []: [ f (),f (),...,f ();.c..,c ] () = W M L, () here f (), =,,..,M (M-fne value) knon me funcons, c k, k=,...,l unknon parameers, hch can seppe change n a parly solay her meanngs. Mahemacal moels () ype ll be calle as ave nerpreaon of surbances (). Lnear escrpon () can be consere as nerpreaon () n funconal area, here funcon se {f (),...,f M ()} s a base, an c s parly solegh facor. In oher ors surbances () presen lnear egh combnaon of knon basc funcons f ( an unknon egh facors c, hch ranomly n perocally-solay change her meanngs []. An example llusraon of equaon () s presene n Fgure. 7
RelSa 08, -8 Ocober 008, Rga, Lava () Fgure. Wavy srucure surbances Propose nerpreaon of surbances () efnely ffers from her raonal ranom reamen. Especally nformaon range conene n equaon () s qualavely fferen from nformaon conene n raonal sasc eas, such as average value, varaon, specral ensy an ohers. Senses of facors c n equaon () are oally unknon (h he excepon ha hey change n perocally-solay). Wavy nerpreaon oesn operae raonal sascal feaures an oesn escrbe hem. So nerpreaon () flls n nformave vacuum n escrpon of surbances hch occur n suspensons of acve vehcles. Especally equaon () allos o escrbng e range of possble avy forms, hch conan any unknon realaon of surbances () n momen. Beses, each separae realaon () n avy nerpreaon can possess s on se of sasc feaures, hanks o hch can be use also o escrbe nonergoc surbng funcons () especally hen each realaon () s a consan ranom value.. Moels of avy surbances Ponng ou sysem of basc funcons {f ()}, s a frs sep n usng avy nerpreaon of surbances as an nsrumen of regulaon sysem. I can be one on he bass of vsual an mahemacal analyss of expermenal recorngs () or by analyss of ynamc characerscs of physcal process hch generae (). In he secon sep proper sae moel for equaon () shoul be specfe. The moel s a fferenal equaon me by funcon (). In oher ors equaon () shoul be consere as knon general soluon of looke for fferenal equaon. Le s suppose ha each chosen funcon f () has Laplace ransform f (s), n he follong ay: Pm (s) f (s) =, () Q (s) n here: P () s,q () s m n - polynomals aequaely m an n grae m n. If c emporarly suppose as permanen values hen he ransformaon of Laplace s equaon () s he follong: M Pm (s) (s) = c f(s) c f (s)... cm f M(s) = c, () Q (s) fnally: n P(s) (s) =, () Q(s) here polynomal of numeraor P(s) nclues facors c, an polynomal of enomnaor Q(s) s he smalles general enomnaor n se of enomnaors polynomals { Qn (s), Q (s),...,q (s)} n n of equaon (). Such nerpreaon M guaranee mnmal se of fnal moel of sae (), hch has he essenals of meanng from pon of ve of coss an apparaus complexy. So le s suppose ha enomnaor polynomal Q(s) n equaon () s follong: Q(s) = s s s... s here M q, () n. From equaon () comes ha surbances () can be reae as nal varable of fconal lnear ynamcal sysem of operaonal ransmsson: G ( s ) =, (6) Q( s ) 8
The 8 h Inernaonal Conference RELIABILITY an STATISTICS n TRANSPORTATION an COMMUNICATION - 008 by nal conons { ( 0), ( 0), ( 0),...}. hen surbances () h akng epenences uner conseraon ()-(), mee he follong homogeneous lnear fferenal equaon h consan parameers:... = 0, (7) here facors q, =,,...,, are knon, because on epen on c an hey are eermne by sysem of basc funcons {f ()}, hch are aken for gven. To ake no conseraon seppe facors changes c n equaon (7) e a o exernal force funcon ω(), hch s a progresson of unknon, ranomly appear pulse funcons h ranomly nensy (sngle, ouble, rple, ec. kn of Drac s funcon). So moel of sae () fnally akes he form: = σ... = ω(). (8) I s mporan ha pulse force funcon ω(), s unknon an nrouce o moel of sae (8) jus only n symbolc ay h he purpose o mahemacal escrpon of seps c n equaon (). Bese momens of appearance ajacen pulse funcons are separae by mnmal posve range μ > 0. So f base funcons f() n equaon () has Laplace ransform n kn () hen h he purpose of fnng sae moel for equaon () s necessary o efne facors {q, q,...,q} from equaons () an (), an nex use general sae moel (8). Dfferenal equaon of he orer of (8) can be presene n he form of sysem of fferenal equaons of he orer of frs. For example equaon (8) can be ren on equvalenly n form of oally observe knon canonc form: =, = σ(), = σ(),. (9).. (), = q q... q σ () here symbolc acvy ω() n equaon (8) s replace n equaon (9) by funcons σ (), =,,...,, hch are progressons of unknon, ranom Drac s funcons, an pon means an operaor /. In general case shoul be expece ha fferenal equaon (8) or sysem of fferenal equaons (9) ll conan varable facors q an/or nonlnear elemens belo,/ ec. So searche sae moel for surbances (), hch have ave srucure, can be presene n he form of one fferenal equaon of he hgh orer: f (,,..., or n form of sysem of equaons of he frs orer: ω = W (,), (,) ( );, ) = ω( ), (0) = (,,... ). () = Z Le s conser ha moel () has he avanage over moel (0) because uses mehos of sae varables. If () s a mul-menson surbance hch conans p componens = (,,..., p ), hen sae moel shoul be eermne for each nepenen componen ().. Examples of Sae Moels for Real Dsurbances n Suspensons of Acve Vehcles Sae moels expresse by equaons (0) an () for real surbances, occurre a leas n shor me nervals n suspensons of acve vehcles, can be eermne on base recorngs of expermenal osclogram presene n Fgure. 9
RelSa 08, -8 Ocober 008, Rga, Lava Example. In case shon n Fgure a surbances mee he follong fferenal equaon: = 0 () c n ha case s a consan. To ake no conseraon unknon ranom, seppe changes c, shoul ae o equaon () he elemen σ(), hch compose of unknon progresson of Drac s funcon. Sae moel of surbances (), presene n Fgure a has fnally he follong form: () = σ. () Example. Dsurbances presene n Fgure b escrbe by equaon () = c c, hch mee he follong fferenal equaon of secon orer: () = ω, () here ω() means unknon progresson of ranom sngle an ouble mpulse of ranomly nensy. Equvalen moel n sysem form (9) s he follong: () (,0 )( ) = () () = 0 σ (), = (6), Example. Dsurbances (), presene n Fgure c. of Laplace ransform: ()= = () ()= = () ()= e = () ()= sn cos sn cos = () ()= e e k k k k k = () Fgure. Dsurbances of avy srucure n suspensons of acve vehcles 0
The 8 h Inernaonal Conference RELIABILITY an STATISTICS n TRANSPORTATION an COMMUNICATION - 008 () s = c c. (7) s s α can be presene n equaon form () as follos: () s = [ c( s α ) cs] s( s α ). (8) In ha case Q(s) = s αs, an from equaons () an (8) follos ha () mee he follong fferenal equaon of secon orer: α = ω (). (9) Equvalen sae moel has he form: = (0) (,0 ) (), = α (), = () σ Example. Dsurbances () n kn of small aves presene n Fgure has he follong Laplace ransform: () s s P () s =. () ( s γ )( s β ) So accorng o equaons ()-(8), () mees he follong fferenal equaon of ffh orer: ( γ β ) ( γβ ) = ω(). () Equvalen presenng he equaon () has a form: () = (,0,0,0,0 ) () = = = (), = (), (), = (), ( γβ ) ( γ β ) (), () () s Example. Impulse ype surbances presene n Fgure e of Laplace ransform : ( s) ( s k s k s k s k ) P = (6) here k are funcons of o knon parameers α β, mee he follong fferenal equaon: k k k k = ω() (7)
RelSa 08, -8 Ocober 008, Rga, Lava or () = (,0,0,0 ) (8) = = = = k (), (), (), k k k here k are knon facors, epene on parameers α β. Conclusons () Achevemens of mcroprocessor echnology n he space of en years gave anoher menson of gal seerng sysems. Conemporary seerng heory canno om problemac surbances hch occur n real complex mulmensonal sysems. Couneracng o surbances s ol an mporan ask n esgnng close seerng sysems. Traonal mehos of regulaon base on eermnsc nerpreaon of surbances or presenng hem by moels of sochasc processes. Frs approach presens oo smplfe unersanng of real phenomenon naure, hle secon reflecs excessve pessmsm an proves o complcae escrpon of real surbances. Ths lecure presens alernave mehos of nefne surbances escrpon, more exac han eermnsc approach an no so complcae as moels of sochasc processes. Dsurbances moels base on ne ave nerpreaon (fg.) can escrbe e class of real, nefne surbances hch occur n suspensons of acve vehcles. (9) he nose nose ave srucure eermnsm Fgure. Place of avy srucure n specrum of nfny Usng a ave meho of surbances moellng an conemporary mehos of sae varables can be bul ne effecve class of close seerng sysems calle as regulaors hch aap o surbances; hose surbances can: a) absorb exernal surbances n auomac ay, b) mnme nfluence of exernal surbances n real me, c) use n opmal ay exernal surbances for conrollng sysem of vehcle suspenson. Usng acve sysems of vbro-nsulaon conrolle by regulaors hch aap o surbances allo no only o monorng vehcles ynamcs n real me bu also mprove rvng comfor.
The 8 h Inernaonal Conference RELIABILITY an STATISTICS n TRANSPORTATION an COMMUNICATION - 008 References. Brason, A., Kho-Ju-Sh. Apple Theory of Opmal Governmen. Mosco: Mr, 97. (In Russan). Kvakernaak, Kh., Svan, Z. Lnear Opmal Governmen Sysems. Mosco: Mr, 977. 60 p. (In Russan). Johnson, C. D. Theory of Dsurbance-Accommoang Conrollers. Chaper n he book, Conrol an Dynamc Sysems; Avances n Theory an Applcaons, Vol., ee by C. T. Leones, Acaemc Press., Inc., Ne York, 976, p. 67.. Johnson, C. D. Accommoaon of Exernal Dsurbances n Lnear Regulaor an Servomechansm Problems, IEEE Trans. Auoma. Conrol, AC-6, 97, pp. 6-6.. Csosk, T.: Zakłócena seroanu ukłau ynamcnego or-poja synoy. Pojay synoe Nr, Ponań, 00. 6. Csosk, T. Dsurbances n he Conrol of a Dynamc Sysem Track-Ralay Vehcle. Zesyy naukoe Polechnk Śląskej Nr 70, ser Transpor., 00.