DSpace at VNU: An application of random process for controlled object identification with traffic delay problem

VN U Journal of Scicnce, M athem atics - Physics 24 (2008) 101-109 An application of random process for controlled object identification w ith traffic delay problem V u Tien Viet* D ep a rim en t o f M athem atics, M echanics, Inform atics, C ollege o f Science, VN Ư 334 N guyen Trai, H anoi, Vietnam R eceived 23 January 2007; received in revised form 20 March 2008 A b stra ct In the articlc proposed an effective method estim ating transfer function model o f controlled plant including dead-time delay, based on slochatstic tim e series o f inpul-output signals The model slm cture is modified with parameters optim ized until the model error b ccom cs ”w h itc-n oisc” series that with inough srnal auto-corrclation function Propose The Real signals w h ich occur in the control process alw ays im lpy influences o f m any random factors, so the D irective O bject Identification Problem is often related to random process M athem atically, th e C ontrolled O bjcct Identification problem is th e problem th at predicts the trend o f R andom Proccss: y{t ) — f { t ^ ĩ i ) -ị- v{t ) , w here t - tim e; u - vector o f non-random input variables; / ( i , ĩi) - regressive function that reflects the trend o f non-random process or is the m odel o f the identification pro b lem ; v{t) - random error T he T heory o f Prediction and Identification has been studied and developed w ith thousands of scientific w orks m ade p u b lic since last century We can find th e fundam ental results o f studies o f statistics and prediction in [1,2], o f kinetics system identification in detail in [3,4 To use linear algebra m ethods, wci often try to changc th e regressive m odels into linear com ­ bination fo n n s o f coefficients: f{t^u) — w here Ci - param eters, - given com ponent functions B y usin g th is m odel, th e Param eter Identification Problem can be solved easily H ow ever, th is m odel is n o t used to solvs the analysis and synthesise pro b lem o f system s and w e have to transform th is m odel into th e fom i o f sets o f state equations (sets o f C auchy differential equations) or tran sfer function fom i There is a close, easy to exchange relation betw een set o f state equations and tran sfer function T h e tra n sfe r fu n ctio n ’s m odel o f controlled object is often in th e follow ing form; rT r \ W { C , s) = bjjiS Òq “h ] ^ ^ —r s ^ ^ ^ ^ /1 \ w here Ó' - com plex num ber, r ^ - the dead tim e delay; r a ^ n - degree o f num erators and denom inators; c = { r , 6o, 6i , bjrv^ ao, a i , a-n} - vector o f param eters to be determ ined In th e classic w o rk s o f identification, all th e authors concentrated on developing identification m ethods based on pure p o ly n o m ial fraction m odels w ith o u t the dead tim e delay com ponents (i.e set Coưesponding author E -m ail: vutienvict.56@i]mail.com 101 102 Vu Tien Viet / VNƯ Journal o f Science, Mathematics - Physics 24 (2008) Ỉ0Ỉ-109 r — 0) In fact, exists r ^ 0, w e n o rm ally tiy to use approxm iate po ly n o m ial fraction m odels w ith h ish e r degrees o f m , n to in crease th e m odel accuracy W ith th is approach, th e object identification problem w ithout dead tim e d e la y is co n sid ered to b e com pletely solved in th eo ry [1,4 In fact, how ever, ap p ly in g th e p u re p o ly n o m ial fraction m ethods to the objects w ith dead tim e delay is reluctant and in effectiv e in co n tro llin g tech n o lo g y p rocesses such as energy, m etallurgy, because m ost o f the ob jects o b v io u sly h ave th e dead tim e delay To have th e necessary m odel accuracy, w e norm ally increase th e deg ree o f p o ly n o m ial fraction to a great value, and therefore m aking the s}Tithesise problem o f sy stem s m ore com plex, even lose its essence D isregarding th e c h ara cteristics o f dead tim e delay o f an object is one o f th e reasons th a t leads to a great num ber o f research d irectio n s o f control th e o ry im practically developed, even caused a ’’crisis” in the previous cen tu ry [5] To accu rately reflect th e controlled object, w e have to consider dead tim e delay as an existing p a m e te r in clu d ed in th e m odel W hereas, clearly, th e m odel is non-Iincar for the param eters In this case, classic m eth o d s are e ith er ineffective or inapplicable Because o f the ab o v e reaso n s, in order to increase th e applicability, w e recom m end a controlled object identification m eth o d b ased on u sin g d ircc tly m odel ( 1) along w ith the dead tim e delay r and other param eters The fo llo w in g m ethod is b ased on considering th e tim e response o f the object as a random data series Estimation of the object m odel from output response data series Suppose the co n tro lled ob ject has w eig h t fiintion w{ t ) w ith cfFect input u{t)~ predeterm ined, output response is m easu red : y { t ) — x { t ) + v { t ) , w h ere v { t ) - additive noise (figure 1) v ) u(t) M t) , • y[J4 ' ^ Fig Linear control system under random effect W ithout loss o f generality , w e restrict v { t ) ^ u { t ) b e in g th e non-in terco irelatio n scalars, w here v{t) is W hite N oise, u { t ) is step p u lse: « { « ) = ( '’ ^ ' [1 " i™ w h en ( 2) Í > Form erly [3,4], so as to solve th e moQol estim atio n problem , w e based on po p u lar relation betw een output response x { t ) o f o b ject and in p u t signal u{t)\ x{t) = [ w { ^ u { t - Ẹ ) d ^ Jo (3) w here w{ t ) is the w e ig h t fu n ctio n From (3), w e esta b lish th e P roblem o f D efin in g w eig h t fiintion w { i ) upon least square condition: \ [ Jo (y{t) - [ \ Jo dt m in (4) Vu Turn Viet / m Journal o f Science, M athematics - Physics 24 (2008) Ỉ0Ỉ - Ỉ 09 103 and then, define the tran sfer function i y ( 6) from th e w eig h t fo ntion w{ t ) If wc param eterize the funtion w{ t ) in the fom i o f lin ear co m b in atio n , the (4) problem w ill bccom c linear to coefficients and can be solved easily H ow ever, in th is w a y it is com plcx to sclcct com ponent functions and causes the problem biiigcn and th erefo re m ::kcs th e problem illconditioncd To avoid this draw back, w e rccom m cnd d ircctly u sin g th e m o d el o f tran sfer function in fom i (.1) and salve the m odel estim ation problem based on th e inverse l.a p la c e transform ation Indeed, if w e co n sid er argum ents in the L.aplace imatỉc dom ain, w e obtain: —rs X ( C , 6-) = VK(C, s ) U{ s ) ^ ^0 + + + -Ị-' + + cijiS w here x { s ) ^ (5 ^ L { x { t ) } , v y (c s ) ^ L { w { t ) } , u { s ) = L{u{t ) }- , s - com plcx variable; L{.} roo - notation fo r Laplacc transform ation; G{ s) = L { g { t ) } = / g{t ) e~' ' ^dt - Laplace transfom iation, Jo from any g{t ) function in real dom ain (g{t) — ^ 0) into co rrcsp o n d in im aac G{ s ) in com plex dom ain A ccording to th e Inverse Laplacc transform ation, in [6 ] w e infcrcd a sim ple form ula to computti tim e response x { t ) from its Laplacc imaac: rgijoo X (C ,s)e^^ds= ^ 7T o = 27rj Jg-j oo P{C, Lo) cos{ujt)(Lj (6 ) w h ere g ' converging ab scissa o f Laplace integral ( if objects are stab le, w e can select ^ > sm all enough, for instance Q — 0.01); X{C,cu jo c ) = —1 , P ( C ,o j i = R e { X ( C ,c j -\' j o c ) } - the real part of Sclect the u p p er lim it (u^m ) o f th e integral w h ic h is b ig enougli, th en trasform into approxim ate sum fonn, w c obtain: x { C , t) = — V P ( C , LUr) COs{uJrt) 7T ^ r -^1 (7) w here M “ the n u m b er o f d iscrete points in frcq u cn cy range: UJ = ^ COMFrom here, w e obtain square error betw een o u u t response and real data: ^‘ { C ) = fT / [y{t) — x{ C, t ) ] ' ^ d t Jo0 mm c w here T - the am ou n t o f tim e to observe the random ou u t d ata series o f real objects R egarding the d iscrete points o f tim e series, w e o b tain m in im izatio n objective function: m in (8) w here N - the n u m b er o f d iscrete points in an interval o f observed tim e: Í = ~ T O bjective fu n ctio n ’s valu e a ^ (C ) is determ ined after a c o m p u tin g process in the follow ing order: c — > W{C, g-\-jio) — ^ X (C , Ơ + jc j) — ^ P{ C^ u j ) — > x { C ^ t ) — ^ (J^(C) Therefore, a ^ (C ) is a com putable function and is continuous and d ifferen tiab le every'w here O n the other hand, is obvious no n -lin ear fu n ctio n s to param eters and esp ecially h ave th e co m p lex cleft (ravine) characteristic W ith these characteristics, th e m ost effective m ethods to solve th e m in im ized problem (8 ) is to apply ”c le ft-o v c f’ optim ization algorithm [7,8 Vu Tien Viet / VNU Jo u ’Tial o f Science, Mathematics - Physics 24 (2008) I0Ỉ-109 104 T he so lu tio n to th e ( ) problem w ith the selected structure ( m , n , q) o f m odel (1) give us an optim al estim atio n a;(C*, t) w ith ĩj{ti) series, and to s e th sr w ith th e optim al tran sfer function W^(C* s) respectively Determ ine the optimal estim ation model A s above, w ith each selected struture ( m n , q), an optim al solution is output response m odel a;(C*, t) and th e W { C * , s) optim al tran sfer function, respectively D epending on the selected (m , n , q) com bination, T h ere are infinite structures o f the m odel So, th e facing problem is to find a (rn, n , q) struture so th a t th e co esp o n d in g solution to th e problem (8 ) b rings out the response x (C * , t), w hich is the p ro p er estim atio n for th e y{ti) tim e series A cco rd in g to [1,2], the m odel is cosidered as a p ro p er estim ation if th e obtained error series betw een the given m odel and tim e series becom o a radom distrib u tio n range in the fonn o f "w hite noise” A ssum e a a significance level, the m odel is considered to be accurate if Goưelation coefticients value ri o f erro r series satisfy th e follow ed condition; nl ^ u^ / V N (9) w h ere U a - is th e lim ited value obeying the norm al d istrib u tio n rules, N - the n u m b er o f data sets o f series O n the o th e r hand, th e (1) mode] is fractional, so if w e increase th e (m , n) degrees its accuracy w ill increase as a result Particularly, q is the n o n static degree o f m odel, it depends on th e behaviour o f output response and is equal to the degree o f th e asym ptote o f output response (J = if the asym ptote o f output respo n se is horizontal asym ptote I i f output has no asym ptote In fact, Ọ < in m o st eases To d efin e th e global optim al estim ation m odel, th e steps to solve th e identification problem arc as follow s: S elect th e degree o f q and fix it from th e output resp o n se’s behaviour E xp lo rativ ely select values o f d en o m in ato r’s degree n, and values o f n om inator’s degree m = n — I sim ultaneously F o r each selected { m, n, q) structure w e solve th e (8 ) estim ation m odel problem W ith th e respective s) and x { c * , t) obtained, w e detcm iin e error scries and check the condition u p o n m odel suitability I f th e condition is satisfied, th e obtained m odel is optim al and th e respective W { C * , s) tran sfer iu n c tio n is th e solution to th e identification problem I f th e m odel is not suitable, w e w ill select other m odels w ith m and n ’s degree gradually increased and reo eat from step E x a m p le S uppose th e output signal o f an im plem ent controlled object is the step p u lse in form th a t tim e, from i ti Vi 0.0 0 - 0 th e output, th e m easured response signal in form o f tim e series 0.2054 0.0060 0.4008 0.2540 0.5962 0.6000 0.7916 1.0641 0.9870 1.3125 1.1824 1.4700 1.3778 1.4126 (2) A t is as follows: 1.5732 1.2500 10 1.7686 1.2191 105 Vu Tien Viet / VNƯ Journal o f Science, Mathematics - Physics 24 (2008) Ỉ 0Ỉ - Ỉ 09 i u i tr in 11 1.9G40 1.0800 12 2.1594 1.0830 13 2.3548 1.0127 14 2.5502 1.0300 IG 2.7457 0.9900 16 2.9411 1.0310 17 3,1365 0.9407 J8 3.3319 0.9900 19 r/2 73 0.9600 20 3.7227 0.9740 21 3.9181 0.9Õ00 22 4.1135 0.9900 23 4.3089 0.9560 24 4.5043 0.9960 25 4.6997 0.9539 26 4.8951 1.0400 27 Õ.0905 0.9680 28 5.2859 1.0510 29 5.4813 0.9900 30 5.6767 1.0211 The G raph o f the tim e series y{t ) obtained from cxperiem cnt is show n in figure Ỉ IV \ i \ ! \ '• ị ị -> ị [ i :ị Fig The curv'e o f output response data scries o f dircclivc object We id en tify the tran sfer function o f objcct b ased on m odel (5) b y solving th e (8 ) problem w ith diftcrcnt structures The im age o f input signal step pulse; u { s ) = / s T he b eh av io u r o f the data series above is co csp o n d in g to the nonstatic degree w here /]V is d e a r ly not satisfied T he second struture, w c choose: m = 0, n = The optim al m odel is: W o(C * s) = - _g - 0,108., ’ ’ l + a i + a2s2' l + , s + 0, 123s2' T he error series b etw een the output response X {C*^ t) and the m easured data is on figure 3-b T he root-m can-square E rror o f th e m odel is Ỡ “ 0, 0682 T his m odel briniĩs out the in c o e ct estim ation because the condition \ri\ ^ U / \ / ĩ is still not satisfied W ith th e third struture, w e choose; m — 1, n — T he optim al m odel is: , 6o( l + ?ii6') , ( + 1,1666') (1 -I- a i s + a 26 '2 ) ( l + a s s ) ■ (1 + 0, s + 0, 0986-^)(l + 0, 6836') ■ T he error series betw een the output response ,X3(C*, t) and the m easured data is on figure 3-c T he root-m can-squarc E rror o f th e m odel is Ỡ = 0, 0342 T h is m odel brings out the nearly e s tim a tio n , th e c o n d itio n u p o n m o d el su ita b ility |r,;| ^ U ẹ / \ Í N is n e a rly sa tisfie d T he fourth struture, w e choose: m = 2, n = The optim al m odel is; (1 + a i s + a S^) {ỉ + a a s + 045^) 1, 001(1 + 0, 716s + 0, 37s2) “ - 0,183 (1 + 0, 202s + 0, s )(l + 0, 5 s + 0, 414s2) T he e o r series b etw een the output response a;4( c * , t) and the m easured data is on figure 3-d T he root-m ean-square E o r o f th e m odel is o' — 0, 0212 T his m odel brings out the co cct estim ation since the condition upon m odel su itab ility Ir^l ^ U i y / \ / N is co m p letely satisfied W hile increasing th e degrees o f m ,n o f th e m odel, th e e o r series is alm ost non-decrcasing In the optim izin g process, th e old coefticients are alm ost invariable, w hereas the added coeftlcicnts a risin g w h ile increasing the degree o f m odel - are alw ays forced to zero b y the algorithm The optim al solution is nearly at a stand still A ccording to th ese results, th e tran sfer function m odel W 4{C*^ s) h as erro r series w h ich is sim ilar to ’’w hite n o ise ” , and sim u ltan eo u sly yields The root-m ean-square E rror m inim um Therefore, w e can considered it as th e global optim al m odel o f object T he respective output response o f the m odel is show n on figure Conclusion The objects w ith dead tim e delay are p o p u lar class o f objects in industrial contro tra n sfe r function has n o n -lin ear property for p aram eters, therefore classic identification m ethods have low eftectiveness Pu Tien lĩet / VNU Journal o f Science, Mathematics - Physics 24 (200H) Ỉ 0Ĩ- Ỉ 09 107 R ecom m ending usin g the transfer function w ith dead tim e delay as the basic m odel and by using the inverse L aplace transfom iation w c obtain th e output response o f the m odel On this basis w c solve the objcct identification problem in the fo n n o f the tim e series estim ation problem based on m easured random d ata o f controlled objcct The rccom m cndcd m ethod in this report enable us to solve the directive objcct identification problem licncrally and effectively un d er the random noise The optim al m odel o f objcct dctcm iincd b y the estim ation m ethod for non-linear random m odel ensures the su itab ility according to pro b ab ility and statistic’s p o in t o f view References |1] G Christian, A Monibrt, Times Deries and Dynamic Models, Cambridge University Press 1997 [2] Nguyen Van IIuu Nguyen fluu Du, Statislic Analysis and Forecast, Hanoi National University' Press, 2003 (in Viet­ namese) 13] P Eykluiil', System Identification Wiley 1974 [4] P I''ykh