Gradient-based step response identification of low-order model for time delay systems Rui Yan, Fengwei Chen, Shijian Dong, Tao Liu*

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1 Gradint-basd stp rspons idntification of low-ordr modl for tim dlay systms Rui Yan, Fngwi Chn, Shijian Dong, ao Liu* Institut of Advancd Control chnology, Dalian Univrsity of chnology, Dalian, 64, P R China * Corrsponding author l: mail: liuroutr@iorg ( Liu) Abstract: In this papr, a stp rspons idntification mthod is proposd to obtain a low-ordr modl for industrial procsss with tim dlay Basd on a classification of th modl pol distribution, th corrsponding tim domain xprssions of th output rspons to a stp chang ar drivd and thrfor, a gradint sarching algorithm is dvlopd to simultanously idntify th rational modl paramtrs togthr with th dlay paramtr h computation ffort can b significantly rducd compard to th xisting tim domain multipl-intgral mthods for modl fitting h convrgnc of th proposd algorithm is analyzd with a strict proof wo illustrativ xampls from th litratur ar shown to dmonstrat th ffctivnss of th proposd mthod Ky Words: Stp rspons idntification, tim dlay, scond-ordr modl, gradint sarching, convrgnc IRODUCIO For control-orintd modl idntification in industrial nginring applications, stp rspons tsts hav bn widly usd owing to its implmntal simplicity and conomy In th past dcads, th idntification of procss modls with tim dlay has rcivd incrasing attntion in viw of th fact that tim dlay is usually involvd with industrial procsss and systm oprations [-3] Early rfrncs (g [4]) proposd th approximation of tim dlay by using Padé approximation and Lagurr xpansion, rsulting in a highr ordr rational transfr function modl that contains mor paramtrs to b stimatd and may caus unaccptabl fitting rror whn th procss rspons has a long tim dlay By fitting a fw rprsntativ points in th transint output rspons to a stp chang, th rfrnc [5] prsntd altrnativ idntification mthods for obtaining a first-ordr-plus-dad-tim (FOPD) or scond-ordr-plus-dad-tim (SOPD) modl Basd on numrical intgral to th tim domain xprssion of stp rspons, th paprs [6, 7] introducd n -fold multipl intgrals to idntify a n -th ordr procss modl ot that ths multipl-intgral basd idntification mthods can giv good accuracy and robustnss in comparison with th prvious mthods, but th computation load is rlativly high and th us of multipl tim intgrals for paramtr stimation may b snsitiv to th tst data lngth In th rcnt yars, a frquncy domain stp idntification mthod was proposd [8] to rduc th computation ffort by introducing a damping factor to th stp rspons for th computation of th Laplac transform A gnralizd xpctation-maximization algorithm was suggstd for robust idntification of linar paramtr varying (LPV) systms with fixd input dlay [9] For idntifying a canonical stat spac modl with stat dlay, a rcursiv last-squars paramtr idntification algorithm was prsntd in [] basd on using a stat filtr o handl th nonlinar stimation problm, a fw numrical optimization mthods wr dvlopd for stimating both th linar modl paramtrs and th dlay paramtr, g hirachical idntification stratgis [], Gradint-basd sarching algorithm [,3], wton-li approachs [4, 5] ot that ths numrical optimization mthods wr dvlopd basd on using prsistnt xcitation signals such as th psudo random binary squnc (PRBS), which may not b xcutabl in many industrial applications In this papr, a stp rspons idntification mthod is proposd to obtain a low-ordr modl for industrial procsss with tim dlay, to furthr xtnd our prvious wor [6] that was limitd to ovrdampd procsss By introducing a nw tim domain xprssion of th procss transfr function modl, a modifid gradint-basd idntification mthod is proposd which can simultanously idntify linar modl paramtrs togthr with th dlay paramtr from th stp rspons data Morovr, no tim intgral is rquird for computation, significantly rducing th computation ffort compard to th rcntly dvlopd multipl-intgral basd idntification mthods Owing to th us of a stp tst, it is vrifid that th cost function of prdiction rror bcoms convx with rspct to all th modl paramtrs in a wid rgion, such that good convrgnc can b guarantd For clarity, th papr is organizd as follows: Sction prsnts th low-ordr procss modls to b idntifid h corrsponding idntification mthod is dtaild in Sction 3, togthr with som guidlins for modl structur slction h convrgnc is analyzd in Sction 4 wo numrical xampls ar shown in Sction 5 Conclusions ar drawn in Sction 6 PROBLEM FORMULAIO For industrial procsss, g multi-componnt blnding ractors and frmntation tans, low-ordr modls, g FOPD and SOPD, ar widly usd for control systm dsign and tuning [7] Without loss of gnrality, th procss modls ar dividd into thr typs according to th distribution of pols, th first on has a singl or rptitiv pols, p Gm() s = m () ( s + ) p /6/$3 c 6 IEEE 6

2 th scond on has two distinct ral pols, p Gm () s = () ( s+ )( s+ ) and th last on has two conjugat pols, p( + ) Gm3 () s = (3) ( s + ) + whr p dnots th procss static gain, L is th procss tim dlay, p,, ar tim constants, and m is th numbr of rptitiv pols or th modl ordr It should b notd that highr-ordr procsss can b ffctivly dscribd by th abov modls [7], [8] Undr a stp tst, th input xcitation with a magnitud of h can b dscribd by ut () = ht, (4) Corrspondingly, th tim domain stp rspons of th procss modls shown in (), (), (3) can b xprssd by ( t L)/ p yt () = y( ) + v( t), t L (5) ( t L)/ ( t L)/ yt () = y( ) + + (6) + v t, t > L () y() t = y( ) cos ( ( t L) ) sin ( ( t L) ) b + + > ( ) ( ) ( t L) ( t L) sin ( t L) v t, t L whr vt () is assumd to b a whit nois with zro man Gnrally, th procss static gain, p, can b dirctly computd from y( ) p = (8) h whr y( ) is obtaind by avraging -5 masurd output valus aftr th procss rspons movs into a stady stat 3 PROPOSED IDEIFICAIO ALGORIEHM o giv a gnral idntification algorithm for th modls shown in (), (), and (3), th idntification algorithm is dtaild for a scond-ordr modl with a zro as xprssd by bs + b Gm4 () s = (9) ( s + ) + Corrspondingly, th tim domain stp rspons is shown into y() t = y( ) cos ( ( t L) ) sin ( ( t L) ) () ( t L) b ( t L) + sin ( ( t L) ) + v( t), t > L o stimat all th unnown modl paramtrs, w dfin a paramtr vctor for stimation, (7) θ = [,, b, L] () h output prdiction rror can b computd by δ(, t θ) = y() t cos( ( t L)) sin( ( t L)) () ( t L) b ( t L) y( ) sin( ( t L)) It is obvious that th unnown tim dlay impd th computation of prdiction rror in () o rsolv th problm, th following cost function is adoptd hrin, ˆ J( θ) = ε ε (3) whr ε = [ δ( t ˆ ˆ, θ),, δ( t, )] θ, ˆ θ dnots an stimat of th unnown paramtrs, yt ˆ( ) is th prdictd output, dnots th collctd data lngth, and t i ( i=,,, ) ar th sampling instants which should b chosn such as L t < t < < t o minimiz th cost function in (3), w comput th first-ordr drivativ for () with rspct to,, b and L, rspctivly, obtaining δ( ti, θ) ( t L) b( t L) ( t L) = y( ) + sin [ ( t L) ] ( t L) ( t L) y( ) cos [ ( t L) ] (4) δ( ti, θ) ( t L) b = ( t L) cos [ ( t L) ] y( ) (5) ( t L) b sin [ ( t L) ] y( )( t L+ ) δ( t i, θ) [ ] ( t L = ) sin ( t L) (6) b δ( ti, θ) ( t L) b = sin [ ( t L) ] y( )( + ) L (7) ( t L) + b cos ( t L) [ ] h Jacobian matrix of () is dfind by δ( ti, θ) δ( ti, θ) δ( ti, θ) δ( ti, θ) δ( ti, θ) Ξ ( ti, θ ) = =,,, θ b L (8) Dnot Ψ ( θ) = [ Ξ( t, θ), Ξ( t, θ),, Ξ( t, )] θ (9) h gradint matrix of J ( θ ) with rspct to θ is thrfor dfind by g( θ) =Ψ ( θ) ε () h corrsponding Hssian matrix is formulatd by H ( θ) =Ψ ( θ) Ψ ( θ) () Hnc, th paramtr vctor θ can b stimatd using a Guass-wton itration approach, i 6 6 8th Chins Control and Dcision Confrnc (CCDC)

3 ˆ ˆ θ = θ H g () whr H and g can b computd from () and () using ˆ θ stimatd from th prvious itration It is wll nown that th standard Guass-wton itration mthod may giv a slow convrgnc rat in th initial itrations, causing a considrabl computation ffort o ovrcom th dficincy, w propos th us of an adaptiv sarching stp for itration basd on th Armijo lin sarching stratgy (s g in [9]), i ˆ m ˆ m J( θ + ρ d) J( θ ) + σρ gd (3) whr d = H g dnots th sarching stp siz, m is th minimal nonngativ intgr satisfying (3) For th convninc of implmntation, w ta σ = ρ and ρ = min{ ρ,8} whr ρ = ( J J + J J), following a similar choic givn in our prvious wor [6] to facilitat th convrgnc A suitabl initial valu is rquird for th Guass-wton itration to avoid convrgnc to an impropr local minimum Howvr, it is wll nown that th convxity of th cost function is rlatd to th typ of input xcitation for idntification [4] o addrss this issu, a graphical study on th rlationship btwn J ( θ ) and th SOPD modl paramtrs is xplord basd on using diffrnt input xcitations Considr a scond-ordr procss with tim dlay Ls dscribd by Gs () = ( s + αs+ ), whr α = and L = 5 h input xcitation is tan as a unity stp chang or a PRBS squnc with th varianc of σ =, rspctivly h numbr of masurd output data is = 3 and th Cost Function Cost Function x α L Figur Plot of th cost function undr a stp tst x 4 3 L 3 4 α Figur Plot of th cost function undr a PRBS xcitation 3 4 sampling priod is s = (s) h plots of J ( θ ) with rspct to α and L ar shown in th Fig and Fig It is sn from Fig that thr is a uniqu minimum, whil Fig has multipl minima h rason lis with that a stp signal contains only low frquncy componnts that provo th output rspons mainly in th low frquncy rang his implis that th cost function is convx in a wid rgion using a stp tst for modl idntification, and th initial choic of modl paramtrs for itration is not vry limitd A small positiv ral numbr is thrfor suggstd to initializ all th modl paramtrs for itration Hnc, th proposd idntification algorithm is summarizd as blow: (i) Collct th input and output obsrvation data, { ut ( i), yt ( i) } ( i=,,, ), from a stp tst (ii) a an initial stimation of θ dnotd by θ with small positiv ral numbrs (iii) Comput δ(, t ˆ θ ) and J ( ˆ θ ) by () and (3) (iv) Dtrmin th sarching stp siz d = H g by computing th gradint matrix g in () and th Hssian matrix H in () (v) Updat ˆ ˆ m θ = θ + μd, whr μ = ρ, and m is th minimal nonngativ intgr that satisfis (3) (vi) End th algorithm if th fitting condition, y( t ) yˆ ( t ) < rr, is satisfid, whr rr is a [ ] i i i= usr spcifid fitting thrshold Othrwis, lt = + and go to stp (iii) With th abov algorithm, b is obtaind by y( )( + ) b = (4) h 4 COVERGECE AALYSIS o analyz th convrgnc proprty of th proposd algorithm, th following proposition is givn accordingly Proposition : Assuming that δ(, t θ ) is Lipschitz continuously diffrntiabl in a nighborhood Ω of th boundd st S = { θ J( θ) J( θ )}, and th Jacobian matrix Ψ ( θ ) in (9) is full row ran togthr with th positiv dfinit Hssian matrix H ( θ ) in (), th proposd algorithm algorithm guarant th uniform convrgnc satisfying lim g( θ) = lim Ψ ( θ) ε( θ) = (5) whr εθ ( ) [ (, ),, (, )] = δt θ δt θ Proof: Sinc δ(, t θ ) is Lipschitz continuously diffrntiabl in th boundd st S, thr follows for som positiv constants β and β that δ(, t θ) β, θ S and t > L (6) 6 8th Chins Control and Dcision Confrnc (CCDC) 63

4 Ξ(, t θ) β, θ S and t > L (7) whr dnots th matrix -norm It can b vrifid that thr xists a constant q > such that Ψ ( θ) = Ψ ( θ) q (8) Owing to that Ψ ( θ ) is full column ran, thr stands for a sufficint larg constant, p >, such that Ψ( θ)z p z (9) Dnot by ω th intrsction angl btwn th ngativ gradint dirction, g, and th Guass-wton sarching dirction of cos( ω ) It can b drivd using (8) and (9) that G d = G G G gd ( d ) Ψ ( θ) Ψ( θ) d = G G G g d Ψ ( θ) Ψ ( θ) d d G G Ψ( θ) d p d G G ( θ ) ( ) G Ψ θ d d q d = Ψ (3) p = > q It follows from (3) that G G g p = g d d = g cos( α ) (3) m G whr p = s s, and s = ρ d a th limit to Eq(3), w gt lim g G G d d = lim g cos( α ) > (3) Hnc, thr is lim g p < (33) * * * Assum θ is a clustr point, but Δ J ( θ ) = g( θ ) From th assumption and th Armijo condition in (3), w hav ˆ m G J( θ ˆ + ρ d ) J( θ ) (34) g s (35) s (36) ot that in th Armijo condition, m is th minimum nonngativ intgr satisfying th inquality (3) So th inquality (3) can b rarrangd as ˆ m G ˆ m G J( θ ρ + d ) J( θ ) > σgρ d (37) From th assumption in (36), w now Q = s ρ hn (37) can b rarrangd as J( ˆ θ ˆ + Qp) J( θ ) > σ g p (38) Q a th limit on both sids of Eq(38), w obtain lim g p σ lim g p (39) It follows that lim g p (4) which is contradictory to (33) Hnc, thr stands lim g p = (4) Sinc p = and { p } is boundd, thr xists a * sub-squnc of { p } that convrgs to P which satisfis (4) According to th proprtis of a boundd sris [], thr follows for { p } with that lim g ( θ) = (4) his complts th proof ot that th convrgnc rsult shown in (5) indicats that th cost function J ( θ ) convrgs to a stady valu whn, that is to say, th stimatd modl paramtrs convrg to th tru valus in cas prfct modl match with th plant, lading to lim θ = lim( θ θ ) = (43) Owing to th us of an instrumntal modl in th proposd algorithm, th itration is isolatd from th masurmnt nois, guaranting good robustnss for convrgnc ot that th tru modl structur of th procss to b idntifid is difficult to b nown in practical applications It is dsirabl to dtrmin th optimal modl structur for rprsnting th procss dynamic rspons charactristics from a stp tst h following hypothsis tsting condition [8], can b tan for choosing th optimal modl ordr, [ yt ˆ( i) yt ( i)] n i= (44) [ yt ˆ( ) yt ( )] i= i i n whr yt ˆ( ) and yt () dnots, rspctivly, th masurd procss output and th modl output to th stp chang, n dnots th currnt modl ordr and n a highr ordr to b vrifid It can b sn from (44) that a highr modl ordr n should b accptd only if th output prdiction rror is largr than on-tnth of that of th currnt modl with n Whn th modl ordr is dtrmind, it is asy to distinguish th typ of th modl pols from th stp rspons tst ot that a SOPD modl with rptitiv pols has an obviously largr curvatur in th transint stp rspons compard to an SOPD modl with two distinct pols, and a SOPD modl with two conjugat pols has obvious oscillation in th transint stp rspons 5 ILLUSRAIO wo xampls from th litratur ar studid to illustrat th prformanc of th proposd mthod h following tim domain fitting critrion is usd to assss stimat accuracy, rr = [ y( t ) ˆ( )] i y t (45) i i = h paramtr stimation rror is assssd by th following critrion, th Chins Control and Dcision Confrnc (CCDC)

5 ERR ˆ θ θ = % (46) θ whr ˆ θ is an stimation of th tru paramtr vctor θ Exampl Considr a scond-ordr procss with a positiv zro studid in th rfrncs [6, 8], s ( 4s+ ) Gs () = 9s + 4s + Basd on a unity stp rspons tst, Liu t al [8] gav a SOPD modl, s Gs ( ) = ( 39989s+ 9998) ( 983s + 395s+ ) and Wang t al [6] obtaind s Gs ( ) = ( 4444s+ ) ( s + 667s+ ) By prforming th sam stp tst, th proposd algorithm givs th procss modl, Gs ( ) = 4s+ s 9s + 4s+ ( ) ( ) in trms of th sampling intrval s = (s), th rspons tim = (s), and th initial valus for itration θ = [,,,] o dmonstrat th robustnss undr diffrnt nois to signal ratio (SR) lvls, i SR=%, %, Mont Carlo tsts ar conductd abl shows th idntification rsult of th proposd algorithm, indicating rapid convrgnc undr ths nois lvls abl lists th avragd itration numbrs whr th intrnal loop indicats th sum of itration numbr for dtrmining m in (3) in ach tst, and th xtrnal loop indicats th itration numbr of sarching stp for convrging to th optimal stimation It can b sn from abl and abl that th proposd algorithm guarants a fast convrgnt spd and maintain good robustnss against masurmnt nois abl 3 shows th stimation rsults of th proposd mthod with rspct to th sampling frquncy f = s (corrsponding to diffrnt data lngths for idntification) undr SR = % h paramtrs shown hr ar th man of Mont Carlo stimation h numbrs in th parnthss ar th standard dviations of stimats It can b sn that good idntification accuracy and consistnt stimation ar obtaind by th proposd algorithm, whil it is not snsitiv to th data lngth collctd for modl idntification abl Idntification rsults undr diffrnt nois lvls SR ˆ ˆ ˆL ERR(%) % % ru abl Avragd itration numbrs undr Mont-Carlo tsts SR Extrnal loop Intrnal loop Succss ratio % 3 5 % 4 5 In addition, th computation load for th proposd mthod is listd in abl 4 in comparison with a multipl intgral mthod givn by Wang t al [6] In abl 4, dnots th numbr of sampling data and K is th itration numbr It can b sn from abl 4 that th proposd algorithm uss obviously lss computational ffort compard to th multipl intgral mthod sinc is much largr than K Exampl Considr th fifth-ordr systm with tim dlay studid by Wang t al [6], 8 s Gs () = 3 ( s+ ) (s+ ) Basd on a stp tst, Wang t al [6] drivd an SOPD modl, abl 3 Paramtr stimation undr diffrnt sampling frquncis Frquncy â ˆL 5Hz 899(±379) 4(±944) -4(±94) 99(±56) (±73) 5Hz 9(±846) 4(±44) -4(±98) 99(±6) 99(±5) 5Hz 899(±5) 4(±8) -4(±685) (±7) (±638) 75Hz 898(±78) 4(±34) -398(±556) (±5) (±499) Hz 9(±87) 4(±33) -4(±543) (±4) (±493) ru abl 4 Comparison of th computation load Mthods Addition Multiplication Proposd K(58+ 4) K(9+ 97) Rf [8] ( )( ) ( + )( + ) + ( + ) ˆb 6 8th Chins Control and Dcision Confrnc (CCDC) 65

6 7 s Gs ( ) = 8 789s + 533s + corrsponding to th man-squard output fitting rror, 4 rr = 57 For comparison, th sam stp tst is prformd for th sampling tim = 5 (s) with th sampling intrval s = (s) An SOPD modl is obtaind by using th proposd algorithm, 59s Gs ( ) = 8 775s s +, -4 corrsponding to rr = 488 hn assum that th output masurmnt nois lvl is SR=%, th proposd mthod givs 49s Gs ( ) = s + 565s +, corrsponding 5 to rr = 37, compard with that of Wang t al [6], 4s Gs ( ) = 789 s s +, corrsponding 4 to rr = 34 h stp rsponss of ths modls ar shown in Fig3, wll dmonstrating good fitting of th proposd mthod Output Proposd Rf [6] im (sc) 6 COCLUSIOS Basd on a classification of th modl pol distribution, a stp rspons idntification mthod has bn proposd for industrial procsss with tim dlay, along with a proof on th convrgnc By dvloping a gradint-basd sarching approach to minimiz th output prdiction rror, th linar modl paramtrs togthr with th dlay paramtr can b simultanously idntifid from th tim domain xprssion of a low-ordr modl rspons to a stp chang h computation ffort can b significantly rducd compard to rcntly dvlopd stp rspons idntification mthods (g [6, 7]) basd on using multipl intgrals to stp rspons data Morovr, th proposd algorithm is not snsitiv to th data lngth collctd for modl idntification, thrfor facilitating practical applications REFERECES Ral procss Figur 3 Stp rspons fitting for Exampl [] H Garnir and L Wang (Ed) Idntification of continuous-tim modls from sampld data Springr-Vrlag, London, 8 [] P C Young, Rcursiv Estimation and im-sris Analysis: An Introduction for th Studnt and Practitionr Springr-Vrlag, Brlin, [3] Liu, Q G Wang, H P Huang, A tutorial rviw on procss idntification from stp or rlay fdbac tst, Journal of Procss Control, Vol 3, o, , 3 [4] P J Gawthrop, M ihtilä, Idntification of tim dlays using a polynomial idntification mthod, Systms & Control Lttrs, Vol 5, o 4, 67-7, 985 [5] H P Huang, M W L, C L Chn, A systm of procdurs for idntification of simpl modls using transint stp rspons, Industrial & Enginring Chmistry Rsarch, Vol 4, o 8, 93-95, [6] Q G Wang, X Guo, Y Zhang, Dirct idntification of continuous tim dlay systms from stp rsponss, Journal of Procss Control, Vol, o 5, 53-54, [7] Liu, F Zhou, Y Yang, F Gao, Stp rspons idntification undr inhrnt-typ load disturbanc with application to injction molding, Industrial & Enginring Chmistry Rsarch, Vol 49, o, 57-58, [8] Liu, F Gao, A frquncy domain stp rspons idntification mthod for continuous-tim procsss with tim dlay, Journal of Procss Control, Vol, o 7, 8-89, [9] X Yang, Y Lu, Z Yan, Robust global idntification of linar paramtr varying systms with gnralisd xpctation maximisation algorithm, IE Control hory & Applications, Vol 9, o 7, 3-, 5 [] Y Gu, F Ding, J Li, Stat filtring and paramtr stimation for linar systms with d-stp stat-dlay, IE Signal Procssing, Vol 8, o6, , 4 [] F Ding, X Liu, J Chu, Gradint-basd and last-squars-basd itrativ algorithms for Hammrstin systms using th hirarchical idntification principl, IE Control hory & Applications, Vol 7, o, 76-84, 3 [] J a, X M Rn, Y Xia, Adaptiv paramtr idntification of linar SISO systms with unnown tim-dlay, Systms & Control Lttrs, Vol 66, 43-5, 4 [3] F Chn, H Garnir, M Gilson, Robust idntification of continuous-tim modls with arbitrary tim-dlay from irrgularly sampld data, Journal of Procss Control, Vol 5, 9-7, 5 [4] K Ji, L Xu, W Xiong, L Chn, wton itrativ algorithm basd modling and proportional drivativ controllr dsign for scond-ordr systms, Journal of Applid Mathmatics and Computing, -6, 4 [5] S W Sung, I B L, Prdiction rror idntification mthod for continuous-tim procsss with tim dlay, Industrial & Enginring Chmistry Rsarch, Vol 4, o4, , [6] R Yan, Liu, F Chn, S Dong, Gradint-basd stp rspons idntification of ovrdampd procsss with tim dlay, Systms Scinc & Control Enginring, Vol 3, o, 54-53, 5 [7] D E Sborg, D A Mllichamp, F Edgar, F J Doyl III, Procss Dynamics and Control, 3nd Edition, Hobon, J: John Wily & Sons, [8] Liu, F Gao, Industrial Procss Idntification and Control Dsign: Stp-tst and Rlay-Exprimnt-Basd Mthods Springr, London UK, [9] S J Wright, J ocdal, umrical Optimization w Yor: Springr, 999 [] B P Rynn, M A Youngson, Linar functional analysis Springr, London UK, th Chins Control and Dcision Confrnc (CCDC)

2008 AP Calculus BC Multiple Choice Exam

2008 AP Calculus BC Multiple Choice Exam 008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl