Variance of Estimates in Dynamic Data Reconciliation

Transcription

1 9 th Eropean Sypos on Copter Aded Proess Engneerng ESCAPE9 J. Jeżowsk and J. hlle (Edtors) 2009 Elsever B.V./Ltd. All rghts reserved. Varane of Estates n Dyna Data Reonlaton Chrstophe Ullrh, Georges Heyen and Carne Gerkens Unversty of Lège, Lass, Allée de la Che 7, B6A, Sart-lan, Lège B-4000, Belg, llrh@lg.a.be Abstrat he ethod prevosly proposed to estate the nertanty of valdated ables n steady state data reonlaton has been etended to dyna data reonlaton. he approah sed n ths artle to estate a posteror anes n the ase of dyna date valdaton s based on the one desrbed n [2] for the statonary ase. Orthogonal olloatons are sed to dsretse ODE. Reslts are presented for an adabat reator wth frst order knet. Keywords: dyna data reonlaton, a posteror anes, orthogonal olloatons. Introdton Effent proess ontorng s a key sse n plant operaton, sne easreent errors are always present. o address ths sse, data valdaton s nowadays rotnely perfored for steady state proesses, bt dyna systes stll present soe hallenges. Data valdaton ses easreent redndany and odel onstrants to rede easreent nertanty and to allate non easred state ables of the syste. A posteror ane for valdated ables opared to raw easreents an be allated for lnear or lnearzed steady state systes. Several ethods are enable to solve the

2 C. Ullrh et al. dyna data reonlaton proble [2]. We se NLP tehnqe and orthogonal olloaton [3] to dsretze the ODE systes, as desrbed n []. 2. Estaton of a posteror anes he algorth ses ovng horzon s sed to lt the sze of the optzaton proble. hs ovng horzon s desrbed by the followng fgre. he valdaton wndow s defned by fve paraeters: - h : easreent freqeny - h 2 : sze of the nterpolaton of the npt ables - h 3 : dsretzaton nterval of the dfferental state ables - h 4 : sze of the ovng wndow - h 5 : the ove of the wndow after optzaton In the ase of orthogonal olloatons, the objetve fnton s: n k, n,,, f nterp nt, f nterp nt s k, s, n n t es es (, j, j) W ( ),, j, j, j j= 0 n zes ( z, j z, j) Wz ( z ),, j, j z, j + nes + (, j, j) W ( ),, j, j, j n (,0, 0 ) R ( ),,0,0 n (,0,0 ) R ( ),,0,0 + + ()

3 Varane of estates n dyna data reonlaton Wth - : the vetor of dfferental state ables; - z: the vetor of algebra ables; - : the vetor of npt ables. hs objetve fnton s sbtted to fve types of onstrants: - the lnk eqatons: they are algebra relatons between all proess ables. hose onstrants have to be satsfed as well at the easreents tes as at the olloatons nodes: A = f ( tj,, z, ) = 0 t j (,,, k ) A = f θ z = 0 θ (2-3) - the relatons between the dfferental state ables and the Lagrange nterpolaton polynoals at all easreent tes of the ovng horzon eept at the ntal tes of the dsretzaton ntervals t :, j k( tj), k, tj t k= 0 B= l = 0 (4) - the lnear nterpolatons of the vales of npt ables between tes t n and t f of the nterpolaton horzon at the other tes of that horzon: tj tn C=, j t,,,,, n t f t = 0 t n j tn t (5) f t t f n θ t C = = 0 (6) k n k, t,,,, n tf t θ n k tf tn - the resdals of the dfferental state eqatons at all olloaton nodes: D= ls ( θk), θ g ( θ,,, ), s k k zk k = 0 θk t (7) n s= 0 - the ontnty onstrants of the dfferental state ables between two dsretzaton ntervals: E = l t l t = 0 k( f) k, k( n) k, k= 0 t, k 0 f dsr nt q = tn, dsr nt q hs onstraned proble an be transfored nto an nonstraned proble sng Lagrange forlaton. he neessary ondton for optalty s epressed by settng to 0 the gradent of the Lagrangan. By lnearsng the eqaton syste as shown n [2], one obtans a lnear relaton between valdated ables and easreents: A A B C C D E ( z z Λ Λ Λ Λ Λ Λ Λ ) M (9) ( P Pz z P F F G 0) = k (8)

4 C. Ullrh et al. Wth F, F and G the onstant ters of the lnear aproaton of the onstrants. One obtans the senstvty atr M whh s the Jaoban atr of the eqaton syste: M = P E E 0 (0) Α Α Α z P A A A Pz z 0 0 P B B = P C E = C C P = + W R Pz = Wz D D D P = + W R z E (-2) As for the statonary estaton [2], posteror anes an be deded fro those last forlas: N 2 N 2 z ( k) = Mk, P, + Mk + N P z z + N 2 k, + N + Nz M P Slar eqatons an be wrtten for npt and algebra ables. (0) 3. Case stdy: an adabat reator wth frst order knet hs reator wth frst order knet s defned by the followng dfferental state eqaton: dca F CA CA k CA dt V = ( ) (4) CA s the state ables whle F and CA are the npts. he proble has no algebra able. he knet onstant k s defned as a onstant of the optzaton proble; so, t an not be optzed.

5 Varane of estates n dyna data reonlaton he paraeters of the wndow have been hosen as follow: h =, h 2 = 4, h 3 = 4, h 4 = 49 and h 5 = 2. he Lagrange nterpolaton polynoals are of the seond order. he proess s sbjet to several npts hanges of the for: d K ( SP ) dt = (5) Fgre : Conentraton profle Fgre 2: Feed flowrate profle As an be seen on fgre and 2, for the onentraton and the feed flowrate profles respetvely, the valdaton allows to rede the nose and the hanges n the profles are very well followed by the valdated vales. Fgres 3 and 4: Unertantes oparson for onentraton and feed flowrate Fgres 3 and 4 show the standard devaton of the easreents and the valdated vales for the onentraton and the feed flowrate respetvely. he anes are reded as well for state ables as for the npts. For the state able, one has the followng reslts: es A pror anes A posteror anes Redton fator

6 C. Ullrh et al. he ane redton s slar for all easreent tes of the valdaton wndow eeptng for the frst tes for whh t s less portant. For the nong flowrate, one has the followng reslts: es A pror anes A posteror anes Redton fator he ane redton s less portant and es ore for the npts along the valdaton wndow. We thnk that t s a onseqene of the way npt ables are defned n the valdaton proble. 4. Conlsons and ftre work he reslts presented n ths artle are for an adabat reator wth a frst order knet. Slar reslts have been obtaned for dfferent systes nldng the eaple desrbed n []. Good redtons for anes of state and algebra ables are obtaned n all ases, bt for npt ables the redtons of the anes are less sgnfant. In the ftre, we plan to eane the nflene of polynoal order sed to odel the npt ables on the ablty of the valdated nertanty. 5. Aknowledgeents he athors are gratefl to the Walloon Regon and the Eropean Soal fnds who o-fnaned ths researh. 6. Referenes [] Leban, M. J., Edgar,. F., Lasdon, L. S., Effent data reonlaton and estaton for dyna proesses sng nonlnear prograng tehnqes, Copters & Cheal Engneerng, Vol. 6, N 0/, , (992) [2] Heyen, G., Maréhal, E., Kaltventzeff, B. (996). Senstvty Callatons and Varane Analyss n Proess Plant Measreent Reonlaton. Copters and Cheal Engneerng 20S, [3] Vlladsen, J., Mhelsen, M. L., Solton of dfferental eqaton odels by polynoal approaton, Prente-Hall, Englewood Clffs, New Jersey, (978)