The Unifying Feature of Projection in Model Order Reduction

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1 he Unfyng Featue of Pojecton n Model Ode Reducton Key Wods: Balanced tuncaton; pope othogonal decomposton; Kylov subspace pojecton; pojecton opeato; eachable and unobsevable subspaces Abstact hs pape consdes the poblem of model ode educton of lnea systems wth the emphass on the common featues of the man appoaches One of these featues s the unfyng ole of opeato pojecton n model educton It s shown how pojectons ae mplemented fo dffeent methods of model educton and what the popetes ae he othe common featue s the subspaces whee pojectons ae defned he man appoaches fo model educton whch ae consdeed n the pape ae balanced tuncaton pope othogonal decomposton and the Lanczos pocedue fom the Kylov subspace methods It s shown that the ange spaces of system gamans fo balanced tuncaton and the ange space of the eachablty and obsevablty matces fo the Lanczos pocedue concde he connecton between balanced tuncaton and the pope othogonal decomposton method s also establshed heefoe the methods fo model educton ae smla n tems of geneal opeatonal pncples and dffe mostly n the techncal mplementaton Seveal numecal examples ae consdeed showng the valdty of the poposed conjectues Intoducton Pnt ISS: 3-6; Onlne ISS: DOI: 55/tc-6-3 he exploaton of many physcal phenomena eques the ceaton of complcated and wth lage numbe of equatons mathematcal models he lage scale models dmenson and complexty s futhe nceased fom the demand of hghe accuacy and pecson n pesentng the system nteelatons he optmzaton and smulaton of such lage scale systems s often unbeaable tas whch detemnes the necessty of usng dffeent appoaches fo system appoxmaton One of the most fequently used appoaches s model ode educton Model ode educton s concened wth eplacng the ognal model wth a lowe ode one whle pesevng n geneal ts nput/output behavo Model ode educton s successfully appled n VLSC electcal ccuts desgn a qualty exploaton molecula dynamcs smulaton deep sea wave popagaton mco electomechancal devces synthess and many othe aeas he methods fo model ode educton can be dvded nto two man goups []: sngula value decomposton (SVD) based methods and methods based on pojecton onto subspaces of Kylov A man epesentatve of the fst goup of methods s balanced tuncaton [5] A dynamcal system s n balanced fom when the eachablty and obsevablty gamans ae dentcal dagonal matces hese states whch coespond to small dagonal elements ae tuncated because they have a small contbuton to the nput/output behavo of the system he educed ode system peseves such popetes le stablty contollabl- K Peev ty and obsevablty [57] Anothe advantage of balanced tuncaton s that t povdes a poy computable uppe bound on the eo of system appoxmaton [97] A dawbac of balanced tuncaton s that the computatonal pocedue s elated to solvng lage scale matx Lyapunov equatons and f the system s of vey hgh ode t ceates computatonal dffcultes Balanced tuncaton s developed n tme and fequency doman n detemnstc and stochastc settngs fo dffeent classes of systems wth the estcton fo system stuctue pesevaton etc [] he pope othogonal decomposton (POD) method ovecomes the numecal dsadvantage of balanced tuncaton by smplfyng to a geat extend the computatons n the numecal pocedues [8] he method delves an othonomal bass fo optmal appoxmaton n quadatc sense of cetan system chaactestcs fom expementally obtaned data Especally useful and effectve numecally s the snapshots appoach [] By ths appoach the state tajectoy s dscetzed n equally dstbuted state ponts called snapshots whch ae futhe employed fo low dmensonal appoxmaton of system states he usage of dscetzed data smplfes the computatons whch ae educed to the geneal algebac opeatons POD s data dependent method and does not eque pevous nowledge of system behavo he devaton of the educed model s obtaned by pojecton on a subspace of the ognal state space he computatonal effectveness allows applyng pope othogonal decomposton to lnea as well as nonlnea system poblems Due to the fact that t eques only standad matx opeatons t can be popely appled even to nfnte dmensonal dynamcal systems Compaed to the balanced tuncaton pocedue howeve pope othogonal decomposton does not guaantee stablty fo the educed system and does not povdes a poy appoxmaton eo bound hee ae dffeent attempts to connect these two methods and to extend POD to balanced tuncaton [9] In [] the concept of empcal gamans s used whch ae obtaned fom expemental data and appoxmate the gaman matces of the dynamcal system he empcal gamans ae easy to compute and gve qute accuate appoxmaton when the ode of the system s vey hgh hs appoach s futhe extended n [6] whee empcal gamans ae calculated by usng othogonal polynomal expansons of the state mpulse esponses of the ognal system and ts dual he othe goup of methods fo model ode educton s the goup of methods based on pojecton onto the subspaces of Kylov wth the two epesentatves: the method of Lanczos [3] and the method of Anold [3] Both methods ae teatve n natue whee at each step of the computatonal pocedues the ode of the appoxmated nfomaton technologes Download Date 7/5/8 8:6 PM

2 system nceases A patcula featue of these methods s that the educed system matces admt a specal canoncal fom whch allows smplfyng the computatons he tansfomaton matces ae othogonal and bult by applyng the Gam Schmdt othogonalzaton pocedue he Anold algothm mplements Galen pojecton on the subspace of Kylov geneated by the columns of the contollablty matx he two sded Lanczos algothm ealzes Petov Galen pojecton onto the Kylov subspaces geneated by the columns and ows of the eachablty and obsevablty matces he Lanczos algothm can be effectvely mplemented n the fequency doman whee the pocedue of moment matchng s successfully appled fo model educton [8] he methods based on pojecton onto subspaces of Kylov ae computatonally effcent and the numecal pocedues fo obtanng the appoxmated systems eque only matx vecto multplcatons wthout matx factozatons o nvesons [] he Lanczos pocedue allows fequency doman mplementaton whee the selecton of fequency ponts leads to moe accuate system appoxmaton n patcula fequency anges he majo dawbac of these methods s the loss of othogonalty f the classcal Gam Schmdt pocedue becomes unstable and the algothms may bea down f cetan matces dung the computatons become sngula [] hs pape consdes the poblem of model ode educton fo lnea systems he goal s to deve some common featues of the man appoaches fo system educton One of these featues s the unfyng ole of pojecton n the pocedues fo model educton It s shown how the pojecton opeato s ealzed nto dffeent educton pocedues and egadless the techncal mplementaton all man appoaches nclude a pocedue of subspace pojecton he othe common featue s the destnaton subspace whee the states of the ognal system ae pojected It s shown that the eachablty and obsevablty opeatos play decsve ole n fomng the stuctue of the pojected subspace he connecton between the system gamans and the coespondng eachablty and obsevablty matces s establshed he close connecton between the man appoaches fo model educton s justfed by seveal numecal smulatons he Pojecton Opeato Pojecton opeatos play an mpotant ole n appoxmaton theoy and fo spectal epesentaton of lnea opeatos A complete chaactezaton of pojecton opeatos and the applcaton can be found elsewhee n the lteatue on opeato theoy fo example n [6] Defnton Assume that V s a vecto space ove the feld of scalas F he pojecton P s a bounded lnea opeato of the vecto space V nto tself whch s dempotent e P = P Poposton Let V be a vecto space ove the feld of scalas F and P s a pojecton defned on V he ange and null spaces of the pojecton opeato ae supplementay lnea subspaces of V e the ntesecton contans only the zeo element and the sum s the whole space: R(P) (P) = {} and R(P) + (P) = V heefoe evey element of the vecto space v V can be unquely expessed as a sum of two elements belongng to the ange and null spaces of a pojecton opeato P: v = v + v whee v R(P) and v (P) Convesely f thee exst two supplementay subspaces of a gven vecto space V then a pojecton can be defned such that evey vecto v can be pojected onto the ange space of the pojecton opeato along ts enel Poposton Let V and V ae two dsjont lnea subspaces of a lnea vecto space V ove the feld of scalas F such that V + V = V hen thee exsts a pojecton P defned on V such that R(P) = V and (P) = V If the vecto space V s a Hlbet space e a complete nne poduct vecto space then pojecton opeatos defned on the Hlbet space cay some addtonal popetes due to the othogonalty popetes of the elements of ths vecto space Defnton A pojecton P on a Hlbet space H s sad to be othogonal f ts ange and null spaces ae othogonal e R(P) (P) Poposton 3 Assume that H s a closed subspace of a Hlbet space H hen fo evey vecto h H thee exst an othogonal pojecton P and unque vectos h and h such that h = h + h whee P(h) = h In ths case h R(P) and h R(P) and theefoe h = h + h Poposton 4 Assume that P s an othogonal pojecton on a closed subspace H of a Hlbet space H hen the followng condtons ae satsfed: ) the pojecton P s a lnea bounded self-adjont opeato satsfyng P = P; ) the nom of the pojecton opeato s ethe zeo o one e ethe P = o P = ; ) the pojecton opeato on the subspace H = H s the opeato (I P) Poposton 4 declaes that the othogonal pojecton opeato s a self-adjont opeato Fo fnte dmensonal eal nne poduct spaces whee the pojecton opeato s a matx wth eal elements ths condton means that all pojecton matces ae symmetc Poposton 5 Assume that H s a closed subspace of a Hlbet space H hen thee exsts a unque othogonal pojecton on H such that R(P) = H Fnally we pesent one of the most mpotant esults concenng othogonal pojectons on Hlbet spaces whch fom the base fo appoxmaton and estmaton theoy on Hlbet spaces heoem (he Othogonal Pojecton heoem) Assume that H s a closed subspace of the Hlbet space H Let P s the othogonal pojecton on the subspace H e R(P) = H hen fo any vecto h H we have h P(h) h x fo all vectos x H In othe wods: h P(h) = nf { h x : x H } x Poposton 6 Assume that P and P ae pojectons defned on a Hlbet space H onto closed subspaces H and nfomaton technologes Download Date 7/5/8 8:6 PM

3 H espectvely hen the followng statements ae equvalent: ) P + P s a pojecton opeato; ) P P = ; ) the pojecton subspaces satsfy the othogonalty condton eh H In the pocedues of model ode educton the state space s a fnte dmensonal space ove the feld of eal numbes and theefoe the pojecton opeato s pesented by a matx wth eal elements 3 he Pojecton Opeato n Model Ode Reducton Consde the stable lnea tme-nvaant dynamcal system: () x(t) = Ax(t) + Bu(t) t () y(t) = Cx(t) x() = x whee x (t) R n u (t) R m and y (t) R p he eachablty and obsevablty gamans at nfnty ae gven by the expesson: () W = e At B B e A t dt and W = e A t C C e A t dt he gamans () can be computed by solvng the followng equatons of Lyapunov: (3) AW + W A + BB = and A W +W A +C C = he balanced tuncaton method s elated to tansfomng the state space coodnates n such a way that the eachablty and obsevablty gamans ae equal dagonal matces he dagonal elements of the gaman matces called Hanel sngula values gve nfomaton about the enegy needed to each o obseve the coespondng states he state vaables elated to lage Hanel sngula values ae easy to each and to obseve and have moe nfluence on the nput/output behavo of the system he state vaables assocated wth small Hanel sngula values have less mpact on the system behavo and theefoe these states can be tuncated fom the system descpton Applyng the squae oot algothm the balancng tansfomaton matces can be computed as follows []: ) pefom Cholesy decompostons on the gaman matces W = U U and W = L L ; ) pefom sngula value decomposton on the poduct U L = W Σ V ; ) compute the smlaty tansfomaton matces to tansfom the system nto a balanced fom: = Σ ½ V L and = UW Σ ½ he system matces of the tansfomed system ae obtaned as A = A B = B C = C and x = x s the state of the balanced system he next step s to tuncate these state vaables whch coespond to small Hanel sngula values Let us assume that the educed system s of ode and theefoe the othe (n ) state vaables ae selected fo tuncaton We patton the smlaty tansfomaton W matces as = and = [V ] whee V W R n and R n (n - ) hen system ode educton s acheved by usng the fst ows of matx and fst columns of matx to obtan the educed system matces as A = W A V B = W B and C = C V he matx P = V W s pojecton snce P = V W V W = V W = P whee W V = I follows fom the defnton of matces W and V Howeve P s oblque (Petov Galen) pojecton and not othogonal (Galen) snce n geneal (V W ) = W V V W he pojected state vecto s obtaned as x = V W x = V x whee x = W s the educed model state vecto of the -th ode system he vecto x s a pojecton of x onto the -dmensonal subspace spanned by the columns of matx V along the enel of W he vecto x = (I n V W ) x belongs to the null space of the pojecton opeato and s the unque vecto such that x = x + x In some sense vecto x s complementay to vecto x and contans nfomaton whch has been emoved fom the ognal state vecto n the pocedue of model ode educton he pope othogonal decomposton (POD) method s also pat of the goup of sngula value decomposton based methods and uses as a statng pont the measuements pefomed on system tajectoes he basc dea s to appoxmate the state tajectoy of the system wth one coespondng to a system of lowe dmenson hs goal s acheved by usng othogonal (Galen) pojecton he pope othogonal decomposton method uses dscetzed tajectoes data called snapshots whch s obtaned fom measuement o smulaton of the state tajectoy he tajectoy data s collected n dscete tme fo the tme moments t t t whch fom the snapshots matx: (4) x x X = xn ( t ) x( t ) x( t ) ( t ) x ( t ) x ( t ) ( t ) x ( t ) x ( t ) n hen a set of othonomal vectos v R n = n ae detemned such that the snapshots data s appoxmated n tems of the othonomal vectos as n x = Σ φ j v j whee x = x(t ) = hese equatons j = ae pesented as follows: (5) [ x x x ] = [ v v ] v n n φ φ φ n φ φ φ n φ φ φn o n matx fom X = V * Φ wth V V = I n he selecton of the othonomal set of vectos s acheved by usng sngula value decomposton of the snapshots data matx nfomaton technologes Download Date 7/5/8 8:6 PM

4 n the fom X = VΣU whee the coeffcents matx s Φ = ΣU If the emphass s on the state tajectoes enegy then the matx XX s computed whch n tems of the SVD decomposton of the data matx can be wtten as XX = VΣU UΣ V = VΣΣ V Futhe the goal s to select only < n vectos such that the appoxmated state tajectoy snapshots x = Σ φ j v j mnmze the data matx j = dffeence X X hs s the case when the sngula values of matx X wth ndex lage than decay apdly hen the state snapshots can be computed by usng only othonomal vectos as follows: φ φ φ [ x φ φ φ x x ] = [ v v v ] (6) φ φ φ o n matx fom X = V * Φ whee X R n V R n and Φ R < n Matx V s bult by selectng the fst columns of the othogonal matx V he pojecton matx s defned as P = V V he equalty P = P s satsfed snce V s an othogonal matx and V V = I he pojecton s othogonal (Galen) snce (V V ) = V V he system matces of the educed ode model ae computed as A = V AV B = V B and C = CV he state vecto of the educed system s computedas x = V x and the pojected state vecto s obtaned as x = V V x = V x whee x s the pojecton of x onto the span columns of V he complementay state vecto x = (I V V ) x (V V ) and s subaddtve to the pojected vecto to estoe the ognal state vecto: x = x + x hee exsts an nteestng ln between balanced tuncaton and pope othogonal decomposton hs ln s establshed by the numecal pocedue of computng the system gamans When the nput to the system s a delta mpulse the state tajectoy s the state mpulse esponse of the system If the system s sngle-nput the state mpulse esponse s x(t) = e At B and the eachablty gaman on the tme nteval [ ] can be computed as: W ( ) = x(t) x (t)dt If the system s mult-nput the state mpulse esponse fo an nput sgnal u(t) j = δ (t)e j j = m can be obtaned as x j (t) = e At B j whee B j s the j-h column of matx B By usng dyadc expanson of matces the eachablty gaman can be obtaned as W m ( ) = x j () t x j ()dt t j= In ths case the snapshots appoach of pope othogonal decomposton can be appled by pattonng the tme nteval [ ] on tme moments t t t = Snce the state tajectoy s Remann ntegable by applyng pocedues of numecal ntegaton the eachablty gaman can be appoxmated by the expes- m son W ( ) x ( t ) x ( t ) j j In tems of the snapshots data matx ths expesson can be pesented n j= = the m X X j j whee X j= j = x j (t ) x j (t ) x j (t ) fom W ( ) Hee we have used the smplest quadatues fomula fo numecal ntegaton namely the ectangula ule In geneal thee exst dffeent appoaches to appoxmate the ntegaton n the gamans expesson Some moe advanced appoach employs the tapezodal ule and educes to the calculaton of the gaman as []: 3 m (7) W ( ) A B B ( A ) 3 μ μ μ μ 4 j= = whee A μ = (A μi n ) (A + μi n ) B μ = (A μi n ) B and μ = Futhe the egenvalue decomposton pocedue can be appled to ths gaman appoxmaton and the Galen pojecton matx can be selected by employng the leadng egenvectos of ths decomposton: Λ U (8) W ( ) [ ] = U U U U U U = Λ + Λ Λ U he pojecton s defned as P = U U and snce matx U s othogonal ts submatx U satsfes U U = I hen the dempotent condton eadly follows P = U U U U = U U = P hs pojecton s othogonal (Galen) snce (U U ) = U U he educed ode system s obtaned as A = U AU B = U B and C = CU Howeve balanced tuncaton pesumes that system eachablty as well as obsevablty popetes ae smultaneously used fo the pupose of model ode educton Smlaly to the eachablty gaman the obsevablty gaman can be computed by usng the state mpulse esponse of the dual system [4]: (9) p(t) = A * p (t) C * u(t) t (9) y(t) = Β * p (t) p() = p whee the sta supescpt denotes the adjont opeato p (t) R n s the dual system state vecto u (t) R p s the dual system nput and y (t) R m s the output he tme of the dual system s unnng bacwads and a change of vaables τ = t s physcally meanngful Snce the system matces ae wth eal elements we can use matx tanspose nstead of the sta supescpt heefoe the dual system equatons can be wtten n the fom: () p(t) = A p (τ) + C u(t) () y(τ) = Β p (τ) Usng the same aguments as fo the eachablty gaman we can pesent the obsevabty gaman as p Wo ( ) Pl Pl whee P = [p (t ) p (t ) p (t )] s l= the dual system state vecto snapshots matx whch s computed fo the l-th component of the nput functon u(t) = δ (t)e hen the usual pocedues fo balancng and balanced state tuncaton can be appled and the educed system can be obtaned by employng Petov Galen pojecton as descbed above n the balanced tuncaton method nfomaton technologes Download Date 7/5/8 8:6 PM

5 Some ntemedate esults between balanced tuncaton and pope othogonal decomposton ae pesented by tansfomng the system nto output nomal fom he output nomal fom s system descpton whch s chaactezed wth dagonal eachablty gaman and dentty obsevablty gaman he numecal pocedue fo tansfomng the system nto output nomal fom ncopoates the followng steps [5]: ) compute the egenvalue decomposton of the system gamans W = R Σ R and W o = R o Σ o R o whee R R = I and Σ = dag{λ λ λ n } ae the coespondng egenvalues; ) compute the matx H = Σ o R o R Σ ; ) compute the sngula value decomposton H = R H Σ H Q H whee R H R H = I and Q H Q H = I; v) compute the smlaty tansfomaton matces fo tansfomng the system nto output nomal fom as = R H Σ o R o and = R o Σ o R H he tansfomed system wll have eachablty gaman W = Σ and obsevablty gaman H W o = I he model educton pocedue poceeds by selectng the fst columns of = [V ] and the fst W ows of = and usng them as tansfomaton matces to obtan A = W A V B = W B and C = C V he pojecton matx s bult as P = V W whee W V = I follows fom the defnton of these matces Futhe (V W ) = W V heefoe the pojecton s Petov Galen and s almost Galen up to scalng by the squae value of the dagonal matx Σ o elements he othe goup of methods fo model ode educton s based on pojecton on the subspaces of Kylov A man epesentatve of ths goup s the method of Lanczos he method of Lanczos s an teatve pocedue fo computng an othonomal bass fo the eachablty subspace of the system We consde hee the sngle-nput sngle-output system case he algothm poceeds by applyng the Gam Schmdt othogonalzaton pocedue to a sequence of vectos gadually fomng the eachablty subspace whee the emanng tem seves as a new decton fo developng the sequence Let us assume that the matx V = [v v v ] conssts of column vectos whch ae othonomal e V V = I hen matx A s tansfomed to a system matx whee the states ae obtaned as a pojecton of the ognal state vectos onto the subspace spanned by the columns of V e H =V AV If spancol{v v v } = spancol{v Av A - v } and f v = B then at each step of the algothm the geneated subspace s pat of the eachablty subspace of the system he teatve pocedue s pesented n the followng fom []: () A V = V H + e whee the emanng tem s obtaned fom the Gam Schmdt othogonalzaton pocedue as follows: = Av () j j j= v Av and theefoe t s othogonal to the columns of V Futhemoe the next vecto fom the sequence can be computed v by the expesson v = + he matx H = V AV whch s obtaned fom the teatve pocess (9) s the system matx of the educed -th ode system It has a specal tdagonal stuctue and s obtaned fom pojecton onto the subspace geneated by the columns of V he pojecton s of Galen type and s computed as P = V V whee V V = I he emanng tem s = (I n V V ) Av and theefoe (V V ) the null space of the pojecton opeato If at cetan step of the algothm the emanng tem s obtaned as = then v + can not be constucted and the Lanczos pocedue comes to an end hs s the case when spancol{v v v } has geneated the whole eachable subspace of the system he poblem wth ths pocedue s that t eles only on the nfomaton fom the eachable subspace fo computng the pojecton matx Fo obtanng close appoxmaton of the nput/output system behavo t s necessay togethe wth the eachablty subspace to consde the set of the obsevable states as well hs condton s accomplshed n the two sded Lanczos algothm he two sded Lanczos algothm conssts of two teatve pocedues: AV = V H + e and A W = W H + g e whee the columns of V span pat of the eachable subspace of the system and the columns of W span pat of the obsevable states heefoe we can wte: spancol (V ) = spancol {B AB A B} and spancol (W ) = spancol {C A C (A ) C } he pojecton matx s computed as P = V W whee W V = I and s oblque (Petov Galen) snce (V W ) = W V V W he pojected state vecto s obtaned as x = V W x = V x whee x = W x s the educed model state vecto of the -th ode system he vecto x = (I n V W )x belongs to the null space of the pojecton opeato and s the unque vecto such that x = x + x 4 he Reachablty and Obsevablty Opeatos fo Lnea Systems Consde the system descbed by equatons () In the pesentaton to follow we consde sgnals wth fnte enegy and theefoe the nput and output sgnals belong to the Hlbet space L ( ) and ts subspaces n he eachablty opeato L ([ ]) : PC R whee PC ([ ] ) L ( ) s the set of pecewse contnuous functons wth fnte enegy defned on the nteval [] maps each admssble nput sgnal u U [] PC([]) nto a ( ) ( )( ) ( ) τ A t τ state vecto by the expesson L u t = e Bu τ d t [] [4] he ange space of the eachablty opeato R(L ) s the set contanng all eachable states of the system and s called the eachable subspace he adjont opeato L * : Rn PC([]) whch satsfes the elaton t nfomaton technologes Download Date 7/5/8 8:6 PM

6 Fgue Unt step esponses of the full ode -----; educed 4 th ode ---; educed nd ode ; by B Fgue Unt step esponses of the full ode -----; educed 4 th ode ---; educed nd ode ; by POD nfomaton technologes Download Date 7/5/8 8:6 PM

7 Fgue 3 Unt step esponses of the full ode -----; educed 4 th ode ---; educed nd ode ; n OF Fgue 4 Unt step esponses of the full ode -----; educed 5 th ode ---; educed 3 d ode ; by L nfomaton technologes Download Date 7/5/8 8:6 PM

8 z L u = L * z u can be detemned fo evey vecto z R n as (L * z)(t) = B e A ( ) z t [] he eachablty gaman on the nteval [] s defned as W At A t ( ) e BB e dt = and s the matx epesentaton of the opeato L L * : Rn R n It s moe convenent to wo wth the gaman than wth the eachablty opeato snce the gaman s a matx and a map between two state vectos whle the eachablty opeato maps elements of the nfnte dmensonal space of admssble nputs nto the fnte dmensonal state space Howeve the eachablty gaman s not state space bass nvaant and ts popetes depend on the bass of system descpton he eachable subspace of the system whch s the ange space of the eachablty opeato concdes wth the ange space of the gaman R(L ) = R(L L * ) heefoe the eachablty condton fo all system states R(L ) = R n s equvalent to the condton R(L L * ) = Rn o detw () Evey eachable state can be obtaned n the ange of the eachablty opeato fo any admssble nput sgnal wth fnte enegy by usng aylo sees expanson as L t A( t τ ) ( u)( t) = e Bu( τ ) dτ = A B t ( t τ ) u ( τ ) dτ and s a ln- =! ea combnaton of the vectos B AB A B fo a fxed tme moment t Fo example n the sngle-nput case f the nput to the system s a delta mpulse then (L δ)(t) = e At B (n the mult-nput case we have (L δe j )(t) = e At B j whee u(t) = δ(t)e j j = m) he expesson e At B can be expanded n aylo sees as e At B = ( At) B = A B =! =! t By the Cayley Hamlton theoem A fo n s a lnea combnaton of n { } = A and theefoe the nfnte sum can be eaanged to contan only a fnte numbe of elements he eachablty matx of the system () s defned as Γ = [B AB A n- B] and fom the above dscusson follows that the ange space of the eachablty opeato s spanned by the columns of matx Γ o equvalently R(L ) = R(L L * ) = R(Γ) hs esult s even moe staghtfowad n the dscete-tme systems case Consde the lnea stable dscetetme system: (3) x( + ) = Fx() + Gu() (3) y() = Cx() + Du() x() = x o he eachablty gaman at nfnty of system (3) s defned as F GG ( F ) W = and the eachablty matx = of system () s X = [G FG F n G] Let us extend the eachablty matx by addng columns wth hghe powes of F and so obtan the extended eachablty matx X = [G FG F n G ] By the Cayley Hamlton theoem follows that the space geneated by the columns of the extended matx s the same as the space geneated by the columns of the ognal matx and ts dmenson s equal to the eachablty matx an Snce W = X X t s clea that the ange spaces of the gaman and the eachablty matx ae the same e R(W ) = R(X ) = R(X) he obsevablty opeato L : R n PC([]) whee PC([]) s the set of pecewse contnuous functons defned on the nteval [] maps each state vecto x R n nto the functon y Y [] PC([]) by the expesson At ( L x )( t) = Ce x o [ ] t [] he null space of the obsevablty opeato (L o ) s the set contanng all state vectos that excte no sgnal at the system output and s called the unobsevable subspace he adjont opeato : PC([]) Rn can be detemned fo evey fnte L * o enegy output sgnal as ( ) A t L y e C y() t o = dt he obsevablty gaman of system () on the tme nteval A t At [] s detemned by the expesson W ( ) e C Ce dt and s the matx epesentaton of the opeato L * L : o o Rn R n Smlaly to the case wth the eachablty gaman t s moe convenent to wo wth the obsevablty gaman than wth the obsevablty opeato snce t s a map between vectos on a fnte dmensonal vecto space Snce lnea opeatos always map zeos nto zeo the null space of the obsevablty gaman s the same as the null space of the obsevablty opeato e (L o ) = (L * L ) o o heefoe the condton that the system () s completely obsevable s (L o ) = {} whch s equvalent to the condton that (L * L ) = {} Equvalently the an of the o o obsevablty gaman has to be equal to the state space dmenson and theefoe det W o ( ) If the system () s completely obsevable the ntal state vecto can be ecoveed fom the elaton x = (L * L o o ) L * y o Snce the output sgnal s of fnte enegy and by usng the aylo sees expanson fomula fo a matx exponent we obtan Lo y = e C y() t dt = ( A ) C y()dt t * A t =! heefoe the ange space of the obsevablty opeato adjont can be spanned by the columns of an nfnte matx O = C A C (A ) n- C hen usng the elaton that [R(L * o )] = (L o ) [4] and theefoe [R(O )] = (O ) we obtan that the unobsevable states of the system ae elements of the enel of matx O and theefoe (O ) = (L ) By the Cayley Hamlton theoem the o enel of matx O s detemned by the fst n tems e CA j j = n and an nfnte dmensonal matx can be eplaced by a fnte dmensonal one O = [C A C (A ) n- C ] As a concluson we obtan that the unobsevable subspace of system () s detemned as (L o ) = (L * L ) = o o (O ) In the dscete-tme system case the obsevablty gaman at nfnty of system () s defned as ( F ) = Wo C CF and the coespondng nfnte = obsevablty matx s O = C F C (F ) n- C It s clea that W o = O O and theefoe by aplyng the Cayley Hamlton theoem t follows dectly that (W o ) = (O ) = (O ) o = nfomaton technologes Download Date 7/5/8 8:6 PM

9 A majo ole fo connectng the eachablty and obsevablty popetes of a dynamcal system plays the dualty pncple All of the esults pesented n the above dscusson can be summazed as a consequence of the dualty pncple he dualty pncple appled to the dynamcal systems () and () states that the ohogonal complement of the eachable subspace of () concdes wth the unobsevable subspace of () [] Futhemoe the system () s eachable f and only f system () s obsevable and vce vesa We have shown that the eachable and unobsevable subspaces of a lnea tme-nvaant stable system and ts dual ae nteelated and theefoe the pojecton on the column and null spaces of the eachablty and obsevablty gamans and matces ae closely connected Regadng the model educton pocedues dffeent methods gve smla esults as concened the pojecton on common subspaces of the state space In ths sense the balanced tuncaton method uses the nfomaton fom both the eachablty and obsevablty gamans and pojects onto the subspace geneated by the egenvectos of the poduct he POD method uses nfomaton fom the state tajectoes and pojects onto a pat of the eachablty subspace of the system he two-sded Lanczos method uses nfomaton fom both the eachablty and obsevablty matces and pojects onto the subspaces geneated by pat of the columns and ows heefoe the pojecton opeato plays a unfyng ole n the model educton pocedues and the ln between the eachable and unobsevable subspaces of the system and ts dual one outlnes the common featues of at fst sght completely dffeent methods fo system appoxmaton 5 umecal Example Consde the lnea stable tme-nvaant system descbed by the followng equatons: x(t) = Ax(t) + Bu(t) t y(t) = Cx(t) x() = x wth the followng system matces: A = 39 B = C = he Hanel sngula values fo the system ae computed as follows: Σ = ( ) Fgue pesents the unt step esponses of the full ode model and the obtaned by balanced tuncaton educed fouth and second ode models It s clealy seen that the step esponses of the full ode and the educed fouth ode models ae almost undstngushable Some dffeence appeas n the step esponse of the educed second ode model and ths dffeence s moe appaent n steady state Fgue pesents the unt step esponses of the full ode model and the obtaned by pope othogonal decomposton based balancng educed fouth and second ode models he tme nteval of appoxmaton s [ ] wth = 8 sec and step sze Δ = sec As n the case of balanced tuncaton the full and fouth ode system ae closely elated and lage dffeence appeas n the step esponse of the educed second ode system ext we exploe the model educton scheme by tansfomng the system nto an output nomal fom he eachablty gaman of the system nto output nomal fom s dagonal matx wth ts elements computed as follows: Σ = ( ) It s not dffcult to notce that the elements of the eachablty gaman Σ ae the squae of the Hanel sngula values he coespondng obsevablty gaman s the dentty matx Fgue 3 pesents the step esponses of the full ode and coespondng educed fouth and second ode models It s evdent that the obtaned esponses ae smla to those obtaned by applyng the balanced tuncaton method hs obsevaton s not supsng and ts explanaton s that both systems have the same Hanel sngula values he step esponses of the full ode system and the educed ffth and thd ode models obtaned by applyng the Lanczos pocedue ae shown n fgue 4 Fom the fgue s clea that the eo between the educed ode systems and the ognal one s lage than n the case of balanced tuncaton Fgue 5 pesents the unt step esponses of the full ode model and the educed fouth ode models obtaned by balanced tuncaton pope othogonal decomposton based balancng output nomal fom based balancng and the Lanczos method It s obvous that the dffeence appeas manly n the step esponse obtaned by applyng the Lanczos pocedue hese esults ae confmed by the computed mean squae eos between the full ode and educed fouth models he elatve mean quadatc eo of appoxmaton between the step esponses of the full ode model and the educed fouth ode models computed by the balanced tuncaton pope othogonal decomposton based balancng output nomal fom based balancng and the Lanczos methods ae obtaned as follows: y y y y B 4 POD 4 = 374 = 566 y y y y y OF = y y y L = 5 nfomaton technologes Download Date 7/5/8 8:6 PM

10 Fgue 5 Unt step esponses full ode -----; educed 4 th ode by B ---; POD ; OF ---; L Fgue 6 Magntude esponses full ode -----; educed 4 th ode by B ---; POD ; OF ---; L nfomaton technologes Download Date 7/5/8 8:6 PM

11 he obtaned elatve eos show that the Lanczos pocedue gves moe naccuate esults than the othe and justfy the theoetcal conjectue that fo low ode systems the balancng pocedues gve most accuate esults hese esults ae confmed n fgue 6 whee the Bode chaactestcs ae pesented fo the full ode and educed fouth ode models obtaned by balanced tuncaton (B) pope othogonal decomposton based balancng (POD) output nomal fom balancng (OF) and the two sded Lanczos method (L) 6 Concluson hs pape consdes the poblem of model ode educton of lnea tme-nvaant stable systems he emphass s placed on the common featues of seveal man methods fo model educton he unfyng featue n the pocedues of model ode educton s pojecton It s shown how pojectons ae computed fo dffeent methods of model educton and what the popetes of the pojecton opeatos ae Anothe common featue n model educton s the detemnaton of the subspaces whee the state space vectos ae pojected It s shown that the ange space of the eachablty gaman and the eachablty matx ae the same and ths obsevaton s elated to the connecton wth the eachablty opeato Smlaly the obsevablty gaman and the obsevablty matx ae assocated wth the obsevablty opeato and theefoe the unobsevable subspace s detemned by the null spaces of these matces It s shown that dffeent methods use dffeent schemes fo model educton as concened the techncal detals of computaton howeve all the methods contan smla chaactestcs detemned by the pojecton on the same subspaces Seveal man methods ae examned namely: balanced tuncaton pope othogonal decomposton based balancng output nomal fom based balancng and the Lanczos pocedue In a numecal example s shown that these methods gve smla esults n model educton although some dffeences ae obseved concenng the accuacy of appoxmaton hs obsevaton can be explaned by the specal featues of these methods appled to a specfc example Refeences Antoulas A Appoxmaton of Lage-Scale Dynamcal Systems Phladelpha SIAM Publ 5 Antoulas A D Soensen and S Gugecn A Suvey of Model Reducton Methods fo Lage-scale Systems Contemp Math AMS Publ Anold W he Pncple of Mnmzed Iteatons n the Soluton of the Matx Egenpoblem Quat Appl Math 95 o Calle F C Desoe Lnea System heoy Y Spnge- Velag 99 5 Chen C Lnea System heoy and Desgn Y Holt Rnehat & Wnston Dunfod J Schwatz Lnea Opeatos Pat I: Geneal heoy Y John Wley & Sons Enns D Model Reducton wth Balanced Realzatons: an Eo Bound and a Fequency Weghted Genealzaton Poceedngs of the 3 d IEEE Confeence on Decson and Contol Las Vegas Gallvan K E Gmme and P Van Dooen Pade Appoxmaton of Lage-scale Dynamc Systems wth Lanczos Method Poceedngs of the 33 d Confeence on Decson and Contol Lae Buena Vsta FL Glove K All Optmal Hanel-om Appoxmatons of Lnea Multvaable Systems and he L -eo Bounds Int J Contol o Gugecn S A Antoulas A Suvey of Model Reducton by Balanced uncaton and Some ew Results Int J Contol 77 4 o Lall S J Masden S Glavaš Empcal Model Reducton of Contolled onlnea Systems Poceedngs of the IFAC Wold Congess Bejng Lall S J Masden S Glavaš A Subspace Appoach to Balanced uncaton fo Model Reducton of onlnea Contol Systems Int J Robust onln Contol Lanczos C An Iteaton Method fo the Soluton of the Egenvalue Poblem of Lnea Dffeental and Integal Opeatos Joun Res at Bu Stand Luenbege D Optmzaton by Vecto Space Methods Y John Wley & sons Mooe B Pncpal Component Analyss n Lnea Systems: Contollablty Obsevablty and Model Reducton IEEE ans Automat Contol AC-6 98 o Peev K Othogonal Polynomals Appoxmaton and Balanced uncaton fo a Lowpass Flte Infomaton echnologes and Contol 3 o Penebo L L Slveman Model Reducton by Balanced State Space Repesentatons IEEE ans Automat Contol vol 7 98 o Pnnau R Model Reducton va Pope Othogonal Decomposton Model Ode Reducton heoy Reseach Aspects and Applcatons Edts Schldes W H van de Vost and J Rommes Beln Spnge Rowley C Model Reducton fo Fluds Usng Balanced Pope Othogonal Decomposton Int J Bfucat Chaos 5 5 o Sovtch L ubulence and the Dynamcs of Coheent Stuctues Pat I II Quat Appl Math Wllcox K J Peae Balanced Model Reducton va he Pope Othogonal Decomposton AIAA Jounal 4 o Manuscpt eceved on 5 Assoc Pof Kamen Peev eceved the engneeng degee n Automatcs fom VMEI Sofa n 985 He eceved the MS degee n Contol systems and dgtal sgnal pocessng and the PhD degee n Contol systems both fom otheasten Unvesty Boston MA USA n 99 and 993 espectvely He became an Assstant Pofesso at the echncal Unvesty of Sofa n 997 In 6 he was pomoted to Assocate Pofesso at the same unvesty Hs aea of scentfc nteests ncludes lnea and nonlnea contol systems obust contol theoy and dynamcal system appoxmaton Assocate Pofesso Peev s a membe of the John Atanasoff Socety of Automatcs and Infomatcs Contacts: Systems and Contol Depatment echncal Unvesty of Sofa 8 Klment Ohds Blvd756 Sofa e-mal: peev@tu-sofabg nfomaton technologes Download Date 7/5/8 8:6 PM