Adaptive rational Krylov algorithms for model reduction

Transcription

1 Proceedings of the European Control Conference 2007 Kos, Greece, July 2-5, 2007 WeD12.3 Adaptive rational Krylov algorithms for model reduction Michalis Frangos and Imad Jaimoukha Abstract The Arnoldi and Lanczos algorithms, which belong to the class of Krylov subspace methods, are increasingly used for model reduction of large scale systems. The standard versions of the algorithms tend to create reduced order models that poorly approximate low frequency dynamics. Rational Arnoldi and Lanczos algorithms produce reduced models that approximate dynamics at various frequencies. Thispaper tackles the issue of developing simple Arnoldi and Lanczos equations for the rational case that allow simple residual error expressions to be derived. Thisin turn permits the development of computationally efficient model reduction algorithms, where thefrequenciesatwhichthedynamicsaretobematchedcanbe updated adaptively resulting in an optimized reduced model. I. INTRODUCTION Consider a linear time-invariant single-input single-output system described by the state-space equations Eẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) (1) where x(t) C n denotesthestatevectorand u(t)and y(t) the scalar input and output signals, respectively. The system matrices A, E C n n areassumedtobelargeandsparse, and B, C C n.theseassumptionsaremetbylargescale models in many applications. The transfer function for the systemin(1)isdenotedas G(s) = C(sE A) 1 B s = [ E A B C 0 To simplify subsequent analysis and design based on the large nth order model in(1), the model reduction problem seeksanapproximate mthordermodeloftheform E m ẋ m (t) = A m x m (t) + B m u(t), y m (t) = C m x m (t) where x m (t) C m, E m, A m C n m, B m, C m C m and m < n.the associated lower order transfer function is denoted by G m (s) = C m (se m A m ) 1 B m s = [ Em A m B m C m 0 Krylov projection methods, and in particular Arnoldi and Lanczos algorithms[1 [5, have been extensively used for model reduction of large scale systems; see [5 [8 and the references therein. Methods related to the work of[9, [10useKrylovsubspacemethodstoobtainbasesforthe controllability and observability subspaces; projecting the original system using these bases results in a reduced system M.Frangos michalis.frangos@imperial.ac.uk and I.Jaimoukha i.jaimouka@imperial.ac.uk are with the Electrical &Electronic Engineering Department, Imperial College London, Exhibition Road, London SW7 2AZ. This work was not supported by any organization. that matches the transfer function and the derivatives of the transfer function of the system evaluated around infinity. The advantage associated to the projection methods using the bases of the controllability and observability subspaces istheexistenceofasetofequationsknownasthestandard Arnoldi and Lanczos equations. The Arnoldi and Lanczos equations can be used to derive algorithms based on the standard Arnoldi and Lanczos algorithms, which improve further the approximations. Implicit restarts are an example [11. Moreover the specific form of the standard equations allows the use of Arnoldi and Lanczos methods directly for solving large Lyapunov equations; see[10,[12. Another important benefit associated to the standard Arnoldi and Lanczos equations is that simple residual error expressions can be derived through them, which provide useful stopping criteria for the algorithms. Unfortunately the standard versions of the Arnoldi and Lanczos algorithms tend to create reduced order models that poorly approximated low frequency dynamics. Rational Arnoldi and Lanczos algorithms[13 [16 are Krylov projection methods which produce reduced models that approximate the dynamics of the system at given frequencies. An existing research problem for the rational case is the efficient and fast selection of the interpolation frequencies to be matched. Such a task requires simple residual expressions and simple equations to describe these algorithms. A drawback of rational methods is that no simple Arnoldi and Lanczos-like equations exist.[14,[15,[17,[18 deal with these challenging problems where they give equations that describe the rational algorithms; however these equations are notinthestandardarnoldiandlanczosform.theworkof thispaperstudiesthesameproblemsbutitisbasedonthe development of Arnoldi and Lanczos-like equations for the rational case in the standard form. Residual error expressions canthenbederivedasinthecaseofstandardarnoldiand Lanczos methods and adaptive algorithms are developed. The paper is organized as follows. Section II briefly describes the approximation problem by moment matching. The standard Lanczos and the rational Lanczos algorithms for moment matching are also described in this section. Section III gives the derivation of the Lanczos and Arnoldilike equations and of the residual errors for the rational case. Section IV suggests new procedures for the efficient computation of the interpolation frequencies to improve the approximation and it gives the pseudo-code for the adaptive algorithm developed in this paper.numerical results to illustrate the adaptive method are also presented and compared with the results of the standard rational method for model reduction and the results of balanced truncation. Finally ISBN:

2 section V gives our conclusions. II. MOMENT MATCHING PROBLEM To simplify the presentation of our results, we only considerthecasewhen E = I n where I n istheidentity matrixandwewillwrite G(s) = C(sI n A) 1 B s = [ A B C 0 The system in (1) can be expanded by Taylor series aroundaninterpolationpoint s 0 Cas G(s) = µ 0 + µ 1 (s s 0 ) + µ 2 (s s 0 ) 2 + wherethetaylorcoefficients µ i areknownasthemoments of the system around s 0 and are related to the transfer functionofthesystemanditsderivativesevaluatedat s 0. The approximation problem by moment matching isto find alowerordersystem G m withtransferfunctionexpanded as G m (s) = ˆµ 0 + ˆµ 1 (s s 0 ) + ˆµ 2 (s s 0 ) 2 + suchthat µ i = ˆµ i,for i = 0,1,...,m. Inthecasewhere s 0 = themomentsarecalledmarkov parametersandaregivenby µ i = CA i B.Themoments aroundanarbitraryinterpolationpoint s 0 Careknown asshiftedmomentsandtheyaredefinedas µ i = C(s 0 I n A) (i+1) B, i = 0,1,... A more general definition of approximation by moment matching is related to rational interpolation. By rational interpolation we mean that the reduced order system matches the moments of the original system at multiple interpolation points. Let V m, W m C n m.byprojectingthestatesofthe high order system with the projector P m = V m (W mv m ) 1 W m, (2) assumingthat W mv m isnonsingular,areducedordermodel isobtained as: [ G m (s) = s Em A m B m C m 0 [ W := m V m W mav m W mb. (3) CV m 0 Acarefulselectionof V m and W m asthebasesofcertain Krylovsubspacesresultsinmomentmatching.For R C n n and Q C n p a Krylov subspace K m (R,Q) is defined as K m (R,Q) = span ([ Q RQ R 2 Q R m 1 Q ). If m = 0, K m (R,Q)isdefinedtobetheemptyset. A. Standard Lanczos algorithm An iterative method for model reduction based on Krylov projections isthe Lanczos process. The Lanczos process, given in algorithm 1, constructs the bases V m = [ v 1,...,v m C n m and W m = [ w 1,...,w m C n m forthekrylovsybspaces K m (A,B)and K m (A,C )respectively,suchthattheyarebiorthogonal,i.e., W mv m = I m. The following equations, referred to as the Lanczos equations[19 hold, AV m = V m A m + v m+1 C Vm B = V m B m A W m = W m A m + w m+1 B W m C = C m W m. Duetobiorthogonality W m+1v m+1 = I m+1, W mv m+1 = V mw m+1 =0,thematrix A m isinthetridiagonalform α 11 α 12. A m = α 21 α αm 1,m, and α m,m 1 α m,m C Vm = w m+1av m = α m+1,m e T m B Wm = v m+1a W m = α m,m+1e T m The bases V m and W m are constructed such that K m (A,B) colsp(v m )and K m (A,C ) colsp(w m ) and so the reduced ordersystem G m (s), defined in (3), matches the first 2mmarkov parameters of G(s)[17. B. Approximation and residual error expressions Theapproximationerror ǫ(s) = G(s) G m (s)inkrylov projection methods can be shown to be ǫ(s) = R C (s) (si n A) 1 R B (s), (4) where R C (s)and R B (s)areknownastheresidualerrors. Thesearegivenby R B (s) = B (si n A)V m X m (s), R C (s) = C (si n A) W m Y m (s). where Y m (s)and X m (s)arethesolutionsofthesystemof equations (si m A m )X m (s) = B m, (si m A m ) Y m (s) = C m, and satisfy the Petrov-Galerkin conditions R B (s) span{w m }, R C (s) span{v m }. i.e., W mr B (s) = V mr C (s) = 0. Using the Lanczos equations, the residual errors can be simplified as (5) R B (s) = v m+1 C Vm (si m A m ) 1 B m (6) R C (s) = w m+1 B W m (si m A m) 1 C m (7) which involves terms related to the reduced order system only. 4180

3 Algorithm 1 Basic Lanczos algorithm 1:Inputs: A,B,C,m,ǫ 2:Initialize: v 0 = 0, w 0 = 0, ˆv 1 = B,ŵ 1 = C 3:If { ŵ 1ˆv 1 < ǫ},stopend ˆv 4: v 1 = 1, w ŵ 1ˆv1 ŵ 1 = ŵ 1 / 1ˆv 1 ŵ 1ˆv 1 5:for j = 1 to m 6: α j,j =w j Av j, α j 1,j =w j 1 Av j, 7: α j,j 1 =w j Av j 1 8: ˆv j+1 = Av j v j 1 α j 1,j v j α j,j 9: ŵ j+1 = A w j w j 1 α j,j 1 w jα j,j 10: α j+1,j = ŵ j+1ˆv j+1 11: If {α j+1,j < ǫ},stopend 12: α j,j+1 = ŵ j+1ˆvj+1 α j+1,j 13: v j+1 = ˆv j+1 /α j+1,j, w j+1 = ŵ j+1 /α j,j+1 14: end 15:Outputs: V m = {v 1,...,v m },W m = {w 1,...,w m } C. The rational Lanczos method The rational Lanczos procedure[14 [16 is an algorithm for constructing biorthogonal bases of the union of Krylov subspaces. Let V m,w m C n m be the bases of such subspacesandlet Pbeaprojectordefinedas P = V m W m. Applyingthisprojectoronthesystemin(1)areducedorder system is obtained with a transfer function as in(3). The nextresultshowsthataproperselectionofkrylovsubspaces will result in reduced order systems that matches the moments of the system at given interpolation frequencies. Theorem1:Let S = {s 1,s 2,...,s K } and S = { s 1, s 2,..., s K}betwosetsofinterpolationpoints.If K K msk ((s k I n A) 1,(s k I n A) 1 B) span{v m } k=1 and K k=1 K m sk (( s ki n A ) 1,( s ki n A ) 1 C ) span{w m } where m sk and m sk are the multiplicities of the interpolationpoints s k and s k respectivelyand K k=1 m s k = K k=1 m s k = mthen: if s k = s k thereducedordersystemmatchesthefirst m sk + m sk shiftedmomentsaroundtheinterpolation point if s k s k the reduced order system matches the first m sk shiftedmomentsaround s k andthefirst m sk shiftedmomentsaround s k respectively assuming the matrices (s k I n A) and ( s k I n A) are invertible. TheproofofTheorem1canbefoundin[17.Arational Lanczos process is given in algorithm 2. For simplicity weassumethat m sk = m sk andalsoitisassumedthat s i s j and s i s j for i j.asmentionedinsection I,a drawback of rational methods is that no Lanczos-like equations exist. This issue has been investigated in[14, [15, [17, [18, where they give equations that describe the rational algorithms; however these equations are not in the standard Lanczos form. The work of this paper studies the same problems but it is based on the development of Lanczos-like equations for the rational case in the standard form. Residual error expressions can then be readily derived as in the case of standard Lanczos methods and adaptive algorithms are developed. Algorithm 2 RationalLanczos algorithm 1: Inputs: S = {s 1,...,s K }, S = { s1,..., s K } 2: A,B,C,ǫ, m sk = m sk 3: Initialize: V m = [,W m = [, i = 0 4: for k = 1 to K 5: if {s k = }, ˆv i+1 = B 6: else 7: ˆv i+1 = (s k I n A) 1 B 8: end 9: if { s k = }, ŵ i+1 = C 10: else 11: ŵ i+1 = (( s k I n A) 1 ) C 12: end 13: If { ŵ iˆv i < ǫ},stopend ˆv 14: v i+1 = i+1, w ŵ i+1 = ŵ i+1 / i+1ˆv i+1 ŵ i+1ˆvi+1 ŵ i+1ˆv i+1 15: V m = [V m v i+1, W m = [W m w i+1 16: i = i : for j = 1 to m sk 1 18: if {s k = }, ˆv i+1 = Av i 19: else 20: ˆv i+1 = (s k I n A) 1 v i 21: end 22: if { s k = }, ŵ i+1 = A w i 23: else 24: ŵ i+1 = (( s k I n A) 1 ) w i 25: end 26: ˆv i+1 = ˆv i+1 V m W mˆv i+1 27: ŵ i+1 = ŵ i+1 W m V mŵ i+1 28: ρ = ŵ i+1ˆv i+1 29: If {ρ < ǫ},stopend ŵ i+1ˆvi+1 30: λ = ρ 31: v i+1 = ˆv i+1 /ρ, w i+1 = ŵ i+1 /λ 32: V m = [ V m v i+1, W m = [ W m w i+1 33: i = i : end 35: end 36:Outputs:V m = {v 1,...,v m },W m = {w 1,...,w m } III. LANCZOS AND ARNOLDI-LIKE EQUATIONS FOR RATIONAL INTERPOLATION Rational interpolation generally provides a more accurate reduced-order model than interpolation around a single point since it is based on the combination of approximations around many interpolation points. The reduced-order model 4181

4 created by rational interpolation can be improved by a proper selection of these points. For this it is necessary to obtain frequency dependent error expressions. The interpolation points could then be the frequencies at which the error expressionsaremaximumusingthe H normasmeasure. The aim is to create rapidly and efficiently accurate reduced order models. In[20 for example they develop an exact expressionforthe H 2 normoftheerrorterm.theysuggest that by interpolating at the negative of the eigenvalues of A with the highest residues of the transfer function G(s) the H 2 isminimized.thismethod howeverrequiresthe computationoftheeigenvaluesofthe Amatrixofthesystem whichcanbeveryslowwhen Aisverylarge. ManyoftheresultsbasedonKrylovsubspacesfollow from the Arnoldi and Lanczos equations of the standard Arnoldi and Lanczos algorithm. No results for extending these equations to the rational Arnoldi or Lanczos case exist in the literature. A main contribution of this work is to derive corresponding equations for the rational version of these algorithms. In this section we derive with minimum additional effort Arnoldi and Lanczos-like equations in the familiar form of the standard Arnoldi and Lanczos equations. The standard version of these equations allow us to have simple residual error expressions for the rational case. A. Derivation of Lanczos equations in the rational case ThefollowingLemmaderivesasetofequationsinthe standard form for the Lanczos process for the rational case. Using these equations, the Lemma also provides expressions for the residuals errors. These expressions will prove crucial in deriving an adaptive rational Lanczos algorithm for choosing the frequencies at which the reduced order system matches the dynamics of the original system such that the approximation error tends to be reduced. Lemma1: Let V m and W m beasdefinedintheorem1 andlet m and m bethemultiplicitiesof in Sand S,respectively.Ifwecompute V m+1 = [V m v m+1 and W m+1 = [W m w m+1 suchthat then [V m A m B span{v m+1 }, [W m (A m ) C span{w m+1 }, W m+1v m+1 = I m+1, AV m = V m A m + v m+1 C Vm, (8) B = V m B m + v m+1 b m, (9) A W m = W m A m + w m+1 B W m, (10) C = C m W m + c m w m+1, (11) where b m = w m+1b, c m = Cv m+1, C Vm = w m+1av m and B Wm = W mav m+1.furthermore, b m = 0if m > 0 and c m = 0if m > 0. Proof: Wefirstprovetheresultfor m = 0.Let V m bedefinedasintheorem1.thenweextend V m to V m+1 = [V m v m+1 suchthat [V m B span{v m+1 } bybiorthogonalising Bagainstallpreviouscolumnsof W m with the Lanczos algorithm. Then the following are true: (s 1 I n A) 1 B span{v 1 } (12) (s k I n A) (i 1) B span{v j 1 }. (s k I n A) i B span{v j } (13). B span{v m+1 } (14) WestartbyprovingtheLemmaforthefirstcolumnof V m. Multiply(12)by (s 1 I n A)fromtheleftandrearrangeto get Av 1 span{v 1,B} (14) AV 1 span{v m+1 }. Weproceedtheproofbyinduction.Weassumethatthe resultholdsforanarbitraryinterpolationpoint s k ofthe K sk krylovsubspaceuptothe (i 1) th multiplicity.we will prove the result for the next multiplicity and hence the proof will be complete. Therefore we assume AV j 1 = A[v 1... v j 1 span{v m+1 } (15) and then we prove that it is true for the next index j. Multiply(13)fromtheleftby (s k I n A)andrearrange to get: Av j span{v j,av j 1 } (15) Av j span{v m+1 }. Combining the above with the assumption made in(15) AV j span{v m+1 }. (16) Thereforeitiseasytoseethattheresultin(16)holdsfor allcolumnsin V m,i.e., AV m span{v m+1 }. (17) To prove (8) we proceed as follows: Since AV m span{v m+1 }thereexistsamatrix Y C (m+1) m such that AV m = V m+1 Y. (18) [ Am Partitioning Y as andmultiplying(18)by W C m+1 Vm fromtheleft,duetobiorthogonalitybetween W m+1 and V m+1 wegetthat A m = W mav m and C Vm = w m+1av m AV m = V m A m + v m+1 C Vm. Similarlytoprove(9)weproceedasfollows:Since B span{v m+1 }thereexistsamatrix Z C (m+1) 1 suchthat Partitioning Z as [ Bm b m B = V m+1 Z. (19) andmultiplying(19)by W m+1 fromtheleftwehavethat B m = W mband b m = w m+1b. B = V m B m + v m+1 b m 4182

5 Thereforewehavethefirststepoftheprooffor(8)and(9). Nextweprovetheresultfor m > 0.Letthebasis V m besuchthat [B AB A p 1 B V m p span{v m } where V m p isasdefinedintheorem1.since Bisalready inthespanof V m itiseasytoseethat(8)wouldholdif {Av 1,...,Av p } span{v m+1 }. (20) Thiscanbeshownbyextending V m to V m+1 =[V m v m+1 such that [V m A p B span{v m+1 } and W m+1v m+1 =I m+1 bybiorthogonalising A p Bagainstallthepreviouscolumns of W m withthelanczosalgorithm.itfollowsthat(20)holds since by construction we have that, { A k span{v1,...,v B k+1 }, for 0 < k < p 1 span{v 1,...,v m+1 }, for k=p 1 whichcompletestheproofof(8)and(9).toprovethelast part,notethatif m > 0then B span{v m }fromwhich itfollowsthat b m = 0. TheproofofEquations(10)and(11)issimilaranditis therefore omitted. B. Simplified Lanczos residual errors WiththeuseofLemma1,theresidualerrors R B (s)and R C (s)in(6)and(7)cannowbesimplifiedfortherational Lanczos procedure as R B (s)=v m+1 {C Vm (si m A m ) 1 B m + b m } (21) R C (s)=w m+1 {B W m ((si m A m ) 1 ) C m + c m} (22) The residual errors in this case can be computed less expensively than the approximation errors, since only quantitiesrelatedtothe m th ordermodelareinvolved,compared to the approximation errors in the initial form which involve quantitiesrelatedtoan n th ordersystem.alsoitisimportant tomentionthatlemma1doesnotalter V m or W m and the only additional cost is an extra iteration of the Lanczos algorithm. Using(8),(9),(10) and(11) we can derive the residual errors in(21) and(22) respectively. R B (s)=b (si n A)V m X m (s) (8) = B V m (si m A m )X m (s) + v m+1 C Vm X m (s) (5),(9) = v m+1 {C Vm (si m A m ) 1 B m + b m }. Thederivationfor R C (s)issimilarandisomitted. C. The Arnoldi-like equations Following similar steps as for the derivation for the rational Lanczos case, we have developed Arnoldi-like equations for the rational Arnoldi case which are stated without proof in the following Lemma. Lemma2: Let V m and W m beasdefinedintheorem1 andlet m and m bethemultiplicitiesof in Sand S, respectively.ifwecomputetheorthogonalbases V m+1 = [V m v m+1 and W m+1 = [W m w m+1 suchthat then [V m A m B span{v m+1 }, V m+1v m+1 = I m+1, [W m (A m ) C span{w m+1 }, W m+1w m+1 = I m+1, AV m =V m Em 1 A m + (v m+1 V m Em 1 W mv m+1 )C Vm, B=V m Em 1 B m + (v m+1 V m Em 1 W mv m+1 )b m, W ma=a m Em 1 W m + B Wm (w m+1 w m+1v m Em 1 W m), C =C m Em 1 W m + c m (w m+1 w m+1v m Em 1 W m) where E m = W mv m, b m = v m+1b, c m = Cw m+1, C Vm = v m+1av m,and B Wm =W maw m+1. D. Simplified Arnoldi residual errors Theresidualerrors R B (s)and R C (s)inthiscasecanbe simplified as R B (s)=(v m+1 V m E 1 m W mv m+1 ) {C Vm (se m A m ) 1 B m + b m } R C (s)=(w m+1 W m (E 1 m ) V mw m+1 ) {B W m ((se m A m ) 1 ) C m + c m}. IV. ADAPTIVE KRYLOV ALGORITHMS FOR MODEL REDUCTION The problem in adaptive rational methods is to decide on the next interpolation point on every iteration. In this section a number of options are described for the selection of the interpolation points. The choice may depend on the computational power or time restrictions required. The proposed methods are based in the idea that the interpolation points should be selected such that the approximation error mentioned in Section I in equation(4) should be minimized ateveryiterationintermsofasuitablenorm.theerror expression in(4) is rewritten below for convenience ǫ(s)=r C (s) (si n A) 1 R B (s). In Section III simple expression terms for the residual errors were derived, involving quantities related to the reduced ordermodelonly.let R C and R B bethefrequencydependent termsoftheresidualerrorsand Band Cthenon-frequency dependent terms of the residual errors. In the Arnoldi case R B (s)=c Vm (se m A m ) 1 B m + b m [ s Em A = m B m, C Vm b m R C(s)=C m (se m A m ) 1 B Wm + c m s Em A =[ m B Wm C m c m B=v m+1 V m Em 1 W mv m+1, C =w m+1 w m+1v m Em 1 W m 4183

6 whereas in the Lanczos case R B (s)=c Vm (si m A m ) 1 B m + b m [ Am B m s = C Vm b m R C(s)=C m (si m A m ) 1 B Wm + c m s Am B =[ Wm, C m B=v m+1, C =w m+1, c m where C Vm, B Wm, b m and c m foreachofthetwomethods areasdefinedinlemmas1and2. Now the error expression in(4) becomes where ǫ(s)= R C(s) C(sI n A) 1 B RB (s) = R C(s)H(s) R B (s) [ H(s) = s A B C 0 As already mentioned the idea is to select an interpolation point at every iteration such that ǫ(s) is small in some norm.ideallywewouldlike,ateveryiteration,tofindthe frequency at which ǫ(s) achieve itsmaximum and then use this frequency as the next interpolation frequency for the Krylov methods. RC (s) and R B (s) are small terms involving quantities related to the reduced order model, but H(s) contains terms related to the original system. Thusacomputationof R C H R B oneveryiterationfor medium and large scale systems iscomputationally very expensive. Therefore instead of using H(s) we suggest using itsapproximation Ĥ(s). The approximation of the error ǫ(s) does not necessarily havetobe ˆǫ(s) = R C (s)ĥ(s) R B (s).theerrorapproximation ˆǫ(s)canbeoneoftheexpressionslistedinTableI. TABLEI ERROREXPRESSIONS. 1 ˆǫ(s)= R B (s) 2 ˆǫ(s)= R C (s) 3 ˆǫ(s)=Ĥ(s) R B (s) 4 ˆǫ(s)=Ĥ(s) 5 ˆǫ(s)= R C (s) Ĥ(s) 6 ˆǫ(s)= R C (s) Ĥ(s) R B (s) In all cases the interpolation points are selected to be the frequencies s C and s C at which one of the approximated error expressions achieve its maximum, i.e., S ={s : ˆǫ(s) = ˆǫ } and S ={s : ˆǫ(s) = ˆǫ } where S and S denotethesetofinterpolationpointsas definedintheorem1.inthecasewhereweareinterested in a range of frequencies, then the interpolated points can beselectedtobethefrequenciesatwhichaweighedapproximatederror ˆǫ W (s) = W oˆǫ(s)w i achieveitsmaximum, where W o,w i arestablefilters. We suggest approximating H(s) using the projection P = Ṽ ( W Ṽ ) 1 W withthearnoldiorlanczosalgorithmsas Ĥ(s) = CṼ (s W Ṽ W AṼ ) 1 W B so that it includes small terms only.note that in the Lanczos case W Ṽ = I. The projection method to approximate H(s) depends on the computational power and the time constrains. Below there is a list of different suggestions for approximating Ĥ(s). On every iteration approximate H(s) by standard rational Arnoldi or Lanczos algorithms using the set offrequencies S and Salreadycomputedsofar.As the algorithm progresses the approximation of H(s) is improved. Insteadofcomputing Ĥ(s)byapplyingrationalinterpolation on H(s) on every iteration which requires long computional time we can perform Arnoldi or Lanczos algorithm to match only at the last frequencies added in Sand S.Thereforeinthiscase: Ṽ =ṽ, and W = w. Ĥ(s)canalsobeprojectedwith Ṽ and Was: Ṽ = [V m ṽ, and W = [W m w. where ṽand wareobtainedasdescribedintheprevious method. As an alternative we suggest to approximate H(s) by usingthealreadycomputedcolumns w m+1 and v m+1 : Ṽ = v m+1, and W =w m+1 Finallywealsosuggestthat Ĥ(s)canbeprojectedwith Ṽ and W asthealreadycomputedbases W m+1 and V m+1 : Ṽ = V m+1, and W = W m+1. A pseudocode for the algorithm for the adaptive rational method, is given in algorithm 3. A. NumericalExamples Figure 1 shows the magnitude of the bode plot of a randomly generated system of order n = 150,the magnitude plot of the reduced order system using the standard rational Arnoldimethodaswellasthemagnitudeoftheerror.In this example it was assumed that there was no previous knowledge on the characteristics of the transfer function of the original system. Therefore the standard Arnoldi rational algorithm was applied to match 4 moments at ten equally spaced(on logarithmic scale) interpolation points S = S = {10 4 j,10 3 j,...,10 5 j}.itcanbeseenthatthe approximation is not satisfactory.figure 2 shows the bode plot of the original system, of the reduced order system using the adaptive algorithm developed in this paper and of 4184

7 Bode Diagram Bode Diagram Magnitude (db) Magnitude (db) Frequency (rad/sec) Frequency (rad/sec) Fig. 1. Bode Diagram of the transfer functions of the actual system (continuous line), of the reduced order system obtained by standard rational Krylov method(dotted line), and of the error between them(dashed line). Theorderoftheoriginalsystemis n=150andofreducedordersystem is m=40. Fig. 2. Bode Diagram of the transfer functions of the actual system (continuous line), of the reduced order system obtained by the adaptive rational Krylov method(dotted line), and of the error between them(dashed line).theorderoftheoriginalsystemis n = 150andofreducedorder system is m=40. the error.it can be clearly seen that the adaptive algorithm has a better performance. It is important to note that in practice there is usually no previous knowledge of the original system frequency response. Applying model approximation using the adaptive algorithm presented in this paper results in reduced models with little redundant information and improves the overall performance of the Krylov reduction methods. It is essential that the extra cost of the adaptive algorithm is small due to speed and storage limitations. In this paper we have developed an adaptive method for a fast computation of the next interpolation frequency using the simple error expressions developed in this paper. Table II gives a comparison of the approximation error G(s) G m (s) ofthemethodsdescribedinthispaper. In all cases the algorithms were performed in real arithmetic[13 and a stable projection was applied to the reduced Algorithm 3 Adaptive rational method pseudo-code 1:Inputs:A,B,C,m 2:Chooseinitialfrequenciestomatch,e.g. s 1 = and s 1 = 3:fori=1tom 4: Compute v m and w m by rational Lanczos (or Arnoldi)methodtomatchat s i and s i. 5: Generate the Lanczos (or Arnoldi) equations as describedby Lemma1(or2). 6: Use one of the error expressions in Table I to computenextinterpolationfrequencies s i+1 and s i+1 7: Iftheinfinitynormoftheerrorexpressionusedis below a threshold stop. 8:end 9:Outputs:A m =W m AV m, E m =W m V m, B m =W m B and C m =CV m system.notethatif Ṽ Cn m isabaseofasubspasethen V = [real{ṽ } imag{ṽ } Rn 2m isalsoabaseof the same subspace. Therefore in this part a complex basis is converted to a real one before projecting the original system.ascanbeseenfromtheresultsintableii,inallthe caseswheretheerrorinvolves Ĥ(s)andaresidualerrorthe approximation error is comparable or better than the error obtained by balanced truncation. Our numerical experience indicatesthatthistendstobethecasewhenthesizeof V m and W m islargeenoughtogiveagoodapproximation, otherwise for very low mbalance truncation gives better results. V. CONCLUSIONS The work of this paper derives Arnoldi and Lanczoslike equations for the rational versions of these algorithms. Simple residual error expressions have also been developed which are used for computing the interpolation frequencies at which the moments of the original system are to be matched. The simple error expressions derived in this paper contain only terms of the reduced order system and this allows fast computations. This selection of the interpolation points reduces the overall approximation error compared to standard rational methods and conventional methods; the adaptive method developed tends to reduce the error in the range of frequencies of interest without a priori knowledge of the original system frequency response. REFERENCES [1 C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Nat. Bur. Stand, vol. 45, pp , [2, Solution of systems of linear equations by minimized iterations, J.Res.Nat.Bur.Stand,vol.49,pp.33 53,1952. [3 W. Arnoldi, The principle of minimized iterations in the solution of matrix eigenvalue problem, Quart. Appl. Math., vol. 9, pp ,

8 TABLEII COMPARISON, n=250, m=50 ErrorExpressionsusingResidualErrorinformationonly S ={s : R B (s) = R B }, S ={s : RC (s) = R C } S = S ={s : R B (s) = R B } S = S ={s : R C (s) = R C } G G m ErrorExpressionsusing Ĥ(s)projectedwith Ṽ = v m+1and W = w m+1 {S = S ={s : Ĥ(s) = Ĥ } S = S ={s : R C (s)ĥ(s) = R S = S ={s : ˆǫ(s) = R CĤ R B } ErrorExpressionsusing Ĥ(s)projectedwith Ṽ = V m+1and W = W m+1 S = S ={s : Ĥ(s) = Ĥ } S = S ={s : R C (s)ĥ(s) = R S = S ={s : ˆǫ(s) = R CĤ R B } ErrorExpressionsusing Ĥ(s)projectedwith Ṽ =[ṽand W =[ w S = S ={s : Ĥ(s) = Ĥ } S = S ={s : R C (s)ĥ(s) = R S = S ={s : ˆǫ(s) = R CĤ R B } ErrorExpressionsusing Ĥ(s)projectedwith Ṽ =[Vm ṽand W =[W m w S = S ={s : Ĥ(s) = Ĥ } S = S ={s : R C (s)ĥ(s) = R S = S ={s : ˆǫ(s) = R CĤ R B } ErrorExpressionsusing Ĥ(s)projectedwith Ṽ and Wcomputedbyrationalinterpolation S = S ={s : Ĥ(s) = Ĥ } S = S ={s : R C (s)ĥ(s) = R S = S ={s : ˆǫ(s) = R CĤ R B } BalancedTruncation RationalKrylov:interpolationpoints S= S={10 4 j,..., 10 5 j}, multiplicity= [4 D. Boley and G. Golub, The nonsymmetric lanczos algorithm and controllability, Syst. Contr. Lett., vol. 16, pp , [5 Y. Saad and H. A. van der Vorst, Iterative solution of linear systems inthe20thcentury, J.Comput.Appl.Math,vol.66,no.0,pp.1 33, [6 A. Antoulas, D. Sorensen, and y. S. Gugercin, A survey of model reduction methods for large-scale systems, Contemp. Math, vol. 280, pp , [7 D. Boley, Krylov space methods on state-space control models, Technical report TR92-18,University of Minnesota, [8 A. Bultheel and M. V. Barel, Padé techniques for model reduction inlinearsystemtheory:asurvey, J.Comput.Appl.Math,vol.14, pp , June [9 I. Jaimoukha and E. Kasenally, Oblique projection methods for largescalemodelreduction, SIAMJ.MatrixAnal.Appl,vol.16,no.2, pp , [10, Krylov subspace methods for solving large lyapunov equations, SIAMJ.MatrixAnal.Appl,vol.31,no.1,pp ,1997. [11, Implicitly restarted krylov subspace methods for stable partial realizations, SIMAX, vol. 18, no. 3, pp , [12 Y. Saad, Numerical solution oflarge Lyapunovequations, in MTNS- 89,vol.3,1990,pp [13 A. Ruhe, The rational krylov algorithm for nonsymmetric eigenvalue problems III: complex shifts for real matrices, BIT, vol. 34, pp , November [14, Rational krylov: A practical algorithm for large sparse nonsymmetric matrix pencils, SIAM J. Sci. Comput., vol. 19, pp , [15 K. Gallivan, E. Grimme, and P. V. Dooren, A rational lanczos algorithm for model reduction, Numer. Algorithms, vol. 12, pp , [16 E. Grimme, K. Gallivan, and P. V. Dooren, A rational lanczos algorithm for model reduction II: Interpolation point selection, Technical report, University of Illinois at UrbanaChampaign., [17 K. Gallivan, Sylvester equations and projection-based model reduction, J.Comput.Appl.Math,vol.162,pp ,2004. [18 H.J.Lee, C.C.Chu, and W.S.Feng, An adaptive-order rational arnoldi method for model-order reductions of linear time-invariant systems. Linear Algebra and Its Applications, vol. 415, pp , [19 V.Papakos and I. Jaimoukha, Implicitly restarted lanczos algorithm for model reduction, Proc. 40th IEEE Conf. Decision Control, Orlando, FL, pp , [20 S. Gugercin and A.C.Antoulas, An H 2 error expression for the lanczos procedure, in Proc. IEEE Conference on Decision and Control,