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Published in Mathematics of Contr ol, Signals, and Systems 16, 297{319, 2004. Gramian based mo del reduction for descriptor systems atjana St yk el Abstract Mo del reduction is of fundamen tal imp ortance in man con trol applications. consider mo del reduction metho ds for linear time-in arian con tin uous-time descriptor systems. These metho ds are based on the balanced truncation tec hnique and closely related to the con trollabilit and observ abilit Gramians and Hank el singular alues of descriptor systems. The Gramians can computed solving generalized Ly apuno equations with sp ecial righ t-hand sides. Numerical examples are giv en. Key ords: Descriptor systems, Gramians, Hank el singular alues, mo del reduction, balanced truncation. In tro duction Consider linear time-in arian con tin uous-time system (0) (1.1) where n;n n;m p;n is the state ector, is the con trol input, is the output and is the initial alue. The um er of state ariables is called the or der of system (1.1). If then (1.1) is standar state sp ac system Otherwise, (1.1) is descriptor system or gener alize state sp ac system Suc systems arise naturally in man applications suc as ultib dy dynamics, electrical circuit sim ulation and semidiscretization of partial dieren tial equations, see [6, 9, 19 ]. will assume throughout the pap er that the encil E is gular i.e., det( E for some In this case E can reduced to the eierstrass canonical form [33 ]. There exist nonsingular matrices and suc that (1.2) where is the iden tit matrix of order is the Jordan blo corresp onding to the nite eigen alues of E is nilp oten and corresp onds to the eigen alue at innit The index of nilp otency of denoted is called the index of the encil E Represen tation (1.2) Institut ur Mathematik, MA 4-5, ec hnisc he Univ ersit at Berlin, Strae des 17. Juni 136, D-10623 Berlin, German Phone: +49 (0)30 314-29292, ax: +49 (0)30 314-79706, E-mail: stykel@math.tu-berlin.de Supp orted Deutsc he orsc ungsgemeinsc haft, Researc Gran ME 790/12-1.

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denes the decomp osition of in to complemen tary deating subspaces of dimensions and corresp onding to the nite and innite eigen alues of the encil E resp ectiv ely The matrices and (1.3) are the sp ctr al pr oje ctions on to the righ and left deating subspaces of E corresp onding to the nite eigen alues. The encil E is called c-stable if it is regular and all the nite eigen alues of E lie in the op en left half-plane. Descriptor systems arising, e.g., from the spatial discretization of partial dieren tial equa- tion ha usually large order while the um er of inputs and the um er of outputs are small compared to Sim ulation or real time con troller design for suc large-scale sys- tems ecomes dicult ecause of storage requiremen ts and exp ensiv computations. In this case mo del order reduction pla ys an imp ortan role. It consists in an appro ximation of the descriptor system (1.1) reduced order system (0) (1.4) where `;` `;m p;` and Note that systems (1.1) and (1.4) ha the same input ). require for the appro ximate system (1.4) to preserv prop erties of the original system (1.1) lik regularit and stabilit It is also desirable for the appro xima- tion error to small. Moreo er, the computation of the reduced order system should umerically stable and ecien t. There exist arious mo del reduction approac hes for standard state space systems suc as balanced truncation [15 24 27 31 37, 38 ], momen matc hing appro ximation [1, 14 18 and optimal Hank el norm appro ximation [15 ]. Surv eys on system appro ximation and mo del reduction can found in [1 13 ]. One of the most eectiv and ell studied mo del reduction tec hniques is balanced truncation whic is closely related to the Ly apuno equations The solutions and of these equations are called the ontr ol lability and observability Gr ami- ans resp ectiv ely The balanced truncation approac consists in transforming the state space system in to balanced form whose the con trollabilit and observ abilit Gramians ecome diagonal and equal, together with truncation of states that are oth dicult to reac and to observ e, see [27 for details. Balanced truncation mo del reduction for descriptor systems has een considered in [26 30 ]. The algorithms presen ted there are based on computing the eierstrass canonical form (1.2) of the encil E Ho ev er, it is ell kno wn [33 that this computational problem is, in general, ill-conditioned in the sense that small erturbations in and ma lead to an inaccurate umerical result. In this pap er generalize con trollabilit and observ abilit Gramians as ell as Hank el singular alues for descriptor systems (Section 2). In Section presen an extension of balanced truncation metho ds [24 37 38 to descriptor systems. These metho ds are based on computing the generalized Sc ur form of the encil E and solving the generalized Sylv ester and Ly apuno equations using umerically stable algorithms. Section con tains umerical examples.

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Descriptor systems Consider the con tin uous-time descriptor system (1.1). It is ell kno wn that if the encil E is regular, is times con tin uously differentiable and is consisten t, i.e., it elongs to the set of onsistent initial onditions =0 (0) then the descriptor system (1.1) has unique con tin uously dieren tiable solution ), see [9], that is giv en d =0 Here tJ (2.1) is fundamental solution matrix of the descriptor system (1.1), and the matrices ha the form (2.2) Clearly for If the initial condition is inconsisten or the input is not sucien tly smo oth (for example, in most con trol problems u(t) is only piecewise con tin uous), then the solution of the descriptor system (1.1) ma ha impulsiv mo des [8 ]. The rational matrix-v alued function sE is called the tr ansfer function of the descriptor system (1.1). quadruple of matrices A; is alization of ). will also often denote realization of sE Tw realizations A; and A; are estricte system quivalent if there exist nonsingular matrices and suc that pair is called system quivalenc tr ansformation haracteristic quan tit of system (1.1) is system invariant if it is preserv ed under system equiv alence transformation. The transfer function is system in arian t, since sE The transfer function is called pr op er if lim !1 and is called strictly pr op er if lim !1 0. Other imp ortan results from the theory of rational functions and realization theory ma found in [9 ].

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2.1 Con trollabilit and observ abilit or descriptor systems there are arious concepts of con trollabilit and observ abilit e.g., [8 42 ]. Denition 2.1. System (1.1) and the triplet A; are called ontr ol lable on the achable set R-c ontr ol lable if rank E A; for all nite (2.3) System (1.1) and the triplet A; are called impulse ontr ol lable I-c ontr ol lable if rank AK n; where the columns of span er (2.4) System (1.1) and the triplet A; are called ompletely ontr ol lable C-c ontr ol lable if (2.3) holds and rank n: (2.5) Observ abilit is dual prop ert of con trollabilit Denition 2.2. System (1.1) and the triplet A; are called observable on the achable set R-observable if rank E for all nite (2.6) System (1.1) and the triplet A; are called impulse observable I-observable if rank n; where the columns of span er (2.7) System (1.1) and the triplet A; are called ompletely observable C-observable if (2.6) holds and rank n: (2.8) Clearly conditions (2.4) and (2.7) are eak er than (2.5) and (2.8), resp ectiv ely Equi- alen algebraic haracterizations of arious concepts of con trollabilit and observ abilit for descriptor systems are presen ted in [8, 9, 42 ]. 2.2 Con trollabilit and observ abilit Gramians Assume that the encil E is c-stable Then the in tegrals pc dt (2.9) and po dt (2.10)

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exist, where is as in (2.1). The matrix pc is called the pr op er ontr ol lability Gr amian and the matrix po is called the pr op er observability Gr amian of the con tin uous-time descriptor system (1.1), see [3 34 ]. The impr op er ontr ol lability Gr amian of system (1.1) is dened ic and the impr op er observability Gr amian of system (1.1) is dened io where are as in (2.2). Note that the improp er con trollabilit and observ abilit Gramians ic and io are, up to the sign, the same as those dened in [3]. If then pc and po are the usual con trollabilit and observ abilit Gramians for standard state space systems [15 43 ]. The prop er con trollabilit and observ abilit Gramians are the unique symmetric, ositiv semidenite solutions of the pr oje cte gener alize ontinuous-time Lyapunov quations pc pc pc pc (2.11) and po po po po (2.12) resp ectiv ely where and are giv en in (1.3), see [34 ]. If E is in eierstrass canonical form (1.2) and if the matrices and are partitioned in blo ks conformally to and then can sho that pc po (2.13) where and satisfy the standard con tin uous-time Ly apuno equations The improp er con trollabilit and observ abilit Gramians are the unique symmetric, ositiv semidenite solutions of the pr oje cte gener alize discr ete-time Lyapunov quations ic ic ic (2.14) and io io io (2.15)

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resp ectiv ely [34 ]. They can represen ted as ic io (2.16) where and satisfy the standard discrete-time Ly apuno equations The con trollabilit and observ abilit Gramians can used to haracterize con trollabilit and observ abilit prop erties of system (1.1). Theorem 2.3. [3 34 Consider the descriptor system (1.1) Assume that E is c-stable. 1. System (1.1) is R-c ontr ol lable if and only if the pr op er ontr ol lability Gr amian pc is ositive denite on the subsp ac im 2. System (1.1) is I-c ontr ol lable if the impr op er ontr ol lability Gr amian ic is ositive denite on the subsp ac er 3. System (1.1) is C-c ontr ol lable if and only if pc ic is ositive denite. 4. System (1.1) is R-observable if and only if the pr op er observability Gr amian po is ositive denite on the subsp ac im 5. System (1.1) is I-observable if the impr op er observability Gr amian io is ositive denite on the subsp ac er 6. System (1.1) is C-observable if and only if po io is ositive denite. Note that the I-con trollabilit (I-observ abilit y) of (1.1) do es not imply that the improp er con trollabilit (observ abilit y) Gramian is ositiv denite on er (on er ). Example 2.4. The descriptor system (1.1) with is I-con trollable and I-observ able. ha ic io and i.e., neither ic nor io are ositiv denite on er er Corollary 2.5. Consider the descriptor system (1.1) wher the encil E is c-stable. 1. System (1.1) is R-c ontr ol lable and R-observable if and only if rank( pc rank( po rank( pc po (2.17) 2. System (1.1) is I-c ontr ol lable and I-observable if rank ic rank( io rank ic io (2.18) 3. System (1.1) is C-c ontr ol lable and C-observable if and only if (2.17) and (2.18) hold. Pr of. The result follo ws from Theorem 2.3 and represen tations (1.2), (2.13) and (2.16).

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2.3 Hank el singular alues The prop er con trollabilit and observ abilit Gramians pc and po as ell as the improp er con trollabilit and observ abilit Gramians ic and io are not system in arian t. Indeed, under system equiv alence transformation the prop er and improp er con trollabilit Grami- ans pc and ic are transformed to pc pc and ic ic resp ectiv ely whereas the prop er and improp er observ abilit Gramians po and io are transformed to po po and io io resp ectiv ely Ho ev er, it follo ws from pc po pc po ic io ic io that the sp ectra of the matrices pc po and ic io are system in arian t. These matrices pla the same role for descriptor systems as the pro duct of the con trollabilit and observ abilit Gramians for standard state space systems [15 43 ]. ha the follo wing result. Theorem 2.6. et E c-stable. Then the matric es pc po and ic io have al and non-ne gative eigenvalues. Pr of. It follo ws from (2.9) and (2.10) that pc and po are symmetric and ositiv semidenite. In this case there exists nonsingular matrix suc that pc po where and are diagonal matrices with ositiv diagonal elemen ts [43 p. 76]. Then pc po Hence, pc po is diagonalizable and it has real and non-negativ eigen alues. Similarly can sho that the eigen alues of ic io are real and non-negativ e. Denition 2.7. Let and the dimensions of the deating subspaces of the c-stable encil E corresp onding to the nite and innite eigen alues, resp ectiv ely The square ro ots of the largest eigen alues of the matrix pc po denoted are called the pr op er Hankel singular values of the con tin uous-time descriptor system (1.1). The square ro ots of the largest eigen alues of the matrix ic io denoted are called the impr op er Hankel singular values of system (1.1). will assume that the prop er and improp er Hank el singular alues are ordered decreas- ingly i.e., The prop er and improp er Hank el singular alues form the set of Hank el singular alues of the con tin uous-time descriptor system (1.1). or the prop er Hank el singular alues are the classical Hank el singular alues of standard state space systems [15 27 ].

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Since the prop er and improp er con trollabilit and observ abilit Gramians are symmetric and ositiv semidenite, there exist Cholesky factorizations pc po ic io (2.19) where the matrices n;n are upp er triangular Cholesky factors. The follo wing lemma giv es connection et een the prop er and improp er Hank el singular alues of system (1.1) and the standard singular alues of the matrices and AR Lemma 2.8. Assume that the encil E in system (1.1) is c-stable. Consider the Cholesky factorizations (2.19) of the pr op er and impr op er Gr amians of (1.1) Then the pr op er Hankel singular values of system (1.1) ar the lar gest singular values of the matrix and the impr op er Hankel singular values of system (1.1) ar the lar gest singular values of the matrix AR Pr of. ha pc po ic io AR AR where and denote, resp ectiv ely the eigen alues and the singular alues ordered decreasingly Mo del reduction In this section consider the problem of reducing the order of the descriptor system (1.1). 3.1 Balanced realizations or giv en transfer function ), there are man dieren realizations [9]. Here are in terested only in particular realizations that are useful in applications. Denition 3.1. realization A; of the transfer function is called minimal if the triplet A; is C-con trollable and the triplet A; is C-observ able. Denition 3.2. realization A; of the transfer function is called alanc if pc po and ic io with diag and diag ). will sho that for minimal realization A; with the c-stable encil E there exists system equiv alence transformation suc that the realization AT (3.1) is balanced.

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Consider the Cholesky factorizations (2.19) of the con trollabilit and observ abilit Grami- ans. ma assume without loss of generalit that the Cholesky factors and ha full ro rank. If A; is C-con trollable and A; is C-observ able, then it follo ws from Corollary 2.5 and Lemma 2.8 that 0, and AR 0, Hence, the matrices ;n and AR ;n are nonsingular. Let AR (3.2) singular alue decomp ositions of and AR where and are orthogo- nal, diag( and diag are nonsingular. Consider the matrices AR (3.3) and (3.4) Since pc po pc po AP obtain that and AR (3.5) Then AR AR i.e., the matrices and are nonsingular and Similarly can sho that the matrices and are also nonsingular and Using (2.19) and (3.2)-(3.5), obtain that the prop er and improp er con trollabilit and observ abilit Gramians of the transformed system (3.1) ha the form pc po ic io Th us, with and as in (3.3) and (3.4), resp ectiv ely is the balancing transfor- mation and realization (3.1) is balanced. Note that just as for standard state space systems [15 27 ], the balancing transformation for descriptor systems is not unique. rom (3.2)-(3.4) nd AT where AR and Th us, the encil E is in eierstrass-lik canonical form. Clearly it is regular, c-stable and has the same index as E

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3.2 Balanced truncation In the previous subsection ha considered reduction of minimal realization to ba- lanced form. Ho ev er, computing the balanced realization ma ill-conditioned as so on as or in (3.2) has small singular alues. In addition, if the realization is not minimal, then the matrix or is singular. In the similar situation for standard state space systems one erforms mo del reduction truncating the state comp onen ts corresp onding to the zero and small Hank el singular alues without signican hanges of the system prop erties, see, e.g., [15 27 37 ]. This pro cedure is kno wn as alanc trunc ation It can also applied to the descriptor system (1.1). The prop er con trollabilit and observ abilit Gramians can used to describ the future output energy dt and the past input energy 1 dt that is needed to reac from 1 the state (0) im when no input is applied for 0. Theorem 3.3. Consider descriptor system (1.1) Assume that the encil E is c-stable and the triplet A; is R-c ontr ol lable. et pc and po the pr op er ontr ol lability and observability Gr amians of (1.1) If im and for then po Mor over, for opt pc we have opt min pc wher is the Hilb ert sp ac of al dimensional ve ctor-functions that ar squar inte gr able on 1 0) and the matrix pc satises pc pc pc pc pc pc pc pc pc pc (3.6) Pr of System (1.1) with im and for has unique solution giv en 0. Then for and, hence, dt po Consider no the minimization of for ). Note that the state (0) of the descriptor system (1.1) with for satises the constrain equation 1 dt: (3.7) 10

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Since system (1.1) is R-con trollable, the matrix in (2.13) is nonsingular and, hence, im im pc In this case there exists ector suc that pc with im Therefore, 1 pc dt pc pc pc pc (3.8) or an input opt with opt pc ), it follo ws from (3.7) and (3.8) that 1 dt Then 1 opt opt dt 2( pc 1 dt 1 dt 1 opt opt dt: Th us, opt minimizes among all inputs that transfer the system from 1 to (0) im Using the second and third equations in (3.6), nd opt 1 pc pc dt pc Remark 3.4. Equations (3.6) imply that pc is symmetric (1 2)-pseudoinv erse [7 of pc It is, in general, not unique, but opt pc and opt pc with im are uniquely dened. Unfortunately ere unable to nd similar energy in terpretation for the improp er con trollabilit and observ abilit Gramians. If the descriptor system (1.1) is not minimal, then it has states that are uncon trollable or/and unobserv able. These states corresp ond to the zero prop er and improp er Hank el sin- gular alues and can truncated without hanging the input-output relation in the sys- tem. Note that the um er of non-zero improp er Hank el singular alues of (1.1) is equal to rank( ic io whic can in turn estimated as rank( ic io min( m; p; This estimate sho ws that if the index of the encil E times the um er of inputs or the um er of outputs is uc smaller than the dimension of the deating subspace of E corresp onding to the innite eigen alues, then the order of system (1.1) can reduced signican tly urthermore, Theorem 3.3 implies that large input energy is required to reac from 1 the state (0) whic lies in an in arian subspace of the prop er con trollabilit Gramian pc corresp onding to its small non-zero eigen alues. Moreo er, if 11

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is con tained in an in arian subspace of the matrix po corresp onding to its small non- zero eigen alues, then the initial alue (0) has small eect on the output energy or the balanced system, pc and po are equal and, hence, they ha the same in arian subspaces. In this case the truncation of the states related to the small prop er Hank el singular alues do es not hange system prop erties essen tially Unfortunately this do es not hold for the improp er Hank el singular alues. If truncate the states that corresp ond to the small non-zero improp er Hank el singular alues, then the encil of the reduced order system ma get nite eigenv alues in the closed righ half-plane, see [26 ]. In this case the appro ximation will inaccurate. Let A; realization (not necessarily minimal) of the transfer function ). Assume that the encil E is c-stable. Consider the Cholesky factorizations (2.19). Let AR 'thin' singular alue decomp ositions of and AR where ], ], and ha orthonormal columns, diag( and diag( +1 with +1 and rank diag( with rank AR ). Then the reduced order realization can computed as sE (3.9) where n;` n;` (3.10) and Note that computing the reduced order descriptor system can in terpreted as erforming system equiv alence transformation suc that sE sE sE where the encil E has the nite eigen alues only all the eigen alues of E are innite, and then reducing the order of the subsystems and with nonsingular and using classical balanced truncation metho ds for con tin uous-time and discrete-time state space systems, resp ectiv ely Clearly the reduced order system (3.9) is minimal and the encil is c-stable. The describ ed decoupling of system matrices is equiv alen to the additiv decomp osition of the transfer function as sp ), where sp sE is the strictly prop er part and sE is the olynomial part of ). The transfer function of the reduced system has the form sp ), where sp and are the reduced order subsystems. Note that ), and, hence, the dierence sp sp 12

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is strictly prop er rational function. Th us, ha the follo wing upp er ound for the -norm of the error system := sup i! i! 2( +1 that can deriv ed as in [15 ]. Here denotes the sp ectral matrix norm. 3.3 Algorithms reduce the order of the descriptor system (1.1) ha to compute the Cholesky factors of the prop er and improp er con trollabilit and observ abilit Gramians that satisfy the pro- jected generalized Ly apuno equations (2.11), (2.12), (2.14) and (2.15). These factors can computed using the gener alize Schur-Hammarling metho [34 35 ]. Let the encil E in generalized real Sc ur form and (3.11) where and are orthogonal, is upp er triangular nonsingular, is upp er triangular nilp oten t, is upp er quasi-triangular and is upp er triangular nonsingular, and let the matrices and (3.12) partitioned in blo ks conformally to and Then one can sho [34 35 that the Cholesky factors of the Gramians of system (1.1) ha the form (3.13) where is the solution of the generalized Sylv ester equation (3.14) the matrices are the Cholesky factors of the solutions pc po of the generalized con tin uous-time Ly apuno equations pc pc )( (3.15) po po (3.16) while and are the Cholesky factors of the solutions ic and io of the generalized discrete-time Ly apuno equations ic ic (3.17) io io (3.18) rom (3.11) and (3.13) obtain that and AR Th us, the prop er and improp er Hank el singular alues of (1.1) can computed from the singular 13

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alue decomp ositions of the matrices and urthermore, it follo ws from (3.10) and (3.13) that the pro jection matrices and ha the form (3.19) with and In this case the matrix co ecien ts of the reduced order system (3.9) are giv en (3.20) In summary ha the follo wing algorithm that is generalization of the squar ot alanc trunc ation metho [24 37 for the con tin uous-time descriptor system (1.1). Algorithm 3.1. Gener alize Squar ot (GSR) metho d. Input: A; such that E is c-stable. Output: duc or der system 1. Compute the gener alize Schur form (3.11) 2. Compute the matric es (3.12) 3. Solve the gener alize Sylvester quation (3.14) 4. Compute the Cholesky factors of the solutions pc and po of quations (3.15) and (3.16) esp ctively. 5. Compute the Cholesky factors of the solutions ic and io of quations (3.17) and (3.18) esp ctively. 6. Compute the 'thin singular value de omp osition wher and have orthonormal olumns, diag and diag( +1 with rank and +1 7. Compute the 'thin singular value de omp osition wher and have orthonormal olumns and diag( with rank 8. Compute 9. Compute the duc or der system as in (3.20) If the original system (1.1) is highly un balanced or if the deating subspaces of the encil E corresp onding to the nite and innite eigen alues are close, then the pro jection matrices and as in (3.19) are ill-conditioned. oid accuracy loss in the reduced order system, squar ot alancing fr metho has een prop osed in [38 for standard state space systems. This metho can generalized for descriptor systems as follo ws. 14

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Algorithm 3.2. Gener alize Squar ot Balancing (GSRBF) metho d. Input: A; such that E is c-stable. Output: duc or der system 1.-7. as in lgorithm 3.1 8. Compute the 'e onomy size' QR de omp ositions L; wher n;` have orthonormal olumns and `;` ar nonsingular. 9. Compute the duc or der system with (3.21) The GSR metho and the GSRBF metho are mathematically equiv alen in the sense that in exact arithmetic they return reduced systems with the same transfer function. It should noted that the reduced order realization A; as in (3.21) is, in general, not balanced and the corresp onding encil is not in eierstrass-lik canonical form. will no discuss the umerical asp ects of Algorithms 3.1 and 3.2. compute the generalized Sc ur form (3.11) can use the QZ algorithm [16 40] the GUPTRI algorithm [10 11 ], or algorithms prop osed in [2 39 ]. solv the generalized Sylv ester equation (3.14) can use the generalized Sc ur metho [23 or its recursiv blo ed mo dication [21 that is more suitable for large problems. The upp er triangular Cholesky factors and of the solutions of the generalized Ly apuno equations (3.15)-(3.18) can determined without computing the solutions themself using the generalized Hammarling metho [20 28 ]. In the general case the generalized Sc ur and Hammarling metho ds are based on the preliminary reduction of the corresp onding matrix encils to the generalized Sc ur form, calculation of the solution of reduced system and bac transformation. Note that the encils E and E in equations (3.14) (3.18) are already in generalized Sc ur form. Th us, need only to solv the upp er (quasi-)triangular matrix equations. Finally the singular alue decomp osition of the matrices and where all three factors are upp er triangular, can computed without forming these pro ducts ex- plicitly see [5 12 17 and references therein. Since the GSR metho and the GSRBF metho are based on computing the generalized Sc ur form, they cost ops and ha the memory complexit ). Th us, these metho ds can used for problems of small and medium size only Moreo er, they do not tak in to accoun the sparsit or an structure of the system and are not attractiv for parallelization. Recen tly iterativ metho ds related to the ADI metho and the Smith metho ha een prop osed to compute lo rank appro ximations of the solutions of standard large- scale sparse Ly apuno equations [25 29 ]. It is imp ortan to extend these metho ds for pro jected generalized Ly apuno equations. This topic is curren tly under in estigations. Remark 3.5. The GSR metho and the GSRBF metho can used to reduce the order of unstable descriptor systems. Firstly compute the additiv decomp osition [22 of the 15

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transfer function ), where sE sE Here the matrix encil E is c-stable and all the eigen alues of the encil E are nite and ha non-negativ real part. Then determine the reduced order system applying the balanced truncation mo del reduction metho to the subsystem ]. Finally the reduced order appro ximation of is giv en ), where is included unmo died. Remark 3.6. The con trollabilit and observ abilit Gramians as ell as Hank el singular alues can also generalized for discrete-time descriptor systems, see [36 for details. In this case an extension of balanced truncation mo del reduction metho ds for suc systems is straigh tforw ard. Numerical examples In this section consider umerical examples to illustrate the eectiv eness of the prop osed mo del reduction metho ds for descriptor systems. All of the follo wing results ere obtained on SunOS 5.8 orkstation with relativ mac hine precision 22 10 16 using the MA TLAB mex-functions based on the GUPTRI routine [10 11 and the SLICOT library routines [4]. Example 4.1. Consider the holonomically constrained planar mo del of truc [32 ]. The linearized equation of motion has the form (4.1) where 11 is the osition ector, 11 is the elo cit ector, is the Lagrange ultiplier, is the ositiv denite mass matrix, is the stiness matrix, is the damping matrix, is the constrain matrix and is the input matrix. System (4.1) together with the output equation forms descriptor system of order 23 with input and 11 outputs. The dimension of the deating subspace corresp onding to the nite eigen alues is 20. The prop er Hank el singular alues of the linearized truc mo del (4.1) are giv en in able 1. All the improp er Hank el singular alues are zero and, hence, the transfer function of (4.1) is strictly prop er. appro ximate system (4.1) mo dels of order 12 computed the GSR metho and the GSRBF metho d. Figure illustrates ho accurate the reduced order mo dels appro ximate the original system. displa the plots of the sp ectral norm of the frequency resp onses of the original system i! and the reduced order systems i! for frequency range [10 10 ]. One can see that the appro ximate system deliv ered the GSR metho diers sligh tly from the original one for high frequencies only while the plots of the full order Av ailable from http://www.cs.umu.se/research/ nla/ singu lar pairs/guptri Av ailable from http://www.win.tue.nl/niconet/ NIC2 /slic ot.h tml 16

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072 10 715 10 11 134 10 16 988 10 962 10 578 10 12 013 10 17 833 10 286 10 925 10 13 392 10 18 847 10 845 10 192 10 14 596 10 19 108 10 10 750 10 10 535 10 15 969 10 20 163 10 10 able 1: Prop er Hank el singular alues of the linearized truc mo del. system and the reduced order system computed the GSRBF metho coincide. Note that the pro jection matrices and in the GSR metho (see (3.19)) ha the condition um ers max min 385 10 max min 965 10 whereas the pro jection matrices in the GSRBF metho giv en and ha orthonormal columns. In Figure compare the absolute appro ximation errors i! i! with the upp er ound whic is the wice sum of the truncated prop er Hank el singular alues 13 20 see that the appro ximation error for the GSRBF metho is considerable smaller than for the GSR metho d. 10 −1 10 10 10 10 10 10 10 −12 10 −11 10 −10 10 −9 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 Frequency (rad/sec) Linearized truck model Full order GSR GSRBF PSfrag replacemen ts i! and i! Figure 1: requency resp onses of the full order system and the reduced order systems. 10 −1 10 10 10 10 10 10 10 −16 10 −15 10 −14 10 −13 10 −12 10 −11 10 −10 10 −9 10 −8 10 −7 10 −6 Frequency (rad/sec) Error systems and error bound Error system, GSR Error system, GSRBF Error bound PSfrag replacemen ts i! i! Figure 2: Error systems and error ound for the linearized truc mo del. Example 4.2. Consider the o of an incompressible uid describing the instationary Stok es equation on square region ]. The spatial discretization of this equation the nite olume metho on uniform staggered grid leads to the descriptor system (4.2) where and are the semidiscretized ectors of elo cities and pressures, resp ectiv ely ;n is the discrete Laplace op erator and ;n is obtained from the discrete gradien op erator discarding the last column [41 ]. or simplicit 17

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is hosen at random and [1 0] ;n or 12, ha 480 and the dimensions of the deating subspaces of the encil in (4.2) corresp onding to the nite and innite eigen alues are 144 and 336. 10 15 20 25 30 10 −30 10 −25 10 −20 10 −15 10 −10 10 −5 10 Proper Hankel singular values Figure 3: Prop er Hank el singular alues for the semidiscretized Stok es equation. 10 −1 10 10 10 10 10 10 10 −16 10 −15 10 −14 10 −13 10 −12 10 −11 10 −10 10 −9 10 −8 Frequency (rad/sec) Error systems and error bound Error system, GSR Error system, GSRBF Error bound PSfrag replacemen ts i! i! Figure 4: Error systems and error ound for the semidiscretized Stok es equation. The 30 largest prop er Hank el singular alues of the semidiscretized Stok es equation (4.2) are giv en in Figure 3. One can see that they deca ery fast, and, hence, system (4.2) can ell appro ximated mo del of lo order. All the improp er Hank el singular alues are zero. appro ximate system (4.2) mo dels of order computed the GSR metho and the GSRBF metho d. The sp ectral norm of the frequency resp onses of the full order and reduced order systems are not presen ted since they ere imp ossible to distinguish. Figure sho ws the plots of the error ound and the sp ectral norm of the error systems i! i! for frequency rang [10 10 ]. see again that the appro ximate system deliv ered the GSRBF metho is etter than the one computed the GSR metho d. Conclusion ha generalized the con trollabilit and observ abilit Gramians as ell as Hank el singular alues for descriptor systems and studied their imp ortan features. Balanced truncation mo del reduction metho ds for descriptor systems ha een presen ted. These metho ds are closely related to the Gramians and deliv er reduced order systems that preserv the regularit and stabilit prop erties of the original system. Moreo er, for these metho ds priori ound on the appro ximation error is ailable. Ac kno wledgemen t. The author ould lik to thank the referees for aluable suggestions. References [1] An toulas C, Sorensen DC, and Gugercin (2001) surv ey of mo del reduction metho ds for large-scale systems, in Structured Matrices in Mathematics, Computer Science and 18

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Engineering, ol. (Olshevsky ed.) Con temp orary Mathematics Series, 280. American Mathematical So ciet Pro vidence, RI, pp. 193{219 [2] Beelen T, an Do oren (1988) An impro ed algorithm for the computation of Kro- nec er's canonical form of singular encil, Linear Algebra Appl., 105:9{65 [3] Bender DJ (1987) Ly apuno v-lik equations and reac habilit y/observ abilit Gramians for descriptor systems, IEEE rans. Automat. Con trol, 32(4):343{348 [4] Benner Mehrmann V, Sima V, an Huel S, arga (1999) SLICOT subroutine library in systems and con trol theory Appl. Comput. Con trol Signals Circuits, 1:499{539 [5] Bo janczyk W, Ew erbring LM, Luk FT, an Do oren (1991) An accurate pro duct SVD algorithm, Signal Pro cess., 25:189{201 [6] Brenan KE, Campb ell SL, etzold LR (1989) The Numerical Solution of Initial-V alue Problems in Dieren tial-Algebraic Equations, Elsevier, North-Holland, New ork [7] Campb ell SL, Mey er CD (1979) Generalized In erses of Linear ransformations, Do er Publications, New ork [8] Cobb (1984) Con trollabilit observ abilit and dualit in singular systems, IEEE rans. Automat. Con trol, 29(12):1076{1082 [9] Dai (1989) Singular Con trol Systems, Lecture Notes in Con trol and Information Sciences, 118. Springer-V erlag, Berlin, Heidelb erg [10] Demmel JW, agstr om (1993) The generalized Sc ur decomp osition of an arbitrary encil B Robust soft are with error ounds and applications. Part I: Theory and algorithms, CM rans. Math. Soft are, 19(2):160{174 [11] Demmel JW, agstr om (1993) The generalized Sc ur decomp osition of an arbitrary encil B Robust soft are with error ounds and applications. Part I: Soft are and applications, CM rans. Math. Soft are, 19(2):175{201 [12] Drmac (2000) New accurate algorithms for singular alue decomp osition of matrix triplets, SIAM J. Matrix Anal. Appl., 21(3):1026{1050 [13] ortuna L, Nunnari G, Gallo (1992) Mo del Order Reduction ec hniques with Appli- cations in Electrical Engineering, Springer-V erlag, London [14] Galliv an K, Grimme E, an Do oren (1996) rational Lanczos algorithm for mo del reduction, Numerical Algorithms, 12(1-2):33{63 [15] Glo er (1984) All optimal Hank el-norm appro ximations of linear ultiv ariable systems and their -errors ounds, In ternat. J. Con trol, 39(6):1115{1193 [16] Golub GH, an Loan CF (1996) Matrix Computations. 3r. ed, The Johns Hopkins Univ ersit Press, Baltimore, London [17] Golub GH, Slna K, an Do oren (2000) Computing the SVD of general matrix pro duct/quotien t, SIAM J. Matrix Anal. Appl., 22(1):1{19 19

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[18] Grimme EJ, Sorensen DC, an Do oren (1996) Mo del reduction of state space systems via an implicitly restarted Lanczos metho d, Numerical Algorithms, 12(1-2):1{31 [19] un ther M, eldmann (1999) CAD-based electric-circuit mo deling in industry I. Mathematical structure and index of net ork equations, Surv eys Math. Indust., 8(2):97{ 129 [20] Hammarling SJ (1982) Numerical solution of the stable non-negativ denite Ly apuno equation, J. Numer. Anal., 2:303{323 [21] Jonsson I, agstr om (2002) Recursiv blo ed algorithms for solving triangular systems Part I: one-sided and coupled Sylv ester-t yp matrix equations, CM rans. Math. Soft are, 28(4):392{415 [22] agstr om B, an Do oren (1992) generalized state-space approac for the additiv decomp osition of transfer function, J. Numer. Linear Algebra Appl., 1(2):165{181 [23] agstr om B, estin (1989) Generalized Sc ur metho ds with condition estimators for solving the generalized Sylv ester equation, IEEE rans. Automat. Con trol, 34:745{751 [24] Laub AJ, Heath MT, aige CC, ard (1987) Computation of system balancing transformations and other applications of sim ultaneous diagonalization algorithms, IEEE rans. Automat. Con trol, 32(2):115{122 [25] Li JR (2000) Mo del Reduction of Large Linear Systems via Lo Rank System Gramians, Ph.D. thesis, Departmen of Mathematics, Massac usetts Institute of ec hnology [26] Liu Q, Sreeram (2000) Mo del reduction of singular systems, in Pro ceedings of the 39th IEEE Conference on Decision and Con trol (Sydney Australia, 2000), 2373{2378 [27] Mo ore BC (1981) Principal comp onen analysis in linear systems: con trollabilit ob- serv abilit and mo del reduction, IEEE rans. Automat. Con trol, 26(1):17{32 [28] enzl (1998) Numerical solution of generalized Ly apuno equations, Adv. Comput. Math., 8(1-2):33{48 [29] enzl (1999/00) cyclic lo w-rank Smith metho for large sparse Ly apuno equations, SIAM J. Sci. Comput., 21(4):1401{1418 [30] erev K, Shafai (1994) Balanced realization and mo del reduction of singular systems, In ternat. J. Systems Sci., 25(6):1039{1052 [31] Safono MG, Chiang (1989) Sc ur metho for balanced-truncation mo del reduc- tion, IEEE rans. Automat. Con trol, 34(7):729{733 [32] Simeon B, Grupp F, uhrer C, Ren trop (1994) nonlinear truc mo del and its treatmen as ultib dy system, J. Comput. Appl. Math., 50:523{532 [33] Stew art GW, Sun JG (1990) Matrix erturbation Theory Academic Press, New ork [34] St yk el (2002) Analysis and Numerical Solution of Generalized Ly apuno Equations, Ph.D. thesis, Institut ur Mathematik, ec hnisc he Univ ersit at Berlin 20

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[35] St yk el (2002) Numerical solution and erturbation theory for generalized Ly apuno equations, Linear Algebra Appl., 349:155{185 [36] St yk el (2003) Input-output in arian ts for descriptor systems, Preprin PIMS-03-1, acic Institute for the Mathematical Sciences, Canada [37] om bs MS, ostleth eite (1987) runcated balanced realization of stable non-minimal state-space system, In ternat. J. Con trol, 46(4):1319{1330 [38] arga (1991) Ecien minimal realization pro cedure based on balancing, in Pro c. of IMA CS/IF Symp osium on Mo delling and Con trol of ec hnological Systems, ol. (EL Moudni A, Borne Tzafestas SG, eds.), pp. 42{47 [39] arga (1998) Computation of coprime factorizations of rational matrices, Linear Algebra Appl., 271:83{115 [40] atkins DS (2000) erformance of the QZ algorithm in the presence of innite eigen- alues, SIAM J. Matrix Anal. Appl., 22(2):364{375 [41] eic ert (1997) Applications of the Theory of Dieren tial-Algebraic Equations to artial Dieren tial Equations of Fluid Dynamics, Ph.D. thesis, ec hnisc he Univ ersit at Chemnitz [42] Yip EL, Sinco ec RF (1981) Solv abilit con trollabilit and observ abilit of con tin uous descriptor systems, IEEE rans. Automat. Con trol, 26(3):702{707 [43] Zhou K, Do yle JC, Glo er (1996) Robust and Optimal Con trol, Pren tice-Hall, Upp er Saddle Riv er 21

Gramian based mo del reduction for descriptor systems atjana St yk el Abstract Mo del reduction is of fundamen tal imp ortance in man con trol applications consider mo del reduction metho ds for linear timein arian con tin uoustime descriptor system ID: 23649

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Published in Mathematics of Contr ol, Signals, and Systems 16, 297{319, 2004. Gramian based mo del reduction for descriptor systems atjana St yk el Abstract Mo del reduction is of fundamen tal imp ortance in man con trol applications. consider mo del reduction metho ds for linear time-in arian con tin uous-time descriptor systems. These metho ds are based on the balanced truncation tec hnique and closely related to the con trollabilit and observ abilit Gramians and Hank el singular alues of descriptor systems. The Gramians can computed solving generalized Ly apuno equations with sp ecial righ t-hand sides. Numerical examples are giv en. Key ords: Descriptor systems, Gramians, Hank el singular alues, mo del reduction, balanced truncation. In tro duction Consider linear time-in arian con tin uous-time system (0) (1.1) where n;n n;m p;n is the state ector, is the con trol input, is the output and is the initial alue. The um er of state ariables is called the or der of system (1.1). If then (1.1) is standar state sp ac system Otherwise, (1.1) is descriptor system or gener alize state sp ac system Suc systems arise naturally in man applications suc as ultib dy dynamics, electrical circuit sim ulation and semidiscretization of partial dieren tial equations, see [6, 9, 19 ]. will assume throughout the pap er that the encil E is gular i.e., det( E for some In this case E can reduced to the eierstrass canonical form [33 ]. There exist nonsingular matrices and suc that (1.2) where is the iden tit matrix of order is the Jordan blo corresp onding to the nite eigen alues of E is nilp oten and corresp onds to the eigen alue at innit The index of nilp otency of denoted is called the index of the encil E Represen tation (1.2) Institut ur Mathematik, MA 4-5, ec hnisc he Univ ersit at Berlin, Strae des 17. Juni 136, D-10623 Berlin, German Phone: +49 (0)30 314-29292, ax: +49 (0)30 314-79706, E-mail: stykel@math.tu-berlin.de Supp orted Deutsc he orsc ungsgemeinsc haft, Researc Gran ME 790/12-1.

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denes the decomp osition of in to complemen tary deating subspaces of dimensions and corresp onding to the nite and innite eigen alues of the encil E resp ectiv ely The matrices and (1.3) are the sp ctr al pr oje ctions on to the righ and left deating subspaces of E corresp onding to the nite eigen alues. The encil E is called c-stable if it is regular and all the nite eigen alues of E lie in the op en left half-plane. Descriptor systems arising, e.g., from the spatial discretization of partial dieren tial equa- tion ha usually large order while the um er of inputs and the um er of outputs are small compared to Sim ulation or real time con troller design for suc large-scale sys- tems ecomes dicult ecause of storage requiremen ts and exp ensiv computations. In this case mo del order reduction pla ys an imp ortan role. It consists in an appro ximation of the descriptor system (1.1) reduced order system (0) (1.4) where `;` `;m p;` and Note that systems (1.1) and (1.4) ha the same input ). require for the appro ximate system (1.4) to preserv prop erties of the original system (1.1) lik regularit and stabilit It is also desirable for the appro xima- tion error to small. Moreo er, the computation of the reduced order system should umerically stable and ecien t. There exist arious mo del reduction approac hes for standard state space systems suc as balanced truncation [15 24 27 31 37, 38 ], momen matc hing appro ximation [1, 14 18 and optimal Hank el norm appro ximation [15 ]. Surv eys on system appro ximation and mo del reduction can found in [1 13 ]. One of the most eectiv and ell studied mo del reduction tec hniques is balanced truncation whic is closely related to the Ly apuno equations The solutions and of these equations are called the ontr ol lability and observability Gr ami- ans resp ectiv ely The balanced truncation approac consists in transforming the state space system in to balanced form whose the con trollabilit and observ abilit Gramians ecome diagonal and equal, together with truncation of states that are oth dicult to reac and to observ e, see [27 for details. Balanced truncation mo del reduction for descriptor systems has een considered in [26 30 ]. The algorithms presen ted there are based on computing the eierstrass canonical form (1.2) of the encil E Ho ev er, it is ell kno wn [33 that this computational problem is, in general, ill-conditioned in the sense that small erturbations in and ma lead to an inaccurate umerical result. In this pap er generalize con trollabilit and observ abilit Gramians as ell as Hank el singular alues for descriptor systems (Section 2). In Section presen an extension of balanced truncation metho ds [24 37 38 to descriptor systems. These metho ds are based on computing the generalized Sc ur form of the encil E and solving the generalized Sylv ester and Ly apuno equations using umerically stable algorithms. Section con tains umerical examples.

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Descriptor systems Consider the con tin uous-time descriptor system (1.1). It is ell kno wn that if the encil E is regular, is times con tin uously differentiable and is consisten t, i.e., it elongs to the set of onsistent initial onditions =0 (0) then the descriptor system (1.1) has unique con tin uously dieren tiable solution ), see [9], that is giv en d =0 Here tJ (2.1) is fundamental solution matrix of the descriptor system (1.1), and the matrices ha the form (2.2) Clearly for If the initial condition is inconsisten or the input is not sucien tly smo oth (for example, in most con trol problems u(t) is only piecewise con tin uous), then the solution of the descriptor system (1.1) ma ha impulsiv mo des [8 ]. The rational matrix-v alued function sE is called the tr ansfer function of the descriptor system (1.1). quadruple of matrices A; is alization of ). will also often denote realization of sE Tw realizations A; and A; are estricte system quivalent if there exist nonsingular matrices and suc that pair is called system quivalenc tr ansformation haracteristic quan tit of system (1.1) is system invariant if it is preserv ed under system equiv alence transformation. The transfer function is system in arian t, since sE The transfer function is called pr op er if lim !1 and is called strictly pr op er if lim !1 0. Other imp ortan results from the theory of rational functions and realization theory ma found in [9 ].

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2.1 Con trollabilit and observ abilit or descriptor systems there are arious concepts of con trollabilit and observ abilit e.g., [8 42 ]. Denition 2.1. System (1.1) and the triplet A; are called ontr ol lable on the achable set R-c ontr ol lable if rank E A; for all nite (2.3) System (1.1) and the triplet A; are called impulse ontr ol lable I-c ontr ol lable if rank AK n; where the columns of span er (2.4) System (1.1) and the triplet A; are called ompletely ontr ol lable C-c ontr ol lable if (2.3) holds and rank n: (2.5) Observ abilit is dual prop ert of con trollabilit Denition 2.2. System (1.1) and the triplet A; are called observable on the achable set R-observable if rank E for all nite (2.6) System (1.1) and the triplet A; are called impulse observable I-observable if rank n; where the columns of span er (2.7) System (1.1) and the triplet A; are called ompletely observable C-observable if (2.6) holds and rank n: (2.8) Clearly conditions (2.4) and (2.7) are eak er than (2.5) and (2.8), resp ectiv ely Equi- alen algebraic haracterizations of arious concepts of con trollabilit and observ abilit for descriptor systems are presen ted in [8, 9, 42 ]. 2.2 Con trollabilit and observ abilit Gramians Assume that the encil E is c-stable Then the in tegrals pc dt (2.9) and po dt (2.10)

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exist, where is as in (2.1). The matrix pc is called the pr op er ontr ol lability Gr amian and the matrix po is called the pr op er observability Gr amian of the con tin uous-time descriptor system (1.1), see [3 34 ]. The impr op er ontr ol lability Gr amian of system (1.1) is dened ic and the impr op er observability Gr amian of system (1.1) is dened io where are as in (2.2). Note that the improp er con trollabilit and observ abilit Gramians ic and io are, up to the sign, the same as those dened in [3]. If then pc and po are the usual con trollabilit and observ abilit Gramians for standard state space systems [15 43 ]. The prop er con trollabilit and observ abilit Gramians are the unique symmetric, ositiv semidenite solutions of the pr oje cte gener alize ontinuous-time Lyapunov quations pc pc pc pc (2.11) and po po po po (2.12) resp ectiv ely where and are giv en in (1.3), see [34 ]. If E is in eierstrass canonical form (1.2) and if the matrices and are partitioned in blo ks conformally to and then can sho that pc po (2.13) where and satisfy the standard con tin uous-time Ly apuno equations The improp er con trollabilit and observ abilit Gramians are the unique symmetric, ositiv semidenite solutions of the pr oje cte gener alize discr ete-time Lyapunov quations ic ic ic (2.14) and io io io (2.15)

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resp ectiv ely [34 ]. They can represen ted as ic io (2.16) where and satisfy the standard discrete-time Ly apuno equations The con trollabilit and observ abilit Gramians can used to haracterize con trollabilit and observ abilit prop erties of system (1.1). Theorem 2.3. [3 34 Consider the descriptor system (1.1) Assume that E is c-stable. 1. System (1.1) is R-c ontr ol lable if and only if the pr op er ontr ol lability Gr amian pc is ositive denite on the subsp ac im 2. System (1.1) is I-c ontr ol lable if the impr op er ontr ol lability Gr amian ic is ositive denite on the subsp ac er 3. System (1.1) is C-c ontr ol lable if and only if pc ic is ositive denite. 4. System (1.1) is R-observable if and only if the pr op er observability Gr amian po is ositive denite on the subsp ac im 5. System (1.1) is I-observable if the impr op er observability Gr amian io is ositive denite on the subsp ac er 6. System (1.1) is C-observable if and only if po io is ositive denite. Note that the I-con trollabilit (I-observ abilit y) of (1.1) do es not imply that the improp er con trollabilit (observ abilit y) Gramian is ositiv denite on er (on er ). Example 2.4. The descriptor system (1.1) with is I-con trollable and I-observ able. ha ic io and i.e., neither ic nor io are ositiv denite on er er Corollary 2.5. Consider the descriptor system (1.1) wher the encil E is c-stable. 1. System (1.1) is R-c ontr ol lable and R-observable if and only if rank( pc rank( po rank( pc po (2.17) 2. System (1.1) is I-c ontr ol lable and I-observable if rank ic rank( io rank ic io (2.18) 3. System (1.1) is C-c ontr ol lable and C-observable if and only if (2.17) and (2.18) hold. Pr of. The result follo ws from Theorem 2.3 and represen tations (1.2), (2.13) and (2.16).

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2.3 Hank el singular alues The prop er con trollabilit and observ abilit Gramians pc and po as ell as the improp er con trollabilit and observ abilit Gramians ic and io are not system in arian t. Indeed, under system equiv alence transformation the prop er and improp er con trollabilit Grami- ans pc and ic are transformed to pc pc and ic ic resp ectiv ely whereas the prop er and improp er observ abilit Gramians po and io are transformed to po po and io io resp ectiv ely Ho ev er, it follo ws from pc po pc po ic io ic io that the sp ectra of the matrices pc po and ic io are system in arian t. These matrices pla the same role for descriptor systems as the pro duct of the con trollabilit and observ abilit Gramians for standard state space systems [15 43 ]. ha the follo wing result. Theorem 2.6. et E c-stable. Then the matric es pc po and ic io have al and non-ne gative eigenvalues. Pr of. It follo ws from (2.9) and (2.10) that pc and po are symmetric and ositiv semidenite. In this case there exists nonsingular matrix suc that pc po where and are diagonal matrices with ositiv diagonal elemen ts [43 p. 76]. Then pc po Hence, pc po is diagonalizable and it has real and non-negativ eigen alues. Similarly can sho that the eigen alues of ic io are real and non-negativ e. Denition 2.7. Let and the dimensions of the deating subspaces of the c-stable encil E corresp onding to the nite and innite eigen alues, resp ectiv ely The square ro ots of the largest eigen alues of the matrix pc po denoted are called the pr op er Hankel singular values of the con tin uous-time descriptor system (1.1). The square ro ots of the largest eigen alues of the matrix ic io denoted are called the impr op er Hankel singular values of system (1.1). will assume that the prop er and improp er Hank el singular alues are ordered decreas- ingly i.e., The prop er and improp er Hank el singular alues form the set of Hank el singular alues of the con tin uous-time descriptor system (1.1). or the prop er Hank el singular alues are the classical Hank el singular alues of standard state space systems [15 27 ].

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Since the prop er and improp er con trollabilit and observ abilit Gramians are symmetric and ositiv semidenite, there exist Cholesky factorizations pc po ic io (2.19) where the matrices n;n are upp er triangular Cholesky factors. The follo wing lemma giv es connection et een the prop er and improp er Hank el singular alues of system (1.1) and the standard singular alues of the matrices and AR Lemma 2.8. Assume that the encil E in system (1.1) is c-stable. Consider the Cholesky factorizations (2.19) of the pr op er and impr op er Gr amians of (1.1) Then the pr op er Hankel singular values of system (1.1) ar the lar gest singular values of the matrix and the impr op er Hankel singular values of system (1.1) ar the lar gest singular values of the matrix AR Pr of. ha pc po ic io AR AR where and denote, resp ectiv ely the eigen alues and the singular alues ordered decreasingly Mo del reduction In this section consider the problem of reducing the order of the descriptor system (1.1). 3.1 Balanced realizations or giv en transfer function ), there are man dieren realizations [9]. Here are in terested only in particular realizations that are useful in applications. Denition 3.1. realization A; of the transfer function is called minimal if the triplet A; is C-con trollable and the triplet A; is C-observ able. Denition 3.2. realization A; of the transfer function is called alanc if pc po and ic io with diag and diag ). will sho that for minimal realization A; with the c-stable encil E there exists system equiv alence transformation suc that the realization AT (3.1) is balanced.

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Consider the Cholesky factorizations (2.19) of the con trollabilit and observ abilit Grami- ans. ma assume without loss of generalit that the Cholesky factors and ha full ro rank. If A; is C-con trollable and A; is C-observ able, then it follo ws from Corollary 2.5 and Lemma 2.8 that 0, and AR 0, Hence, the matrices ;n and AR ;n are nonsingular. Let AR (3.2) singular alue decomp ositions of and AR where and are orthogo- nal, diag( and diag are nonsingular. Consider the matrices AR (3.3) and (3.4) Since pc po pc po AP obtain that and AR (3.5) Then AR AR i.e., the matrices and are nonsingular and Similarly can sho that the matrices and are also nonsingular and Using (2.19) and (3.2)-(3.5), obtain that the prop er and improp er con trollabilit and observ abilit Gramians of the transformed system (3.1) ha the form pc po ic io Th us, with and as in (3.3) and (3.4), resp ectiv ely is the balancing transfor- mation and realization (3.1) is balanced. Note that just as for standard state space systems [15 27 ], the balancing transformation for descriptor systems is not unique. rom (3.2)-(3.4) nd AT where AR and Th us, the encil E is in eierstrass-lik canonical form. Clearly it is regular, c-stable and has the same index as E

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3.2 Balanced truncation In the previous subsection ha considered reduction of minimal realization to ba- lanced form. Ho ev er, computing the balanced realization ma ill-conditioned as so on as or in (3.2) has small singular alues. In addition, if the realization is not minimal, then the matrix or is singular. In the similar situation for standard state space systems one erforms mo del reduction truncating the state comp onen ts corresp onding to the zero and small Hank el singular alues without signican hanges of the system prop erties, see, e.g., [15 27 37 ]. This pro cedure is kno wn as alanc trunc ation It can also applied to the descriptor system (1.1). The prop er con trollabilit and observ abilit Gramians can used to describ the future output energy dt and the past input energy 1 dt that is needed to reac from 1 the state (0) im when no input is applied for 0. Theorem 3.3. Consider descriptor system (1.1) Assume that the encil E is c-stable and the triplet A; is R-c ontr ol lable. et pc and po the pr op er ontr ol lability and observability Gr amians of (1.1) If im and for then po Mor over, for opt pc we have opt min pc wher is the Hilb ert sp ac of al dimensional ve ctor-functions that ar squar inte gr able on 1 0) and the matrix pc satises pc pc pc pc pc pc pc pc pc pc (3.6) Pr of System (1.1) with im and for has unique solution giv en 0. Then for and, hence, dt po Consider no the minimization of for ). Note that the state (0) of the descriptor system (1.1) with for satises the constrain equation 1 dt: (3.7) 10

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Since system (1.1) is R-con trollable, the matrix in (2.13) is nonsingular and, hence, im im pc In this case there exists ector suc that pc with im Therefore, 1 pc dt pc pc pc pc (3.8) or an input opt with opt pc ), it follo ws from (3.7) and (3.8) that 1 dt Then 1 opt opt dt 2( pc 1 dt 1 dt 1 opt opt dt: Th us, opt minimizes among all inputs that transfer the system from 1 to (0) im Using the second and third equations in (3.6), nd opt 1 pc pc dt pc Remark 3.4. Equations (3.6) imply that pc is symmetric (1 2)-pseudoinv erse [7 of pc It is, in general, not unique, but opt pc and opt pc with im are uniquely dened. Unfortunately ere unable to nd similar energy in terpretation for the improp er con trollabilit and observ abilit Gramians. If the descriptor system (1.1) is not minimal, then it has states that are uncon trollable or/and unobserv able. These states corresp ond to the zero prop er and improp er Hank el sin- gular alues and can truncated without hanging the input-output relation in the sys- tem. Note that the um er of non-zero improp er Hank el singular alues of (1.1) is equal to rank( ic io whic can in turn estimated as rank( ic io min( m; p; This estimate sho ws that if the index of the encil E times the um er of inputs or the um er of outputs is uc smaller than the dimension of the deating subspace of E corresp onding to the innite eigen alues, then the order of system (1.1) can reduced signican tly urthermore, Theorem 3.3 implies that large input energy is required to reac from 1 the state (0) whic lies in an in arian subspace of the prop er con trollabilit Gramian pc corresp onding to its small non-zero eigen alues. Moreo er, if 11

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is con tained in an in arian subspace of the matrix po corresp onding to its small non- zero eigen alues, then the initial alue (0) has small eect on the output energy or the balanced system, pc and po are equal and, hence, they ha the same in arian subspaces. In this case the truncation of the states related to the small prop er Hank el singular alues do es not hange system prop erties essen tially Unfortunately this do es not hold for the improp er Hank el singular alues. If truncate the states that corresp ond to the small non-zero improp er Hank el singular alues, then the encil of the reduced order system ma get nite eigenv alues in the closed righ half-plane, see [26 ]. In this case the appro ximation will inaccurate. Let A; realization (not necessarily minimal) of the transfer function ). Assume that the encil E is c-stable. Consider the Cholesky factorizations (2.19). Let AR 'thin' singular alue decomp ositions of and AR where ], ], and ha orthonormal columns, diag( and diag( +1 with +1 and rank diag( with rank AR ). Then the reduced order realization can computed as sE (3.9) where n;` n;` (3.10) and Note that computing the reduced order descriptor system can in terpreted as erforming system equiv alence transformation suc that sE sE sE where the encil E has the nite eigen alues only all the eigen alues of E are innite, and then reducing the order of the subsystems and with nonsingular and using classical balanced truncation metho ds for con tin uous-time and discrete-time state space systems, resp ectiv ely Clearly the reduced order system (3.9) is minimal and the encil is c-stable. The describ ed decoupling of system matrices is equiv alen to the additiv decomp osition of the transfer function as sp ), where sp sE is the strictly prop er part and sE is the olynomial part of ). The transfer function of the reduced system has the form sp ), where sp and are the reduced order subsystems. Note that ), and, hence, the dierence sp sp 12

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is strictly prop er rational function. Th us, ha the follo wing upp er ound for the -norm of the error system := sup i! i! 2( +1 that can deriv ed as in [15 ]. Here denotes the sp ectral matrix norm. 3.3 Algorithms reduce the order of the descriptor system (1.1) ha to compute the Cholesky factors of the prop er and improp er con trollabilit and observ abilit Gramians that satisfy the pro- jected generalized Ly apuno equations (2.11), (2.12), (2.14) and (2.15). These factors can computed using the gener alize Schur-Hammarling metho [34 35 ]. Let the encil E in generalized real Sc ur form and (3.11) where and are orthogonal, is upp er triangular nonsingular, is upp er triangular nilp oten t, is upp er quasi-triangular and is upp er triangular nonsingular, and let the matrices and (3.12) partitioned in blo ks conformally to and Then one can sho [34 35 that the Cholesky factors of the Gramians of system (1.1) ha the form (3.13) where is the solution of the generalized Sylv ester equation (3.14) the matrices are the Cholesky factors of the solutions pc po of the generalized con tin uous-time Ly apuno equations pc pc )( (3.15) po po (3.16) while and are the Cholesky factors of the solutions ic and io of the generalized discrete-time Ly apuno equations ic ic (3.17) io io (3.18) rom (3.11) and (3.13) obtain that and AR Th us, the prop er and improp er Hank el singular alues of (1.1) can computed from the singular 13

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alue decomp ositions of the matrices and urthermore, it follo ws from (3.10) and (3.13) that the pro jection matrices and ha the form (3.19) with and In this case the matrix co ecien ts of the reduced order system (3.9) are giv en (3.20) In summary ha the follo wing algorithm that is generalization of the squar ot alanc trunc ation metho [24 37 for the con tin uous-time descriptor system (1.1). Algorithm 3.1. Gener alize Squar ot (GSR) metho d. Input: A; such that E is c-stable. Output: duc or der system 1. Compute the gener alize Schur form (3.11) 2. Compute the matric es (3.12) 3. Solve the gener alize Sylvester quation (3.14) 4. Compute the Cholesky factors of the solutions pc and po of quations (3.15) and (3.16) esp ctively. 5. Compute the Cholesky factors of the solutions ic and io of quations (3.17) and (3.18) esp ctively. 6. Compute the 'thin singular value de omp osition wher and have orthonormal olumns, diag and diag( +1 with rank and +1 7. Compute the 'thin singular value de omp osition wher and have orthonormal olumns and diag( with rank 8. Compute 9. Compute the duc or der system as in (3.20) If the original system (1.1) is highly un balanced or if the deating subspaces of the encil E corresp onding to the nite and innite eigen alues are close, then the pro jection matrices and as in (3.19) are ill-conditioned. oid accuracy loss in the reduced order system, squar ot alancing fr metho has een prop osed in [38 for standard state space systems. This metho can generalized for descriptor systems as follo ws. 14

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Algorithm 3.2. Gener alize Squar ot Balancing (GSRBF) metho d. Input: A; such that E is c-stable. Output: duc or der system 1.-7. as in lgorithm 3.1 8. Compute the 'e onomy size' QR de omp ositions L; wher n;` have orthonormal olumns and `;` ar nonsingular. 9. Compute the duc or der system with (3.21) The GSR metho and the GSRBF metho are mathematically equiv alen in the sense that in exact arithmetic they return reduced systems with the same transfer function. It should noted that the reduced order realization A; as in (3.21) is, in general, not balanced and the corresp onding encil is not in eierstrass-lik canonical form. will no discuss the umerical asp ects of Algorithms 3.1 and 3.2. compute the generalized Sc ur form (3.11) can use the QZ algorithm [16 40] the GUPTRI algorithm [10 11 ], or algorithms prop osed in [2 39 ]. solv the generalized Sylv ester equation (3.14) can use the generalized Sc ur metho [23 or its recursiv blo ed mo dication [21 that is more suitable for large problems. The upp er triangular Cholesky factors and of the solutions of the generalized Ly apuno equations (3.15)-(3.18) can determined without computing the solutions themself using the generalized Hammarling metho [20 28 ]. In the general case the generalized Sc ur and Hammarling metho ds are based on the preliminary reduction of the corresp onding matrix encils to the generalized Sc ur form, calculation of the solution of reduced system and bac transformation. Note that the encils E and E in equations (3.14) (3.18) are already in generalized Sc ur form. Th us, need only to solv the upp er (quasi-)triangular matrix equations. Finally the singular alue decomp osition of the matrices and where all three factors are upp er triangular, can computed without forming these pro ducts ex- plicitly see [5 12 17 and references therein. Since the GSR metho and the GSRBF metho are based on computing the generalized Sc ur form, they cost ops and ha the memory complexit ). Th us, these metho ds can used for problems of small and medium size only Moreo er, they do not tak in to accoun the sparsit or an structure of the system and are not attractiv for parallelization. Recen tly iterativ metho ds related to the ADI metho and the Smith metho ha een prop osed to compute lo rank appro ximations of the solutions of standard large- scale sparse Ly apuno equations [25 29 ]. It is imp ortan to extend these metho ds for pro jected generalized Ly apuno equations. This topic is curren tly under in estigations. Remark 3.5. The GSR metho and the GSRBF metho can used to reduce the order of unstable descriptor systems. Firstly compute the additiv decomp osition [22 of the 15

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transfer function ), where sE sE Here the matrix encil E is c-stable and all the eigen alues of the encil E are nite and ha non-negativ real part. Then determine the reduced order system applying the balanced truncation mo del reduction metho to the subsystem ]. Finally the reduced order appro ximation of is giv en ), where is included unmo died. Remark 3.6. The con trollabilit and observ abilit Gramians as ell as Hank el singular alues can also generalized for discrete-time descriptor systems, see [36 for details. In this case an extension of balanced truncation mo del reduction metho ds for suc systems is straigh tforw ard. Numerical examples In this section consider umerical examples to illustrate the eectiv eness of the prop osed mo del reduction metho ds for descriptor systems. All of the follo wing results ere obtained on SunOS 5.8 orkstation with relativ mac hine precision 22 10 16 using the MA TLAB mex-functions based on the GUPTRI routine [10 11 and the SLICOT library routines [4]. Example 4.1. Consider the holonomically constrained planar mo del of truc [32 ]. The linearized equation of motion has the form (4.1) where 11 is the osition ector, 11 is the elo cit ector, is the Lagrange ultiplier, is the ositiv denite mass matrix, is the stiness matrix, is the damping matrix, is the constrain matrix and is the input matrix. System (4.1) together with the output equation forms descriptor system of order 23 with input and 11 outputs. The dimension of the deating subspace corresp onding to the nite eigen alues is 20. The prop er Hank el singular alues of the linearized truc mo del (4.1) are giv en in able 1. All the improp er Hank el singular alues are zero and, hence, the transfer function of (4.1) is strictly prop er. appro ximate system (4.1) mo dels of order 12 computed the GSR metho and the GSRBF metho d. Figure illustrates ho accurate the reduced order mo dels appro ximate the original system. displa the plots of the sp ectral norm of the frequency resp onses of the original system i! and the reduced order systems i! for frequency range [10 10 ]. One can see that the appro ximate system deliv ered the GSR metho diers sligh tly from the original one for high frequencies only while the plots of the full order Av ailable from http://www.cs.umu.se/research/ nla/ singu lar pairs/guptri Av ailable from http://www.win.tue.nl/niconet/ NIC2 /slic ot.h tml 16

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072 10 715 10 11 134 10 16 988 10 962 10 578 10 12 013 10 17 833 10 286 10 925 10 13 392 10 18 847 10 845 10 192 10 14 596 10 19 108 10 10 750 10 10 535 10 15 969 10 20 163 10 10 able 1: Prop er Hank el singular alues of the linearized truc mo del. system and the reduced order system computed the GSRBF metho coincide. Note that the pro jection matrices and in the GSR metho (see (3.19)) ha the condition um ers max min 385 10 max min 965 10 whereas the pro jection matrices in the GSRBF metho giv en and ha orthonormal columns. In Figure compare the absolute appro ximation errors i! i! with the upp er ound whic is the wice sum of the truncated prop er Hank el singular alues 13 20 see that the appro ximation error for the GSRBF metho is considerable smaller than for the GSR metho d. 10 −1 10 10 10 10 10 10 10 −12 10 −11 10 −10 10 −9 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 Frequency (rad/sec) Linearized truck model Full order GSR GSRBF PSfrag replacemen ts i! and i! Figure 1: requency resp onses of the full order system and the reduced order systems. 10 −1 10 10 10 10 10 10 10 −16 10 −15 10 −14 10 −13 10 −12 10 −11 10 −10 10 −9 10 −8 10 −7 10 −6 Frequency (rad/sec) Error systems and error bound Error system, GSR Error system, GSRBF Error bound PSfrag replacemen ts i! i! Figure 2: Error systems and error ound for the linearized truc mo del. Example 4.2. Consider the o of an incompressible uid describing the instationary Stok es equation on square region ]. The spatial discretization of this equation the nite olume metho on uniform staggered grid leads to the descriptor system (4.2) where and are the semidiscretized ectors of elo cities and pressures, resp ectiv ely ;n is the discrete Laplace op erator and ;n is obtained from the discrete gradien op erator discarding the last column [41 ]. or simplicit 17

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is hosen at random and [1 0] ;n or 12, ha 480 and the dimensions of the deating subspaces of the encil in (4.2) corresp onding to the nite and innite eigen alues are 144 and 336. 10 15 20 25 30 10 −30 10 −25 10 −20 10 −15 10 −10 10 −5 10 Proper Hankel singular values Figure 3: Prop er Hank el singular alues for the semidiscretized Stok es equation. 10 −1 10 10 10 10 10 10 10 −16 10 −15 10 −14 10 −13 10 −12 10 −11 10 −10 10 −9 10 −8 Frequency (rad/sec) Error systems and error bound Error system, GSR Error system, GSRBF Error bound PSfrag replacemen ts i! i! Figure 4: Error systems and error ound for the semidiscretized Stok es equation. The 30 largest prop er Hank el singular alues of the semidiscretized Stok es equation (4.2) are giv en in Figure 3. One can see that they deca ery fast, and, hence, system (4.2) can ell appro ximated mo del of lo order. All the improp er Hank el singular alues are zero. appro ximate system (4.2) mo dels of order computed the GSR metho and the GSRBF metho d. The sp ectral norm of the frequency resp onses of the full order and reduced order systems are not presen ted since they ere imp ossible to distinguish. Figure sho ws the plots of the error ound and the sp ectral norm of the error systems i! i! for frequency rang [10 10 ]. see again that the appro ximate system deliv ered the GSRBF metho is etter than the one computed the GSR metho d. Conclusion ha generalized the con trollabilit and observ abilit Gramians as ell as Hank el singular alues for descriptor systems and studied their imp ortan features. Balanced truncation mo del reduction metho ds for descriptor systems ha een presen ted. These metho ds are closely related to the Gramians and deliv er reduced order systems that preserv the regularit and stabilit prop erties of the original system. Moreo er, for these metho ds priori ound on the appro ximation error is ailable. Ac kno wledgemen t. The author ould lik to thank the referees for aluable suggestions. References [1] An toulas C, Sorensen DC, and Gugercin (2001) surv ey of mo del reduction metho ds for large-scale systems, in Structured Matrices in Mathematics, Computer Science and 18

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Engineering, ol. (Olshevsky ed.) Con temp orary Mathematics Series, 280. American Mathematical So ciet Pro vidence, RI, pp. 193{219 [2] Beelen T, an Do oren (1988) An impro ed algorithm for the computation of Kro- nec er's canonical form of singular encil, Linear Algebra Appl., 105:9{65 [3] Bender DJ (1987) Ly apuno v-lik equations and reac habilit y/observ abilit Gramians for descriptor systems, IEEE rans. Automat. Con trol, 32(4):343{348 [4] Benner Mehrmann V, Sima V, an Huel S, arga (1999) SLICOT subroutine library in systems and con trol theory Appl. Comput. Con trol Signals Circuits, 1:499{539 [5] Bo janczyk W, Ew erbring LM, Luk FT, an Do oren (1991) An accurate pro duct SVD algorithm, Signal Pro cess., 25:189{201 [6] Brenan KE, Campb ell SL, etzold LR (1989) The Numerical Solution of Initial-V alue Problems in Dieren tial-Algebraic Equations, Elsevier, North-Holland, New ork [7] Campb ell SL, Mey er CD (1979) Generalized In erses of Linear ransformations, Do er Publications, New ork [8] Cobb (1984) Con trollabilit observ abilit and dualit in singular systems, IEEE rans. Automat. Con trol, 29(12):1076{1082 [9] Dai (1989) Singular Con trol Systems, Lecture Notes in Con trol and Information Sciences, 118. Springer-V erlag, Berlin, Heidelb erg [10] Demmel JW, agstr om (1993) The generalized Sc ur decomp osition of an arbitrary encil B Robust soft are with error ounds and applications. Part I: Theory and algorithms, CM rans. Math. Soft are, 19(2):160{174 [11] Demmel JW, agstr om (1993) The generalized Sc ur decomp osition of an arbitrary encil B Robust soft are with error ounds and applications. Part I: Soft are and applications, CM rans. Math. Soft are, 19(2):175{201 [12] Drmac (2000) New accurate algorithms for singular alue decomp osition of matrix triplets, SIAM J. Matrix Anal. Appl., 21(3):1026{1050 [13] ortuna L, Nunnari G, Gallo (1992) Mo del Order Reduction ec hniques with Appli- cations in Electrical Engineering, Springer-V erlag, London [14] Galliv an K, Grimme E, an Do oren (1996) rational Lanczos algorithm for mo del reduction, Numerical Algorithms, 12(1-2):33{63 [15] Glo er (1984) All optimal Hank el-norm appro ximations of linear ultiv ariable systems and their -errors ounds, In ternat. J. Con trol, 39(6):1115{1193 [16] Golub GH, an Loan CF (1996) Matrix Computations. 3r. ed, The Johns Hopkins Univ ersit Press, Baltimore, London [17] Golub GH, Slna K, an Do oren (2000) Computing the SVD of general matrix pro duct/quotien t, SIAM J. Matrix Anal. Appl., 22(1):1{19 19

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[18] Grimme EJ, Sorensen DC, an Do oren (1996) Mo del reduction of state space systems via an implicitly restarted Lanczos metho d, Numerical Algorithms, 12(1-2):1{31 [19] un ther M, eldmann (1999) CAD-based electric-circuit mo deling in industry I. Mathematical structure and index of net ork equations, Surv eys Math. Indust., 8(2):97{ 129 [20] Hammarling SJ (1982) Numerical solution of the stable non-negativ denite Ly apuno equation, J. Numer. Anal., 2:303{323 [21] Jonsson I, agstr om (2002) Recursiv blo ed algorithms for solving triangular systems Part I: one-sided and coupled Sylv ester-t yp matrix equations, CM rans. Math. Soft are, 28(4):392{415 [22] agstr om B, an Do oren (1992) generalized state-space approac for the additiv decomp osition of transfer function, J. Numer. Linear Algebra Appl., 1(2):165{181 [23] agstr om B, estin (1989) Generalized Sc ur metho ds with condition estimators for solving the generalized Sylv ester equation, IEEE rans. Automat. Con trol, 34:745{751 [24] Laub AJ, Heath MT, aige CC, ard (1987) Computation of system balancing transformations and other applications of sim ultaneous diagonalization algorithms, IEEE rans. Automat. Con trol, 32(2):115{122 [25] Li JR (2000) Mo del Reduction of Large Linear Systems via Lo Rank System Gramians, Ph.D. thesis, Departmen of Mathematics, Massac usetts Institute of ec hnology [26] Liu Q, Sreeram (2000) Mo del reduction of singular systems, in Pro ceedings of the 39th IEEE Conference on Decision and Con trol (Sydney Australia, 2000), 2373{2378 [27] Mo ore BC (1981) Principal comp onen analysis in linear systems: con trollabilit ob- serv abilit and mo del reduction, IEEE rans. Automat. Con trol, 26(1):17{32 [28] enzl (1998) Numerical solution of generalized Ly apuno equations, Adv. Comput. Math., 8(1-2):33{48 [29] enzl (1999/00) cyclic lo w-rank Smith metho for large sparse Ly apuno equations, SIAM J. Sci. Comput., 21(4):1401{1418 [30] erev K, Shafai (1994) Balanced realization and mo del reduction of singular systems, In ternat. J. Systems Sci., 25(6):1039{1052 [31] Safono MG, Chiang (1989) Sc ur metho for balanced-truncation mo del reduc- tion, IEEE rans. Automat. Con trol, 34(7):729{733 [32] Simeon B, Grupp F, uhrer C, Ren trop (1994) nonlinear truc mo del and its treatmen as ultib dy system, J. Comput. Appl. Math., 50:523{532 [33] Stew art GW, Sun JG (1990) Matrix erturbation Theory Academic Press, New ork [34] St yk el (2002) Analysis and Numerical Solution of Generalized Ly apuno Equations, Ph.D. thesis, Institut ur Mathematik, ec hnisc he Univ ersit at Berlin 20

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[35] St yk el (2002) Numerical solution and erturbation theory for generalized Ly apuno equations, Linear Algebra Appl., 349:155{185 [36] St yk el (2003) Input-output in arian ts for descriptor systems, Preprin PIMS-03-1, acic Institute for the Mathematical Sciences, Canada [37] om bs MS, ostleth eite (1987) runcated balanced realization of stable non-minimal state-space system, In ternat. J. Con trol, 46(4):1319{1330 [38] arga (1991) Ecien minimal realization pro cedure based on balancing, in Pro c. of IMA CS/IF Symp osium on Mo delling and Con trol of ec hnological Systems, ol. (EL Moudni A, Borne Tzafestas SG, eds.), pp. 42{47 [39] arga (1998) Computation of coprime factorizations of rational matrices, Linear Algebra Appl., 271:83{115 [40] atkins DS (2000) erformance of the QZ algorithm in the presence of innite eigen- alues, SIAM J. Matrix Anal. Appl., 22(2):364{375 [41] eic ert (1997) Applications of the Theory of Dieren tial-Algebraic Equations to artial Dieren tial Equations of Fluid Dynamics, Ph.D. thesis, ec hnisc he Univ ersit at Chemnitz [42] Yip EL, Sinco ec RF (1981) Solv abilit con trollabilit and observ abilit of con tin uous descriptor systems, IEEE rans. Automat. Con trol, 26(3):702{707 [43] Zhou K, Do yle JC, Glo er (1996) Robust and Optimal Con trol, Pren tice-Hall, Upp er Saddle Riv er 21

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