Optimal RH 2 - and RH -Approximation of Unstable Descriptor Systems

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1 Optimal RH - and RH -Approximation of Unstable Descriptor Systems Marcus Köhler August, 1 arxiv:13.595v [math.oc] 1 Aug 1 Stability perserving is an important topic in approximation of systems, e.g. model reduction. If the original system is stable, we often want the approximation to be stable. But even if an algorithm preserves stability the resulting system could be unstable in practice because of round-off errors. Our approach is approximating this unstable reduced system by a stable system. More precisely, we consider the following problem. Given an unstable linear time-invariant continuous-time descriptor system with transfer function G, find a stable one whose transfer function is the best approximation of G in the spaces RH and RH, respectively. Explicit optimal solutions are presented under consideration of numerical issues. 1 Introduction We deal with linear time-invariant descriptor systems described by differential algebraic equations see e.g. [KM6], [Dai89] for more details Eẋt = Axt + But yt = Cxt + Dut 1.1 with matrices E, A, B, C, D R n n R n n R n m R p n R p m and the corresponding transfer function s CsE A 1 B +D. In many applications such as model reduction see [ASG1] for an overview an further references or system identification see [Lju99] one wishes to approximate systems such that the transfer functions of the original and the approximated system are as close as possible. An important property of an approximation technique is preservation of stability cf. e.g. balanced truncation [ASG1], hankel norm approximation [Glo84]. However, not every method has this property in general Arnoldi method, Lanczos method [ASG1]. Moreover, a stable approximation can be unstable in practice [BFF98]. This happens in computer implementations because of round-off errors. One way out is restarting the algorithm with 1

2 changed parameters, e.g. interpolation points in Lanczos method. The disadvantage is at least doubled computational complexity. Another approach is to modify the algorithm directly e.g. partial Padé-via-Lanczos method [BF1]. In this paper our approach for stability perserving is to consider the computed unstable approximation of the original system. Given is an unstable descriptor system S 1.1 with transfer function G. Find a stable descriptor system whose transfer function is the best approximation of G in the spaces RL and RL of real rational functions on the imaginary axis ir in the Lebesgue spaces L and L, respectively. We call this system an optimal RH -approximation and optimal RH -approximation of S, respectively. For special cases this problem was already solved. The characterization of the unique optimal RH -approximation of a standard system i.e. E = I in 1.1 with D = is a simple consequence of the Paley-Wiener theorem. The set of all suboptimal RH - approximations of standard systems was determined in [GGLD9] with j-spectral factorizations to solve model matching problems. However, no optimal ones were explicitly given. An explicit representation of a nonunique optimal RH -approximation of minimal antistable standard systems was presented in the proof of [Glo84, Theorem 6.1]. This theorem was only used as an auxiliary result for a more general problem, the optimal hankel norm approximation. That representation requires the possibly ill-conditioned balanced minimal realization of G. This numerical issue was discussed in [SCL9]. Our approach for approximating an unstable descriptor system S in a numerically reliable way is the following. First, we decompose S into a stable and antistable part by combining [GL96, Theorem 7.7.] and [KD9, Section 4.1]. We show that it is sufficient to replace the antistable part by its approximation to get an approximation of S. This was e.g. proposed in [Fra87, Section 8.3] for standard systems and w.r.t. RL. The stable part is the unique optimal RH -approximation of S. Our representation of the optimal RH -approximation only requires at most one singular value decomposition. For discret-time standard systems [Mar] presented an approximation method similar to ours. However, in contrast to our approach they used a balanced minimal realization of the antistable part. The paper is organized as follows. In Section we summarize some results regarding the theory of matrices and matrix pairs. We introduce the setting and recall some system transformations, e.g. balanced realization. Section 3 states and solves both approximation problems. Section 4 presents an algorithm solving our problem and some numerical examples. Preliminaries We define C > := {s C; Res > } and analogously C, C <, C. The set ir is the imaginary axis. For a function f we denote its domain by Df and for M Df the image of M under f by f[m]. The kernel of a matrix A C n m is denoted by kera, the adjoint by A, the spectrum by σa and the Frobenius norm by A F = tracea A.

3 The matrix I n R n n is the identity matrix and n m R n m the zero matrix. The vector e i R n is the ith unit vector. Definition.1. We call E, A R n n R n n a matrix pair. A scalar λ C is called a generalized eigenvalue of E, A, if detλe A =. If there exists v such that Ev = and Av, then is called a generalized eigenvalue of E, A. We denote the set of all eigenvalues of E, A by σe, A C { } and the resolvent set by ρe, A := C\σE, A. If ρe, A =, then E, A is called singular otherwise regular. We denote the set of all regular matrix pairs by P n := {E, A R n n R n n ; E, A regular}. Remark.. If the matrix pair is regular, then the set of eigenvalues is finite. Otherwise every λ C is an eigenvalue. Details about the generalized eigenvalue problem can be found in [Ste7]. To prepare decompositions of systems, we combine the results in [GL96, Theorem 7.7.] and [KD9, Section 4.1] to get the following theorem. Theorem.3. Let E, A P n and M 1, M C { } be two disjoint sets such that σe, A = M 1 M. Then there exist orthogonal U, V R n n such that E1 E UEV, UAV = A1 A,,.1 E 3 A 3 σe 1, A 1 = M 1 and σe 3, A 3 = M. Moreover, there exist R, L R m p such that The matrices P := I L I A 1 R LA 3 = A E 1 R LE 3 = E. satisfy U, Q := V I R I E1 P EQ, P AQ = E 3 A1,. A 3. Remark.4. We get.1 e.g. by applying the generalized Schur decomposition to E, A. Equation. is called generalized or coupled Sylvester equation see [KD9]. We recall the following known result which can be used to test regularity of matrices. Theorem.5 [ZDG96, Section.3]. Let A C n n, B C n m, C C m n, D C m m and M := A B C D. If A is regular, then we have detm = detd CA 1 B deta. If D is regular, then we have detm = deta BD 1 C detd. The matrices D CA 1 B and A BD 1 C are called the Schur complement of A and D in M, respectively. Definition.6. For n, p, m N we define the sets of systems compare 1.1 S n,p,m := P n R n m R p n R p m, S n,p,m := {E, A, B, C, D S n,p,m ; ir ρe, A}, S + n,p,m := {E, A, B, C, D S n,p,m ; C ρe, A}, S n,p,m := {E, A, B, C, D S n,p,m ; C ρe, A, E regular}, 3

4 where S + n,p,m and S n,p,m are the set of the stable and antistable systems, respectively. We call a system S = E, A, B, C, D S n,p,m a standard system if E = I and a descriptor system otherwise. For S i = E i, A i, B i, C i, D i S ni,p,m, i {1, }, and regular P, Q R n 1 n 1 we define the operations E1 A1 B1 S 1 S :=,,, C E A B 1 C, D1 + D, P S 1 Q := P E 1 Q, P A 1 Q, P B 1, C 1 Q, D 1, P S 1 := P S 1 I n1, S 1 Q := I n1 S 1 Q. Some of the following definitions and theorems for standard systems can be given for descriptor systems, too. However, we restricted ourselves to the specializations we need here. Definition.7 [Dai89, Definition -6.1]. Let S = E, A, B, C, D S n,p,m. Then GS: ρe, A s CsE A 1 B + D R p m is called the transfer function of S. We denote the set of all p m transfer functions by T p,m := {GS; S n N S n,p,m}. A system S Sñ,p,m is called a realization of GS, if GSs = G Ss for all s DGS DG S. We denote the set of all realizations of GS by SGS. A realization S 1 S n1,p,m of GS is said to be minimal, if every other realization S S n,p,m of GS fulfils n n 1. Remark.8. Transfer functions are meromorphic. Obviously, GS 1 S = GS 1 +GS and GP S Q = GS hold. We want to transform a system S = E, A, B, C, D into S = Ẽ, Ã, B, C, D with GS = G S, in particular σe, A = σẽ, Ã, such that Ẽ, Ã, B, C, D have a special structure, e.g. to decompose systems into a stable and antistable part. Note that all matrices stay real under the presented transformations. Definition.9 [Dai89, Definition 1-3.1]. Two systems S i = E i, A i, B i, C i, D i S n,p,m, i {1, }, are called restricted system equivalent for short r.s.e., S 1 S, if there exist regular P, Q R n n such that P S 1 Q = S. Lemma.1. For every S = E, A, B, C, D S n,p,m there exist S w S, k {,..., n} and ν N such that Ik J BJ S w =,,, C N I n k B J C N, D, N and N ν =. Moreover, DGS = ρj = ρe, A holds and ν 1 s DGS: GSs = C J si J 1 B J + D s i C N N i B N..3 i= 4

5 Proof. We apply Theorem.3 to E, A with M 1 = σe, A\{ }, M = { } and get regular E 1, A 3, P, Q as in the theorem. Choose P := E 1 1 I P, Q := I A 1 Q 3 and set S w := P S Q. The matrix N := E 3 is nilpotent because σn, I = { } and thus σn = {}. Equation.3 follows from GS = GS w Remark.8 and C N sn I 1 B N = ν 1 i= si C N N i B N Neumann series. Theorem.11 [ZDG96, Theorem 3.1, Theorem 3.17]. Let S = I, A, B, C, D S n,p,m. Then A 11 A 13 S I n, A 1 A A 3 A 4 A 33, A 43 A 44 B 1 B, C 1 C 3, D, where A ii R n i n i, n i {,..., n} for i {1,, 3, 4} and S M := I, A 11, B 1, C 1, D is a minimal realization of GS. This decomposition is known as the Kalman decomposition. All minimal realizations of GS are r.s.e. to S M. Now we introduce the controllability and observability Gramians. Later these matrices will play a central role in our problem of computing an optimal RH -approximation. The next theorem follows easily from [Pen98, Corollary ]. Theorem.1 [Pen98], Lyapunov equation. Let S = E, A, B, C, D S n,p,m. Then there exist X c, X o R n n such that AX c E + EX c A + BB =, A X o E + E X o A + C C =..4 They are unique and symmetric. We denote them by C S := X c controllability Gramian of system S and O S := X o observability Gramian. Lemma.13. Let S = E, A, B, C, D S + n,p,m and E be regular. Let P, Q R n n be regular. Then C S = QC P S Q Q and O S = P O P S Q P hold. Proof. We prove this only for C S since O S is analogously. By definition of C P S Q = P AQC P S Q Q E P + P EQC P S Q Q A P + P BB P, = P AQC P S Q Q E + EQC P S Q Q A + BB P hold. Due to the regularity of P and Q and the uniqueness of the solution of.4 Theorem.1 the assertion is proved. 5

6 Theorem.14 [Glo84, Theorem 4.3]. Let S = I, A, B, C, D S n,p,m. Then there exist S b S, r N, h R such that Σ1 Σ C Sb =, O hi Sb =, h / σσ r hi 1 Σ. r If Σ 1 = Σ are diagonal, then S b is known as the balanced realization of GS. To check whether λ C is an eigenvalue of E, A, we need the following easy generalization of a known result for standard systems e.g. [Mac4, Theorem c]. Lemma.15. Let G T p,m have a realization S = E, A, B, C, D S n,p,m. Then for every finite λ σe, A at least one of the following three statements holds: i λ is a pole of G, ii rank λe A B λe A < n, iii rank < n. C Proof. Let S w S be as in Lemma.1. We set S J := I, J, B J, C J, D S k,p,m. By Lemma.1 we know that λ σe, A\ { } λ σj. By [Mac4, Theorem c] one of the following three statements holds: i λ is a pole of GS J, ii rank λi J B J < k, iii λi J rank < k. Equation.3 shows that i i. By the regularity of λn I we have rank λi J BJ λi J B J + n k = rank = rank λe A B λn I B N and analogously rank λi J C J iii. This finishes the proof. + n k = rank λe A C. Thus ii ii and iii Definition.16 [ZDG96, Section 4.3]. We approximate unstable systems with respect to the following spaces and corresponding norms. Let p, m N. RL p m := {G T p,m ; ir DG, G < }, G := sup Giω ; { RH p m := G RL p m ; G analytic on C > }; RL p m := {G T p,m ; ir DG, G < }, G := { RH p m := G RL p m ; G analytic on C > }. We omit the indices, if there is no risk of confusion. ω R 1 π C J Giω F dω; 6

7 Remark.17. These spaces are often introduced as a subset of socalled real rational functions, i.e. entrywise a quotient of polynomials with real coefficients. Actually, every real rational function is a transfer function see [Dai89, Theorem -6.3] and vice versa, since Gs i,j = e i CsE A 1 Be j +D i,j is the Schur complement of se A in Ms := se A Bej e i C D i,j and thus Gs i,j = detgs i,j = detms detse A be elements of one of these spaces with SF = SG, e.g. s 1 s 1 by Theorem.5. Let F and G s 1s. and s s By definition they are equal on the cofinite set DF DG. Therefore we identify them as the same function. With this and by the identity theorem for analytic functions the spaces are indeed normed spaces. Theorem.18 [Fra87, Section.3]. Let G RL be analytic on C < and G + RH. Then G + G + = G + G + holds. The following known result characterizes these spaces in terms of realizations of transfer functions. Theorem.19 [Fra87, Section.3]. Let G T p,m with ir DG. Then the following hold: a G RL I, A, B, C, D SG, b G RH I, A, B, C, D SG S + n,p,m, c G RL I, A, B, C, SG, d G RH I, A, B, C, SG S + n,p,m. 3 Optimal stable approximations 3.1 Problem statement In this paper, we deal with the following problem. Approximation-Problem AP q. Let q {, }. For S S n,p,m find S S+ n,p,m such that n N GS G S q = inf Ŝ n N S+ n,p,m GS GŜ q. In the next subsections we show that this problem is solvable. We also present an explicit solution in Theorem 3. for q = and Theorem 3.16 for q =. We do not consider systems S = E, A, B, C, D S n,p,m where E, A has imaginary eigenvalues. If there is no imaginary pole of GS, then there exists a realization S S n,p,m of GS and we solve AP q for S. Otherwise, for every Ŝ n N S+ n,p,m we have GS GŜ q = and hence AP q is not solvable. 7

8 First, we show that solving AP q for a descriptor system we can equivalently solve AP q for its antistable part. This was e.g. proposed in [Fra87, Section 8.3] but only for standard systems and q =. We need Theorem 3.1 to use some results in [Glo84] and [SCL9] which are only applicable for antistable standard systems. Theorem 3.1. Let S S n,p,m. Then there exist S + = E +, A +, B +, C +, D S + n +,p,m and S = E, A, B, C, S n,p,m such that S S + S. Let q {, } and γ. Then the following two are equivalent: i S n N S + n,p,m : GS G S q γ, ii S n N S + n,p,m : GS G S q γ. If S satisfies ii, then S S + satisfies i. Proof. Theorem.3 assures the existence of such systems S + and S. Let S S + ñ,p,m satisfy i. Then we have γ GS G S q = GS G S GS + q. Thus Ŝ := S E +, A +, B +, C +, D satisfies ii compare Remark.8. The fact that Ŝ n N S+ n,p,m follows easily from the definition of. Let S S + ñ,p,m fulfils ii. Then analogously S S + satisfies i. 3. Optimal RH -approximations Theorem 3.. Let S S n,p,m. We use the notations of Theorem 3.1. Then S + solves AP. More precisely, we have inf Ŝ n N S+ n,p,m GS GŜ = GS GS + = GS. The solution is unique in the following sense: If S is another solution, then it is also a realization of GS +. Proof. Let Ŝ n N S+ n,p,m with GS GŜ RL. Since GS RL Theorem.19 we have GS + GŜ RH. By Theorem.18 we obtain GS GŜ = GS +GS + GŜ = GS + GS + GŜ GS. The lower bound is attained, if and only if Ŝ SGS +, e.g. if Ŝ = S +. Remark 3.3. In the proof of Theorem 3. it can be seen that the function GS + is even the best approximation in the Hardy space H. The reason is that GS is also orthogonal to H see the more general statement of Theorem.18 in [Fra87, Section.3]. 8

9 3.3 Optimal RH -approximations with balanced realization For a minimal antistable standard system S the problem AP was already solved in [Glo84]. Combining [Glo84, Theorem 6.1] and its proof based on [Glo84, Theorem 6.3], we obtain the following result. We reformulate the statements in terms of realizations instead of transfer functions. Note also the interchange of the stable and antistable system which is easy to show. Theorem 3.4 [Glo84]. Let S S n,p,m be a minimal realization such that I A11 A S =, 1 B1, I r A 1 A B Σ1 Σ C S =, O σ 1 I S = r σ 1 I r, C 1 C, D,, σ 1 / σσ 1 Σ, where σ 1 := max σc S O S compare Theorem.14. Then there exists U R p m with B = C U. Set Γ := Σ 1 Σ σ 1I, Ã := Γ 1 σ 1A 11 + Σ A 11 Σ 1 + σ 1 C 1 UB 1, C := C1 Σ 1 σ 1 UB 1, B := Γ 1 Σ B 1 σ 1 C 1 U, D := D + σ1 U. Then S := I, Ã, B, C, D S + n r,p,m holds and inf Ŝ n N S+ n,p,m GS GŜ = GS G S = σ 1. Remark 3.5. The value σ 1 is the greatest hankel singular value of GS see [Fra87, Section 5.1] for further details. As stated in [Glo84, Theorem 6.1], the function G S is even the best approximation in the more general Hardy space H. Because of Theorem 3.1 we are now able to solve AP, if S is a descriptor system. The disadvantage of algorithms based on Theorem 3.4 is that we need a balanced minimal realization of GS to compute an optimal solution. 3.4 Optimal RH -approximations without balanced realization Some results of [Glo84], concerning optimal hankel norm approximation, were generalized in [SCL9] to nonminimal stable standard systems. We use [SCL9, Theorem 1] and its proof to solve AP for general S S n,p,m. Among some modifications, one of our main tasks is to prove that the optimal solution lies indeed in S + n,p,m, since [SCL9] only considers the poles of the resulting transfer function. First, we need to compute the greatest hankel singular value σ 1 of GS with the formula given in Theorem 3.4 which is restricted to minimal standard systems. However, we 9

10 avoid the determination of a minimal realization. To the author s knowledge, it was never shown that the nonzero hankel singular values of GS equal the roots of the nonzero eigenvalues of C S O S independently of the realization S of GS. This equality was only proved for minimal realizations cf. e.g. [Fra87, Section 5.1, Theorem 3]. Lemma 3.6. Let S = I, A, B, C, D S n,p,m be a minimal realization of G T p,m and let S = Ẽ, Ã, B, C, D S ñ,p,m be a realization of G. Then σ C S O S \ {} = σ C SẼ O SẼ \ {}. In particular, the equation σ1 = max σ C SẼ O SẼ holds for all S S ñ,p,m SG. Proof. We set S I := Ẽ 1 S. By Theorem.11 there exists a regular T Rñ ñ such that A 11 A 13 B 1 S T := T S I T 1 = I, A 1 A A 3 A 4 A 33, B, C 1 C 3, D, A 43 A 44 where S 1 := I, A 11, B 1, C 1, D is a minimal realization of G. The condition σẽ, Ã = σẽ 1 Ã C > implies A11 A11 A σ C A 1 A >, σ 13 C A >, σa 11 C >. 33 X1 X By Theorem.1 there exist unique solutions Y1 Y X X and 4 Y Y 4 of A11 X1 X X1 X A 1 A X + A11 X 4 X + X 4 A 1 A A11 A 13 Y1 Y Y1 Y A 33 Y + A11 A 13 Y 4 Y Y 4 A 33 Thus X 1 = C S1, Y 1 = O S1, B1 B B 1 B =, C C1 + 1 C3 C 3 =. X 1 X Y 1 Y C ST = X X 4 and O S T = Y Y 4. By Theorem.11 and Lemma.13 there exists a regular P R n n with C S1 = P 1 C S P, 1

11 O S1 = P O S P. We conclude σ C SẼ O SẼ \ {} = σ C SI O SI \ {} = σ T CST T T O ST T 1 \ {} X 1 Y 1 X 1 Y = σ C ST O ST \ {} = σ X Y 1 X Y \ {} = σ X 1 Y 1 \ {} = σ CS1 O S1 \ {} = σ CS O S \ {}. The next theorem is one of our main results. Together with Theorem 3.1 it gives suboptimal solutions of AP and, with an additional regularity-condition, an optimal solution. Theorem 3.14 solves AP, if the regularity-condition is not fulfilled. Theorem 3.7. Let S = E, A, B, C, D S n,p,m. Choose γ σ 1 and set R S,γ := O S EC S E γ I, E S,γ := E R S,γ, B S,γ := E O S B, A S,γ := A R S,γ C C S,γ, C S,γ := CC S E. If E S,γ, A S,γ is regular, e.g. if γ > σ 1, then S S,γ := E S,γ, A S,γ, B S,γ, C S,γ, D S + n,p,m and S S,γ fulfils σ 1 GS GS S,γ γ. Proof. First, let E = I. In this and the following proofs we use the equation A S,γ = A R S,γ C C S,γ = R S,γ A B S,γ B = γ A + O S AC S, 3.1 which holds because of A R S,γ C C S,γ = γ A A O S C S C CC S = γ A + O S AC S = γ A O S C S A O S BB = R S,γ A B S,γ B. Thus A S,γ is indeed the same as in [SCL9, Theorem 1]. To apply [SCL9, Theorem 1], we first construct an antistable system S with a hankel singular value greater than γ. This method was proposed in [Glo84, Remark 8.4]. Let γ 1 > γ. We set S 1 := 1, 1 γ 1, γ 1, γ 1, and A B C D S := I, 1 γ,,, S 1 γ 1 γ 1 n+1,p+1,m+1. Then we have CS OS C S =, O γ S =. 1 γ 1 Thus γ 1 is the greatest hankel singular value of GS and σ 1 γ < γ 1. If we apply [SCL9, Theorem 1] to S with K = RH, then the transfer function of ES,γ AS,γ BS,γ CS,γ D S S,γ = γ1, γ 1 γ γ 1 + γ1 3, γ1, γ1, 11

12 is a suboptimal solution, i.e. GS GS S,γ γ. Note that [SCL9, Theorem 1] does not state, that for every K RH with K 1 the resulting transfer function is a suboptimal solution. However, this was shown in the proof see also [Glo84, Remark 8.4]. With GS GSS,γ GS =, GS GS 1 S,γ = GS S1,γ we conclude GS GS S,γ GS GS S,γ γ. Now we show that S S,γ S + n,p,m. [SCL9, Theorem 1] states that GS S,γ has at most one pole λ in C counting multiplicities. The pole of GS S1,γ is λ = γ 1 γ > with γ γ 1 +γ1 3 γ1 γ >, γ γ 1 + γ1 3 >. Thus λ = λ and GS S,γ has no poles in C. Let λ C. We use Lemma.15 to show that λ / σe S,γ, A S,γ. Let v R n satisfy λe S,γ A S,γ v = and C S,γ v =. Hence with A S,γ = A E S,γ C C S,γ we get λi A E S,γ v =. Because of σ A C < we conclude E S,γ v =. Thus s C: se S,γ A S,γ v = se S,γ + A E S,γ + C C S,γ v =. Since the matrix pair E S,γ, A S,γ is regular we have v =. Now let v R n satisfy λe S,γ A S,γv = and B S,γv =. Hence λi + AE S,γv = follows with A S,γ = AE S,γ BB S,γ. Again we deduce v =. Therefore λ / σe S,γ, A S,γ and thus S S,γ S + n,p,m follows. Finally we consider general regular E R n n. We already proved that the assertion holds for S E 1 S,γ. With C E 1 S = C S and O E 1 S = E O S E Lemma.13 we conclude E E 1 S,γ = E O S EC S γ I, B E 1 S,γ = E O S EE 1 B, A E 1 S,γ = A E E E 1 S,γ C C E 1 S,γ, C E 1 S,γ = CC S. Now we see that S S,γ = S E 1 S,γ E. Thus the assertion also holds for S S,γ S E 1 S,γ. If γ > σ 1, the matrix pair E S,γ, A S,γ is regular, since R S,γ, E and thus E S,γ are regular. We formulate some equivalent conditions for the regularity-condition of E S,σ1, A S,σ1 in Theorem 3.7. Corollary 3.8. The following three statements are equivalent. i E S,σ1, A S,σ1 is regular, ii σ ES,σ1, A S,σ1 C< { }, iii A S,σ1 is regular. Proof. The implication i ii follows from Theorem 3.7. The statement ii implies / σ E S,σ1, A S,σ1 and thus ES,σ1 + A S,σ1 is regular. Finally, iii implies / σ E S,σ1, A S,σ1, therefore ρ ES,σ1, A S,σ1 and thus i holds. 1

13 To get optimal solutions if E S,σ1, A S,σ1 is singular, we analyze SS,γ in Theorem 3.7 for γ σ 1 +. For the proofs we use the realization S b S of GS as in Theorem.14 with h = σ 1 because we will take advantage of the special structure of S Sb,σ 1. First we investigate the relation between S S,γ and S Sb,γ. Lemma 3.9. Let S = E, A, B, C, D S n,p,m. Let P, Q R n n be regular. Then S P S Q,γ = Q T S S,γ P T holds for all γ R. Proof. We set S := P S Q. By Lemma.13 the equations C S = QC SQ, O S = P O SP hold. This implies B S,γ = E O S B = E P O SP B = Q B S,γ, C S,γ = CC S E = CQC SQ E = C S,γ P, E S,γ = E O S EC S E γ E = E P O SP EQC SQ E γ E = Q E S,γ P, A S,γ = A E E S,γ + C C S,γ = A E Q E S,γ P + C C S,γ P = Q A S,γ P. Lemma 3.1. Let S b = I, A b, B b, C b, D S be as in Theorem.14 with h = σ 1. Let T 1 R n n be regular such that T 1 S E 1 T1 1 I A11 A = S b =, 1 B1,, C I r A 1 A 1 C, D. Then the following three statements hold: a B B = C C, b kerb = kerc, c E S,σ1, A S,σ1 singular kerc {} kerb {}. Proof. We get B B = C C by adding the equations σ 1 A σ 1 A + B B = and σ 1 A + σ 1A C C = resulting from.4. Therefore v kerb B v = B v = = v B B v = v C C v C v = v kerc. B By Lemma 3.9 and with Γ := Σ Σ 1 σ1 I we have S Sb,σ 1 = T1 E S S,σ1 T1 Γ A =, 11 Γ C1 C 1Σ 1 σ 1 C1 C Σ B A 1 Γ C C 1Σ 1 σ 1 C C, 1, C σ 1 B 1 Σ 1 σ 1 C,. Therefore E S,σ1, A S,σ1 is singular, if and only if ESb,σ 1, A Sb,σ 1 is singular. 13

14 Let kerc {}. Then there exists v with C v =, i.e. for all s C se Sb,σ 1 A Sb,σ 1 = v That means E Sb,σ 1, A Sb,σ 1 is singular. sγ + A 11 Γ + C1 C 1Σ 1 σ 1 C1 C A 1 Γ + C C 1Σ 1 σ 1 C C = v. Let kerc = {}. Then C C is regular. Let s C. The matrix se Sb,σ 1 A Sb,σ 1 is singular, if and only if the Schur complement si + A 11 + C 1 C 1 Σ 1 Γ 1 C 1 C C C 1 A 1Γ + C C 1 Σ 1 Γ 1 Γ =: si MΓ of se Sb,σ 1 A Sb,σ 1 in σ 1 C C is singular Theorem.5. This is equivalent to s σm. Hence E S,σ1, A S,σ1 has a finite number of eigenvalues and consequently is regular. Similar to the proof of [SCL9, Theorem 1] we want to eliminate the singular part of the system S S,σ1 and show that the resulting system is an optimal solution. Lemma Let E S,σ1, A S,σ1 be singular. Then there exists a regular T R n n such that Ẽ11 T E S S,σ1 T 1 Ã11 B1 =,,, C1, D, where Ẽ11, Ã11 is regular. Proof. Our proof is based on the proof of [SCL9, Theorem 1]. Let S b, T 1 and Γ be as in Lemma 3.1. By Lemma 3.1 we have d :=dimkerc >. Therefore there exists an orthogonal W R r r such that Ĉ p d = C W and kerĉ = {} e.g. singular value decomposition. We define B B 3 := B W with appropriate dimensions. By Lemma 3.1 we have B 3 =. Now we set T := I W, conclude 14

15 A 1 Γ C C 1Σ 1 = σ 1 B B1 from 3.1 and compute Σ B 1 I Σ B T B Sb,σ 1 = T C Sb B b = 1 = σ W σ 1 B 1 B =: d m C Sb,σ 1 T 1 = C b O Sb T 1 = C 1 Σ 1 σ 1 C W = C 1 Σ 1 σ 1 Ĉ p d Γ T E Sb,σ 1 T 1 I Γ I = W W = =: d d T A Sb,σ 1 T 1 A = T 11 Γ C1 C 1Σ 1 σ 1 C1 C σ 1 B B1 σ 1 C C T 1 A 11 Γ C 1 C 1Σ 1 σ 1 C 1 Ĉ = σ 1 B B1 σ 1 Ĉ Ĉ =: d d =: B1 d m C1 p d Ẽ11 d d Ã11. d d Since Ĉ Ĉ is regular, there exists s C such that the Schur complement si + A 11 + C1 C 1 Σ 1 Γ 1 σ 1 C1 ĈĈ Ĉ 1 B B1 Γ 1 Γ of sẽ11 Ã11 in σ 1 Ĉ Ĉ is regular. Theorem.5 states that sẽ11 Ã11 is regular. Thus Ẽ11, Ã11 is regular. With T := T T 1 the proof is finished. Lemma 3.1. We set S 1 := Ẽ11, Ã11, B 1, C 1, D. For every ω R with iω ρẽ11, Ã11 the unequality GSiω GS 1 iω σ 1 holds. Moreover, GS 1 has no poles in C. Proof. First we show that GS S,γ s γ σ 1+ GS 1 s for all s C except for a finite number. Let T be as in Lemma We rewrite GS S,γ s for γ > σ 1 and s C as GS S,γ s = GT E S S,γ T 1 s = C1 T E se S,γ A S,γ T 1 1 B1 + D sẽ11 = C1 Ã11 where holds with 1 B1 + σ1 γ si + T E A T 1 E S,γ = E S,σ1 + E S,γ E S,σ1 = E S,σ1 + σ 1 γ E, A S,γ = A S,σ1 σ 1 γ A. + D, 15

16 We set M1 M M 3 M 4 := T E A T 1 with appropriate dimensioned M i. In addition, let s / σm 4 σẽ11, Ã11. Then L γ := sẽ11 Ã11 + σ 1 γ si + M 1 σ 1 γ M si M 4 1 M 3 is the Schur complement of T E se S,γ A S,γ T 1 = s Ẽ 11 Ã11 + σ1 γ si + M 1 σ1 γ M σ1 γ M 3 σ1 γ si + M 4 in σ1 γ si + M 4. By Theorem.5 the matrix L γ is regular because se S,γ A S,γ is regular. The functions γ L γ and thus γ GS S,γ s = C 1 L 1 γ B 1 + D are continuous in σ 1,. Note that L σ1 = sẽ11 Ã11 is regular. We obtain GS S,γ s = C 1 L 1 γ Therefore for all iω / σm 4 σẽ11, Ã11 GSiω GS 1 iω = B 1 + D γ σ 1+ C 1 sẽ11 Ã11 1 B1 + D = GS 1 s. lim GSiω GS S,γiω γ σ 1 + lim γ = σ 1. γ σ 1 + Now we show that GS 1 has no poles in C. By [Fra87, Section.3] we have Gs G for all G RH and s C. Thus we get the upper estimate GS S,γ s GS S,γ s GSs + GSs γ + GS. for all s C. Hence for all s C \σẽ11, Ã11 σm 4 we have GS 1 s = Thus GS 1 has no poles in C. lim GS S,γs γ σ 1 + lim γ + GS γ σ 1 + = σ 1 + GS. Theorem The system S 1 given in Lemma 3.1 solves AP. Proof. We use the notations of the proof of Lemma By Theorem 3.7 and Lemma 3.1 it remains to prove that σẽ11, Ã11 C < { }. Let s C. We apply Lemma.15 x1 because s is not a pole of GS 1. Let x = satisfy = C 1 x and = s Ẽ 11 Ã11x s = Γ + A 11 Γ + C 1 C 1Σ 1 σ 1 B B1 x x1 σ 1 C1 Ĉ σ 1 Ĉ. 3. Ĉ x 16

17 Hence x = C1 s Ẽ = 11 Ã11 = C Sb,σ 1 T 1 x x, 3.3 = T s E Sb,σ 1 A Sb,σ 1 T 1 = T s E Sb,σ 1 + A b E S b,σ 1 + C b C S b,σ 1 T 1 x x 3.3 = T s I + A b E S b,σ 1 T 1 Due to S b S S n,p,m we have s / σ A b and the equation Γ = E Sb,σ 1 T 1 x = T E Sb,σ 1 T 1 x = follows. Thus we get x 1 = by the regularity of Γ. Now we deduce 3. σ 1 Ĉ Ĉx = x = x 1 x x. from the regularity of Ĉ Ĉ. Thus x =. Analogously every x R n d which fulfils = s Ẽ 11 Ã11 x and = B 1 x equals zero. Thus s / σẽ11, Ã11 and σẽ11, Ã11 C < { }. The optimal solution given in the theorem above is still based on the balanced realization. Our second main contribution is the following result, which only needs one singular value decomposition. Theorem Let S = E, A, B, C, D S n,p,m. Suppose E S,σ1, A S,σ1 given in Theorem 3.7 is singular. Then there exist orthogonal U, V R n n with UA S,σ1 V = Every regular U, V R n n satisfying 3.4, fulfil UE S,σ1 V = Ê11 Ê 1, UB S,σ1 = The system S := Ê11, Â11, B 1, Ĉ1, D solves AP. Â11  1 and regular  B1, C S,σ1 V = Ĉ1 Ĉ. Proof. Let T R n n be as in Lemma Then we obtain T E E S,σ1 T 1, T E A S,σ1 T 1 = Ẽ11, Ã11 17

18 with Ã11 R n d n d and σẽ11, Ã11 C < { }. In particular, Ã11 is regular. The matrices A S,σ1 and T E A S,σ1 T 1 have the same rank. By singular value decomposition there exist regular orthogonal U, V R n n such that Â11 Â UA S,σ1 V = 1 with regular Â11 R n d n d. Let U, V R n n be regular and satisfy 3.4. We define Ê11 Ê UE S,σ1 V =: 1 B1, UB S,σ1 =:, C S,σ1 V =: Ê 1 Ê B Ĉ1 Ĉ. For all s C the matrices sê11 Â11 and sẽ11 Ã11 are regular. Thus we have sê11 Â11 sê1 Â1 [R n ] = UE T 1 s Ẽ 11 Ã11 T V [R n ] sê1 sê = UE T 1 [ R n d {} d] = R n d {} d. 3.5 We obtain the last equality because holds for all s C and in particular for s =. That implies Ê1 =, Ê = and finally σê11, Â11 = σẽ11, Ã11 which means S S + n d,p,m. Furthermore, we have B = because B1 [R B m ] = B1 UE T 1 [R m ] 3.5 = R n d {} d. Finally, we show that GS = GS 1. Let s ρê11, Â11 and Y 1 := C 1 sẽ11 Ã11 1. We conclude with Y := Y 1 R p n and X := X 1 X := Y T E U 1 that Ĉ1 Ĉ = C1 T V = Y s Ẽ 11 Ã11 T V = X In particular X 1 = Ĉ1sÊ11 Â11 1. That implies s Ê 11 Â11 sê1 Â1. GS 1 s = C 1 sẽ11 Ã11 1 B1 = Y 1 B1 = B1 Y 1 Y = B1 Y 1 Y T E U 1 = B1 X 1 X = X 1 B1 = Ĉ1sÊ11 Â11 1 B1 = GS s. By Theorem 3.13 we get GS GS = σ 1. Under the additional condition E = I we can use the less complex Schur decomposition to attain 3.4 in Theorem

19 Lemma In the situation of Theorem 3.14, where E = I, we obtain regular matrices U and V := U, such that 3.4 holds, by applying the Schur decomposition to A S,σ1. Proof. Let T R n n be as in Lemma Then we obtain T ES,σ1 T 1, T A T 1 Ẽ11 Ã11 S,σ1 =, with Ã11 R n d n d and σẽ11, Ã11 C < { }. In particular, à 11 is regular. The matrices A S,σ1 and T A S,σ1 T 1 have the same eigenvalues counting multiplicities. By the Schur decomposition there exists a regular in particular an orthogonal U R n n such that Â11 UA S,σ1 U 1  = 1 Ã11 = UT 1 T U 1,  where Â11 R n d n d is regular and σâ = {}. This implies rankâ11 = rankã11 = rankua S,σ1 U 1 and thus  =. In summary, we obtain the following result for AP. Theorem Let S S n,p,m. Then there exist S + S + n +,p,m and S = E, A, B, C, S n,p,m such that S S + S. We set σ 1 := max σe O S E C S, then We define the matrices inf Ŝ n N S+ n,p,m GS GŜ = σ 1. R S,σ 1 := O S E C S E σ 1I, E S,σ 1 := E R S,σ1, B S,σ 1 := E O S B, A S,σ 1 := A R S,σ1 C C S,σ 1, C S,σ 1 := C C S E. If E S,σ 1, A S,σ 1 is regular, then S + E S,σ 1, A S,σ 1, B S,σ 1, C S,σ 1, solves AP. If E S,σ 1, A S,σ 1 is singular, then there exist orthogonal matrices U, V R n n with UA S,σ 1 V = Every regular U, V R n n satisfying 3.6, fulfil UE S,σ 1 V =, UB S,σ 1 = Ê11 Ê 1 Â11  1 and regular  B1 The system S + Ê11, Â11, B 1, Ĉ1, solves AP., C S,σ 1 V = Ĉ1 Ĉ. 19

20 4 Application Algorithm We rewrite our results in an algorithmic manner. Note that the solution of AP is necessary to compute the solution of AP and thus no extra computation is needed. Input: S S n,p,m Output: S q S + n q,p,m which solves AP q for q {, }. 1. Decompose S into S S n,p,m and S + S + n +,p,m as in Theorem 3.1 with the help of Theorem.3.. The system S := S + solves AP. 3. Determine C := C S E, Õ := E O S and σ 1 = max σõ C. 4. Compute R := Õ C σ 1 I, Ê := E R, B := ÕB, Â := A R C Ĉ, Ĉ := C C. 5. If Ê, Â is regular test with Corollary 3.8, then AP is solved by S := S + Ê, Â, B, Ĉ,. 6. If Ê, Â is singular, then determine U, V Rn n SVD such that Â11 UÂV = Â 1 with regular Â11. Ê11 Compute Ê11, B 1, Ĉ1 of UÊV = Ê 1, U B B1 =, ĈV = Ĉ1 Ĉ, then AP is solved by S := S + Ê11, Â11, B 1, Ĉ1,. For the numerical issues of step 1 we refer to [KD9] and of step 3 to [Pen98]. Both approximation techniques does not increase the order of the system, i. e. n q n. For optimal RH -approximation we even have n < n. Another advantage is the less computational complexity compared to RH -approximation. The behaviour of the transfer functions GS and GS at infinity is the same, i. e. GS = GS. This does not hold for optimal RH -approximation in general. However, the suboptimal approximation S + S S,γ choosing γ > σ 1 in Theorem 3.7 has this property. Since is regular, we have E S,γ GS = GS + = GS + + GS S,γ = GS + S S,γ. Examples The following benchmarks are performed in Matlab R. We test the building model Figure 1 and the clamped beam model Figure, see [CD] for details. Both models are described by stable standard systems with one input and one output, i.e. m = p = 1.

21 We apply shifted Arnoldi method to reduce the systems. The parameters interpolation point s C { } and reduced order k N are based on the examples in [ASG1]. The computed reduced systems are unstable as listed in Table 1. Thus we apply the algorithm presented above. Since the transfer functions of the original systems and hence of the reduced systems vanish at infinity an optimal RH -approximation would result in huge relative errors at high frequencies. That is why we compute suboptimal approximations i. e. γ > σ 1 to match the original transfer functions at infinity. model order model reduction reduced unstable max. real part method order poles of unstable poles building 48 Arnoldi, s = RH RH, γ = 1.1 σ clamped 348 Arnoldi, s = beam RH RH, γ = 1.1 σ Table 1: Summary of the results The error between the original systems and the reduced systems of Arnoldi method does not significantly chance after applying our stabilization algorithm as depicted in Figure 1 and Figure. 4 The norm of Giω Error w.r.t. o rig inal Arnoldi Arnoldi Arnoldi RH 6 Arnoldi RH Arnoldi RH 1 Arnoldi RH Giω in db 8 Giω Kiω in db frequency ω frequency ω Figure 1: Frequency responses of the reduced building model left and the error systems right References [ASG1] A. C. Antoulas, D. C. Sorensen, and S. Gugercin. A survey of model reduction methods for large-scale systems. Contemporary Mathematics, 8:193 19, 1

22 8 The norm of Giω 4 Error w.r.t. original Arnoldi Arno ldi Arnoldi RH 4 Arnoldi RH Arnoldi RH Arnoldi RH Giω in db 4 Giω Kiω in db frequency ω frequency ω Figure : Frequency responses of the reduced clamped beam model left and the error systems right [BF1] [BFF98] [CD] 1. Zhaojun Bai and Roland W. Freund. A partial padé-via-lanczos method for reduced-order modeling. Linear Algebra and its Applications, : , 1. Zhaojun Bai, P. Feldmann, and R. W. Freund. How to make theoretically passive reduced-order models passive in practice. In Proc. Custom Integrated Circuits Conf the IEEE 1998, pages 7 1, Younès Chahlaoui and Paul Van Dooren. A collection of benchmark examples for model reduction of linear time invariant dynamical systems,. [Dai89] L. Dai. Singular Control Systems. Springer-Verlag New York, [Fra87] B. A. Francis. A Course in H Control Theory, volume 88 of Lecture Notes in Control and Information Sciences. Springer-Verlag Berlin Heidelberg, [GGLD9] M. Green, K. Glover, D. Limebeer, and J. Doyle. A J-spectral factorization approach to H control. SIAM Journal on Control and Optimization, 8: , 199. [GL96] G. H. Golub and C. F. Van Loan. Matrix Computations. The Johns Hopkins University Press, [Glo84] K. Glover. All optimal hankel-norm approximations of linear multivariable systems and their L -error bounds. International Journal of Control, 39: , [KD9] B. Kågström and P. Van Dooren. A generalized state-space approach for the additive decomposition of a transfer matrix, 199. [KM6] P. Kunkel and V. Mehrmann. Differential-Algebraic Equations - Analysis and Numerical Solution. EMS Publishing House, 6.

23 [Lju99] [Mac4] [Mar] [Pen98] [SCL9] [Ste7] L. Ljung. System Identification - Theory For the User, volume. Prentice Hall, U. Mackenroth. Robust Control Systems - Theory and Case Studies. Springer- Verlag Berlin Heidelberg, 4. Jorge Mari. Modifications of rational transfer matrices to achieve positive realness. Signal Processing, 84: ,. T. Penzl. Numerical solution of generalized lyapunov equations. Advances in Computational Mathematics, 8:33 48, M. G. Safonov, R. Y. Chiang, and D. J. N. Limebeer. Optimal hankel model reduction for nonminimal systems. IEEE Transactions on Automatic Control, 354:496 5, 199. G. W. Stewart. On the sensitivity of the eigenvalue problem Ax = λbx. SIAM Journal on Numerical Analysis, 9: , 197. [ZDG96] K. Zhou, J. Doyle, and K. Glover. Robust and Optimal Control, volume 1. Prentice Hall,