Realization-independent H 2 -approximation

Transcription

1 Realization-independent H -approximation Christopher Beattie and Serkan Gugercin Abstract Iterative Rational Krylov Algorithm () of [] is an effective tool for tackling the H -optimal model reduction problem. However, so far it has relied on a first-order state-space realization of the model-to-be-reduced. In this paper, by exploiting the Loewner-matrix approach for interpolation, we develop a new formulation of that only requires transfer function evaluations without access to any particular realization. This, in turn, extends to H approximation of irrational, infinite-dimensional dynamical systems. We also introduce a residue-correction step in that adjusts the vector residues to minimize the H error at the end of each cycle using a new set of necessary and sufficient conditions for H optimality. This new step further improves the convergence speed and performance of. Three numerical examples illustrate the effectiveness of the proposed methods. I. INTRODUCTION Dynamical systems are the basic framework for modeling and control of a wide variety of complex systems having scientific interest and industrial value. Examples include signal propagation and interference in electric circuits, heat transfer and temperature control in various media, and behavior of micro-electro-mechanical systems. For a collection of such examples, we refer the reader to [] and [6]. Direct numerical simulation of the associated models has been one of few available means for studying complex underlying physical phenomena. However, the ever present need for improved accuracy drives the inclusion of ever more detail in the modeling stage, leading inevitably to ever larger scale, ever more complex dynamical systems. Simulations in such largescale settings often lead to unmanageably large demands on computational resources, creating the main motivation for model reduction. Simply stated, the goal is to produce a relatively low order system approximation with input/output behavior very close to the original one. We consider stable multiple-input/multiple-output (MIMO) linear dynamical systems described via transfer functions, H(s), that are assumed to be meromorphic functions with poles in the open left half-plane (hence analytic in the open right halfplane). We have particular interest in cases where the order of H(s) (total number of poles) is very large, perhaps on the order of 0 5 or more. The methods we develop here only require the ability to evaluate H(s) at selected points, s C. Notably, we do not require access to any particular realization of H(s). We assume H(s) to be an -function where denotes the set of p m matrix-valued functions, H(s), with This work was supported in part by the NSF Grants DMS C. Beattie and S. Gugercin are affiliated with the Department of Mathematics, Virginia Tech, Blacksburg, VA, , USA. {beattie, gugercin}@math.vt.edu components, h ij (s), that are analytic for s in the open right half plane, Re(s) > 0, and such that for each fixed Re(s) x > 0, h ij (x + ıy) is square integrable as a function of y (, ) in such a way that sup x>0 h ij (x + ıy) dy <. is a Hilbert space. Indeed, if G(s) and H(s) are - functions then the -inner product can be defined as G, H H def π trace ( G(ıω) H(ıω) T ) dω () with an associated norm defined as ( def + / H H H(ıω) F dω), () π where M F M, M F and M, N F trace ( M N ) T defines the Frobenius norm and Frobenius inner product, respectively. Notice that if G(s) and H(s) represent real dynamical systems then G, H H H, G H and G, H H itself must be real. Our goal is to find, for a given order r, a reduced model described by H r (s) : C r (se r A r ) B r (3) with A r, E r R r r, B r R r p, and C r R m r, such that H r (s) is an optimal approximation to H(s): H H r H min H G r G H. (4) r:stable Since finding global minimizers can be very difficult, the more modest goal of finding local minimizers is more usual. Optimal approximation in this sense has been investigated extensively; see for instance [6], [3], [4], [5], [5], [7] for Lyapunov-based methods and [0], [], [0], [4], [6], [8], [7], [3], [4] for interpolation-based methods. These methods, however, require access to a standard first-order realization for H(s), a drawback we will overcome in this paper. Although the Lyapunov and interpolation frameworks are theoretically equivalent [], we focus on the latter as it presents significant computational advantages. Interpolatory optimality conditions for systems with a single scalar-valued input and a single scalar-valued output ( SISO systems), were introduced by Meier and Luenberger [0] and have been recently extended to MIMO systems by [], [6], [4]. In [] we introduced an efficient interpolatory model reduction algorithm, the Iterative Rational Krylov Algorithm (), that generates a reduced model satisfying first-order optimality conditions. This method

2 has proved remarkably effective in many diverse problems. For many small-order problems used as model reduction benchmarks, it will typically find the global optimum among numerous local minimizers (though this behavior is not guaranteed). For modest order examples, consistently yields a better reduced order approximation than does balanced truncation. This comparison is significant since balanced truncation will yield reduced order models having small and often near-optimal error [], [5]; generally, balanced truncation is viewed as the gold standard for model reduction. More importantly, due to the numerical efficiency of, it has been successfully applied to systems having hundreds of thousands degrees of freedom; see, for example, [5] for an application in highenergy efficient building design where is applied to construct a 30 th order optimal reduced-model for a system of dimension bigger than The purpose of this paper is threefold. First, we develop an implementation of that only requires transfer function evaluations without access to any particular realization. This would allow, in turn, extending optimal model reduction to irrational, infinite-dimensional dynamical systems. Second, by incorporating an easy-to-compute least-squares minimization step in the implementation of, we further improve the convergence speed of, especially for the cases where the input and output dimension are large. Finally. this least-squares minimization step leads to a new set of necessary and sufficient conditions for a constrained optimization problem. We restrict ourselves to reduced models having simple poles which we label as λ, λ,... λ r. The corresponding residue of H r (s) at each λ i is matrix valued and rank one: res[h r (s), λ i ] c i b T i, for nontrivial c i and b i. H r (s) can be represented directly in terms of its poles and residues: H r (s) k c k b T k (5) For systems of this form we have Theorem : [4] Suppose that H r (s) has the form given in (5) and that both H r (s) and H(s) are transfer functions associated with real dynamical systems. Then and H, H r H H H r trace ( res[h( s)h T r (s), λ k ] ) k c T k H( λ k ) b k (6) k H c T k H( λ k )b k (7) k + c k λ k,l k λ c l b l b k. l For H r (s) as in (6), one can also write the first-order optimality conditions directly in terms of the poles and residues of H r (s). Theorem : [] Let H r (s) in (6) be the best rth order approximation of H(s) with respect to the norm. Then H( λ k )b k H r ( λ k )b k, c T k H( λ k ) c T k H r ( λ k ), (8) and c T k H ( λ k )b k c T k H r( λ k )b k, for each k,,..., r. Theorem states that if H r (s) is a solution to the optimal- approximation problem (4), then it is a bitangential Hermite interpolant to H(s) at the mirror images of the its own (reduced-order) poles along directions determined by its (vector) residues. The rest of the paper is organized as follows: In Section II, we introduce TF-, an optimal model reduction method which only uses transfer function evaluations and can be applied without accessing any specific realization of H(s). In Section III, we incorporate a residue correction step in TF- which re-assigns residue directions at each step of TF- so as to minimize the for the current reduced-order poles. In both sections, numerical examples illustrate the effectiveness of the proposed methods. II. OPTIMAL MODEL REDUCTION FROM TRANSFER FUNCTION EVALUATIONS [] has provided an efficient way of constructing reduced order models satisfying the first-order - optimality conditions. However, so far, it has been applied only for the special case of first-order realizations, i.e. H(s) C(sE A) B. In this section, we will extend the optimal model reduction by to approximating any -transfer function H(s) without any specific constraint on the structure. H(s) can correspond to an infinite dimensional (irrational) transfer function, can contain delays or can correspond to a second-order dynamical system H(s) C(s M + sg + K) B. The vital point towards this goal is to understand what [] performs in the intermediate steps in terms of the transfer function. Recall the first-order conditions for -optimality in (8): H r (s) should be a bitangential Hermite interpolant to H(s) at the mirror images of the poles of H r (s). For the first-order case H(s) C(sE A) B, uses projection-based interpolatory model reduction to construct such an H r (s) at every step; for details, see []. Thus, the goal is to be able to construct an H r (s) that is an Hermite bitangential interpolant to H(s), which does not necessarily correspond to a firstorder state-space model. The Loewner matrix framework for interpolation by Mayo and Antoulas [9] is the perfect tool for this goal. A. Loewner-matrix approach for interpolation [9] Given H(s), interpolation points {s,..., s r } and tangential directions {b,..., b r } and {c,..., c r }, we would like to construct a reduced model H r (s) C r (se r A r ) B r

3 that satisfies, for k,,..., r, H(s k )b k H r (s k )b k, c T k H(s k ) c T k H r (s k ), (9) and c T k H (s k )b k c T k H r(s k )b k The framework of [9] only requires evaluating H(s) and H (s) at s k without any constraint on the structure of H(s): Construct (E r) i,j : ct i (H(s i) H(s j)) b j s i s j if i j, (0) c T i H (s i)b i if i j ct i (s ih(s i) s jh(s j)) b j if i j (A r) i,j : s i s j, () c T i [sh(s)] ssi b i if i j and C r [H(s )b,..., H(s r)b r], B r c T H(s ). c T r H(s r) () Then the reduced model H r (s) C r (se r A r ) B r satisfies (9). E r from (0) and A r from () are called, respectively, the Loewner matrix and the shifted Loewner matrix associated with H(s). The construction in (0)-() assumes that A r s i E r is invertible for i,..., r. If this is not the case, one uses the singular value decomposition of A r s i E r to truncate the redundant data; for details, see [9], []. B. using transfer function evaluations The Loewner-matrix framework of Section II-A gives us the necessary tools to extend to to general H(s) settings. Our algorithm will be an iterative interpolation method as at the end producing a reduced model satisfying the first-order optimality conditions. Let H r (s) r c ib T i i s λ i be the current reduced model. Then, the next iterate will be a Hermite bitangential interpolant to H(s) at the interpolation points s i λ i with the tangential directions b i and c i. This step will be achieved using the Loewner matrix approach outlined above and the process will be repeated until convergence. A brief sketch of the resulting iteration is given below in Algorithm TF-. Step a) creates an rth order bitangential Hermite interpolant to H(s) at the current interpolation points and tangent directions. Steps b) and c) determine a pole-residue representation of the current rth order model which defines the next set of interpolation points and potentially also the next set of tangent directions if the residue correction step of Step d) is omitted. The optional Step d) is described in Section III. Notice that upon convergence, the interpolation points are the mirror images of the poles of H r (s) and the interpolation tangent directions are residue directions of H r (s); thus the first-order optimality conditions (8) are satisfied. As for, this algorithm is a fixed point iteration and a ALGORITHM: TF- using transfer function evaluations Given: a transfer function H(s); reduction order r; initial interpolation points {s,..., s r} and tangent directions {b,..., b r} and {c,..., c r} (chosen to be closed under conjugation). Return: H r(s) C r(se r A r) B r, a reduced system realization satisfying -optimality conditions (8). ) repeat until convergence a) Construct E r, A r, C r and B r as in (0)-(). b) Compute A r x i λ i E r x i and yi A r λ i yi E r with yi E rx j δ ij where yi and x i are left and right eigenvectors associated with λ i. c) s i λ i, b T i yi B r, and c i C r x i, for i,..., r. d) (optional) Residue Correction; see Section III. ) Construct E r, A r, C r and B r as in (0)-(). similar convergence analysis can be mustered. These details are left for a separate work. We note that for the special case of generic first-order realizations H(s) C(sE A) B, TF- is theoretically equivalent to regular even though the numerical implementation are different, in other words, TF- contains as a special case. C. A one dimensional heat equation: The example taken from [] analyzes the temperature distribution on a semi-infinite rod. Let x 0 describe the half-line for the rod and let y(z, t) denote the temperature at position x and time t. Then, the temperature distribution is given by y t y 0. (3) x Following [], we assume that the temperature is controlled at x 0 and we are interested in the temperature at x. Using the initial conditions y(x, 0) 0 for x 0 and the boundary condition y(, t) 0 for t 0, one obtains the transfer function representing the dynamics from y(0, t) to y(, t) as H(s) Y (, s) Y (0, s) s e, where Y (, s) denote the Laplace transform of y(, t). Note that H(s) e s is an -function. Equivalently the impulse response can be obtained as h(t) 4πt 3 e 4t, t 0. Now, our goal is to construct an optimal reduced model approximation for H(s) e s directly without any discretization (in space), i.e. we will not discretize the PDE in (3) to obtain a state-space description; we will directly apply TF- to H(s) e s. We choose r 6. The main costs of the algorithm are simply evaluating

4 H(s) e s and H (s) s e s, and solving an 6 6 generalized eigenvalue problem; both of which are trivial computational tasks. Compare this with reducing a large-scale semi-discretized version of the original PDE, for example. Since this example is single-input/single-output, all the tangential directions are simply b i c i. Initial shifts are chosen as 6 real logarithmically spaced points between 0 5 and 0 3. Then, Algorithm TF- converges to the optimal interpolation points Impulse Responses h(t) and h r (t) h(t) h r (t) s, ± 3.96ı, s , s , s , s leading to H r (s) C r (se r A r ) B r where H r(s) s5 3.86s s s s s s s s s s We note that since we use H(s) e s evaluations directly, our 6 th -order rational approximation is an exact Hermite interpolant to H(s) e s. This is in direct contrast to the methods where H(s) e s is first approximated by a high-order rational function approximation which is then fed to generic model reduction techniques such as balanced truncation. In those instances, the final reduced order model is no longer an exact interpolant to the original (irrational) transfer function. The (time domain) impulse responses of the original model and the reduced model are depicted in Figure illustrating that the reduced model is virtually indistinguishable from the original one. Santarelli [] has also approximated this model using an L optimization approach. Figure 6.4 on page 764 of [] depicts the impulse response comparison using the method of [] with r 0. Even with r 0 (recall that we have used r 6), the reduced model of [] shows a bigger deviation from the original model for t < 0.5; indeed the approximation is poor around t 0. This is expected since the method of [] is restrictive in the sense that it pre-assumes a specific structure on the reduced order poles (such as one distinct pole repeated r times or two distinct poles repeated k and r k times etc). In our case, we do not enforce any conditions on either reduced order poles or the residues. We let reduced order poles vary through out the iteration to align themselves in a way to minimize the resulting error. III. RESIDUE CORRECTION typically converges quickly for systems having a modest number of input or output dimensions. This has been discussed and illustrated via several examples in the literature; see, for example, [], [], [9]. Convergence may The vector fitting method [3] also allows construction of reduced models using only transfer function evaluations. However, this approach produces an H r(s) with a nontrivial D r term, H r(s) C r(se r A r) B r +D r. Thus leading to an unbounded error. For example, we have applied the vector fitting method to reduce H ( s) e s, obtaining a reduced model having a small, but nonzero D r value (D r ). We do not consider vector fitting in this paper as it is more appropriate for an H approximation setting time (s) Fig.. Impulse responses of the original and reduced model slow down or become erratic when the number of both inputs and outputs grows large. This could be anticipated since when m and p are large, the number of decision variables in the minimization problem grow as well; overall there are r(m + p) degrees of freedom in H r (s). We consider here the effect of adding an additional correction step that adjusts only the vector residues at the end of each cycle of Algorithm TF-. In particular, suppose that the set of reduced order poles, {λ, λ,... λ r }, at the end of any cycle of Algorithm TF- consists of r distinct values in the open left half-plane. We view these values as fixed for the time being. Consider the set of rth order p m transfer functions: P r Hr H H r(s) r c k b T k k s λ k for c k C p and b k C m ; H r is real analytic. The condition H r is real analytic implies that we may assume without loss of generality that the vector residues, {c k } and {b k } are closed sets under conjugation with the same conjugation symmetry as the set of (fixed) poles {λ, λ,... λ r }. We are interested in finding a reduced order system H r P r that has an optimal adjustment of vector residues such that H H r H min H r P r H H r H (4) Notice that the objective function (7) is biquadratic with respect to the residue directions, that is, quadratic with respect to each of c k and b k, separately. This observation suggests the use of alternating least squares with respect to left and right families of residue directions in order to solve (4) within each cycle of Algorithm TF-. No additional transfer function evaluations are necessary and the added computational effort is negligible; added complexity per cycle will not exceed O(r 3 ).

5 Suppose for the moment, that reduced order poles, {λ, λ,... λ r }, and right residue directions, {b, b,... b r }, are fixed. We assume that only the left residue directions, {c, c,... c r }, are allowed to vary. For C [c, c,... c r ] C r r, define the linear map M : C r r P r as M(C) k c k b T k C and seek C C r r that solves k e k b T k, min H M(C). (5) C C r r C C r r may be characterized via the pseudoinverse of M: C M (H). We obtain an expression for M via a singular value decomposition of M as follows: Define the r r Cauchy matrix: [ b i M b ] j. (6) λ i λ j Since the reduced order poles, {λ, λ,... λ r }, are in the left half-plane, M is Hermitian positive definite, and Mu j σj u j, for unit norm eigenvectors, u j, and values σ... σ r > 0. Now define Φ ij e i u T j. Note that {Φ ij} is an orthonormal set in C r r with respect to the Frobenius norm: Φ ij, Φ kl F δ ik δ jl, hence {Φ ij } constitutes an orthonormal basis for C r r. Furthermore, {M(Φ ij )} ij is an orthogonal set in : M(Φ ij), M(Φ kl ) H ν Φ ije νb T ν s λ ν, µ Φ kl e µb T µ s λ µ ( ) trace e iu b ν b µ j e ν e T µ u l e T k λ ν λ µ µ,ν u j Mu l δ ik σ j δ jl δ ik We find that Φ ij is a right singular vector for M and ( ) G ij (s) u T j e k e i b T k M(Φ ij ) σ j σ j k is a left singular vector for M, both associated with the corresponding singular value σ j. Thus, M(C) σ j G ij (s) Φ ij, C F i,j and the pseudoinverse of M is directly available as M (H) Φ ij G ij, H σ H j with G ij, H H i,j H, G ij H (u T j σ e k) trace ( H( λ k )b k e T i j k (u j e k ) e T i H( λ k ) b k. σ j k ) The Cauchy matrix, M, may be extremely poorly conditioned leading to singular values, σ j, for M that may be very close to zero. Instead of using the full singular value decomposition of M to solve the best approximation problem (5), one may wish to truncate the SVD of M and solve a regularized version of (5): For truncation index, ρ r, define the truncated operator M ρ (C) i j σ j G ij (s) Φ ij, C F. Then the solution to the regularized problem is given by C M ρ(h) i,k k min H M ρ(c) C C r r H. (7) i,k j Φ ij σj (u j e k ) e T i H( λ k ) b k e i e T i H( λ k ) b k (e T k ūj) σ j j j H( λ k ) b k (e T k ūj) σ j Denoting now Y [ H( λ )b,..., H( λ r )b r ], Σ ρ diag(σ,..., σ ρ ), and U ρ [ u, u,..., u ρ ], we have C Y U ρ Σ ρ U T ρ. Notice that when ρ r then C YM. We summarize these findings in a theorem: Theorem 3: Let H(s) be an -function. For the fixed reduced order poles, {λ, λ,... λ r }, and right residue directions, {b, b,... b r }, H r (s) r k s λ k c k b T k minimizes H H r H if and only if H( λ i )b i H r ( λ i )b i, for i,,..., r. Let C [c,..., c r ] denote the left-residue directions that minimizes H H r H. Then, C YM where M and Y are as defined above. Note that once the poles and the right-residues are fixed, the optimality conditions become necessary and sufficient. This theorem extends the Gaier s result in [7] for SISO systems to MIMO systems we are considering here. An analogous development may be followed to determine the best right residues, given fixed poles and fixed left residues. Toward that end, suppose that reduced order poles, {λ, λ,... λ r }, and left residue directions, {c, c,... c r }, are fixed. The right residue directions, {b, b,... b r }, are allowed to vary: let B [b, b,... b r ] C r r and define the linear map N : C r r P r as N(B) k c k b T k k u T j u T j c k e T k B T.

6 We seek B C r r that solves min H N(B). (8) B C r r Analogous to the previous development, B C r r may be characterized via the pseudoinverse of N: B N (H), which in turn can be represented through a singular value decomposition of N. Define [ c i N c ] j (9) λ i λ j and note that N is Hermitian positive definite, and Nv j ςj v j, for orthonormal eigenvectors, {v j }, and values ς... ς r > 0. Now define Ψ ij e i vj T. Note that {Ψ ij} is an orthonormal set in C r r with respect to the Frobenius norm: Ψ ij, Ψ kl F δ ik δ jl, hence {Ψ ij } constitutes an orthonormal basis for C r r. Furthermore, {N(Ψ ij )} ij is an orthogonal set in N(Ψ ij), N(Ψ kl ) H : ν c ν e T ν Ψ T ij s λ ν, µ c µe T µ Ψ T kl s λ µ ) ( trace e k vl T c T µ c ν e µ e T ν v je T i λ µ λ ν µ,ν v T l N v j δ ik v l N vj δ ik ς j δ jl δ ik We find that Ψ ij is a right singular vector for N and ( ) F ij (s) v T j e k c k e T i N(Ψ ij ) ς j ς j k is a left singular vector for N, both associated with the corresponding singular value ς j. Thus, N(B) ς j F ij (s) Ψ ij, B F i,j and the pseudoinverse of N is directly available as N (H) Ψ ij F ij, H ς H j with F ij, H H i,j H, F ij H (vj T ς e k) trace(h( λ k )e ic T k ) j k (vj e k ) c kh( λ k )e i. ς j k The Cauchy matrix, N, may be extremely poorly conditioned leading to singular values, ς j, for N that may be very close to zero. Instead of using the full singular value decomposition of N to solve the best approximation problem (8), one may wish to truncate the SVD of N and solve a regularized version of (8): For truncation index, ρ r, define the truncated operator N ρ (B) ς j F ij (s) Ψ ij, B F. i j Then the solution to the regularized problem is given by B N ρ(h) min H N ρ(b) B C r r H. (0) i,k j i,k j k j Ψ ij ςj (vj e k ) c kh( λ k )e i e i e T i H( λ k ) T c k (e T k v j) ς j H( λ k ) T c k (e T k v j) ς j Denoting now Z [ H( λ ) T c,..., H( λ r ) T c r ], Σ ρ diag(ς,..., ς ρ ), and V ρ [ v, v,..., v ρ ], we have B Z V Σ ρ ρ Vρ T. Notice that when ρ r then B ZN. An analogous result to Theorem 3 holds here as well: Theorem 4: Let H(s) be an -function. For the fixed reduced order poles, {λ, λ,... λ r }, and left residue directions, {c, c,... c r }, H r (s) r k s λ k c k b T k minimizes H H r H if and only if c T i H( λ i ) c T i H r ( λ i ), for i,,..., r. Let B [b,..., b r ] denote the left-residue directions that minimizes H H r H. Then, B ZN where N and Z are as defined above. In order to solve the residue correction subproblem: min Hr P H H r r H, we alternate making optimal adjustments on left vector residues and right vector residues: RESIDUE CORRECTION [Step d) in TF-] Given: evaluations of the transfer function:{h( λ i)} r i; a pole-residue representation of a current reduced order model: H r(s) r c k b T k k s λ k ; a truncation index ρ: ρ r. Return: An updated reduced order model H r(s) r k having the same poles as H r(s) with vector residues adjusted so that H H r H min Hr P r v T j v T j c k b T k s λ k H H r H ) repeat until convergence a) Evaluate Y [H( λ )b,..., H( λ r)b r] b) Evaluate the Cauchy matrix (6); find Mu j σj u j, Σ ρ diag(σ,..., σ ρ ) and U ρ [ u, u,..., u ρ ]; c) Calculate [c, c,... c r ] Y U ρ Σ ρ U T ρ. d) Evaluate Z [H( λ ) T c,..., H( λ r) T c r] e) Evaluate the Cauchy matrix (9); find Nv j ςj v j, Σ ρ diag(ς,..., ς ρ ) and V ρ [ v, v,..., v ρ ]; f) Calculate [b, b,... b r ] Z V Σ ρ ρ Vρ T. g) Equilibrate magnitudes of left/right residues: τ k ck b k ; c k τ k c k ; b k τ k b k. ) c k c k ; b k b k.

7 A. Linearized Shallow Water Equations The full-order model, taken from [8], represents the D linearized shallow water equations with tidal forcing used in the modeling of St. Louis Bay. It has 58 inputs (corresponding to 9 wind-forecast locations) and 5 outputs (corresponding to surface elevations at 5 measurement stations). A finite element discretization leads to a model of the form H(s) C(sE A) B with E, A R , B R and C R Hence, in this case the original model is indeed a rational function itself with degree n 363; and and TF- are theoretically the same even though their implementations are different. We reduce the order to r 0 using both with and without the residue correction step. Figure shows the comparison for the initial interpolation point selection logspace( 6,, 0). The top plot in Figure depicts the relative error during the iteration. While the with the residue correction converges only after 3 steps, regular takes 8 steps to converge; hence the residue correction improves the convergence of even further. The missing data points in for k and k correspond to intermediate unstable models., after the third step, successfully corrects these models and converges to a stable system. This has been a common observation for. Even for the cases that an intermediate unstable model has developed, correct these reduced order poles (or equivalently the interpolation points) and leads to a stable reduced model. Interestingly, for this example with the residue correction step avoids these unstable intermediate steps. We note that at the end both methods converge to the same reduced model. The bottom plot in Figure shows the relative distance in the -norm between the interpolation points from k th and (k + ) th step, further illustrating the faster convergence in with the residue correction Evolution of the Relative H error Convergence of the interpolation points k: Iteration index Fig.. Evolution of We repeat the same numerical study for r 0, now using logspace( 5, 0, 0) as the starting interpolation points. The results are shown in Figure 3. The first observation is that both versions of (with and without the residue correction) converge to the same stable reduced model obtained in the previous case. While the regular takes 6 steps to converge, with residue correction converges after only 5 steps and avoids the intermediate unstable models thus further illustrating the effectiveness of the residue correction algorithm Evolution of the Relative H error Convergence of the interpolation points k: Iteration index Fig. 3. Evolution of B. International Space Station R Module For the previous model, even though the convergence behavior was different, with and without the residue correction have converged to the same reduced model. In this example, we illustrate that the residue correction step not only can improve convergence but also can lead to a smaller error. To illustrate this, we use one of the benchmark models for model reduction, namely the International Space Station R Module with m 3 inputs and p 3 outputs []. As in the previous example, the original model is a rational function itself having a degree n 70 with a corresponding transfer function H(s) C(sI A) B with A R 70 70, B R 70 3 and C R For r 0,, 4,..., 30, we construct a degree-r rational approximant using with and without residue correction using the same initalization. The top plot in Figure 4 shows the relative error vs r for both cases. As the plot illustrates even though for some r values both methods converge to the same reduced model, in most of the cases with the residue correction converges to a smaller error value. This is better illustrated in the lower plot of Figure 4 where we depict the ratio of the error from to the error from with the residue correction. Except for the r, r 6 and r 6 cases where both methods yielded the same error, the with the residue correction has outperformed. For the r 0 case, the Only even r values are considered since the model originates from a large-scale second-order model H(s) C (s M + sg + K) B.

8 gain is almost double with an error ratio of 0.543, an almost 50% improvement. 0 0 Evolution of the Relative H error vs r H error ratios vs r r: Reduced order Fig. 4. Relative error as r varies IV. CONCLUSION By incorporating the Loewner-matrix approach in, we have developed an optimal approximation method, namely TF-, that only uses transfer function evaluations without access to any particular realization; thus extending optimal model reduction to irrational, infinitedimensional dynamical systems such as systems with delays. Moreover, we have introduced a residue-correction step in that adjusts the vector residues at the end of each cycle of TF- so as to minimize the error for the current pole selection. Several numerical examples have been used to illustrate the effectiveness of the proposed methods. REFERENCES [] A.C. Antoulas. Approximation of large-scale dynamical systems. Society for Industrial Mathematics, 005. [] A.C. Antoulas, C.A. Beattie, and S. Gugercin. 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