Data-Driven Identification of Rayleigh-Damped Second-Order Systems
aa r X i v : . [ m a t h . O C ] O c t Data-Driven Identification of Rayleigh-DampedSecond-Order Systems
Igor Pontes Duff, Pawan Goyal, and Peter Benner
Abstract
In this paper, we present a data-driven approach to identify second-ordersystems, having internal Rayleigh damping. This means that the damping matrixis given as a linear combination of the mass and stiffness matrices. These systemstypically appear when performing various engineering studies, e.g., vibrational andstructural analysis. In an experimental setup, the frequency response of a systemcan be measured via various approaches, for instance, by measuring the vibrationsusing an accelerometer. As a consequence, given frequency samples, the identifi-cation of the underlying system relies on rational approximation. To that aim, wepropose an identification of the corresponding second-order system, extending theLoewner framework for this class of systems. The efficiency of the proposed methodis demonstrated by means of various numerical benchmarks.
In this paper, we discuss a data-driven identification framework for a class of second-order ( SO ) systems of the form: Σ SO : = (cid:26) M Ü x ( t ) + D Û x ( t ) + Kx ( t ) = Bu ( t ) , y ( t ) = Cx ( t ) , (1) Igor Pontes Duff, Pawan Goyal are with the Max Planck Institute for Dynamics ofComplex Technical Systems, Sandtorstraße 1, 39106 Magdeburg, Germany; (e-mail: \protect\T1\textbraceleftpontes,goyalp\protect\T1\[email protected] ).Peter Benner is with the Max Planck Institute for Dynamics of Complex Technical Systems,Sandtorstr. 1, 39106 Magdeburg, Germany, and also with the Technische Universität Chem-nitz, Faculty of Mathematics, Reichenhainer Straße 41, 09126 Chemnitz, Germany (email:[email protected]). 1 Igor Pontes Duff, Pawan Goyal, and Peter Benner where x ( t ) ∈ R n is the state vector, u ( t ) ∈ R m are the inputs, y ( t ) ∈ R p are the out-puts or measurements, and M , D , K ∈ R n × n are, respectively, the mass matrix, thedamping matrix and the stiffness matrix, B ∈ R n × m and C ∈ R p × n . For simplicity, weaddress the problem for single-input single-output (SISO) systems, i.e., m = p = SO systems (1) by Σ SO = ( M , D , K , B , C ) . Moreover,we assume a zero inhomogeneous condition, i.e., x ( ) = Û x ( ) =
0. Hence, by meansof the Laplace transform, the input-output behavior of the system Σ SO is associatedwith the transfer function as follows: H SO ( s ) = C (cid:16) s M + s D + K (cid:17) − B . (2)Furthermore, throughout the paper, we assume the proportional Rayleigh dampinghypothesis, i.e., the damping matrix D is given by a linear combination of the massand stiffness matrices: D = α M + β K , (3)for α, β ≥
0. This hypothesis is often considered in several engineering applica-tion, where the damper is numerically constructed in order to avoid non-dampenedoscillations, see [17] for more details.In the past twenty years, model order reduction of SO systems has been investigatedextensively; see for instance [18, 7, 21] for balancing-type methods, and [6, 4, 26, 3]for moment matching and H -optimality based methods. Recently, the authors in[22] provided an extensive comparison among common methods for SO modelorder reduction applied to a large-scale mechanical artificial fishtail model. In allof the above-mentioned works, the authors suppose that they have access to thematrices, defining the original systems and the reduced-order systems are constructedvia Petrov-Galerkin projections. Thus, the main goal is to find projection matrices V , W ∈ R n × r , leading to the SO reduced-order system ˆH SO ( s ) = ˆC (cid:16) s ˆM + s ˆD + ˆK (cid:17) − ˆB , (4)with ˆM = W T MV , ˆD = W T DV , ˆK = W T KV , ˆB = W T B and ˆC = CV .However, it is not necessary that the realization is given or is feasible to obtain;thus, we suppose that the original system realization may not be available. Instead, weassume to have access only to frequency domain data, e.g., arising from experimentsor numerical simulations. More precisely, we are interested in solving the followingproblem. ata-Driven Identification of Rayleigh-Damped Second-Order Systems 3 Problem 1 ( SO data-driven identification) Given interpolation data {( σ i , ω i )| σ i ∈ C and ω i ∈ C , i = , . . . , ρ } , (5)construct a SO realization Σ SO = ( M , D , K , B , C ) of appropriate dimensions,satisfying the proportional Rayleigh damping hypothesis, i. e., D = α M + β K , whose transfer function H SO ( s ) : = C ( s M + s D + K ) − B satisfies the interpo-lation conditions, i.e., H SO ( σ i ) = ω i , i = , . . . ρ. (6)Problem 1 corresponds to an identification problem which aims at determininga SO realization that not only interpolates at given measurements, but also satisfiesthe Rayleigh damping hypothesis. A similar problem for time-delay systems wasstudied in [20] and [23]. Furthermore, we would like to mention that a data-drivenapproach for structured non-parametric systems has been studied in [24]. However,the construction of the structured reduced-order system is not a straightforward task.The purpose of this paper is thus to extend the application domain of the Loewnerframework established in [15, 16] to SO systems. With this aim, a new SO Loewnerframework is developed, yielding a Rayleigh damped SO system of the form (2) thatinterpolates at given frequency measurements.The rest of the paper is organized as follows. Section 2 recalls some preliminaryresults on the rational interpolation Loewner framework proposed in [16]. Section 3presents an extension of these results to the class of Rayleigh damped SO systems.The section is divided into two parts. The first one assumes the knowledge ofthe Rayleigh damping parameters, α and β , and derives the Loewner matrices for SO systems. The second part presents a heuristic procedure, originally proposed in[23] in the context of time-delay systems, enabling us to estimate the parameters α and β . Finally, Section 4 illustrates the proposed framework by numerical examplesand Section 5 concludes the paper. In this section, we briefly recall the Loewner framework [16]. A first-order ( FO )system Σ FO = ( E , A , B , C ) is a dynamical system of the form: Σ FO : = (cid:26) E Û x ( t ) = Ax ( t ) + Bu ( t ) , x ( ) = , y ( t ) = Cx ( t ) , (7) Igor Pontes Duff, Pawan Goyal, and Peter Benner with E , A ∈ R n × n , B ∈ R n × m and C ∈ R p × n , and the leading dimension n is theorder of the system. For clarity of exposition, we focus for now on the single-inputsingle-output (SISO) case, i.e., when m = p =
1. The system (7) is associated withthe transfer function given by H FO ( s ) = C ( s E − A ) − B . (8)There exist several MOR techniques for first-order systems such as explicit mo-ment matching [27, 25], implicit moment matching using Krylov subspaces [10, 13],Sylvester equations based method [12], extensions for MIMO systems [11]. We referthe reader to the books [2, 5] for more details. However, our goal lies in the iden-tification of linear systems using only the frequency data. Hence, the identificationproblem, in its SISO form, is stated as follows. Problem 2 (First-order data-driven model reduction)
Given interpolationdata {( σ i , ω i )| σ i ∈ C and ω i ∈ C , i = , . . . , ρ } (9)construct a minimal-order realization Σ = ( E , A , B , C ) of appropriate dimen-sions, whose transfer function H FO ( s ) = C ( s E − A ) − B satisfies the interpola-tion conditions H FO ( σ i ) = ω i , i = , . . . ρ. (10)A wide range of methods has been developed to solve Problem 2, e.g., vectorfitting [14], the AAA algorithm [19] and the Loewner framework [16]. In this paper,we focus on the latter approach and, in what follows, we recall some of the resultscontained therein. Firstly, we assume that the number of interpolation data is even,i.e., ρ = ℓ , and as a result, the data can be partitioned in two disjoint sets as follows:right interpolation set P r : {( λ i , w i )| λ i ∈ C and w i , i = , . . . , ℓ } , and (11a)left interpolation set P l : {( µ j , v j )| µ j ∈ C and v j ∈ C , j = , . . . , ℓ } . (11b)Using this partition, we associate the following Loewner matrices. Definition 1 (Loewner matrices [16])
Given the right P r and left P l interpo-lation sets, we associate them with the Loewner matrix L and shifted Loewnermatrix L σ given by L = © « v − w µ − λ · · · v − w ℓ µ − λ ℓ ... . . . ... v ℓ − w µ ℓ − λ · · · v ℓ − w ℓ µ ℓ − λ ℓ ª®®®¬ , L σ = © « µ v − λ w µ − λ · · · µ v − λ ℓ w ℓ µ − λ ℓ ... . . . ... µ ℓ v ℓ − λ w µ ℓ − λ · · · µ ℓ v ℓ − λ ℓ w ℓ µ ℓ − λ ℓ ª®®®¬ . (12) ata-Driven Identification of Rayleigh-Damped Second-Order Systems 5 Remark 1
The Loewner matrix L was introduced in [1]. As shown therein, its useful-ness derives from the fact that its rank is equal to the order of the minimal realization H FO satisfying the interpolation conditions in (10). Hence, it reveals the complexityof the reduced-order model solving Problem 2.Next, let us introduce the following matrices associated with the interpolationproblem as follows: (cid:26) Λ = diag ( λ , . . . , λ ℓ ) ∈ C ℓ × ℓ ˆH ( Λ ) = (cid:2) w . . . w ℓ (cid:3) T ∈ C ℓ × and (cid:26) M = diag ( µ , . . . , µ ℓ ) ∈ C ℓ × ℓ ˆH ( M ) = (cid:2) v . . . v ℓ (cid:3) T ∈ C ℓ × (13)Also, let ∈ R ℓ × be the column vector with all entries equal to one. Hence, theLoewner matrices satisfy the following Sylvester equations M L − L Λ = ˆH ( M ) T − ˆH ( Λ ) T , and (14a) M L σ − L σ Λ = M ˆH ( M ) T − ˆH ( Λ ) Λ . (14b)An elegant solution for Problem 2 based on the Loewner pair ( L , L σ ) was proposedin [16]. This is summarized in the following theorem. Theorem 1 (Loewner framework [16])
Let L and L σ be the Loewner matri-ces associated with the partition in (13) . If ( L σ , L ) is a regular pencil with no µ i or λ j being an eigenvalue, then the matrices ˆE = − L , ˆA = − L σ , ˆB = V , ˆC = W , provides a realization ˆ Σ FO = ( ˆE , ˆA , ˆB , ˆC ) for a minimal order interpolant ofProblem 2, i.e., the transfer function ˆH FO ( s ) = W ( s L σ − L ) − V satisfies the interpolation conditions in (10) . Theorem 1 allows to obtain a FO system ˆH = ( ˆE , ˆA , ˆB , ˆC ) whose transfer functioninterpolates right and left data as stated in Problem 2. However, when more datathan necessary are provided, then the hypothesis of Theorem 1 may not be satisfied.Hence, a singular-value decomposition (SVD) based procedure has been proposedin [16] to find an FO system interpolating the frequency data.Next, recall that a SO system Σ SO = ( M , D , K , B , C ) can be written as a first-orderrealization, for instance, as follows: H SO _ FO ( s ) = C( s E − A) − B , where Igor Pontes Duff, Pawan Goyal, and Peter Benner E = (cid:20) I 00 M (cid:21) , A = (cid:20) − K − D (cid:21) , B = (cid:20) (cid:21) and C = (cid:2) C 0 (cid:3) . As a consequence, the classical Loewner framework presented in Section 2 can beemployed to find a first-order realization. However, the intrinsic SO structure will notbe preserved in the identified model. But the classical Loewner framework yields aninformation about the order of a SO realization fitting the data, which is outlined inthe following remark. Remark 2 (
Order of SO model) Let us suppose that the frequency data in Problem2 and let L be a Loewner matrix given in (12) constructed with this data. Then, theorder of the SO system fitting the data equals rank ( L ) .In the following section, we discuss an extension of the Loewner framework forthe class of Rayleigh damped SO systems. This section contains our main contribution, which presents an extension of theLoewner framework to the class of SO Rayleigh damped systems (1). Here, we alsoassume that the number of interpolation data is even, i.e., ρ = ℓ , and the datais partitioned into two disjoint sets as in (11a) and (11b). Moreover, the data isorganized into the matrices Λ , ˆH ( Λ ) , M , ˆH ( M ) as in (13). This section is dividedinto two parts. In the first one, we assume to have a priori knowledge of the Rayleighdamping parameters α and β and we derive the equivalents of the Loewner matrices(12) and the Theorem 1 to the class of SO Rayleigh damped systems. The secondpart is dedicated to proposing a heuristic procedure to estimate the parameters α and β using the frequency data available. In what follows, we assume that Problem 1 has a minimal order r solution H ⋆ SO , givenby H ⋆ SO ( s ) = C ⋆ (cid:16) s M ⋆ + s D ⋆ + K ⋆ (cid:17) − B ⋆ , (15)with D ⋆ = α M ⋆ + β K ⋆ . Here, we also assume that the coefficients α and β fromthe Rayleigh-Damped hypothesis are known. Then, later in this section, we willshow how to construct a realization equivalent to H ⋆ SO ( s ) that only depends on thefrequency data. To that aim, let us first recall a result from [4] enabling projection-based structured preserving model reduction. ata-Driven Identification of Rayleigh-Damped Second-Order Systems 7 Theorem 2 (Structure preserving SO model reduction [4]) Consider the SO trans-fer function H SO ( s ) as given in (2) . For given interpolation points λ i and µ i , i ∈ { , . . . , ℓ } , let the projection matrices V and W be as follows: V = h (cid:0) λ M + λ D + K (cid:1) − B , . . . , (cid:0) λ ℓ M + λ ℓ D + K (cid:1) − B i (16a) W = h (cid:0) µ M + µ D + K (cid:1) − T C T , . . . , (cid:0) µ ℓ M + µ ℓ D + K (cid:1) − T C T i (16b) Hence, the reduced-order model ˆ H SO ( s ) constructed by Petrov-Galerkin projectionas in (4) satisfies the interpolation conditions H SO ( λ i ) = ˆH SO ( λ i ) and H SO ( µ i ) = ˆH SO ( µ i ) , for i = , . . . , ℓ . The above theorem allows us to construct a SO reduced-order model by interpolation.Let us apply this theorem to the SO system H ⋆ SO ( s ) (15). For this, we will construct thematrix V using the interpolation points in Λ , and the matrix W using the interpolationpoints in M . As a consequence, V and W are, respectively, the solutions of thefollowing matrix equations M ⋆ V Λ + D ⋆ V Λ + K ⋆ V = B ⋆ T , and (17a) M W T M ⋆ + M W T D ⋆ + W T M = C ⋆ , (17b)Multiplying the equations on the left (17a) and (17b) on the left by W T and V T ,respectively, one obtains W T M ⋆ V Λ + W T D ⋆ V Λ + W T K ⋆ V = W T B ⋆ T , M W T M ⋆ V + WD ⋆ V M + WK ⋆ V = C ⋆ V T . If we set ˆM = W T M ⋆ V , ˆD = W T D ⋆ V , ˆK = W T K ⋆ V , (18a) ˆB = W T B ⋆ = ˆH ( M ) , and ˆC = C ⋆ V = ˆH ( Λ ) T , (18b)then the SO system ˆH SO = ( ˆM , ˆD , ˆK , ˆB , ˆC ) is the reduced-order model obtained byTheorem 2, satisfying the interpolation conditions from Problem 1. Hence, we canrewrite the above equations as follows: ˆM Λ + ˆD Λ + ˆK = ˆH ( M ) T , M ˆM + M ˆD + ˆK = ˆH ( Λ ) T . Moreover, if we apply the Raylegh-Damped hypothesis, i.e., ˆD = α ˆM + β ˆK , weobtain Igor Pontes Duff, Pawan Goyal, and Peter Benner ˆM (cid:16) Λ + α Λ (cid:17) + ˆK ( β Λ + I ) = ˆH ( M ) T , (20a) (cid:16) M + α M (cid:17) ˆM + ( β M + I ) ˆK = ˆH ( Λ ) T . (20b)Notice that the above equations can be solved for ˆM and ˆK . However, in order tohave an analytic expression for the matrices of the reduced-order system in a similarway as for the Loewner matrices (12), we need to introduce the following change ofvariables: L SO : = −( I + β M ) ˆM ( I + β Λ ) , L SO σ : = ( I + β M ) ˆK ( I + β Λ ) , (21a) B SO : = ( I + β M ) ˆH ( M ) , and C SO : = ˆH ( Λ ) T ( I + β Λ ) . (21b)Notice that the two realizations ˆ Σ SO = ( ˆM , α ˆM + β ˆK , ˆK , ˆB , ˆC ) and ˆ Σ Loew SO = (− L SO , − α L SO + β L SO σ , L SO σ , B SO , C SO ) are equivalent, i.e., they represent the same transfer function. Hence, the realization ˆH Loew SO also satisfies the interpolation conditions from Problem 1. Additionally, by asimple computation, we obtain that the matrices L SO and L SO σ satisfy the followingequations L SO F ( Λ ) + L SO σ = −D( M ) ˆH ( M ) T , F ( M ) L SO + L SO σ = − ˆH ( Λ ) T D( Λ ) , where , for a given matrix Ω , F ( Ω ) : = ( I + β Ω ) − ( Ω + α Ω ) and D( Ω ) : = ( I + β Ω ) .As a consequence, L SO F ( Λ ) − F ( M ) L SO = ˆH ( Λ ) T D( Λ ) − D( M ) ˆH ( M ) T , (22a) L SO σ F ( Λ ) − F ( M ) L SO σ = ˆH ( Λ ) T N( Λ ) − N( M ) ˆH ( M ) T , (22b)where, for a given matrix Ω , N( Ω ) : = ( Ω + α Ω ) . Notice that the Sylvester equa-tions (22) are equivalent to (14a) for the case of SO systems. Hence, using thoseequation, one can derive analytic expressions of L SO and L SO σ . Definition 2 ( SO Loewner matrices)
Let us suppose α and β are known andlet d ( s ) : = + s β, n ( s ) : = s + α s , and f ( s ) : = n ( s ) d ( s ) , be scalar functions. Then the SO Loewner matrices, namely, the SO Loewnermatrix L SO and the shifted Loewner matrix L SO σ are given by ata-Driven Identification of Rayleigh-Damped Second-Order Systems 9 L SO = © « d ( µ ) v − d ( λ ) w f ( µ )− f ( λ ) · · · d ( µ ) v − d ( λ ℓ ) w ℓ f ( µ )− f ( λ ℓ ) ... . . . ... d ( µ ℓ ) v ℓ − d ( λ ) w f ( µ ℓ )− f ( λ ) · · · d ( µ ℓ ) v ℓ − d ( λ ℓ ) w ℓ f ( µ ℓ )− f ( λ ℓ ) ª®®®¬ , (23) L SO σ = © « n ( µ ) v − n ( λ ) w f ( µ )− f ( λ ) · · · n ( µ ) v − n ( λ ℓ ) w ℓ f ( µ )− f ( λ ℓ ) ... . . . ... n ( µ ℓ ) v ℓ − n ( λ ) w f ( µ ℓ )− f ( λ ) · · · n ( µ ℓ ) v ℓ − n ( λ ℓ ) w ℓ f ( µ ℓ )− f ( λ ℓ ) ª®®®¬ . (24)Moreover, by construction L SO = −( I + β M ) ˆM ( I + β Λ ) = −( I + β M ) W T M ⋆ V ( I + β Λ ) . Thus, the following remark holds.
Remark 3
If we have sufficient interpolation data, then rank ( V ) = rank ( W ) = r . As a consequence, the rank of the SO Loewner matrix L SO gives us the order of theRayleigh damped SO minimal realization interpolating the points, sincerank (cid:0) L SO (cid:1) = rank (cid:16) W T M ⋆ V (cid:17) = rank (cid:0) M ⋆ (cid:1) = order of the minimal SO interpolant . We are now able to state the analogue result to Theorem 1 for Rayleigh-damped SO systems. Theorem 3 ( SO data-driven identification) Assume that µ i , λ j for all i , j = , . . . , ℓ . Additionally, suppose that ( s + α s ) L SO + ( β s + ) L SO σ is invertible forall s = { λ , . . . , λ ℓ } ∪ { µ , . . . , µ ℓ } . Then ˆM = − L SO , ˆK = L SO σ , ˆB = ( I + β Λ ) − V SO ˆC = W SO ( I + β M ) , and ˆK = α ˆM + β ˆK satisfy the interpolation conditions from Problem 1. We now consider the case where more data than necessary are provided, whichis realistic for applications. In this case, the assumptions of the above theorem arenot satisfied; thus, one needs to project onto the column span and the row span of alinear combination of the two Loewner matrices. More precisely, let the followingassumption be satisfied:rank (cid:0) (cid:2) L SO L SO σ (cid:3) (cid:1) = rank (cid:18) (cid:20) L SO L SO σ (cid:21) (cid:19) = r (25)Then, we consider the compact SVDs (cid:2) L SO L SO σ (cid:3) = Y ρ Σ l ˜ V T and (cid:20) L SO L SO σ (cid:21) = ˜ W Σ r X T ρ . (26)Using the projection matrices V ρ and W ρ , we are able to remove the redundancy inthe data by means of the following result. Theorem 4
The SO realization ˆ Σ SO = ( ˆM , ˆD , ˆK , ˆB , ˆC ) of a minimal interpolant ofProblem 1 is given as: ˆM = − Y T ρ L SO X ρ , ˆK = − Y T ρ L SO σ X ρ , ˆD = α ˆM + β ˆK , (27a) ˆB = Y T ρ ˆB Loew , and ˆC = ˆC Loew X ρ . (27b) Depending on whether r in (25) is the exact or approximate rank, we obtain eitheran interpolant or an approximate interpolant of the data, respectively. In the previous section, we have shown how to construct a SO realization for giventransfer function measurements and a priori knowledge of the parameters α and β from the Rayleigh-damped hypothesis. However, there are several cases, where exactvalues of α and β are not known but we rather can have a hint of the range for theparameters, i.e., α ∈ R α and β ∈ R β . Therefore, as done for delay systems in [23],we also propose a heuristic optimization approach to obtain the parameters α and β for SO systems, satisfying the Rayleigh-damped hypothesis. For this purpose, wesplit the data training D training and test set D test , e.g., in the ratio 80:20. Hence, weideally aim at solving the optimization as follows:min α ∈R α , β ∈R β J ( α, β ) (28)where J ( α, β ) : = Õ ( σ k , v k )∈D test (cid:13)(cid:13) ˆH SO ( σ k α, β ) − v k (cid:13)(cid:13) + Õ ( µ k , w k )∈D test (cid:13)(cid:13) ˆH SO ( µ k α, β ) − w k (cid:13)(cid:13) , where ˆH SO is constructed using only the training data. However, the optimizationproblem (28) is non-convex, and solving it is a challenging task. Therefore, weseek to solve a relaxed problem. For this purpose, in the paper, we make a 2-Dgrid for the parameters α and β in given intervals. Then, we seek to determinethe parameters on the grid where the function J ( α, β ) is minimized. Nonetheless,solving the optimization problem (28) needs future investigation and so we leave itas a possible future research problem. ata-Driven Identification of Rayleigh-Damped Second-Order Systems 11 In this section, we illustrate the efficiency of the proposed methods via severalnumerical examples, arising in various applications. All the simulations are done ona CPU 2.6 GHz Intel ® Core™i5, 8 GB 1600 MHz DDR3, MATLAB ® At first, we discuss an artificial example to illustrate the proposed method. Let usconsider a SO system of order n = Σ SO = ( M , D , K , B , C ) whose matrices are givenby: M = (cid:20) (cid:21) , K = (cid:20) (cid:21) , D = α M + β M , and B T = C = (cid:2) (cid:3) , with α = .
01 and β = .
02. We collect 20 samples ( σ j , ˆH SO ( σ j )) , for σ j ∈ ι [ − , ] logarithmically spaced. Then, we construct the FO and SO Loewnermatrices in (12) and (23), receptively.In Figure 1, we plot the decay of the singular values of the L and L SO matrices.It can be observed that rank ( L ) = (cid:0) L SO (cid:1) =
2, as expected. Indeed, thedemo system has a minimal SO realization of order 2 and a minimal FO realizationof order 4. By applying the SVD procedure, we construct two reduced-order modelsof order 2, one for FO and the other for SO . We compare the transfer functions ofthe original and reduced-order systems, and the results are plotted in Figure 2. Thefigure shows that the error between the original and SO reduced-order system is ofthe level of machine precision, which means that the SO approach has recovered anequivalent realization of the original model. Additionally, the FO reduced system oforder 2 was not able to mimic the same behavior of the original system, showingthat a larger order is required in this case. Let us now consider the building model from the SLICOT library [8]. It describesthe displacement of a multi-storey building, for example, during an earthquake. It isa FO system of order r =
48, whose dynamics comes from a mechanical system. TheRayleigh damping coefficients here are α ≈ . β ≈ . H ( i ω ) , with ω ∈ [ , ] . Then, webuild the FO and SO Loewner matrices in (12) and (23), receptively. Additionally,using the heuristic procedure in Subsection 3.2, we constructed the reduced model FO Loewner SO Loewner2 4 6 8 1010 − − − R e l a ti v e s i ngu l a r v a l u e s Fig. 1
Demo example: Decay of the singular values for the FO and SO Loewner matrices.10 − Freq(s) M a gn it ud e − − − Freq(s)
Fig. 2
Demo example: The figure on the left shows the Bode plot of the original system and the FO and SO reduced-order models. The figure on the right shows the Bode plot of the error between theoriginal and reduced-order systems. assuming we do not know a priori the parameters α and β . After this procedure, weobtain α ∗ = .
495 and β ∗ = . FO Loewner matrix, the SO Loewner matrix for the original parameters α and β , and the SO Loewner matrix forthe estimated parameters α and β . The decay of the singular values for the SO Loewnermatrix with original parameters is faster than for the FO Loewner matrix. However,for the SO Loewner matrix with estimated parameters, the decay of singular valuesstarts fast and then becomes slower. This shows that if the parameters α and β arenot well identified, a higher reduced-order will be needed to interpolate the data. Byapplying the SVD procedure, we construct three reduced-order models of order 16.We compare the transfer functions of the original and reduced-order systems, and theresults are plotted in Figure 4. This figure shows that for the SO Loewner approach(original parameters or with estimated parameters) outperform the classical Loewnerframework. ata-Driven Identification of Rayleigh-Damped Second-Order Systems 13Original system FO Loewner SO Loewner SO Loewner opt. par.0 20 40 60 8010 − − − R e l a ti v e s i ngu l a r v a l u e s Fig. 3
Build example: Decay of the singular values for the FO Loewner matrix and for SO Loewnermatrices.10 − − Freq(s) M a gn it ud e − − − Freq(s)
Fig. 4
Build example: The figure on the left shows the Bode plot of the original system and the FO and SO reduced-order models. The figure on the right shows the Bode plot of the error between theoriginal and reduced-order systems. As the last example, we consider the artificial fishtail model presented in [22]. Thismodel comes from a finite-element discretization of the continuous mechanics modelof an artificial fishtail. After discretization, the finite-dimensional system has a SO realization of order 779 , α = . · − , β = · − . It is a MIMO system, but for the numericalapplication, here we consider only the first transfer function, i.e., from u to y .For this example, we collect 200 samples H ( i ω ) , with ω ∈ [ , ] . Then, webuild FO and SO Loewner matrices in (12) and (23), receptively. Additionally, wealso compute the reduced model using the heuristic procedure in Subsection 3.2, forwhich we obtain the estimated parameters α ∗ ≈ . · − and β ∗ ≈ · − .In Figure 3, we plot the decay of the singular values of the FO Loewner matrix,the SO Loewner matrix for the original parameters α and β , and the SO Loewnermatrix for the estimated parameters α ∗ and β ∗ . By applying the SVD procedure, weconstruct three reduced-order models of order 8. We compare the transfer functionsof the original and reduced-order systems, and the results are plotted in Figure 4. This FO Loewner SO Loewner SO Loewner opt. par.0 20 40 60 8010 − − R e l a ti v e s i ngu l a r v a l u e s Fig. 5
Fishtail example: Decay of the singular values for the FO Loewner matrix and for SO Loewnermatrices. 10 − Freq(s) M a gn it ud e − − − − Freq(s)
Fig. 6
Fishtail example: The figure on the left shows the Bode plot of the original system and the FO and SO reduced-order models. The figure on the right shows the Bode plot of the error betweenthe original and reduced-order systems. figure shows that the SO Loewner approach with original parameters and SO Loewnerwith estimated parameters outperform the classical Loewner framework.
In this paper, we have studied the problem of the identification of Rayleigh-dampedsecond-order systems from frequency data. To that aim, we propose modified SO Loewner matrices which are the key tools to construct a realization interpo-lating the given data. Additionally, in the case of redundant data, an SVD-basedscheme is presented to construct reduced-order models. Moreover, a heuristic opti-mization problem is sketched to estimate the damping parameters. Finally, we haveillustrated the efficiency of the proposed approach in some numerical examples, andwe compared the results with the classical Loewner framework. ata-Driven Identification of Rayleigh-Damped Second-Order Systems 15
Acknowledgement
This work was supported by
Deutsche Forschungsgemeinschaft (DFG) , Collabora-tive Research Center CRC 96 "Thermo-energetic Design of Machine Tools".
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