Generation of Linear Models using Simulation Results

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1 4. IMACS-Symosium MATHMOD, Wien, , Generation of Linear Models using Simulation Results Georg Otte, Sven Reitz, Joachim Haase Fraunhofer Institute for Integrated Circuits, Branch Lab Design Automation Dresden Zeunerstraße 38, D Dresden, Germany Phone: +49 (0) , Fax +49 (0) Abstract: A two ste aroach for the generation of linear models using imulse resonse values comuted by simulation will be described. In the first ste a model of a "middle" order is determined using a realization based method. In the second ste the order of this model will be reduced using nown model order reduction methods. The objective of this aer is to show, that this ind of modeling can be realized by a combination of well nown methods and leads to accurate models of very low order. 1. Introduction In many alications very fast models of as low order as ossible are required. In the last time very owerful methods have been nown which are used for the order reduction of high dimensional models [1]. Basing on the nowledge of the matrices Σ= ( A, BCD,, ) of the high dimensional model these methods determine a suitable subsace and roject transient rocesses of the system into this subsace. On the owerful methods the subsace is determined using balanced truncation or as a Krylov-subsace, resectively. If the matrices Σ= ( A, BCD,, ) are not available but only the oututs of the system, e.g. comuted via simulation, these methods cannot be used. The aim of this aer is the generation of linear time discrete models from simulation results. Given the inut and outut values of a rocess the determination of a model is called an identification tas. But the existing identification methods do not use all advantages of simulated data. For rocess identification there are established solutions, nown as for examle the "rediction error method PEM" or the "instrumental variable method IVM", resectively. These identification methods are develoed for the determination of models from exerimental, i.e. disturbed measurements. They are secialized for the handling of disturbances in the rocess outut which occur in all real measurements. But if there is a high number of inuts and oututs or a high systems order their numerical roerties are very oor []. In the case of nearly undisturbed data lie simulation results these methods are without advantage. Recently develoed subsace-methods [], [3], [4] have better numerical roerties. Otimization stes are not needed. The multi-inut multi-outut case can be handled in the same way as the single-inut single-outut one. Initial values are not needed. An imortant advantage in ractice is the simle determination of the model structure by only one arameter, the order of the model [6]. If it is ossible to determine the ste-resonse or imulse-resonse directly, simlified (basic-) versions of these methods can be used. These are much more robust and their alication area can be extended to large systems. Our aroach to determine time discrete state-sace-models directly from imulse resonse values follows this idea. Methods used here are closely related to the realization theory []. Partial realizations that are realizations of lower order aroximate the given rocess data. The aroximation is excellent at the beginning of a transient rocess, but it is not so good at the end. Often the artial realization has to be of relative high ("middle") order to obtain models with small static errors. Therefore it will be suggested in such cases to reduce the order of the analytic "middle" order model in a second ste with nown standard methods lie modal or balanced truncation, with or without singular erturbation, resectively [5]. This aer is organized lie this: In section the basic relations between system matrices, observability, controllability, Hanel matrices and the imulse resonse are resented. The construction of a suitable Hanel matrix from given imulse resonse values and the determination of the model from these Hanel matrix will be discussed. In section 3 methods for model reduction as second ste of modeling are mentioned. Section 4 contains examles and section 5 concludes the aer. 436

2 . First ste: Realization based determination of a model Suose Σ= ( A, BCD,, ) is a linear time discrete system x = 1 Ax + + Bu y = Cx + Du Here x is an N-dimensional state vector, u is a m-dimensional inut vector and y is a q-dimensional outut vector. The matrix A is N N, B is N m, C is q N and D is q m. To this system belong an observability matrix O, a controllability matrix R and a Hanel matrix H. 1 C CB CAB CA B CA O 1 CAB CA B CA B =, R = B AB A B and R H = O = 1 CA 1 CA CA B CA B If is full controllable and observable and if N, all these matrices are of the same ran N. 1 The elements of the Hanel matrix are m matrices g = CA B, 1, g1 g g3 g g3 g 4 H = g3 g4 g5 which are equal to the imulse resonse values of for 1. If the g are nown, e.g. as a simulation result, matrices H can be constructed without exlicit nowledge of the matrices A,B,C,D and the Order N. Given the g the first tas is to assemble a Hanel matrix H having a sufficient high numeric ran N n. The second tas is to determine the matrices Σ= ( A, BCD,, ), n N of an aroximate system. n.1. Construction of a Hanel matrix and choosing the model order If the system matrices and the system order are nown, the Hanel matrix of correct order N can be constructed easily. Lie in all identification rocesses, these information is not available here. In our case the matrix H is to assemble by a sufficient number of bloc-columns and bloc-rows of the g by numeric exeriments. It will be assumed that there exists a large number K of instants of the imulse resonse{ g, = 1, K}. Thus the numeric ran of H is not limited by the columns number but it can be limited by the rows number. To find a sufficient number of rows, these can be increased ste by ste until the ran does not more grow or it grows only slowly, Fig 1 numeric ran of Hanel matrices of different rows (examle) Fig singular values of Hanel matrices of a system with a high number of oututs (examle) caused by small numeric disturbances. In this way a Hanel matrix of sufficient numeric ran can be established. Fig. 1 shows the ran values versus row bloc numbers of an examle. In such cases, the order of the model looed for can be chosen as high as the ran. 437

3 In other alications, if there exits a very high number of oututs, may be, many thousands, one bloc row can be sufficient. The numeric ran of the Hanel matrix can be as high as the outut number. Here the model order cannot be chosen as high as the ran. Fig. shows the singular values of such an Hanel matrix of an examle with aroximately 100 oututs. As can be seen, the use of more bloc rows does not really change the relations. The large number of singular values which are very small and do not decrease further on is caused by numeric disturbances. In such cases the model order should be chosen near the bottom of the left art of the grah, before the bli. In other cases, if the ran of the Hanel matrix in the range of interest increases ermanently but slowly and the singular values decrease regularly slow, too, the sufficient ran and the system order are to determine by numerical exeriments. The choice of a relative high model order is not critical. Suerfluous states will be removed in the next ste... Determination of the model from Hanel matrix From H it is ossible to comute two matrices Oˆ and R ˆ with any full ran factorization, so that H = OR ˆ ˆ. Different factorizations lead to different realizations of the same system. A well suitable factorization method is the singular value decomosition (svd) which calculates matrices U,S,V with H = USV. Furthermore, the matrices Oˆ and Rˆ calculated by svd are well suitable to carry out an aroximation of the Hanel matrix if the numeric ran is too large. In S 0 V H = ( U Uneg ) USV U = + negsnegvneg 0 S. neg V neg the comonents indexed by "neg" are connected to small singular values si, n< i Nn. These comonents are negligible. This can be done, by setting si = 0, n< i Nn. From the aroximation H of ran n of the Hanel matrix H H =USV aroriate observability and controllability matrices can be found, e.g.: O = U S and R = SV. The aroximations O and R contain on the to the matrices B and C. In MATLAB notation B = R (1: n,1: m) (m: number of inuts) C = O (1: q,1: n) (q: number of oututs) Because of the shift roerty of O one gets O(1: end q,1: n) A = O( q + 1: end,1: n) A + = O(1: end q,1: n) O( q + 1: end,1: n) where ( ) + denotes the Moore-Penrose seudo inverse. These inverse is to build for a matrix of the low order n. Directly from given imulse resonse there follows D = g0. So the system Σ is determined. 3. Second ste: Order reduction Using the shortly described method, the demand of a small static error leads to models of relatively high ("middle") order n. In most alications these order can be reduced further on without significant reducing the accuracy of the model. The model matrices are nown now. This enables the alication of well established order reduction methods lie truncated modal or balanced realization, with or without singular erturbation, resectively. The main roerties of these methods are summarized in [5]. For a truncation of the model it is to transform in a suitable form. The state sace vector has to be divided in a x1 retained ν-vector x1 and a (n-ν)-vector x that can be discarded. x = x artitioned according these reordering A11 A1 B1 A = A1 A B = B By omitting the states x the following lower order system is obtained. C = [ C1 C].The matrices A, BC, have to be 438

4 ξ + 1 = A11ξ + Bu 1 η = C1ξ + Du These elimination of x can cause a static error. Therefore in the second way, the singular erturbation aroximation will be used, to correct the static behavior. Only the "dynamic" of the omitted states will be neglected. Setting x =, 1 x and eliminating +, x from the remaining equations yields (if ( E A) is nonsingular) 1 1 ξ + 1 = ( A11+ A1( E A) A1) ξ + ( B1+ ( E A) B) u 1 1 η = ( C1+ C( E A) A1) ξ + ( D+ C( E A) B) u The truncation of modal realizations is common in engineering ractice. In this case the eigenvalues of A have to be comuted and ordered by asects of ractice (e.g. ordered so that λ i, Re( λ i ) or Im( λ i ), resectively, is nondecreasing). For the balancing truncation a state transformation is needed, which arranges the (transformed) states according to their amount of energy transformation from inut to outut, i.e. according to their transfer significance. So, the states are omitted, which are most nonsignificant. It is very difficult to redict, which method in a secial alication gives the better result. This question is to answer by numerical exeriments. Thus, the order of the model which is determined in the first ste can be reduced in a second ste. Finally one gets a model of the order ν with v< n< N. 4. Examles By three examles it will be shown that the two-ste method yields very comact models and wors reliable. 4.1 An examle from MATLAB The first examle "gasf" was taen from the MATLAB Control System Toolbox, described in get_start.df,.6. The original system contains 4 oututs, 6 inuts, and 5 states. It is characterized by values with great differences (static factors from 10-5 to ) between amlifications of different inut-outut branches. The imulse resonse instants were obtained using the ste und diff functions. The start of the first ste is the construction of the Hanel matrix, the svd and the choice of order n=17. This setting is motivated by the grah of the singular values of the Hanel matrix, see the vertical blac bar in fig rel. Singulärwerte versch. Hanelmatrizen über der Aroximationsordnung 1 Bloczeile Bloczeilen 4 Bloczeilen 7 Bloczeilen Bloczeilen 0 Bloczeilen Ordnung Fig.3 Relative singular values of the Hanel matrices, constructed with a different number of bloc rows In the following only the ste resonses with the largest and with the smallest magnitude are considered. From the figures 4 and 5 you can see the good corresondence between the original and the both aroximations. Only at time ste =0 the second aroximation of order 1 doesnt match the green circle (the original). For the order reduction in the second ste modal truncation (MATLAB function modreal) was used. By choosing the order ν=1 already for the model 1 the corresondence of the ste resonses in the finish art will be very oor, comare fig

5 3 x 105 Teilsystem "MBs51" - ElemNr.: 1.5 Original reduz. Modell 1 Ordnung 17 reduz. Modell Ordnung 1 rel Fehlernorm: ? O ut m d i dimt? Fig. 4 Ste resonse from inut #1 to outut #1, ste resonse with maximal magnitude, the left diagram shows original und both aroximations at the start of ste resonse 0 x 10-5 Teilsystem "MBs56" - ElemNr.: 4-1 -? O ut m d i Original reduz. Modell 1 Ordnung 17 reduz. Modell Ordnung 1 rel Fehlernorm: dimt? Fig. 5 Ste resonse from inut #6 to outut #4, ste resonse with minimal magnitude, the left diagram shows original und both aroximations at the start of ste resonse x Teilsystem "MBs56" - ElemNr.: 4 Original reduziertes Modell rel. stat. Fehler: ? O ut m d i dimt? Fig. 6 Ste resonse from inut #6 to outut #4, original and aroximation of order 1 already in the first ste This examle demonstrates the robustness of the first modeling ste. It is caable to wor on systems with extreme different static branches. the advantage of the second ste for reduction the model order further on. 440

6 4. Temerature distribution of an analog Amlifier-IC Fig. 7 shows the locations of critical oints, where heat is roduced (blac arrows). The temerature distribution is of interest in 10 oints, mared in fig. 8 on the surface of the IC. The arrows show the locations of the temerature sensible inut stages (in our model the oututs) numbered by 36 and 41. inut 1 inut outut #36 outut 41 Fig. 7 Energy entry oints (ower transistors) Fig.8 10 outut-oints on the surface (magenta color) The ste resonses between the (normalized) two heat ower inuts of the ower stages and the 10 mared nodes were calculated by an FEM model, consisting of nodes. Fig.9 Relative singular values of the Hanel matrices, constructed with a different number of bloc rows. The bloc row consists of 10 rows. Outut No. #36 Outut No. #41 I n u t 1 I n u t Fig. 10 Temerature ste resonses of oututs #36 and #41 caused by temerature stes of the ower stages. 441

7 The order of the first model was chosen as n=1, comare fig. 9. For the order reduction in the second ste balanced truncation (MATLAB functions balreal and modred) was used. The order was reduced to ν=15. The ste resonses of the second model agree with the original and the first model aart from the first three / four time instants. In this examle already the first model is of comarative small order. Nevertheless these order can be reduced further on. The corresondence of the model and the original remains aart from the first time oints. 4.3 A micro system acceleration sensor Fig. 11 shows the scheme of a micro system acceleration sensor. The FEM model describing the outut signal as a function of the acceleration consists of 5365 nodes with a degree of freedom of 3. Therefore the system is of order Fig. 11 Scheme of a micro system acceleration sensor Fig. 1 Relative singular values of the Hanel matrices, constructed with a different number of bloc rows The singular values of the Hanel matrix in fig. 1 suggest an order n=8. In the second ste the order can be reduced to ν= without loss of accuracy. The ste resonse oututs of the original and of the two models are shown in fig. 13. Fig. 14 shows zoomed the starting sector of the ste resonse. Both of the models match the oints given. The order reduction in the second ste was carried out by modal truncation (MATLAB function modreal) Fig. 13 Ste resonses of original and models Fig. 14 Origin sector of the Ste resonses A difference between the models determined is visible in the Bode-lot, see fig. 15. As can be seen, they differ on very high frequencies at the end of the lot. 44

8 Fig. 15 Bode lot of the two accelerator models of order n=8 and order ν= 5. Conclusions The combination of well nown aroaches to calculate a artial realization via factorization of the Hanel matrix using svd and model order reduction by modal or balanced truncation yields a owerful and fast tool to generate time discrete state sace models from given imulse resonse values. The two-ste aroach yields in most alications significantly better results than the modeling of order ν already in the first ste. The method wors well on large systems too. The models are accurate and of very low order. 6. References 1. Antoulas, A.C. and Sorensen, D. C., Aroximation of large-scale dynamical systems: An overview Technical Reort, February 001, htt://www-ece.rice.edu/~aca/mtns00.df. Viberg, M., Subsace-based Methods for the Identification of Linear Time-invariant Systems Automatica, (1995) Vol. 31 No van Overschee, P. and De Moor, B., A Unifying Theorem for three Subsace System Identification Algorithms Automatica, (1995) Vol. 31 No Ljung, L., System Identification, Theory for the User, nd ed. Prentice Hall PTR, Uer Saddle River, NJ, Green, M. and Limebeer, D. J. N., Linear Robust Control Prentice Hall, Englewood Cliffs, New Jersey 0763, Varga, A., Subsace Identifiation htt:// Jan