This paper addresses issues related to the order reduction of systems with multiple input/output ports. The order reduction is divided up into two steps. The first step is the standard order reduction method based on the multipoint approximation of system matrices by applying Krylov subspace. The second step is based on the rejection of the weak part of a system. To recognise the weak system part, Lyapunov equations are used. Thus, this paper introduces efficient solutions of the Lyapunov equations for port to port subsystems.BMBF/01 M 3169 

John, W.

Ludwig, Stefan

Mathis, Wolfgang

Radić-Weissenfeld, Ljubica J.

Advances in Radio Science

Institutionelles Repositorium der Leibniz Universität Hannover

Adv. Radio Sci., 6, 129–132, 2008www.adv-radio-sci.net/6/129/2008/© Author(s) 2008. This work is distributed underthe Creative Commons Attribution 3.0 License.Advances inRadio ScienceModel order reduction of linear time invariant systemsLj. Radic´-Weissenfeld1,2, S. Ludwig1,2, W. Mathis1, and W. John21Leibniz University of Hannover, Institute of Electromagnetic Theory, Hannover, Germany2Advanced System Engineering (ASE), Fraunhofer Institute for Reliability and Microintegration (IZM), Paderborn, GermanyAbstract. This paper addresses issues related to the orderreduction of systems with multiple input/output ports. Theorder reduction is divided up into two steps. The first stepis the standard order reduction method based on the multi-point approximation of system matrices by applying Krylovsubspace. The second step is based on the rejection of theweak part of a system. To recognise the weak system part,Lyapunov equations are used. Thus, this paper introduces ef-ficient solutions of the Lyapunov equations for port to portsubsystems.1 IntroductionThe order reduction of a linear time invariant system is ap-plied in almost all fields of electrical engineering. The useof order reduced models for test simulations of complex sys-tems is a lot easier than utilising full order models. This isdue to the fact that the lower order transfer function can beanalysed more easily. Therefore, order reduction algorithmsare standard techniques in the integrated circuits communityfor analysis, approximation and simulation of models arisingfrom interconnect and electromagnetic structure analysis.The combination of order reduction algorithms, derivedfrom a Krylov subspace projection, and singular value basedmethods are successfully used for the order reduction of pas-sive linear electrical networks. The Krylov subspace projec-tion method is commonly used for models with a low numberof input/output ports. The singular value based methods aresuccessfully used for models of a modest order, i.e. a fewhundred. The singular value based method is used for mod-els which matrix representation allows the inversion opera-tion. The classical approach is to use the Krylov subspaceCorrespondence to: Lj. Radic´-Weissenfeld(radic@tet.uni-hannover.de)projection to more easily obtain the balanced model for sin-gular value based methods of an order reduction.In this work, these two order reduction methods are com-bined in two steps. The first step of order reduction algorithmis based on the use of the Krylov subspace projection forthe selection of dominant moments, which are found throughmoment matching. In the second step, the order reducedmodel is further reduced by rejecting its weak states. A stan-dard matrix inversion is avoided in the second step of theorder reduction. This combination of order reduction meth-ods keeps low magnitude and phase errors of order reducedsystems.2 Standard two-step order reductionA two-step order reduction method usually consist of theKrylov subspace projection method (Boley, 1994) and thebalanced truncation method (Antoulas and Sorensen, 2001).The Krylov subspace projection produces a system of an or-der which is proportional to the number of input/output ports.Moreover, the order of the reduced system depends on thenumber of expansion points and matched moments. There-fore for systems with a high number of input/output ports, theorder of the order reduced system may be much higher thannecessary. The required part of the system can be reducedby employing a Lyapunov balanced based order reductionmethod.Models taken from an integrity circuit (IC) analysis as anIC conducted emission model of p input/output ports, can bedescribed by using MNA as an MIMO systemCsX = GX + BUV = LX, (1)where U,V∈Rp×1 are input and output vectors, while C,G∈RN×N , B, LT ∈RN×p are system matrices of the orderPublished by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V.130 Lj. Radic´-Weissenfeld et al.: Model order reduction of linear time invariant systemsN . After the order reduction by the Krylov subspace projec-tion the order reduced system is described byC˜sX˜ = G˜X˜ + B˜UV = L˜X˜, (2)where C˜, G˜ ∈ Rq×q , B˜, L˜T ∈Rq×p are order reduced sys-tem matrices of the order qN . Now the Lyapunov equa-tions are created from the order reduced system described inEq. (2) by inversion of the matrix C˜. This is possible as longas the relevant matrix is not singular. Unfortunately this ma-trix is normally singular, but due to numerical inaccuracy itsdeterminant differs from zero. For the invertible matrix C˜ theLyapunov equations areC˜−1G˜P+ P(C˜−1G˜)T = −B˜B˜TQC˜−1G˜+ (C˜−1G˜)T Q = −L˜T L˜. (3)The balancing can take place if above equations have uniquesolutions, commonly known as observability P and control-lability Q Grammians.3 Proposed two-step order reductionThe proposed two-step order reduction algorithm consists ofthe Krylov subspace projection and truncation of the weakpart of a Lyapunov balanced system. The difference in re-spect to the standard approach, described in the previous sec-tion, is going to be discussed in the following.It is well known that the order of a system reduced by ap-plying the Krylov subspace projection strongly depends onthe number of its input/output ports. There are publishedworks that deal with this topic, such as Feldmann (2004),Silva et al. (2006) and Ludwig et al. (2008). In Feldmann(2004) a system of a high number of input/output ports is de-scribed so that the input/output ports are not directly involvedin the order reduction of a system. Authors in Silva et al.(2006) introduce the correlation matrix with respect to whichthe weak coupled input/output ports are easily recognised bylooking at the input signal. For this work, the method intro-duced in Ludwig et al. (2007) is employed. The proposedmethod reduces the number of inputs/outputs, with respectto the model functionality. In contrast to Silva et al. (2006)the input and output signals are not needed. Moreover, theproposed method in Ludwig et al. (2007) can be used forsystems with bidirectional ports, for which the method de-scribed in Feldmann (2004) is not the most suitable. In Lud-wig et al. (2007), it is suggested to replace all independentsources that have the same waveform with the same numberof controlled sources and only one port.Controlled sources and the RLCG part of the system canbe efficiently reduced by the projection of a system matriceson the Krylov subspace. The system matrices are matricesobtained by MNA. They do not have a full rank, so that theirorder N exceeds the amount of state space variables. Theorder reduced system is described by Eq. (2).In the second-step of the order reduction, either a wholesystem or a part of a system that describes a port to port re-lation can be reduced. To achieve an efficient reduction of asystem that was reduced in the first-step, it is divided up intoport to port subsystems, so that a subsystem between any portl and any port m of the system is described by the followingtransfer functionH˜ lm =q∑k=1r˜lmksd˜k − 1+ sH˜lm∞ + H˜lm0. (4)Poles 1d˜k of all port to port transfer functions originate frommatrices C˜ and G˜. These two matrices are equal for all portto port transfer functions, so that all port to port transfer func-tions have equal poles. Since transfer functions have differ-ent B˜ and L˜ matrices, they have also different residues r˜lmk .Each port to port transfer function has also different constantH˜lm0 and a polynomial function of the degree one H˜lm∞, be-cause these two parts of a transfer function originate from theresidues.The rational and a polynomial function of the degree onein the Eq. (4) build a state space of a system. Therefore, onlythe rational part of the transfer function has an impact on theorder of the sate space matrices. Hence, in the second-stepof the order reduction only the part of the system that buildsa state space is affected by the order reduction. Namely, in-stead of reducing the system matrices, the state space of thesystem is reduced in the second-step. The state space matri-ces are created from the rational part of the Eq. (4) asA =1/d˜1 0 · · · 00 1/d˜2 · · · 0....... . ....0 0 · · · 1/d˜q , b =11...1 ,clm =[r˜lm1 r˜lm2 · · · r˜lmq]. (5)Lyapunov matrix equations are nowAP+ PAT = −bbTQlmA+ AT Qlm = −cTlmclm. (6)Since matrices A and b are independent from the port to porttransfer functions, the controllability Grammian P has to becalculated only once for all port to port transfer functions.Because the matrix A is diagonal and b consists of ones, thecontrollability Grammian is a symmetric matrixP = −d˜2112d˜11d˜11d˜22d˜11+d˜22 . . .d˜11d˜qqd˜11+d˜qqd˜11d˜22d˜11+d˜22d˜2222d˜22. . .d˜qqd˜22+d˜qq....... . ....d˜11d˜qqd˜11+d˜qqd˜22d˜qqd˜22+d˜qq . . .d˜2qq2d˜qq. (7)Adv. Radio Sci., 6, 129–132, 2008 www.adv-radio-sci.net/6/129/2008/Lj. Radic´-Weissenfeld et al.: Model order reduction of linear time invariant systems 131EquivalentIC conductedemission modelFig. 1. An IC conducted emission model that consists of two pins,and one added independent current source at the one port, that re-places twelve controlled sources.Due to the fact that the observability Grammian dependson the residues of the Eq. (4) it differs for each port to porttransfer function exposing the symmetry asQlm = −r˜2lm1d˜2112d˜11. . . r˜lm1˜rlmqd˜11d˜qqd˜11+d˜qqr˜lm1˜rlm2d˜11d˜22d˜11+d˜22 . . . r˜lm2˜rlmqd˜qqd˜22+d˜qq.... . ....r˜lm1˜rlmqd˜11d˜qqd˜11+d˜qq . . . r˜2lmqd˜2qq2d˜qq. (8)The complexity of the proposed solutions of Lyapunov equa-tions is O(n2), while the complexity of the standard solutionis O(n3).By using the singular value decomposition of the productof controllability and observability Grammians the transfor-mation matrix T for the Lyapunov balancing can be calcu-lated (Li et al., 1999). With the change of the state spacevariables X = TX̂ in Eq. (1) the balanced system is ob-tained. Now the Schmidt-Mirsky theorem (Antoulas andSorensen, 2001) can be used to reduce the state space ma-trices of the balanced system. The transfer function Eq. (4)after the second-step of order reduction is now˜˜H lm = c˜lm(sI+ A˜)−1b˜+ sH˜lm∞ + H˜lm0. (9)where A˜∈Rn×n, and b˜, c˜T ∈Rn×1. After second-step of theorder reduction nqN holds.4 ExamplesThe proposed two-step order reduction will be validated byusing an IC conducted emission model. The model is anelectrical representation of IC behaviour on high frequencies(Steinecke et al., 2006; Gstoettner et al., 2006). The systemof the IC conducted emission model is a MIMO, where acertain amount of input/output ports are engaged with cur-rent sources. The remaining ports are used for the voltagesupply of the chip.The observed IC conducted emission model consists oftwo supply pins and twelve current sources. The current0 10 20-600-400-2000Amplitude(dB)IndexSVD Z1125Fig. 2. Hankel singular values of Z11 transfer function.5 10 15 20 25-600-400-2000Amplitude(dB)IndexSVD Z33Fig. 3. Hankel singular values of Z33 transfer function.sources are collected into one node, so that the IC conductedemission model has three inputs/output ports. The system isdescribed by six transfer functions in mean of Z-parameters.In Fig. 1 ports nPIN1 and nPIN2 are supply pins, while portnblock1 is the equivalent current source.The low order of the system depicted in Fig. 1 can beachieved by using only one expansion point and one mo-ment. However, the transfer function of the order reducedsystem does not match the original transfer function even inlow frequencies.So, if the order reduction is split into two steps, the or-der reduced system exhibits more similarities to the origi-nal. Namely, in the first-step the original transfer function ofthe IC conducted emission model is projected on the Krylovsubspace, by utilising three expansion points and three mo-ments. The order of the projected system matrices is 27. Nowthe transfer function of the reduced system is split into ratio-nal and nonrational parts as in Eq. (4). The rational part ofthe transfer function contains 26 poles. Therefore, the coeffi-cients of the Lyapunov equations are state space matrices ofthe order 26. The order reduction of the state space matricesdepends on the singular values of the system.Singular values of the transfer functions Z11 and Z33 areshown in Figs. 2 and 3. The singular values show that thestate space matrices that build the transfer functions Z11 andZ33 can be reduced to the order one. Moreover, the flow ofthe phase and magnitude of the transfer functions Z11 andZ33 suggests that the transfer functions can be created by us-ing only one pole, since magnitude and phase have a mono-tone relation to the frequency. However the order one of thesystem matrices from Eq. (1) cannot be achieved by usingwww.adv-radio-sci.net/6/129/2008/ Adv. Radio Sci., 6, 129–132, 2008132 Lj. Radic´-Weissenfeld et al.: Model order reduction of linear time invariant systems10010210410610810101051010|Z11|Frequency (Hz)originalMOR: first-stepMOR: second-step1001021041061081010-100-500Phase(grad) originalMOR: first-stepMOR: second-stepFrequency (Hz)Fig. 4. Magnitude and phase of original (solid line), reduced in first-step (solid plus line) and in second-step (dotted line) Z11 transferfunction.10010210410610810101001020|Z33|Frequency (Hz)originalMOR: first-stepMOR: second-step1001021041061081010-100-500Phase(grad)Frequency (Hz)originalMOR: first-stepMOR: second-stepFig. 5. Magnitude and phase of original (solid line), reduced in first-step (solid plus line) and in second-step (dotted line) Z33 transferfunction.only the projection of the system matrices on the Krylov sub-space without a loss in system information. By using thesuggested two-step order reduction algorithm the system ofthe IC conducted emission model can be reduced to the orderone, while preserving both magnitude or phase characteris-tics in the whole frequency range of interest.The results of order reduction are plotted in Figs. 4 and5. The negligible magnitude and phase differences betweenoriginal and order reduced transfer functions occur for bothtransfer functions. The good matching between original andorder reduced transfer functions is the consequence of goodselection of the expansion points in the first-step and goodselection of the transformation matrix in the second-step ofthe order reduction.5 ConclusionsThe first-step of the order reduction is pure projection of eachsystem matrix on the Krylov subspace. For implementationof the second-step of the algorithm, the transfer function isdescribed in mean of residues and poles. This description en-ables the partitioning of the transfer function onto the ratio-nal, constant and as the case may be a polynomial functionof the degree one. Since the constant and the polynomialfunctions are not of importance for a system order, only therational part of the transfer function is consider in the second-step of the algorithm. In this way the state space matrices ofthe system are reduced instead of the system matrices, whatenables the creation of very low order system.The two-step order reduction algorithm is validated by re-ducing the order of the systems of the IC conducted emis-sion models. The proposed algorithm shows better resultsthan standard order reduction algorithm based on projectionof the system matrices on the Krylov subspace.Acknowledgements. Represented research and development workis carried out in the frame of the MEDEA+ project 2 A701PARACHUTE project (Parasitic Extraction and Optimization forEfficient Microelectronic System Design and Application). Thisparticular research is supported by the BMBF – Bundesministeriumfuer Bildung und Forschung Federal Republic of Germany undergrant 01 M 3169 E. The responsibility for this publication is heldby authors only.ReferencesAnatoulas, A. C. and Sorensen, D. C.: Approximation of large-scaledynamica systems: An overview, I. J. A. M., 11(5), 1093–1121,2001.Boley, D. L.: Krylov Space Methods on State-space Control Mod-els, Circ. Syst. Signal Pr., 13(6), 733–758, 1994.Feldmann, P.: Model Order Reduction of Systems with Large Num-bers of Terminals, Proceedings of the Design, Automation andTest in Europe Conference and Exhibition, Paris, France, 2, 944-947, 2004.Gstoettner, A., Steinecke T., and Huemer, M.: Activity Based HighLevel Modeling of Dynamic Switching Currents in Digital ICModules, EMC Zurich 2006, Singapore, 2006.Li, J.-R., Wang, F. and White, J.: An Efficient Lyapunov Equation-Based Approach for Generating Reduced-Order Models of In-terconnect, Proceedings of 36th Design Automation Conference,New Orleans, LA, USA, 1–6, 1999.Ludwig, S., Radic´-Weissenfeld, Lj., Mathis, W., and John, W.:Model Order Reduction of Integrated Circuit Conducted Emis-sion Models, Proceedings of the EMC Compo, Torino, Italy,November 2007, 215-219, 2007.Ludwig, S., Radic´-Weissenfeld, Lj., Mathis, W., and John, W.: Ef-ficient Network Description of IC Conducted Emission Modelsfor Model Reduction, Adv. Radio Sci., in press, 2008.Silva, J. M., Vilena, J. F., Flores, P., and Miguel, L.: OutstandingIssues in Model Order Reduction, Proceedings of the ScientificComputing in Electrical Engineering, Sinaia, Romania, Septem-ber 2006, 11, 139-152, 2006.Steinecke, T., Goekcen, M., and Gstoettner, A.: High-Accuracy Eis-sion Simulation Models for VLSI Chips including Package andPrinted Circuit Board, EMC Compo 2006, Torino, Italy, 41-46,2006.Adv. Radio Sci., 6, 129–132, 2008 www.adv-radio-sci.net/6/129/2008/

Model order reduction of linear time invariant systems

This paper addresses issues related to the order reduction of systems with multiple input/output ports. The order reduction is divided up into two steps. The first step is the standard order reduction method based on the multipoint approximation of system matrices by applying Krylov subspace. The second step is based on the rejection of the weak part of a system. To recognise the weak system part, Lyapunov equations are used. Thus, this paper introduces efficient solutions of the Lyapunov equations for port to port subsystems

Radi-Weissenfeld, L.

Ludwig, S.

Mathis, W.

Fraunhofer-ePrints

Lj. Radić-Weissenfeld

S. Ludwig

W. Mathis

W. John

Directory of Open Access Journals

Adv. Radio Sci., 6, 129–132, 2008www.adv-radio-sci.net/6/129/2008/© Author(s) 2008. This work is distributed underthe Creative Commons Attribution 3.0 License.Advances inRadio ScienceModel order reduction of linear time invariant systemsLj. Radi ć-Weissenfeld1,2, S. Ludwig1,2, W. Mathis1, and W. John21Leibniz University of Hannover, Institute of Electromagnetic Theory, Hannover, Germany2Advanced System Engineering (ASE), Fraunhofer Institute for Reliability and Microintegration (IZM), Paderborn, GermanyAbstract. This paper addresses issues related to the orderreduction of systems with multiple input/output ports. Theorder reduction is divided up into two steps. The first stepis the standard order reduction method based on the multi-point approximation of system matrices by applying Krylovsubspace. The second step is based on the rejection of theweak part of a system. To recognise the weak system part,Lyapunov equations are used. Thus, this paper introduces ef-ficient solutions of the Lyapunov equations for port to portsubsystems.1 IntroductionThe order reduction of a linear time invariant system is ap-plied in almost all fields of electrical engineering. The useof order reduced models for test simulations of complex sys-tems is a lot easier than utilising full order models. This isdue to the fact that the lower order transfer function can beanalysed more easily. Therefore, order reduction algorithmsare standard techniques in the integrated circuits communityfor analysis, approximation and simulation of models arisingfrom interconnect and electromagnetic structure analysis.The combination of order reduction algorithms, derivedfrom a Krylov subspace projection, and singular value basedmethods are successfully used for the order reduction of pas-sive linear electrical networks. The Krylov subspace projec-tion method is commonly used for models with a low numberof input/output ports. The singular value based methods aresuccessfully used for models of a modest order, i.e. a fewhundred. The singular value based method is used for mod-els which matrix representation allows the inversion opera-tion. The classical approach is to use the Krylov subspaceCorrespondence to: Lj. Radić-Weissenfeld(radic@tet.uni-hannover.de)projection to more easily obtain the balanced model for sin-gular value based methods of an order reduction.In this work, these two order reduction methods are com-bined in two steps. The first step of order reduction algorithmis based on the use of the Krylov subspace projection forthe selection of dominant moments, which are found throughmoment matching. In the second step, the order reducedmodel is further reduced by rejecting its weak states. A stan-dard matrix inversion is avoided in the second step of theorder reduction. This combination of order reduction meth-ods keeps low magnitude and phase errors of order reducedsystems.2 Standard two-step order reductionA two-step order reduction method usually consist of theKrylov subspace projection method (Boley, 1994) and thebalanced truncation method (Antoulas and Sorensen, 2001).The Krylov subspace projection produces a system of an or-der which is proportional to the number of input/output ports.Moreover, the order of the reduced system depends on thenumber of expansion points and matched moments. There-fore for systems with a high number of input/output ports, theorder of the order reduced system may be much higher thannecessary. The required part of the system can be reducedby employing a Lyapunov balanced based order reductionmethod.Models taken from an integrity circuit (IC) analysis as anIC conducted emission model ofp input/output ports, can bedescribed by using MNA as an MIMO systemCsX = GX + BUV = LX, (1)where U,V∈Rp×1 are input and output vectors, whileC,G∈RN×N , B, LT ∈RN×p are system matrices of the orderPublished by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V.130 Lj. Radíc-Weissenfeld et al.: Model order reduction of linear time invariant systemsN . After the order reduction by the Krylov subspace projec-tion the order reduced system is described byC̃sX̃ = G̃X̃ + B̃UV = L̃X̃, (2)whereC̃, G̃ ∈ Rq×q , B̃, L̃T ∈Rq×p are order reduced sys-tem matrices of the orderqN . Now the Lyapunov equa-tions are created from the order reduced system described inEq. (2) by inversion of the matrix̃C. This is possible as longas the relevant matrix is not singular. Unfortunately this ma-trix is normally singular, but due to numerical inaccuracy itsdeterminant differs from zero. For the invertible matrixC̃ theLyapunov equations areC̃−1G̃P + P(C̃−1G̃)T = −B̃B̃TQC̃−1G̃ + (C̃−1G̃)T Q = −L̃T L̃ . (3)The balancing can take place if above equations have uniquesolutions, commonly known as observabilityP and control-lability Q Grammians.3 Proposed two-step order reductionThe proposed two-step order reduction algorithm consists ofthe Krylov subspace projection and truncation of the weakpart of a Lyapunov balanced system. The difference in re-spect to the standard approach, described in the previous sec-tion, is going to be discussed in the following.It is well known that the order of a system reduced by ap-plying the Krylov subspace projection strongly depends onthe number of its input/output ports. There are publishedworks that deal with this topic, such asFeldmann(2004),Silva et al.(2006) andLudwig et al. (2008). In Feldmann(2004) a system of a high number of input/output ports is de-scribed so that the input/output ports are not directly involvedin the order reduction of a system. Authors inSilva et al.(2006) introduce the correlation matrix with respect to whichthe weak coupled input/output ports are easily recognised bylooking at the input signal. For this work, the method intro-duced inLudwig et al. (2007) is employed. The proposedmethod reduces the number of inputs/outputs, with respectto the model functionality. In contrast toSilva et al.(2006)the input and output signals are not needed. Moreover, theproposed method inLudwig et al. (2007) can be used forsystems with bidirectional ports, for which the method de-scribed inFeldmann(2004) is not the most suitable. InLud-wig et al. (2007), it is suggested to replace all independentsources that have the same waveform with the same numberof controlled sources and only one port.Controlled sources and the RLCG part of the system canbe efficiently reduced by the projection of a system matriceson the Krylov subspace. The system matrices are matricesobtained by MNA. They do not have a full rank, so that theirorderN exceeds the amount of state space variables. Theorder reduced system is described by Eq. (2).In the second-step of the order reduction, either a wholesystem or a part of a system that describes a port to port re-lation can be reduced. To achieve an efficient reduction of asystem that was reduced in the first-step, it is divided up intoport to port subsystems, so that a subsystem between any portl and any portm of the system is described by the followingtransfer functionH̃ lm =q∑k=1r̃lmksd̃k − 1+ sH̃lm∞ + H̃lm0. (4)Poles 1̃dk of all port to port transfer functions originate frommatrices̃C andG̃. These two matrices are equal for all portto port transfer functions, so that all port to port transfer func-tions have equal poles. Since transfer functions have differ-ent B̃ andL̃ matrices, they have also different residuesr̃lmk.Each port to port transfer function has also different constantH̃lm0 and a polynomial function of the degree onẽHlm∞, be-cause these two parts of a transfer function originate from theresidues.The rational and a polynomial function of the degree onein the Eq. (4) build a state space of a system. Therefore, onlythe rational part of the transfer function has an impact on theorder of the sate space matrices. Hence, in the second-stepof the order reduction only the part of the system that buildsa state space is affected by the order reduction. Namely, in-stead of reducing the system matrices, the state space of thesystem is reduced in the second-step. The state space matri-ces are created from the rational part of the Eq. (4) asA =1/d̃1 0 · · · 00 1/d̃2 · · · 0....... . ....0 0 · · · 1/d̃q , b =11...1 ,clm =[r̃lm1 r̃lm2 · · · r̃lmq]. (5)Lyapunov matrix equations are nowAP + PAT = −bbTQlmA + AT Qlm = −cTlmclm. (6)Since matricesA andb are independent from the port to porttransfer functions, the controllability GrammianP has to becalculated only once for all port to port transfer functions.Because the matrixA is diagonal andb consists of ones, thecontrollability Grammian is a symmetric matrixP = −d̃2112d̃11d̃11d̃22d̃11+d̃22. . .d̃11d̃qqd̃11+d̃qqd̃11d̃22d̃11+d̃22d̃2222d̃22. . .d̃qqd̃22+d̃qq....... . ....d̃11d̃qqd̃11+d̃qqd̃22d̃qqd̃22+d̃qq. . .d̃2qq2d̃qq. (7)Adv. Radio Sci., 6, 129–132, 2008 www.adv-radio-sci.net/6/129/2008/Lj. Radić-Weissenfeld et al.: Model order reduction of linear time invariant systems 131EquivalentIC conductedemission modelFig. 1. An IC conducted emission model that consists of two pins,and one added independent current source at the one port, that re-places twelve controlled sources.Due to the fact that the observability Grammian dependson the residues of the Eq. (4) it differs for each port to porttransfer function exposing the symmetry asQlm = −r̃2lm1d̃2112d̃11. . . r̃lm1̃rlmqd̃11d̃qqd̃11+d̃qqr̃lm1̃rlm2d̃11d̃22d̃11+d̃22. . . r̃lm2̃rlmqd̃qqd̃22+d̃qq.... . ....r̃lm1̃rlmqd̃11d̃qqd̃11+d̃qq. . . r̃2lmqd̃2qq2d̃qq. (8)The complexity of the proposed solutions of Lyapunov equa-tions isO(n2), while the complexity of the standard solutionisO(n3).By using the singular value decomposition of the productof controllability and observability Grammians the transfor-mation matrixT for the Lyapunov balancing can be calcu-lated (Li et al., 1999). With the change of the state spacevariablesX = TX̂ in Eq. (1) the balanced system is ob-tained. Now the Schmidt-Mirsky theorem (Antoulas andSorensen, 2001) can be used to reduce the state space ma-trices of the balanced system. The transfer function Eq. (4)after the second-step of order reduction is now˜̃H lm = c̃lm(sI + Ã)−1̃b + sH̃lm∞ + H̃lm0. (9)whereÃ∈Rn×n, andb̃, c̃T ∈Rn×1. After second-step of theorder reductionnqN holds.4 ExamplesThe proposed two-step order reduction will be validated byusing an IC conducted emission model. The model is anelectrical representation of IC behaviour on high frequencies(Steinecke et al., 2006; Gstoettner et al., 2006). The systemof the IC conducted emission model is a MIMO, where acertain amount of input/output ports are engaged with cur-rent sources. The remaining ports are used for the voltagesupply of the chip.The observed IC conducted emission model consists oftwo supply pins and twelve current sources. The current0 10 20-600-400-2000Amplitude(dB)IndexSVD Z1125Fig. 2. Hankel singular values ofZ11 transfer function.5 10 15 20 25-600-400-2000Amplitude(dB)IndexSVD Z33Fig. 3. Hankel singular values ofZ33 transfer function.sources are collected into one node, so that the IC conductedemission model has three inputs/output ports. The system isdescribed by six transfer functions in mean ofZ-parameters.In Fig. 1 portsnPIN1 andnPIN2 are supply pins, while portnblock1 is the equivalent current source.The low order of the system depicted in Fig.1 can beachieved by using only one expansion point and one mo-ment. However, the transfer function of the order reducedsystem does not match the original transfer function even inlow frequencies.So, if the order reduction is split into two steps, the or-der reduced system exhibits more similarities to the origi-nal. Namely, in the first-step the original transfer function ofthe IC conducted emission model is projected on the Krylovsubspace, by utilising three expansion points and three mo-ments. The order of the projected system matrices is 27. Nowthe transfer function of the reduced system is split into ratio-nal and nonrational parts as in Eq. (4). The rational part ofthe transfer function contains 26 poles. Therefore, the coeffi-cients of the Lyapunov equations are state space matrices ofthe order 26. The order reduction of the state space matricesdepends on the singular values of the system.Singular values of the transfer functionsZ11 andZ33 areshown in Figs.2 and3. The singular values show that thestate space matrices that build the transfer functionsZ11 andZ33 can be reduced to the order one. Moreover, the flow ofthe phase and magnitude of the transfer functionsZ11 andZ33 suggests that the transfer functions can be created by us-ing only one pole, since magnitude and phase have a mono-tone relation to the frequency. However the order one of thesystem matrices from Eq. (1) cannot be achieved by usingwww.adv-radio-sci.net/6/129/2008/ Adv. Radio Sci., 6, 129–132, 2008132 Lj. Radíc-Weissenfeld et al.: Model order reduction of linear time invariant systems10010210410610810101051010|Z11|Frequency (Hz)originalMOR: first-stepMOR: second-step1001021041061081010-100-500Phase(grad) originalMOR: first-stepMOR: second-stepFrequency (Hz)Fig. 4. Magnitude and phase of original (solid line), reduced in first-step (solid plus line) and in second-step (dotted line)Z11 transferfunction.10010210410610810101001020|Z33|Frequency (Hz)originalMOR: first-stepMOR: second-step1001021041061081010-100-500Phase(grad)Frequency (Hz)originalMOR: first-stepMOR: second-stepFig. 5. Magnitude and phase of original (solid line), reduced in first-step (solid plus line) and in second-step (dotted line)Z33 transferfunction.only the projection of the system matrices on the Krylov sub-space without a loss in system information. By using thesuggested two-step order reduction algorithm the system ofthe IC conducted emission model can be reduced to the orderone, while preserving both magnitude or phase characteris-tics in the whole frequency range of interest.The results of order reduction are plotted in Figs.4 and5. The negligible magnitude and phase differences betweenoriginal and order reduced transfer functions occur for bothtransfer functions. The good matching between original andorder reduced transfer functions is the consequence of goodselection of the expansion points in the first-step and goodselection of the transformation matrix in the second-step ofthe order reduction.5 ConclusionsThe first-step of the order reduction is pure projection of eachsystem matrix on the Krylov subspace. For implementationof the second-step of the algorithm, the transfer function isdescribed in mean of residues and poles. This description en-ables the partitioning of the transfer function onto the ratio-nal, constant and as the case may be a polynomial functionof the degree one. Since the constant and the polynomialfunctions are not of importance for a system order, only therational part of the transfer function is consider in the second-step of the algorithm. In this way the state space matrices ofthe system are reduced instead of the system matrices, whatenables the creation of very low order system.The two-step order reduction algorithm is validated by re-ducing the order of the systems of the IC conducted emis-sion models. The proposed algorithm shows better resultsthan standard order reduction algorithm based on projectionof the system matrices on the Krylov subspace.Acknowledgements.Represented research and development workis carried out in the frame of the MEDEA+ project 2A701PARACHUTE project (Parasitic Extraction and Optimization forEfficient Microelectronic System Design and Application). Thisparticular research is supported by the BMBF – Bundesministeriumfuer Bildung und Forschung Federal Republic of Germany undergrant 01 M 3169 E. The responsibility for this publication is heldby authors only.ReferencesAnatoulas, A. C. and Sorensen, D. C.: Approximation of large-scaledynamica systems: An overview, I. J. A. M., 11(5), 1093–1121,2001.Boley, D. L.: Krylov Space Methods on State-space Control Mod-els, Circ. Syst. Signal Pr., 13(6), 733–758, 1994.Feldmann, P.: Model Order Reduction of Systems with Large Num-bers of Terminals, Proceedings of the Design, Automation andTest in Europe Conference and Exhibition, Paris, France, 2, 944-947, 2004.Gstoettner, A., Steinecke T., and Huemer, M.: Activity Based HighLevel Modeling of Dynamic Switching Currents in Digital ICModules, EMC Zurich 2006, Singapore, 2006.Li, J.-R., Wang, F. and White, J.: An Efficient Lyapunov Equation-Based Approach for Generating Reduced-Order Models of In-terconnect, Proceedings of 36th Design Automation Conference,New Orleans, LA, USA, 1–6, 1999.Ludwig, S., Radíc-Weissenfeld, Lj., Mathis, W., and John, W.:Model Order Reduction of Integrated Circuit Conducted Emis-sion Models, Proceedings of the EMC Compo, Torino, Italy,November 2007, 215-219, 2007.Ludwig, S., Radíc-Weissenfeld, Lj., Mathis, W., and John, W.: Ef-ficient Network Description of IC Conducted Emission Modelsfor Model Reduction, Adv. Radio Sci., in press, 2008.Silva, J. M., Vilena, J. F., Flores, P., and Miguel, L.: OutstandingIssues in Model Order Reduction, Proceedings of the ScientificComputing in Electrical Engineering, Sinaia, Romania, Septem-ber 2006, 11, 139-152, 2006.Steinecke, T., Goekcen, M., and Gstoettner, A.: High-Accuracy Eis-sion Simulation Models for VLSI Chips including Package andPrinted Circuit Board, EMC Compo 2006, Torino, Italy, 41-46,2006.Adv. Radio Sci., 6, 129–132, 2008 www.adv-radio-sci.net/6/129/2008/

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Model order reduction of linear time invariant systems

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