By the rising complexity and miniaturisation of the device's dimensions, the density of the conductors increases considerably. Referring to this, locally transient interactions between single physical values become apparent. Therefore, for the investigation and optimisation of integrated circuits it is essential to develop suitable models and simulation surroundings which allow for memory and timeefficient calculation of the behaviour. By means of the dynamic reconstruction theory and the radial basis functions nets the so-called black box models are provided. The description of black box models is derived from the input and output behaviour or so-called time series of a dynamic system. Concerning the time series, the black box model adapts its parameters via the extended Kalman filter. This paper provides a modelling approach that enables fast and efficient simulations.BMBF/01M3169 EInfineon Technologies AG/01M 3169 

Fischer, C.

Hedayat, C.

Hilleringmann, U.

John, W.

Kazemzadeh, Reza

Wiegand, C.

Advances in Radio Science

By the rising complexity and miniaturisation of the device's dimensions, the density of the conductors increases considerably. Referring to this, locally transient interactions between single physical values become apparent. Therefore, for the investigation and optimisation of integrated circuits it is essential to develop suitable models and simulation surroundings which allow for memory and timeefficient calculation of the behaviour. By means of the dynamic reconstruction theory and the radial basis functions nets the so-called black box models are provided. The description of black box models is derived from the input and output behaviour or so-called time series of a dynamic system. Concerning the time series, the black box model adapts its parameters via the extended Kalman filter. This paper provides a modelling approach that enables fast and efficient simulations

Kazemzadeh, R.

Fraunhofer-ePrints

Macro-modelling via radial basis functionen nets

By the rising complexity and miniaturisation of the device&apos;s dimensions, the density of the conductors increases considerably. Referring to this, locally transient interactions between single physical values become apparent. Therefore, for the investigation and optimisation of integrated circuits it is essential to develop suitable models and simulation surroundings which allow for memory and time-efficient calculation of the behaviour. By means of the dynamic reconstruction theory and the radial basis functions nets the so-called black box models are provided. The description of black box models is derived from the input and output behaviour or so-called time series of a dynamic system. Concerning the time series, the black box model adapts its parameters via the extended Kalman filter. This paper provides a modelling approach that enables fast and efficient simulations

C. Wiegand

C. Fischer

R. Kazemzadeh

C. Hedayat

W. John

U. Hilleringmann

Directory of Open Access Journals

Adv. Radio Sci., 6, 139–143, 2008www.adv-radio-sci.net/6/139/2008/© Author(s) 2008. This work is distributed underthe Creative Commons Attribution 3.0 License.Advances inRadio ScienceMacro-modelling via radial basis functionen netsC. Wiegand1,2, C. Fischer1, R. Kazemzadeh2,3, C. Hedayat1,2, W. John2, and U. Hilleringmann11University of Paderborn, Department of Electrical Engineering, Sensor Technology Group, Germany2Fraunhofer Institute for Reliability and Microintegration (IZM) Advanced System Engineering (ASE), Paderborn, Germany3Leibniz University of Hannover Institute of Electromagnetic Theory (TET), Hannover, GermanyAbstract. By the rising complexity and miniaturisation ofthe device’s dimensions, the density of the conductors in-creases considerably. Referring to this, locally transient in-teractions between single physical values become apparent.Therefore, for the investigation and optimisation of inte-grated circuits it is essential to develop suitable models andsimulation surroundings which allow for memory and time-efficient calculation of the behaviour. By means of the dy-namic reconstruction theory and the radial basis functionsnets the so-called black box models are provided. The de-scription of black box models is derived from the input andoutput behaviour or so-called time series of a dynamic sys-tem. Concerning the time series, the black box model adaptsits parameters via the extended Kalman filter. This paperprovides a modelling approach that enables fast and efficientsimulations.1 IntroductionThe increasing simulation time is a big limiting factor dur-ing the design and the development of HDI/HDP systems.Hence, it is more and more crucial to provide models whichallow for quick and accurate simulations (Wiegand et al.,2007b,a; Stievano, 2002). The classical modelling approach,based on the transistor level circuits and the symbolic anal-ysis, requires unreasonably long computation times. Theblack box modelling (BBM) via radial basis functions (RBF)nets for integrated circuits is an adequate solution that pro-vides robust and fast simulations. In the following, a methodis presented for developing mathematical models to reducethe very growing simulation times for transient analysis. Thismodelling approach is based on the dynamic reconstructionof dynamic systems. That is, a mathematical formulationneeds to be found, that describes the dynamics of the systemto be examined as accurately as possible without necessarilyconsidering the physical characteristics of the circuit.Correspondence to:C. Wiegand(christopher.wiegand@pb.izm.fraunhofer.de)In Sect.2 the modelling flow for developing black box mod-els is discussed, although the radial basis functions nets arethe main focus of all considerations. Section3 describes thearchitecture of the RBF nets. The parameter adaptation, i.e.the extended Kalman filter, is explained in Sect.4. In Sect.5a NOR element with two inputs and three outputs is exam-ined. Then, an example of a diode is considered and the RBFmodel is parameterised in such a way so as to allow for thetemperature to be regarded as a further input.2 Modelling flowIt is necessary for the developing of black box models bymeans of radial basis functions nets to choose a suitable timeseries (measurements or simulations), so that the model is asclose to the original system as possible. Thus, the trainingset must comprise a high information content with which areliable model can be provided. The next step in the mod-elling chain (see Fig.1) is the embedding (Sauer et al., 1991)of this time series in the RBF net, i.e. the choosing of centres.This can be done with the help of the so-called locally reg-ularised orthogonal least square (Chen et al., 1996,?; Chenaet al., 1990a) or the K-Means-Cluster-Algorithm (Sing et al.,2003). Within this work the centres are determined by theclustering procedure, because the initialisation of the remain-ing parameters of the net can be determined via least squaremethods. Afterwards the parameter adaptation (parameterestimation) is achieved by means of the extended Kalmanfilter (EKF) (Haykin, 2001; Ahmida and Charef, 2002). Thefinal step is the Validation, i.e., assuming that the error be-tween the model and the system is in the expected order ofmagnitude, a robust simulation model is provided. If thiscase does not occur, it has to be assessed, whether the train-ing signal was chosen appropriately since the centres werepossibly not picked wisely, or the parameter adaptation di-verged or remained in a local minimum.Published by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V.140 C. Wiegand et al.: Macro-modelling via radial basis functionen netsFig. 1. Modelling flow.3 Radial basis functions netsWith using RBF networks, it is possible to represent anycontinuous nonlinear function. Because of this importantfeature, they are utilised to model nonlinear dynamicsystems and integrated circuits. Radial basis functions netsare special feedforward neural networks consisting of threelayers with a hidden layer.Input layer: definition of the number of inputs, i.e.within the regressor vector the number of inputs is fixed.Hidden layer: signal processing viap radial basis functions(Gaussian) or so-called neurons. Furthermore the embeddingof the data occurs just in the hidden layer.Output layer: definition of the number of outputs. Regardingweighted transitions the neurons outgoing signals are addedup within every output.Figure 2 shows a schematic representation of a typicalRBF network with multiple inputs and multiple outputs(MIMO). The response of a MIMO radial basis functions netFig. 2. Schematic representation of a multiple input and multipleoutput radial basis functions net.can be formulated as follows:ŷ1...ŷm =θ11 · · · θ1p.... . ....θm1 · · · θmpφ1(ζ1)...φp(ζp) (1)whereφi(ζi) denotes theith gaussian activation function inthe hidden layer and can be written asφi(ζi) = e−ζi mit ζ = (x − ci)T 6−1i (x − ci) (2)andx is the regressor vector and is denoted byx =[xT1 , · · · , xTn]T. (3)Within the regressor vector then inputs are de-fined; m and n describe the number of outputs andthe number of basis functions;ci is the ith centre and6i = diag(σ i) = diag([σ 21i, · · · , σ2ri])represents the shapeof the ith basis function. Equation1 can be described in thefollowing way:ŷ1...ŷm =θT1...θTmφ1(ζ1)...φp(ζp)ŷ = 2φ. (4)In a standard design, the RBF net provides a static input andoutput behaviour. For temporal processing, time delays haveto be introduced to convert the static RBF network into adynamic network. There are only very few possibilities toinclude memory into the architecture. For example discreteAdv. Radio Sci., 6, 139–143, 2008 www.adv-radio-sci.net/6/139/2008/C. Wiegand et al.: Macro-modelling via radial basis functionen nets 141time FIR1/IIR-Filters2 can be used instead of the weightsor global feedback paths (Howlett, 2001a,b) could be intro-duced. A simple method to create dynamics can be expressedby:xi(t) = [ui(t − 1), · · · , ui(t − d1),yi(t − 1), · · · , yi(t − d2)]T ,(5)whered1 andd2 denote the input embedding dimension andthe output embedding dimension. By making such changes,a state space model can be described (Wiegand et al., 2007a).The first order derivation can be employed to transform thediscrete time representation into a continuous state spacemodel.4 Parameter estimationGenerally, the parameters of a RBF net can be determinedby a huge number by algorithms, for example, by the lo-cally regularised least square algorithm with centre selection(Chena et al., 1990a; Chen et al., 1990b, 1996). Nevertheless,non-linear optimisation methods must then be employed toadapt the shapes6i of the basis functions. Within this work,the extended Kalman filter (EKF) is used for parameter iden-tification of non-linear systems. In accordance to this, allparameters are determined iteratively by the EKF, includingthose which are in the argument of the gaussian function. Forthe initialisation and the improvement of the convergence,the K-Means cluster algorithms is used to determine the cen-tres. In addition, this cluster algorithm delivers an facile pos-sibility for the calculation of the shapes (widths) of the basisfunctions, to be able to determine the weights by means ofleast square methods.4.1 K-Means cluster algorithmThe algorithm is an iterative procedure to divide a set oftraining data into groups (cluster). The only available in-formation is the training data and the number of the clus-ters. This is also known as unsupervised learning. A randomchoice of the cluster centresc(0) within the training setTserves as the starting point. Next, in every iteration stepk,every set of trainig data is allocated to the nearest situatedcluster centre and thus the subsetsMi are obtained. Theidentification of the new centresci(k + 1) =1Ni∑x∈Mix (6)is accomplished by means of average determination via thesetsMi = {x ∈ T : ci(k) = argminc(k)‖x − c(k)‖2}, (7)1FIR: Finite Impulse Response2IIR: Infinite Impulse ResponseFig. 3. NOR-Circuit with two inputs (VA und VB ) and three out-puts (Vout1, Vout2 undVout3). The buffers of the respective outputsare four cascaded inverters. Also, the outputs are concluded withdifferent complex impedances (Zi , wherei∈[1, 2, 3]).whereasNi denotes the number of elements which are in thesetMi . An initialisation of the matrices6 can take place af-ter the determination of the centres, i.e. after the last iterationstep:σi,j =√√√√ 1Ni∑x∈Mi(xj − ci,j (K))2 (8)σ i = [σi,1, · · · , σi,r ]T (9)6i = (diag(σ i))2 (10)With the calculated initial centres and the widths of the ba-sis functions, it is possible to determine the weights, i.e., byutilising the least square methods.4.2 Extended Kalman FilterIteratively, the EKF estimates the parameter vectorwk =[θT1 , · · · , θTp , σT1 , · · · , σTp , cT1 , · · · , cTp]T. (11)usingwk = wk−1 + Kke (12)whereKk is the Kalman gain.Kk = P k−1H k[Rk + HTk P k−1H k]−1(13)e=y−ŷ is the error between the observed outputsy (targetvector) and the estimated outputsŷ of the RBF network;H k = ∇wf (x, w) is the matrix of the partial derivative off (x, w). For this reason, when considering thej th output,the derivations of j (x, w) are related respectively to the pa-rametersci , θji and6i :www.adv-radio-sci.net/6/139/2008/ Adv. Radio Sci., 6, 139–143, 2008142 C. Wiegand et al.: Macro-modelling via radial basis functionen netsFig. 4. Input and output signals of the simulated system and model.The output voltages marked with a hat denote the outputs of theRBF model.∂fj (x, w)∂ci= 2θji6−1i (x − ci) e−ζ (14)∂fj (x, w)∂θji= e−ζ (15)∂fj (x, w)∂σ i= 2θji6̃−1i sq(x − ci)e−ζ (16)where6̃i is defined as6̃i = diag(σ̃ i) = diag([σ 31i, · · · , σ3ri])(17)withsq(x − ci) = [(x1 − c1i)2, · · · , (xr − cri)2]T . (18)Rk and P k denote the variance of the measurement noiseand the error covariance matrix. The error covariance matrixis a positive definite symmetricν×ν matrix, whereν is thenumber of parameters being updated and can be determinedby:P k =[I − KkHTk]P k−1 + q0I . (19)Fig. 5. Identity lines of the diode and the model for different tem-peratures.According toAhmida and Charef(2002), q0 regulates theallowed random step towards the gradient. For fast parameterestimation, the training data is subdivided into smaller sets.These sets are used to adjust the parameters of the RBF net.Afterwards, a test is performed as well as an update of theregressor vector.5 SimulationsIn the following section, two different nonlinear systems areconsidered.The first example corresponds to a two input and three outputNOR-Circuit (see Fig.3). In the second example a diodeis envisaged, in which the ability of parameterisation of theRBF nets is illustrated.5.1 Example 1: NOR CircuitAt first a NOR circuit is considered, which was built-upand simulated by using the Agilent Advanced Design Sys-tem (ADS) environment. This circuit was implemented withtwo inputs and three outputs with different complex loads,from which a MIMO system arises. The circuit is imple-mented within 130 nm technology derived from an ADS-implemented Philips library. The radial basis functions nethas been trained using a set of 1000 data samples of the trape-zoidal input signals and the corresponding output signals.The RBF net is developed with 15 centres (selected via thepresented cluster algorithm) and the regressor vector is de-fined as follows:Adv. Radio Sci., 6, 139–143, 2008 www.adv-radio-sci.net/6/139/2008/C. Wiegand et al.: Macro-modelling via radial basis functionen nets 143x(k) =[uT (k), uT (k − 1), yT (k − 1)]T(20)u(k) = [u1(k), u2(k)]T (21)y(k) = [y1(k), y2(k), y3(k)]T (22)In Fig. 4 the input signals (VA undVB ) as well as the re-sponse of the ADS-system (Vout1, Vout2 und Vout3) and theRBF model (̂Vout1, V̂out2 undV̂out3) are shown. The model isable to represent the dynamic as well as the logical behaviourof the system. The mean square error (MSE) is about 4·10−5.5.2 Example 2: diodeThe system under consideration is a diode (SISO), but theRBF net was implemented with a second input. This secondinput is virtual and corresponds to the parameterisation of themodel taking the temperature drift into account. This meansthat the temperature is considered at the modelling level asif it were an other input. This is therefore a simple examplethat shows the ability to parameterise the RBF nets. The mostsimple identity equation isy(k) = Is(eqUinkT − 1), (23)whereUin denotes the input voltage,k andqstand for theBoltzmann constant and the elementary charge.Is is thesaturation current and for simulation purposesIs was fixedto 5·10−3 A. During the training, the temperature was var-ied between 18◦C to 33◦C. The training signal, consisting of1000 values, had a sine-shaped course and formed, in combi-nation with the temperature, the two-dimensional regressor.For conclusive testing, four characteristic curves were sim-ulated for 21◦C, 24◦C, 27◦C und 30◦C (Fig. 5). The meansquare error is in the order of magnitude 10−6.6 ConclusionsIn this paper it was explained how a MIMO RBF net can begenerated by means of the K-Means cluster algorithm andthe extended Kalman filter. It became apparent that it is pos-sible to model physical parameters in the same fashion. Toachieve a noticeable gain in speed, the systems to be mod-elled must be substantially more complex. Nevertheless, itwas shown that the RBF nets are able to carry out precise andfast system-level simulations. The future efforts will concen-trate on the application of the findings in a industrial envi-ronment.Acknowledgements.The reported R+D work was carried out withinthe MEDEAplus A701 PARACHUTE (Parasitic Extraction and Op-timization for Efficient Microelectronics System Design and Ap-plication) project. This particular research was supported by theBMBF (Bundesministerium fur Bildung und Forschung) of FederalRepublic of Germany University of Paderborn under grant 01M3169 E and Infineon Technologies AG under grant 01M 3169 A.The views expressed in this paper are those of the authors only.ReferencesChen, S., Billings, S. A., Cowan, C. F. N., and Grant, P. M.: Non-linear system identification using radial basis functions, Int. J.Control, 21, 2513-2539, 1990a.Chen, S., Billings, S. A., Cowan, C. F. N., and Grant, P. M.: Prac-tical identification of NARMAX models using radial basis func-tions, Int. J. Control, 52, 1327-1350, 1990b.Chen, S., Chng, E. S., and Alkadhimi, K.: Regularized orthogo-nal least squares algorithm for constructing radial basis functionnetworks, Int. J. Control, 64, 829-837, 1996.Haykin, S.: Kalman Filtering and Neural Networks, John Wiley &Sons, Inc., New York, Chichester, Weinheim, Brisbane, Singa-pore, Toronto, 2001.Howlett, R. J. and Jain, L. C.: Radial Basis Funcrion Networks1 – Recent Developments in Theory and Application, Physica-Verlag, 2001a.Howlett, R. J. and Jain, L. C.: Radial Basis Funcrion Networks 2 –New Advances in Design, Physica-Verlag, 2001b.Sauer, T., Yorke, J. A., and Casdagli, M.: Embedology, J. Stat.Phys., 65, 579-616, 1991.Sing, J. K., Basu, D. K., Nasipuri, M., and Kundu, M.: Improved K-means Algorithm in the Design of RBF Neural Networks, TEN-CON 2003, Conference on Convergent Technologies for Asia-Pacific Region, 2, 841-845, 2003.Stievano, I. S., Maio, I. A., and Canavero, F. G.: Parametric Macro-models of Digital I/O Ports, IEEE T. Adv. Packaging, SpecialSelection on EPEP01, 25(2), 255–264, 2002.Wiegand, C., Hedayat, C., John, W., Radic-Weissenfeld, L., andHilleringmann, U.: Nonlinear Identification of Complex Systemsusing Radial Basis Function Networks and Model Order Reduc-tion, IEEE International Symposium on EMC, Honolulu Hawaii,USA, 2007a.Wiegand, C., Radic-Weissenfeld, L., Hedayat, C., John, W.: BlackBox Model and Singular Value Based Model Order Reduction,18th International Zurich Symposium on EMC, 2007b.Ahmida, Z. and Charef, A.: Nonlinear Systems Modelling UsingRBF Neural Networks: A Ransdom Learning Approach to theResource Allocating Network Algorithm, Proceedings if the 10thMediterranean Conference on Control and Automation, 2002.www.adv-radio-sci.net/6/139/2008/ Adv. Radio Sci., 6, 139–143, 2008

Institutionelles Repositorium der Leibniz Universität Hannover

Adv. Radio Sci., 6, 139–143, 2008www.adv-radio-sci.net/6/139/2008/© Author(s) 2008. This work is distributed underthe Creative Commons Attribution 3.0 License.Advances inRadio ScienceMacro-modelling via radial basis functionen netsC. Wiegand1,2, C. Fischer1, R. Kazemzadeh2,3, C. Hedayat1,2, W. John2, and U. Hilleringmann11University of Paderborn, Department of Electrical Engineering, Sensor Technology Group, Germany2Fraunhofer Institute for Reliability and Microintegration (IZM) Advanced System Engineering (ASE), Paderborn, Germany3Leibniz University of Hannover Institute of Electromagnetic Theory (TET), Hannover, GermanyAbstract. By the rising complexity and miniaturisation ofthe device’s dimensions, the density of the conductors in-creases considerably. Referring to this, locally transient in-teractions between single physical values become apparent.Therefore, for the investigation and optimisation of inte-grated circuits it is essential to develop suitable models andsimulation surroundings which allow for memory and time-efficient calculation of the behaviour. By means of the dy-namic reconstruction theory and the radial basis functionsnets the so-called black box models are provided. The de-scription of black box models is derived from the input andoutput behaviour or so-called time series of a dynamic sys-tem. Concerning the time series, the black box model adaptsits parameters via the extended Kalman filter. This paperprovides a modelling approach that enables fast and efficientsimulations.1 IntroductionThe increasing simulation time is a big limiting factor dur-ing the design and the development of HDI/HDP systems.Hence, it is more and more crucial to provide models whichallow for quick and accurate simulations (Wiegand et al.,2007b,a; Stievano, 2002). The classical modelling approach,based on the transistor level circuits and the symbolic anal-ysis, requires unreasonably long computation times. Theblack box modelling (BBM) via radial basis functions (RBF)nets for integrated circuits is an adequate solution that pro-vides robust and fast simulations. In the following, a methodis presented for developing mathematical models to reducethe very growing simulation times for transient analysis. Thismodelling approach is based on the dynamic reconstructionof dynamic systems. That is, a mathematical formulationneeds to be found, that describes the dynamics of the systemto be examined as accurately as possible without necessarilyconsidering the physical characteristics of the circuit.Correspondence to: C. Wiegand(christopher.wiegand@pb.izm.fraunhofer.de)In Sect. 2 the modelling flow for developing black box mod-els is discussed, although the radial basis functions nets arethe main focus of all considerations. Section 3 describes thearchitecture of the RBF nets. The parameter adaptation, i.e.the extended Kalman filter, is explained in Sect. 4. In Sect. 5a NOR element with two inputs and three outputs is exam-ined. Then, an example of a diode is considered and the RBFmodel is parameterised in such a way so as to allow for thetemperature to be regarded as a further input.2 Modelling flowIt is necessary for the developing of black box models bymeans of radial basis functions nets to choose a suitable timeseries (measurements or simulations), so that the model is asclose to the original system as possible. Thus, the trainingset must comprise a high information content with which areliable model can be provided. The next step in the mod-elling chain (see Fig. 1) is the embedding (Sauer et al., 1991)of this time series in the RBF net, i.e. the choosing of centres.This can be done with the help of the so-called locally reg-ularised orthogonal least square (Chen et al., 1996,?; Chenaet al., 1990a) or the K-Means-Cluster-Algorithm (Sing et al.,2003). Within this work the centres are determined by theclustering procedure, because the initialisation of the remain-ing parameters of the net can be determined via least squaremethods. Afterwards the parameter adaptation (parameterestimation) is achieved by means of the extended Kalmanfilter (EKF) (Haykin, 2001; Ahmida and Charef, 2002). Thefinal step is the Validation, i.e., assuming that the error be-tween the model and the system is in the expected order ofmagnitude, a robust simulation model is provided. If thiscase does not occur, it has to be assessed, whether the train-ing signal was chosen appropriately since the centres werepossibly not picked wisely, or the parameter adaptation di-verged or remained in a local minimum.Published by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V.140 C. Wiegand et al.: Macro-modelling via radial basis functionen netsFig. 1. Modelling flow.3 Radial basis functions netsWith using RBF networks, it is possible to represent anycontinuous nonlinear function. Because of this importantfeature, they are utilised to model nonlinear dynamicsystems and integrated circuits. Radial basis functions netsare special feedforward neural networks consisting of threelayers with a hidden layer.Input layer: definition of the number of inputs, i.e.within the regressor vector the number of inputs is fixed.Hidden layer: signal processing via p radial basis functions(Gaussian) or so-called neurons. Furthermore the embeddingof the data occurs just in the hidden layer.Output layer: definition of the number of outputs. Regardingweighted transitions the neurons outgoing signals are addedup within every output.Figure 2 shows a schematic representation of a typicalRBF network with multiple inputs and multiple outputs(MIMO). The response of a MIMO radial basis functions netFig. 2. Schematic representation of a multiple input and multipleoutput radial basis functions net.can be formulated as follows:yˆ1...yˆm =θ11 · · · θ1p.... . ....θm1 · · · θmpφ1(ζ1)...φp(ζp) (1)where φi(ζi) denotes the ith gaussian activation function inthe hidden layer and can be written asφi(ζi) = e−ζi mit ζ = (x − ci)T6−1i (x − ci) (2)and x is the regressor vector and is denoted byx =[xT1 , · · · , xTn]T. (3)Within the regressor vector the n inputs are de-fined; m and n describe the number of outputs andthe number of basis functions; ci is the ith centre and6i = diag (σ i) = diag([σ 21i, · · · , σ 2ri]) represents the shapeof the ith basis function. Equation 1 can be described in thefollowing way:yˆ1...yˆm =θT1...θTmφ1(ζ1)...φp(ζp)yˆ = 2φ. (4)In a standard design, the RBF net provides a static input andoutput behaviour. For temporal processing, time delays haveto be introduced to convert the static RBF network into adynamic network. There are only very few possibilities toinclude memory into the architecture. For example discreteAdv. Radio Sci., 6, 139–143, 2008 www.adv-radio-sci.net/6/139/2008/C. Wiegand et al.: Macro-modelling via radial basis functionen nets 141time FIR1/IIR-Filters2 can be used instead of the weightsor global feedback paths (Howlett, 2001a,b) could be intro-duced. A simple method to create dynamics can be expressedby:xi(t) = [ui(t − 1), · · · , ui(t − d1),yi(t − 1), · · · , yi(t − d2)]T ,(5)where d1 and d2 denote the input embedding dimension andthe output embedding dimension. By making such changes,a state space model can be described (Wiegand et al., 2007a).The first order derivation can be employed to transform thediscrete time representation into a continuous state spacemodel.4 Parameter estimationGenerally, the parameters of a RBF net can be determinedby a huge number by algorithms, for example, by the lo-cally regularised least square algorithm with centre selection(Chena et al., 1990a; Chen et al., 1990b, 1996). Nevertheless,non-linear optimisation methods must then be employed toadapt the shapes 6i of the basis functions. Within this work,the extended Kalman filter (EKF) is used for parameter iden-tification of non-linear systems. In accordance to this, allparameters are determined iteratively by the EKF, includingthose which are in the argument of the gaussian function. Forthe initialisation and the improvement of the convergence,the K-Means cluster algorithms is used to determine the cen-tres. In addition, this cluster algorithm delivers an facile pos-sibility for the calculation of the shapes (widths) of the basisfunctions, to be able to determine the weights by means ofleast square methods.4.1 K-Means cluster algorithmThe algorithm is an iterative procedure to divide a set oftraining data into groups (cluster). The only available in-formation is the training data and the number of the clus-ters. This is also known as unsupervised learning. A randomchoice of the cluster centres c(0) within the training set Tserves as the starting point. Next, in every iteration step k,every set of trainig data is allocated to the nearest situatedcluster centre and thus the subsets Mi are obtained. Theidentification of the new centresci(k + 1) = 1Ni∑x∈Mix (6)is accomplished by means of average determination via thesetsMi = {x ∈ T : ci(k) = argminc(k)‖x − c(k)‖2}, (7)1FIR: Finite Impulse Response2IIR: Infinite Impulse ResponseFig. 3. NOR-Circuit with two inputs (VA und VB ) and three out-puts (Vout1, Vout2 und Vout3). The buffers of the respective outputsare four cascaded inverters. Also, the outputs are concluded withdifferent complex impedances (Zi , where i∈[1, 2, 3]).whereas Ni denotes the number of elements which are in thesetMi . An initialisation of the matrices 6 can take place af-ter the determination of the centres, i.e. after the last iterationstep:σi,j =√√√√ 1Ni∑x∈Mi(xj − ci,j (K))2 (8)σ i = [σi,1, · · · , σi,r ]T (9)6i = (diag(σ i))2 (10)With the calculated initial centres and the widths of the ba-sis functions, it is possible to determine the weights, i.e., byutilising the least square methods.4.2 Extended Kalman FilterIteratively, the EKF estimates the parameter vectorwk =[θT1 , · · · , θTp , σ T1 , · · · , σ Tp , cT1 , · · · , cTp]T. (11)usingwk = wk−1 +Kke (12)where Kk is the Kalman gain.Kk = P k−1H k[Rk +H Tk P k−1H k]−1(13)e=y−yˆ is the error between the observed outputs y (targetvector) and the estimated outputs yˆ of the RBF network;H k = ∇wf (x,w) is the matrix of the partial derivative off (x,w). For this reason, when considering the j th output,the derivations of fj (x,w) are related respectively to the pa-rameters ci , θji and 6i :www.adv-radio-sci.net/6/139/2008/ Adv. Radio Sci., 6, 139–143, 2008142 C. Wiegand et al.: Macro-modelling via radial basis functionen netsFig. 4. Input and output signals of the simulated system and model.The output voltages marked with a hat denote the outputs of theRBF model.∂fj (x,w)∂ci= 2θji6−1i (x − ci) e−ζ (14)∂fj (x,w)∂θji= e−ζ (15)∂fj (x,w)∂σ i= 2θji6˜−1i sq(x − ci)e−ζ (16)where 6˜i is defined as6˜i = diag (σ˜ i) = diag([σ 31i, · · · , σ 3ri])(17)withsq(x − ci) = [(x1 − c1i)2, · · · , (xr − cri)2]T . (18)Rk and P k denote the variance of the measurement noiseand the error covariance matrix. The error covariance matrixis a positive definite symmetric ν×ν matrix, where ν is thenumber of parameters being updated and can be determinedby:P k =[I −KkH Tk]P k−1 + q0I . (19)Fig. 5. Identity lines of the diode and the model for different tem-peratures.According to Ahmida and Charef (2002), q0 regulates theallowed random step towards the gradient. For fast parameterestimation, the training data is subdivided into smaller sets.These sets are used to adjust the parameters of the RBF net.Afterwards, a test is performed as well as an update of theregressor vector.5 SimulationsIn the following section, two different nonlinear systems areconsidered.The first example corresponds to a two input and three outputNOR-Circuit (see Fig. 3). In the second example a diodeis envisaged, in which the ability of parameterisation of theRBF nets is illustrated.5.1 Example 1: NOR CircuitAt first a NOR circuit is considered, which was built-upand simulated by using the Agilent Advanced Design Sys-tem (ADS) environment. This circuit was implemented withtwo inputs and three outputs with different complex loads,from which a MIMO system arises. The circuit is imple-mented within 130 nm technology derived from an ADS-implemented Philips library. The radial basis functions nethas been trained using a set of 1000 data samples of the trape-zoidal input signals and the corresponding output signals.The RBF net is developed with 15 centres (selected via thepresented cluster algorithm) and the regressor vector is de-fined as follows:Adv. Radio Sci., 6, 139–143, 2008 www.adv-radio-sci.net/6/139/2008/C. Wiegand et al.: Macro-modelling via radial basis functionen nets 143x(k) =[uT (k),uT (k − 1), yT (k − 1)]T(20)u(k) = [u1(k), u2(k)]T (21)y(k) = [y1(k), y2(k), y3(k)]T (22)In Fig. 4 the input signals (VA und VB ) as well as the re-sponse of the ADS-system (Vout1, Vout2 und Vout3) and theRBF model (Vˆout1, Vˆout2 und Vˆout3) are shown. The model isable to represent the dynamic as well as the logical behaviourof the system. The mean square error (MSE) is about 4·10−5.5.2 Example 2: diodeThe system under consideration is a diode (SISO), but theRBF net was implemented with a second input. This secondinput is virtual and corresponds to the parameterisation of themodel taking the temperature drift into account. This meansthat the temperature is considered at the modelling level asif it were an other input. This is therefore a simple examplethat shows the ability to parameterise the RBF nets. The mostsimple identity equation isy(k) = Is(eqUinkT − 1), (23)where Uin denotes the input voltage, k and qstand for theBoltzmann constant and the elementary charge. Is is thesaturation current and for simulation purposes Is was fixedto 5·10−3 A. During the training, the temperature was var-ied between 18◦C to 33◦C. The training signal, consisting of1000 values, had a sine-shaped course and formed, in combi-nation with the temperature, the two-dimensional regressor.For conclusive testing, four characteristic curves were sim-ulated for 21◦C, 24◦C, 27◦C und 30◦C (Fig. 5). The meansquare error is in the order of magnitude 10−6.6 ConclusionsIn this paper it was explained how a MIMO RBF net can begenerated by means of the K-Means cluster algorithm andthe extended Kalman filter. It became apparent that it is pos-sible to model physical parameters in the same fashion. Toachieve a noticeable gain in speed, the systems to be mod-elled must be substantially more complex. Nevertheless, itwas shown that the RBF nets are able to carry out precise andfast system-level simulations. The future efforts will concen-trate on the application of the findings in a industrial envi-ronment.Acknowledgements. The reported R+D work was carried out withinthe MEDEAplus A701 PARACHUTE (Parasitic Extraction and Op-timization for Efficient Microelectronics System Design and Ap-plication) project. This particular research was supported by theBMBF (Bundesministerium fur Bildung und Forschung) of FederalRepublic of Germany University of Paderborn under grant 01M3169 E and Infineon Technologies AG under grant 01M 3169 A.The views expressed in this paper are those of the authors only.ReferencesChen, S., Billings, S. A., Cowan, C. F. N., and Grant, P. M.: Non-linear system identification using radial basis functions, Int. J.Control, 21, 2513-2539, 1990a.Chen, S., Billings, S. A., Cowan, C. F. N., and Grant, P. 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S., Maio, I. A., and Canavero, F. G.: Parametric Macro-models of Digital I/O Ports, IEEE T. Adv. Packaging, SpecialSelection on EPEP01, 25(2), 255–264, 2002.Wiegand, C., Hedayat, C., John, W., Radic-Weissenfeld, L., andHilleringmann, U.: Nonlinear Identification of Complex Systemsusing Radial Basis Function Networks and Model Order Reduc-tion, IEEE International Symposium on EMC, Honolulu Hawaii,USA, 2007a.Wiegand, C., Radic-Weissenfeld, L., Hedayat, C., John, W.: BlackBox Model and Singular Value Based Model Order Reduction,18th International Zurich Symposium on EMC, 2007b.Ahmida, Z. and Charef, A.: Nonlinear Systems Modelling UsingRBF Neural Networks: A Ransdom Learning Approach to theResource Allocating Network Algorithm, Proceedings if the 10thMediterranean Conference on Control and Automation, 2002.www.adv-radio-sci.net/6/139/2008/ Adv. Radio Sci., 6, 139–143, 2008

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Macro-modelling via radial basis functionen nets

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