We propose an algorithm for the identification of guaranteed stable parameterized macromodels from sampled frequency responses. The proposed scheme is based on the standard Sanathanan-Koerner iteration in its parameterized form, which is regularized by adding a set of inequality constraints for enforcing the positiveness of the model denominator at suitable discrete points. We show that an ad hoc aggregation of such constraints is able to stabilize the iterative scheme by significantly improving its convergence properties, while guaranteeing uniformly stable model poles as the parameter(s) change within their design range

Bradde, T.

De Stefano, M.

Grivet-Talocia, S.

Zanco, A.

English

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On stabilization of parameterized macromodeling

Zanco A.

Grivet-Talocia S.

Bradde T.

De Stefano M.

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

04 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEOn stabilization of parameterized macromodeling / Zanco, A.; Grivet-Talocia, S.; Bradde, T.; De Stefano, M.. -ELETTRONICO. - (2019), pp. 1-4. ((Intervento presentato al convegno 23rd IEEE Workshop on Signal and PowerIntegrity, SPI 2019 tenutosi a Chambery, France nel 18-21 June, 2019.OriginalOn stabilization of parameterized macromodelingPublisher:PublishedDOI:10.1109/SaPIW.2019.8781639Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2752697 since: 2019-09-18T14:38:19ZInstitute of Electrical and Electronics Engineers Inc.On stabilization of parameterized macromodelingAlessandro Zanco, Stefano Grivet-Talocia, Tommaso Bradde, Marco De StefanoDept. Electronics and Telecommunications, Politecnico di Torino, Torino, Italy{alessandro.zanco,stefano.grivet,tommaso.bradde,marco.destefano}@polito.itAbstract—We propose an algorithm for the identification ofguaranteed stable parameterized macromodels from sampled fre-quency responses. The proposed scheme is based on the standardSanathanan-Koerner iteration in its parameterized form, which isregularized by adding a set of inequality constraints for enforcingthe positiveness of the model denominator at suitable discretepoints. We show that an ad hoc aggregation of such constraintsis able to stabilize the iterative scheme by significantly improvingits convergence properties, while guaranteeing uniformly stablemodel poles as the parameter(s) change within their design range.I. INTRODUCTIONSignal and power integrity assessment through numericalsimulation heavily relies on efficient and accurate modelsof all system parts that have an influence on the qualityof signals. In particular, electrical interconnects and theirparasitics pose significant challenges due to their complexity,both in terms of geometry, material characteristics, as well asresulting frequency responses. For this reason, macromodelingalgorithms [1] have gained tremendous popularity due to theirability to produce accurate and reliable simulation models,which run very efficiently on legacy circuit solvers.This paper focuses on a particular algorithm [2], [3] for pro-ducing linear macromodels from sampled frequency responses,while also embedding in a closed form the dependence onone or more additional parameters related to geometry ormaterial properties. The resulting macromodels can be usedto perform sensitivity, what-if, statistical, and optimizationstudies in a pre-layout design phase. With respect to traditional(univariate) macromodeling, the multivariate (parameterized)macromodeling schemes pose additional challenges for whatconcerns stability and passivity, which must be guaranteeduniformly throughout the space where the external parametersare defined.Several approaches that are able to enforce stability andpassivity by construction exist [4], [5]. Almost invariably,these methods achieve parameterization by post-processingthrough dedicated interpolation schemes a possibly large setof non-parameterized macromodels computed for fixed param-eter configurations. Both the interpolation schemes and thecoordinate system in which such interpolation is performedcan indeed be optimized and targeted to preserve stability andpassivity [6].We address a different class of algorithms, which do notrequire such two-step procedure. A parameterized model isconstructed in a single step by processing an entire set of fre-quency responses avaiable, e.g., from a field solver parametericsweep [2]. However, if no special countermeasures are taken,uniform stability (and passivity) are not enforced and theresulting model might be unusable. In this work, we proposea regularized iterative scheme based on the ParameterizedSanathanan-Koerner (PSK) iteration [3], [7]. In addition toprovably guaranteed uniform stability, this algorithm demon-strates significantly improved convergence properties, thanksto a set of accumulated inequality constraits that are adaptivelyformulated and aggregated at each iteration. The performanceof proposed approach is illustrated on one simple antennastructure.II. BACKGROUND AND MOTIVATIONWe consider a P -port electrical/electromagnetic structureparameterized by ρ independent parameters ϑ1, . . . , ϑρ col-lected in vector ϑ ∈ Θ ⊂ Rρ. All parameters are nor-malized so that Θ is a ρ-dimensional hypercube. We aimat the construction of a reduced-order model whose P × Pscattering responses H(s;ϑ) approximate the true systemresponse H˘k,m = H˘(sk;ϑm) assumed to be available fromsome measurement or field solver at k¯ discrete frequenciessk = jωk and m¯ parameter samples ϑm.The model is constructed by enforcingH(jωk;ϑm) ≈ H˘k,m, k = 1, . . . , k¯, m = 1, . . . , m¯, (1)based on the standard rational model structureH(s;ϑ) =N(s,ϑ)D(s,ϑ)=∑n¯n=0Rn(ϑ)ϕn(s)∑n¯n=0 rn(ϑ)ϕn(s), (2)where ϕn(s) = (s− qn)−1 is the partial fraction basis relatedto the n-th ”basis” pole qn. Numerator and denominatorcoefficients Rn(ϑ) and rn(ϑ) depend on the parameter ϑthrough suitable multivariate basis function expansions (see [2]for details), e.g. through orthogonal or trigonometric polyno-mials. Model identification is carried out through the so-calledParameterized Sanathanan-Koerner (PSK) iteration [8]min∥∥∥∥∥Nµ(jωk,ϑm)− Dµ(jωk,ϑm) H˘k,mDµ−1(jωk,ϑm)∥∥∥∥∥2F(3)where the denominator D0 is initialized at 1 and µ = 1, 2, . . .is the iteration index. This scheme is a simple iteratively re-weighted least squares fit, which stops when the estimates ofthe denominator coefficients stabilize.In its simplest form, the PSK iteration is not able to enforcemodel stability. In fact, even though the initial basis poles qnare stable, the parameterization of denominator coefficientsrn(ϑ) may lead the poles of H(s;ϑ) (the zeros of D(s,ϑ)) tohave positive real part. In [3], [7] it has been demonstrated thatmodel stability for all ϑ ∈ Θ can be guaranteed by ensuringthe denominator D(s,ϑ) to be a passive immittance functionsatisfying the following Positive-Realness (PR) condition<{D(jω,ϑ)} > 0, ∀ω, ∀ϑ ∈ Θ (4)where operator <{·} extracts the real part. This propertymotivated the introduction of a stability-preserving PSK al-gorithm presented in [3], [7], which enforces (4) at discretefrequency/parameter points by embedding a set of linearinequality constraints<{Dµ(jω,ϑ)} > 0, ∀(jω,ϑ) ∈ Vµ−1 (5)when solving (3) at each iteration µ, where the setVµ−1 = {(jωµ−1r ,ϑµ−1r ), r = 1, . . . , r¯µ−1} (6)collects the location of all negative local minima ofDµ−1(jω,ϑ) in (ω,ϑ) ∈ [0,+∞)×Θ. These are extracted ina pre-processing phase by applying a parameterized passivitycheck [9] to the denominator function Dµ−1(s,ϑ) availablefrom previous iteration µ − 1. The constraints (5) facilitatethe enforcement of uniform PR conditions through the PSKiterations, by constraining the new denominator estimate beingcomputed to be PR at these points. The effectiveness of thisapproach was illustrated by the several test cases documentedin [3], [7].The main problem of the above stabilized PSK iteration isthe possible lack of convergence. The constraints (5) are effec-tive only at their precise location (jωµ−1r ,ϑµ−1r ), but nothingprevents the denominator to become negative at completelydifferent locations at the next iteration. Moreover, it may bethe case that the denominator coefficients do not stabilize, withundesired erratic behaviors through iterations. We would likeeach coefficient update to vanish as iterations progress,dµn(ϑ) =∣∣rµn(ϑ)− rµ−1n (ϑ)∣∣ µ→∞−−−−→ 0 ∀n, ∀ϑ. (7)However, as pointed out in [10], there is no guarantee that suchconvergence condition occurs. The left panels of Fig. 1 illus-trate this problem on an H-shaped antenna test case (see belowfor details on this structure). The top-left panel confirms thatthe denominator coefficients never stabilize through iterations.Although the model responses are very accurate (middle-leftpanel), the lack of uniform PR-ness of the model denominatoris not able to constrain the poles to have a negative real part(bottom-left panel).We remark that this problem is well-known even in non-parameterized passive macromodeling, for which applicationof non-convex formulations of the passivity constraints maylead to non-converging passivity enforcement schemes [1].III. ROBUST ITERATIONSA solution to the above-described convergence issues isavailable through the so-called “robust iterations”, which wereintroduced in [11] for non-parameterized passivity enforce-ment (see [1] for a complete treatment). In this work, we applya similar but more efficient procedure to the parameterizedsetting, with the objective of enforcing uniform model stability.The same strategy can of course be applied to the enforcementof uniform passivity [9], which is not of interest in this work.The main idea of [11] is as follows. Let us consider againthe PR denominator violation points Vµ−1 used to define thePR constraints for iteration µ, as defined in (5) and (6). Let usassume that the µ-th iteration is performed, that the PR-ness ofthe denominator is checked again, and that the correspondingnew PR violations are collected in set Vµ. If we aggregateall constraints in a cumulative set Vµ−1 ∪ Vµ and we usethe latter instead of Vµ−1 to repeat (5), we see that thisiteration µ will include constraints both where the denominatoris detected to be real negative, and also where the denominatorwill become real negative. This strategy has thus a predictiveor look-ahead property. The main disatvantage is however anincreased (approximately threefold) computational cost, sinceeach iteration is repeated three times.In this work, we propose a similar strategy that aggregatesconstraints using a look-behind strategy. In essence, eachiteration µ enforces PR constraints through<{Dµ(jω,ϑ)} > 0, ∀(jω,ϑ) ∈ ∪µ−1i=0 Vi, (8)implying that constraints are placed at the PR violations ofthe denominator at current and all previous iterations. Thisstrategy avoids by construction oscillating behaviors, where aviolation that is eliminated at some iteration reappears everytwo or more iterations.The left column of Fig. 2 shows stability violation locationson the parameter space for three successive fitting iterationsfor the same antenna example of Fig. 1, where a red to blackrelative color scale is used to represent the extent of eachlocal violation (in terms of minimum real part of the modeldenominator). It can be noted that, despite the constraints (5)enforced at each of the highlighted points, new violationsreappear at the same locations at subsequent iterations, thusimpairing convergence. Conversely, the right column of Fig. 2depicts the accumulated constraints in case of proposed robustimplementation, where each panel represents constraints fromprevious iterations using transparent dots. The resulting set canbe interpreted as an iterative refinement of the parameter space,which zooms in the regions characterized by the stabilityviolations that are not eliminated in one step. This providesa clear illustration that the aggregation of all constraints isequivalent to narrowing the feasible set of the optimizationproblem, which is obtained as the intersection of all feasiblesets at all previous iterations.IV. RESULTSWe demonstrate the performance of the look-behind robustiterations on an H-shaped antenna (number of ports P = 1),already used above to illustrate convergence issues of thestandard PSK method. The structure, whose geometry isdescribed in full details in [12], is parameterized by its lengthL, which varies in the range L = [7, 10] mm, and the aperturewidth W ∈ [1.5, 1.8] mm. Through repeated EM simulations,Fig. 1: Macromodeling results using standard PSK iteration (left panels) and proposed look-behind robust iterations (rightpanels) applied to an H-shaped antenna. Top panels: evolution of coefficient updates through iterations. Middle panels: selectedmodel responses for few parameter configurations, compared with raw data. Bottom panels: parameterized model poles.we gather a set of m¯ = 16 scattering responses, withinthe frequency band [0, 5.5] GHz, for different parametervalues uniformly distributed on a structured bi-dimensionalgrid. The model is synthesized with n¯ = 10 poles andChebychev polynomials as parameter basis functions withorders {3, 3} and {2, 2} for numerator and denominatorexpansions, respectively.The results of proposed robust implementation are depictedin the right panels of Fig. 1. The top-right panel shows thatthe coefficient updates at each subsequent iteration becomesmaller and smaller, as a clear evidence of convergence. Incontrast with the results from the standard implementation(top-left panel), where denominator coefficients change signif-icantly as µ increases, the new robust algorithm successfullyconverges after 11 iterations. Thanks to the aggregate PRconstraints (8) the model results uniformly stable, as thebottom-right panel confirms by showing a sweep of the modelpoles in the parameter space. Model accuracy is not com-promised, as depicted in the middle-right panel. The worst-case relative error of the stable model obtained by proposedapproach is 3.6 ·10−4, whereas the corresponding error for theunconstrained identification is 1.6 · 10−4.Iteration 17 8 9 101.51.551.61.651.71.751.8Standard7 8 9 101.51.551.61.651.71.751.8ProposedIteration 27 8 9 101.51.551.61.651.71.751.8StandardIteration 37 8 9 101.51.551.61.651.71.751.8StandardFig. 2: Location in the parameter space Θ of stability con-straints for three successive fitting iterations. Left column:standard approach of [7] with independent stability constraintsat each iteration. Right column: proposed robust approach,with constraints accumulation through iterations (violationextent is represented with a red-to-black color scale, withtransparent dots representing the constraints at previous itera-tions).V. CONCLUSIONIn this paper, we presented a robust, stability preserving,implementation of the well known Parameterized Sanathanan-Koerner iteration, which brings major improvements in termsof convergence capabilities with respect to previous works [3],[7]. The presented strategy aims at avoiding that the modelcoefficients undergo oscillatory behaviors as the fitting itera-tions proceed, undermining the overall convergence. This isachieved by stacking stability constraints as the number ofiterations grows through a look-behind strategy. The propsedapproach is illustrated on a relevant antenna test-case.REFERENCES[1] S. Grivet-Talocia and B. Gustavsen, Passive macromodeling: Theory andapplications. John Wiley & Sons, 2015, vol. 239.[2] P. Triverio, S. Grivet-Talocia, and M. S. Nakhla, “A parameterizedmacromodeling strategy with uniform stability test,” IEEE Transactionson Advanced Packaging, vol. 32, no. 1, pp. 205–215, 2009.[3] S. Grivet-Talocia and R. Trinchero, “Behavioral, parameterized, andbroadband modeling of wired interconnects with internal discontinu-ities,” IEEE Transactions on Electromagnetic Compatibility, vol. 60,no. 1, pp. 77–85, 2018.[4] F. Ferranti, L. Knockaert, and T. Dhaene, “Parameterized s-parameterbased macromodeling with guaranteed passivity,” IEEE Microwave andWireless Components Letters, vol. 19, no. 10, pp. 608–610, 2009.[5] ——, “Passivity-preserving parametric macromodeling by means ofscaled and shifted state-space systems,” IEEE Transactions on Mi-crowave Theory and Techniques, vol. 59, no. 10, pp. 2394–2403, 2011.[6] E. R. Samuel, L. Knockaert, F. Ferranti, and T. Dhaene, “Guaranteedpassive parameterized macromodeling by using sylvester state-spacerealizations,” IEEE Transactions on Microwave Theory and Techniques,vol. 61, no. 4, pp. 1444–1454, 2013.[7] M. De Stefano, S. Grivet-Talocia, T. Bradde, and A. Zanco, “A frame-work for the generation of guaranteed stable small-signal bias-dependentbehavioral models,” in 2018 13th European Microwave IntegratedCircuits Conference (EuMIC). IEEE, 2018, pp. 142–145.[8] C. Sanathanan and J. Koerner, “Transfer function synthesis as a ratioof two complex polynomials,” IEEE transactions on automatic control,vol. 8, no. 1, pp. 56–58, 1963.[9] A. Zanco, S. Grivet-Talocia, T. Bradde, and M. De Stefano, “Enforc-ing passivity of parameterized lti macromodels via hamiltonian-drivenmultivariate adaptive sampling,” IEEE Transactions on Computer-AidedDesign of Integrated Circuits and Systems, 2018.[10] S. Lefteriu and A. C. Antoulas, “On the convergence of the vector-fittingalgorithm,” IEEE Transactions on Microwave Theory and Techniques,vol. 61, no. 4, pp. 1435–1443, 2013.[11] B. 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On stabilization of parameterized macromodeling

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