Sensitivity Analysis using Data-Driven Parametric Macromodels

Krishnan Chemmangat, Francesco Ferranti, Luc Knockaert, and Tom DhaeneDepartment of Information Technology, Ghent University - IBBT,

Gaston Crommenlaan 8, Bus 201, B-9050, Gent, Belgium.email: {krishnan.cmc, francesco.ferranti, luc.knockaert, tom.dhaene}@intec.ugent.be

Abstract

An accurate parametric macromodeling method which buildsthe parameterized frequency behavior of systems from fre-quency data samples is presented. The method aims to calcu-late parametric sensitivity responses of the model with respectto design parameters over the entire design space. A judiciouslychosen interpolation scheme is used to parameterize state-spacematrices such that parametric sensitivities can be computed an-alytically. The modeling capability of the proposed method isvalidated by a pertinent numerical example.

Introduction

The use of parametric macromodels as an approximation ofcomplex systems for design space exploration, design optimiza-tion etc., is becoming increasingly important because of theirefficiency in terms of reduction in computation time with re-spect to the original complex system. It is equally importantthat these parametric macromodels can accurately calculate theparametric sensitivity responses over the entire design space ofinterest, implying that this information could be used in dif-ferent stages of the design process such as sensitivity analysis,gradient-based design optimization etc.

One of the most common approaches in calculating local sen-sitivities is the adjoint variable method [1, 2].The main attrac-tiveness of this approach is that sensitivity information can beobtained from at most two systems analyses regardless of thenumber of designable parameters. However, these methods in-volve the calculation of system matrix derivatives, which aremost frequently estimated by means of finite difference approx-imations.

In recent years, there has been ongoing research on paramet-ric macromodeling based on the interpolation of systems [3, 4].These methods interpolate a set of frequency dependent uni-variate models, called root macromodels, over the parameterspace, yielding a complete parametric macromodel. As an ex-tension of these methods, some techniques have been presentedin [5, 6], where instead of interpolating the root macromod-els, the corresponding state-space matrices are interpolated toobtain a parametric macromodel. These methods allow a pa-rameterization of both poles and residues, thereby providing animproved modeling capability as compared to [3, 4].

This paper proposes a parametric macromodeling technique,which builds parametric sensitivity responses efficiently and ac-curately over the entire design space of interest. With a properchoice of the interpolation scheme, i.e. at least continuouslydifferentiable, the state-space matrices are parameterized asin [5, 6] to build parametric sensitivity macromodels which areable to describe parametric sensitivities analytically. Pertinentnumerical results validate the proposed parametric macromod-eling approach.

Parametric Sensitivity MacromodelingThe macromodeling process starts with a set of data sam-

ples {(𝑠, �⃗�)𝑘,H(𝑠, �⃗�)𝑘}𝐾𝑡𝑜𝑡

𝑘=1 which depends on frequency andseveral other design parameters. From these data samples, fre-quency dependent rational models in pole-residue form are builtfor all grid points in the design space by means of the VectorFitting (VF) technique [7], yielding the rational model

R�⃗�𝑘(𝑠) =

𝑁𝑃∑𝑛=1

c�⃗�𝑘𝑛

𝑠− 𝑎�⃗�𝑘𝑛+ d�⃗�𝑘 (1)

The rational model in (1) has 𝑁𝑃 poles with 𝑎�⃗�𝑘𝑛 , c�⃗�𝑘𝑛 andd�⃗�𝑘 representing poles, residues and the constant term respec-tively at the design point �⃗�𝑘 = (𝑔

(1)𝑘1

, ..., 𝑔(𝑁)𝑘𝑁

). A pole-flippingscheme is used to enforce strict stability [7] and passivity en-forcement can be accomplished using one of the robust stan-dard techniques [8, 9] resulting in stable and passive rationalunivariate macromodels called root macromodels.

Each of these root macromodels R�⃗�𝑘(𝑠), corresponding to aspecific design space point �⃗�𝑘, is converted from a pole-residueform into a state-space form:

R�⃗�𝑘(𝑠) = C�⃗�𝑘(𝑠I−A�⃗�𝑘)−1B�⃗�𝑘 +D�⃗�𝑘 (2)

The state-space matrices A�⃗�𝑘,B�⃗�𝑘,C�⃗�𝑘,D�⃗�𝑘 in (2) are param-eterized with the help of different interpolation schemes whichare at least continuously differentiable over rectangular grids.The continuous differentiability of the interpolation scheme en-sures that the derivatives are smooth and sufficiently accurate.Here we have investigated two different interpolation schemes,namely Cubic Spline (CS) interpolation and Piecewise CubicHermite Interpolation (PCHIP) which are briefly described inthe sequel.

Cubic Spline (CS) Interpolation In this method a cubic splinepolynomial 𝑠𝑖(𝑥) is built for each interval 𝑥𝑖 ≤ 𝑥 ≤ 𝑥𝑖+1,𝑖 = 1, . . . , 𝑛 of the input-output data set (𝑥𝑖, 𝑦𝑖)

𝑛𝑖=1. The coef-

ficients of each of these polynomials are calculated by impos-ing the first and second order derivative continuity at the datapoints along with a 𝑛𝑜𝑡-𝑎-𝑘𝑛𝑜𝑡 end condition [10]. Once thesecoefficients are computed, the derivatives of the overall splineinterpolation function can be calculated analytically in terms ofits coefficients. If the data to be interpolated happen to be ma-trices, each entry of the matrices is independently interpolated.

The univariate CS interpolation can be extended to higherdimensions by means of a tensor product implementation [10].

Piecewise Cubic Hermite Interpolation (PCHIP) ThePCHIP method is a monotonic shape preserving interpolationscheme. As in the CS interpolation, each data interval is mod-eled by a cubic polynomial with additional constraints to pre-serve the monotonicity locally [11]. An extension to higher

dimension can be performed by a tensor product implementa-tion [10]. As in the CS interpolation case, the derivatives arecalculated analytically. Since PCHIP is a bounded interpola-tion scheme it works better for non-smooth datasets, whereasCS could result in overshoots or oscillatory behavior. How-ever, PCHIP is only continuous in the first derivatives, unlikeCS which is continuous in the second derivatives, which effectsthe smoothness of the derivatives obtained from PCHIP [11].

Parametric Sensitivity MacromodelsThe set of root macromodel state-space matrices A�⃗�𝑘,B�⃗�𝑘,

C�⃗�𝑘,D�⃗�𝑘 is interpolated entry-wise and the multivariate mod-els A(�⃗�),B(�⃗�),C(�⃗�),D(�⃗�) are built, yielding a parametricmacromodel over the entire design space,

R(𝑠, �⃗�) = C(�⃗�)(𝑠I−A(�⃗�))−1B(�⃗�) +D(�⃗�). (3)

A parametric macromodel of sensitivity responses is obtainedby differentiating (3) with respect to the design parameters �⃗�,i.e.:

∂

∂�⃗�R(𝑠, �⃗�) =

∂C(�⃗�)

∂�⃗�(𝑠I−A(�⃗�))−1B(�⃗�) +

C(�⃗�)(𝑠I−A(�⃗�))−1 ∂A(�⃗�)

∂�⃗�(𝑠I−A(�⃗�))−1B(�⃗�) +

C(�⃗�)(𝑠I−A(�⃗�))−1 ∂B(�⃗�)

∂�⃗�+

∂D(�⃗�)

∂�⃗�(4)

In (4), ∂∂�⃗�R(𝑠, �⃗�) is a function of the parameterized matrices,

A(�⃗�),B(�⃗�),C(�⃗�),D(�⃗�) and their derivatives, computed effi-ciently and analytically using the CS and PCHIP interpolationschemes.

A schematic of the proposed method is shown in Fig. 1.

Numerical ExampleA coaxial cable is modeled with cross section shown in Fig.

2. The relative permittivity 𝜖𝑟𝑒𝑙 of the dielectric is chosen equalto 2.5. The impedance matrix 𝑍(𝑠, 𝑎, 𝐿) of the model is cal-culated analytically [12] as a function of the radius of the innerconductor 𝑎 and the length 𝐿, in addition to frequency, on agrid of 150× 15× 15 samples (𝑓𝑟𝑒𝑞, 𝑎, 𝐿). The correspondingranges of these parameters are shown in Table 1.

Parameter Min MaxFrequency (𝑓𝑟𝑒𝑞) 10 KHz 2 GHzInner radius (𝑎) 2 mm 3 mmLength (𝐿) 100 mm 110 mm

Table 1: Design parameters of the coaxial cable

A set of stable and passive root macromodels has been builtfor 8 values of 𝑎 and 8 values of 𝐿 using VF. The remainingdata are used for validation. The number of poles 𝑁𝑃 for eachroot macromodel is 18, selected using an error-based bottom upapproach. Each root macromodel has been converted to state-space form and the state-space matrices have been interpolatedby the CS and PCHIP interpolation methods. Next, the para-metric sensitivity of 𝑍(𝑠, 𝑎, 𝐿) with respect to 𝑎 and 𝐿 has

Figure 1: Schematic of parametric sensitivity macromodeling.

Figure 2: Cross section of the coaxial cable.

been computed by means of the derivatives (4) of the trivariatemacromodel and the analytical model computed from [12]. Theaccuracy of the model and its derivatives for the two interpola-tion methods are measured in terms of the weighted rms-errordefined as:

ERMS(�⃗�) =

√∑𝑃 2

𝑖=1

∑𝐾𝑠

𝑘=1 ∣𝑤𝑍𝑖(𝑠, �⃗�)(𝑅𝑖(𝑠𝑘, �⃗�)− 𝑍𝑖(𝑠𝑘, �⃗�))∣2

𝑃 2𝐾𝑠.

(5)In (5) 𝑃 is the number of ports, 𝐾𝑠 is the number of frequencysamples and 𝑤𝑍𝑖

= ∣𝑍𝑖(𝑠𝑘, �⃗�)∣−1 is the weighting function forthe error. The worst case rms-error over the validation grid ischosen to assess the accuracy and the quality of the parametricsensitivity macromodels

EMaxRMS = max

�⃗�ERMS(�⃗�), �⃗� ∈ validation grid (6)

The maximum weighted rms-error calculated from (6) for themodel and its sensitivities is tabulated in Table 2.

MethodQuantity CS PCHIPZ(𝑠, 𝑎, 𝐿) 0.0054 0.0054∂Z(𝑠,𝑎,𝐿)

∂𝑎 0.0061 0.0325∂Z(𝑠,𝑎,𝐿)

∂𝐿 0.0119 0.0194

Table 2: Modeling accuracy of the proposed method

Fig. 3 shows the magnitude of 𝑍11 as a function of frequencyand 𝑎 for 𝐿 = 105 𝑚𝑚, while Fig. 4 shows the magnitude ofthe corresponding parametric sensitivity ∂Z11

∂𝑎 obtained by theCS scheme. Fig. 5 shows the magnitude of 𝑍12 as a func-tion of frequency and 𝐿 for 𝑎 = 2.5 𝑚𝑚, while Fig. 6 showsthe magnitude of the corresponding parametric sensitivity ∂Z12

∂𝐿obtained by the CS scheme. Fig. 7 compares the magnitude of∂Z11

∂𝑎 obtained by the analytical model, and the CS and PCHIPmethods for the values 𝑎 = 2.5 𝑚𝑚 and 𝐿 = 105 𝑚𝑚, whichhave not been used for the generation of the root macromodels.Fig. 8 shows the magnitude of ∂Z12

∂𝐿 for the same values of 𝑎and 𝐿. A very good agreement between the methods can beobserved.

00.5

11.5

2

2

2.5

310

−5

100

105

1010

Frequency [GHz]

Inner Radius, a [mm]

|Z11

| [Ω]

Figure 3: Magnitude of 𝑍11 for 𝐿 = 105 𝑚𝑚.

00.5

11.5

2

2

2.5

310

0

105

1010

Frequency [GHz]

Inner Radius, a [mm]

| ∂Z 11

/∂a |

[Ω/m

]

Figure 4: Magnitude of ∂𝑍11

∂𝑎 (CS) for 𝐿 = 105 𝑚𝑚.

Fig.9 shows the rms-error distribution of the parametricmacromodel using the CS interpolation scheme with respect tothe analytical model of [12] over the complete design space.

00.5

11.5

2

100

105

11010

0

102

104

106

108

Frequency [GHz]

Cable length, L [mm]

|Z12

| [Ω]

Figure 5: Magnitude of 𝑍12 for 𝑎 = 2.5 𝑚𝑚.

00.5

11.5

2

100

105

11010

−5

100

105

1010

Frequency [GHz]

Cable length, L [mm]

| ∂Z 12

/∂L |

[Ω/m

]

Figure 6: Magnitude of ∂𝑍12

∂𝐿 (CS) for 𝑎 = 2.5 𝑚𝑚.

0 0.5 1 1.5 210

1

102

103

104

105

106

107

108

109

Frequency [GHz]

| ∂Z 11

/∂a |

[Ω/m

]

AnalyticalCSPCHIP

Figure 7: Magnitude of ∂𝑍11

∂𝑎 for 𝑎 = 2.5 𝑚𝑚 and 𝐿 = 105𝑚𝑚.

0 0.5 1 1.5 210

1

102

103

104

105

106

107

108

Frequency [GHz]

| ∂Z 12

/∂L |

[Ω/m

]

AnalyticalCSPCHIP

Figure 8: Magnitude of ∂𝑍12

∂𝐿 for 𝑎 = 2.5 𝑚𝑚 and 𝐿 = 105𝑚𝑚.

100102

104106

108110

2

2.5

310

−4

10−3

10−2

Length, L [mm]Inner radius, a [mm]

RMS−

Erro

r

Figure 9: RMS-Error of 𝑍 for the entire design space (CS).

100102

104106

108110

2

2.5

310

−4

10−3

10−2

Length, L [mm]Inner radius, a [mm]

RMS−

Erro

r

Figure 10: RMS-Error of ∂𝑍∂𝐿 for the entire design space (CS).

Fig.10 shows the rms-error distribution of the parametric sensi-tivity macromodel of ∂𝑍

∂𝐿 with respect to the analytical model.Similar results were obtained for other cases in Table 2. We notethat a good accuracy is achieved by both interpolation methods,but the CS scheme leads to a lower average error, probably dueto the continuity of the second derivatives.

ConclusionsThis paper presents a new macromodeling technique for ac-

curately calculating parametric sensitivity responses from fre-quency sampled data. The parameterization of the model andcalculation of the parametric sensitivities is based on the inter-polation of state-space matrices with a judicious choice of the

interpolation scheme. The proposed method has been validatedusing a pertinent numerical example, thereby demonstrating itsmodeling capability.

AcknowledgmentsThis work was supported by the Research Foundation Flan-

ders (FWO).

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