A Perturbation Scheme for Passivity Verification and Enforcement

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[33], [34], [40], temperature or biasing conditions [35], etc. H˘ (s; ϑ) ∈ CP ×P is the scattering matrix of the system at well- Few approaches are able to provide parameterized models defined ports. The accent ˘ will be used to label the original that are uniformly passive througout the parameter domain. or “true” system responses, that are generally unknown. These approaches impose some constraints on the model struc- The “true” response is assumed to be available, either via ture and/or the model identification procedure. For instance, direct measurements or via numerical calculation based on a we can cite the interpolatory approaches that first construct first-principle model (e.g., the numerical solution of Maxwell’s univariate models for fixed parameter values, and then recover equations for an electromagnetic system) at a set of fixed a multivariate model by interpolating such “grid” or “root” frequencies fk, k = 1,..., k¯ over a given frequency band macromodels. If the root macromodels are passive and if [fmin,fmax], with f1 = fmin and fk¯ = fmax, and for a set of a passivity-preserving interpolation scheme is used, also the fixed parameter values ϑm, m =1,..., m¯ spanning the range multivariate model will be uniformly passive [37]–[40]. This [ϑmin, ϑmax]. We will denote this characterization as approach has two main problems. First, passivity-preserving ˘ ˘ ¯ interpolation schemes are overconstrained and may lead to Hk,m = H(j2πfk; ϑm), k =1,..., k, m =1,..., m.¯ (1) inaccurate behavior of the interpolant. This problem is usually Since a closed-form dependence of the true system response addressed by increasing the number of root macromodels, on s and ϑ is not available, there comes the need of construct- thus degrading the efficiency of the identification process. ing a parameterized model that fits or interpolates the above Second, the multivariate model is cast as a linear combination data points, and that can be solved efficiently in frequency and of root macromodels. Therefore, if each root macromodel time domain. We assume here the following model structure is characterized by a given order n¯, the multivariate model N n¯ will have a larger order Ln¯, where L is the number of root (s, ϑ) Rn(ϑ) ϕn(s) H(s; ϑ)= = n=0 . (2) macromodels contributing to the interpolation. On one hand, D(s, ϑ) n¯ r (ϑ) ϕ (s) P n=0 n n this model structure may not be related to the physics of the underlying system (if n¯ natural frequencies are required, this The frequency-dependent basisP functions ϕn(s) are partial should be true for all parameter values). On the other hand, fractions based on a prescribed set of distinct n¯r real poles R− ′ ′′ C− model complexity is larger than strictly necessary. qn ∈ and n¯c complex pole pairs pn = pn ± jpn ∈ , The approach that we follow in this work is different. where ϕ0(s)=1, We embed the parameter dependence through a dedicated −1 ϕn(s) = (s − qn) , for n =1,..., n¯r, (3) model structure, based on a suitable expansion into frequency- dependent and parameter-dependent basis functions [32]–[35]. and −1 ∗ −1 Thus, model structure is general and can be tuned by selecting ϕn¯r +2ν−1(s) = (s − pν ) + (s − pν ) −1 ∗ −1 (4) the appropriate basis functions for the specific application ϕn¯r +2ν (s) = j(s − pν ) − j(s − pν ) at hand. Model coefficients are then identified through a for ν = 1,..., n¯ , where ∗ denotes the complex conjugate. Generalized Sanathanan-Koerner iteration [41]. This approach c The (real-valued) numerator and denominator coefficients are is standard [30], [32]. further expressed as The novel contribution of this work is twofold. First, we devise an adaptive sampling scheme in the parameter space ℓ¯ ℓ¯ based on some special features of the Hamiltonian eigenspec- Rn(ϑ)= Rn,ℓ ξℓ(ϑ), rn(ϑ)= rn,ℓ ξℓ(ϑ) (5) trum. This scheme is able to pinpoint all passivity violations Xℓ=1 Xℓ=1 throughout the frequency and parameter plane. Second, we where the basis functions ξℓ(ϑ) are responsible for reproducing extend a known passivity enforcement scheme for univariate the variations induced by the external parameter ϑ. We remark models based on singular value perturbation to the multi- that the adopted model structure is the same as discussed variate case. The result is an iterative passivity enforcement in [1] and originally introduced in [32], [36], [37], [39], [40], scheme for multivariate macromodels that is able to guarantee where polynomials, piecewise polynomials, or trigonometric uniform passivity as well as model accuracy with constant polynomials were used as basis functions ξℓ. This work model order throughout the parameter space. To the best of makes no a-priori assumption on the specific choice of basis Authors’ knowledge, this is the first passivity verification and functions, which should be defined considering the particular enforcement scheme that is applicable to multivariate models. application at hand. We focus this work on the case of a single external parameter, Model structure (2) guarantees that H(s; ϑ) is a rational or equivalently on bivariate macromodels. The extension to the function of s, with implicitly parameterized poles and residues muldimensional case will be documented in a future report. (explicit parmeterization of model poles is to be avoided, due to the possibly non-smooth behavior in case of bifur- II. PRELIMINARIES AND NOTATION cations [36]). Thanks to the assumed linear dependence of both model numerator N(s, ϑ) and denominator D(s, ϑ) on the Let us consider a generic LTI multiport structure with respective coefficients R and r , the latter can be easily ˘ n,ℓ n,ℓ transfer matrix H(s; ϑ), where s is the Laplace variable, and evaluated during a model identification step by enforcing the ϑ ∈ Θ = [ϑmin, ϑmax] is some external parameter, e.g., fitting condition some material or geometrical characteristic of the underlying physical system. Throughout this work, it is assumed that H(j2πfk; ϑm) ≈ H˘ k,m, k =1,..., k,¯ m =1,..., m.¯ (6) 3

in least squares sense. This is achieved here through a linear by determining the correction ∆Rn,ℓ that is required to ensure relaxation of (6) known as (Generalized) Sanathanan-Koerner that the perturbed scattering response of the parameterized (GSK) iteration [1], [32], [41]. This approach transforms model H(s; ϑ) is uniformly Bounded Real in the prescribed the nonlinear least-squares problem arising from (6) into an parameter range. Note that we do not perturb the denomina- iterative sequence of weighted linear least squares problem, tor coefficients,b since we would like to preserve the model whose solution is straightforward. This procedure is standard poles, i.e., the zeros of the denominator D(s, ϑ). This choice in parameterized model identification and is not further dis- is standard in practically all passivity enforcement methods cussed here. The Reader is referred to [1] for more details applicable to univariate models. on the GSK iteration, and to [44], [45] for a discussion Setting up the model perturbation requires a precise local- on its convergence properties. We finally remark that, in ization of the regions in the frequency-parameter plane where order to guarantee uniqueness in the model representation, we the Bounded Realness condition (8) is violated. This problem normalize the model coefficients by setting r0,1 =1. is addressed in Section V, which first casts some known results for univariate models in the parameterized framework III. PROBLEM STATEMENT (Section V-A) and then extends the check to the multivariate The above-mentioned GSK iteration is not able to enforce case (Section V-B). Section VI presents the main passivity model and passivity by construction, since no explicit passivity enforcement scheme. In particular, Section VI-A constructs constraints are enforced during model identification. We recall the algebraic constraints that, when iteratively enforced, lead that the model is passive for a given parameter value ϑ if and to removal of all passivity violations; Section VI-B constructs only if the corresponding scattering matrix H(s; ϑ) is Bounded a cost function whose minimization will ensure preservation Real [12], [13]: of model accuracy during perturbation; and Section VI-C 1) H(s; ϑ) is regular for Re {s} > 0, presents the main iterative passivity enforcement scheme. All 2) H∗(s; ϑ)= H(s∗; ϑ), these developments require the construction of a parameter- H dependent descriptor realization of the model, which is dis- 3) IP − H (s; ϑ)H(s; ϑ) ≥ 0 for Re {s} > 0, H cussed next. where is the Hermitian transpose, and IP is the identity matrix of size P . Condition 1) is related to (asymptotic) stability, which is here assumed a priori (as can be readily IV. DESCRIPTOR REALIZATIONS veryfied by a suitable parameter sweep of the parameter- Starting from model (2), we define the following descriptor dependent model poles), whereas condition 2) ensuring a realization Ex˙ = A(ϑ)x + Bu real impulse response is automatically satisfied thanks to the (10) assumed model structure (2). Condition 3), implying no energy ( y = C(ϑ)x gain from the model throughout the open right complex plane, RN+P is instead more difficult to check and to enforce. Note that it where x ∈ denotes internal generalized states with RP is sufficient to check condition 3) only on the imaginary axis N =nP ¯ , and u, y ∈ are incident and reflected scat- s = jω, tering waves at the model ports. The descriptor matrices are constructed as I HH H R (7) P − (jω; ϑ) (jω; ϑ) ≥ 0 ∀ω ∈ , I 0 A B E = N N,P , A(ϑ)= 0 0 , which in turn is equivalent to 0 0 C (ϑ) D (ϑ) P,N P,P 2 2 σmax{H(jω; ϑ)}≤ 1, ∀ω ∈ R, (8) 0N,P C(ϑ)= C1(ϑ) D1(ϑ) , B(ϑ)= , where σ {·} extracts the largest singular value of its matrix −IP (ϑ) max argument. (11) Several solutions are in fact available for checking and where 0J,K denotes the all-zero matrix block of size enforcing (8) on univariate models that depend only on fre- J × K. In (11), the matrices {A0, B0, C1(ϑ), D1(ϑ)} and quency, which in our setting would correspond to the model {A0, B0, C2(ϑ), D2(ϑ)} provide regular state-space realiza- N instantiated for a fixed value of the external parameter ϑ. tions of model numerator (s, ϑ) and “extended” denominator D For a comprehensive review of such methods, the Reader is (s, ϑ)IP , respectively. Individual matrices in these realiza- referred to [1] and to [26]. What we are interested in this tions are here defined as work is the uniform passivity of the parameterized model A0 = blkdiag{A0r, A0c} T (12) H(s; ϑ) throughout the parameter range, so that the above B = BT , BT Boudned Realness conditions, in particular (8), will hold 0 0r 0c T ∀ϑ ∈ [ϑmin, ϑmax]. where is the matrix transpose, with The approach that is pursued here can be regarded as n¯r A0r = blkdiag{qnIP }n=1 an extension to the multivariate case, of existing standard n¯c p′ I p′′I approaches valid for non-parameterized systems. We start from A = blkdiag n P n P 0c −p′′I p′ I some initial non-passive model H(s; ϑ), and we perturb its n P n P n=1 (13) numerator coefficients as T B0r = 1,..., 1 ⊗ IP R R R (9) T n,ℓ = n,ℓ + ∆ n,ℓ ∀n,ℓ B0c = 2, 0,...,2, 0 ⊗ IP

b 4 where ⊗ is the Kronecker product, and Im {λ} = ω σ C (ϑ)= R (ϑ) ··· R (ϑ) σ¯ 1 1 n¯ ω 1 2 σ = 1 C2(ϑ)= IP r1(ϑ) ··· IP rn¯(ϑ) (14) ω1 D (ϑ)= R (ϑ) 1 0 Re {λ} D2(ϑ)= IP r0(ϑ) −ω1 It is straightforward to show that, with the above definitions, ω − 2 ω the transfer matrix associated to the descriptor form (10) 0 ω1 ω¯1 ω2 H(s; ϑ)= C(ϑ)(sE − A(ϑ))−1B (15) Fig. 1. Graphical illustration of Theorem 1. Left panel: finite SHH eigenvalues matches (2). We leave this tedious verification to the Reader, (empty dots highlight purely imaginary eigs); right panel: singular value ω see also [32]. trajectories, which cross the unit threshold at frequencies i corresponding to purely imaginary SHH eigenvalues. Local singular value maxima σ¯i occur at frequencies ω¯i. For this example, P = 2, ν = 2, Ip = {0, 2} and V. CHECKING PASSIVITY OF PARAMETERIZED MODELS Inp = {1}. Based on the descriptor form (10) of the model transfer function, we can formulate an algorithm for checking its that satisfy (17) for some v 6= 0. Of course, all these uniform passivity, with the objective of a precise localization eigenvalues depend on the considered parameter value ϑ. of eventual passivity violations. Therefore, the (finite part of the) SHH eigenspectrum is parameter-dependent A. Passivity Check of Univariate Models Λ(ϑ)= {λ(ϑ)= β(ϑ)/α(ϑ): α(ϑ) 6=0}. (19) Let us consider the univariate model obtained from (15) by “freezing” the parameter ϑ. Checking the passivity of such We remark that, due to the SHH structure of the pencil, the a model is a standard problem [1]. The most effective tool set Λ(ϑ) for each ϑ is symmetric with respect to both real to perform this check is the so-called Hamiltonian matrix and imaginary axis, so that eigenvalues are either occurring in associated to the model [16], [17], [23], which in the case of pairs {λ, −λ} if they are purely real or purely imaginary, or ∗ ∗ our descriptor form becomes a Skew-Hamiltonian/Hamiltonian quadruples {λ, −λ, λ , −λ } if they are complex-valued with (SHH) matrix pencil, also denoted Generalized Hamiltonian a nonvanishing real and imaginary part [46]–[48]. Pencil [18]–[21]. We define the two block-matrices Based on Theorem 1, if Λ(ϑ) includes a purely imaginary eigenvalue jω0, then the transfer matrix of the model has a A(ϑ) BBT E 0 M(ϑ)= T T , K = T singular value σ(jω0, ϑ)=1. We conclude that the singular C C A 0 E − (ϑ) (ϑ) − (ϑ) value trajectory for fixed ϑ in a neighborhood of jω crosses (16) 0 the unit threshold, which is the critical condition leading to a where M(ϑ) has Hamiltonian structure and K is skew- passivity violation, see (8). This violation will be for ω>ω Hamiltonian, forming the SHH matrix pencil (M(ϑ), K). We 0 if σ increases with frequency, and vice versa. Figure 1 il- have the following result: lustrates the relationship between imaginary SHH eigenvalues Theorem 1. Assume that the pencil (A(ϑ), E) has no purely and localized passivity violations based on singular value imaginary eigenvalues. Then, the following conditions are trajectories. equivalent: As proposed in [23], we define the two sets • σ =1 is a singular value of H(jω0; ϑ) with |ω0| < +∞; ν(ϑ) • jω0 is an eigenvalue of the pencil (M(ϑ), K), i.e., there χ(ϑ)= {ωi(ϑ)}i=1 , χ¯(ϑ)= {0} ∪ χ(ϑ) ∪{+∞} (20) exist some vector v0 6= 0 such that collecting the frequencies ωi(ϑ) > 0 corresponding to the the M(ϑ)v0 = jω0Kv0 ν(ϑ) purely imaginary SHH eigenvalues λi(ϑ) = jωi(ϑ) with positive (finite) imaginary part, sorted in ascending order. The The proof is straightforward and follows the same flow as the augmented set χ¯(ϑ) includes also the DC point ω = 0 and proof of Theorem 1 in [16]. See also [1], [19], [21]. 0 the infinte frequency ω = +∞. The elements of χ¯(ϑ) thus Matrix K is singular, since E is singular, with induce a subdivision of the positive frequency axis [0, +∞) dim ker(E)= P and dim ker(K)=2P . Therefore, we expect into ν(ϑ)+1 disjoint subbands that the eigenspectrum of the pencil (M(ϑ), K) includes at least 2P infinite eigenvalues. We recall that infinite eigenval- Ωi(ϑ) = (ωi(ϑ),ωi+1(ϑ)), i =0,...,ν(ϑ), (21) ues are characterized as pairs (α, β) such that where we defined ω0(ϑ)=0 and ων(ϑ)+1(ϑ)=+∞. Since αM(ϑ)v = βKv (17) all intersections of some singular value trajectory with the threshold σ = 1 is captured in the set χ(ϑ), each subband for some vector v, with α = 0. These eigenvalues will be can be flagged either as locally passive (if all singular disregarded in the following, and we will consider only the Ωi(ϑ) values are less than 1 in this band) or locally not passive finite eigenvalues of the pencil as (otherwise), by splitting the corresponding index sets into λ = β/α, α 6=0 (18) i ∈ Ip(ϑ) and i ∈ Inp(ϑ), respectively. The worst-case 5 passivity violation for each non-passive band is defined as the maximum singular value σ¯ (ϑ), occurring at some frequency i ω¯i(θµ) ω¯i(θµ+1) ωi(θµ) ωi(θµ+1) ω¯i(ϑ) ∈ Ωi(ϑ) with i ∈ Inp(ϑ). Numerical estimates of such σ¯i(ϑ) σmax(jω, ϑ) > 1 local maxima (¯ωi(ϑ), σ¯i(ϑ)) are easily determined through local sampling. See Figure 1 for a graphical illustration. ϑ

ψ(θµ+1) = 0 B. Uniform Passivity Check of Parameterized Models θµ+1 ν(θµ+1) = 2 The procedure discussed in Section V-A allows a passivity Γµ,i ψ(θµ) = 0 check of univariate models throughout the frequency axis, θµ ν(θµ) = 2 without any need of sampling the singular values of the transfer function, but requiring only an algebraic determination of some SHH eigenvalues. In this Section, we enrich this test by extending its scope to a uniform passivity check throughout (a) ω the parameter range ϑ ∈ Θ. The main tool that we consider is the auxiliary function ψ(ϑ), defined as ϑ | Re {λ(ϑ)} | ψ(ϑ) = min (22) ψ(θµ+1) = 0 λ(ϑ)∈Λ(ϑ) θµ+1 ρ(ϑ) ν(θµ+1) = 2 Γµ,i where ∈ is the spectral radius of ψ(θµ) = 0 ρ(ϑ) = maxλ(ϑ) Λ(ϑ) |λ(ϑ)| θµ the SHH pencil (16), computed considering only the finite ν(θµ) = 4 eigenvalues. This function vanishes when the set χ(ϑ) is non- empty, denoting the presence of purely imaginary eigenvalues. Conversely, if χ(ϑ) is empty and there are no imaginary ω eigenvalues, so that the univariate model is passive for the con- (b) sidered value of ϑ, the function ψ(ϑ) measures the distance of the SHH eigenspectrum from the imaginary axis, normalized ϑ to the largest eigenvalue magnitude. This normalization makes ψ(θµ+1) > 0 θµ+1 the test less sensitive to the finite numerical precision in the ν(θµ+1) = 0 eigenvalue computation, as well as independent on the units ψ(θµ) = 0 θµ and/or normalizations used for the frequency variable. ν(θµ) = 2 The proposed check performs an adaptive sampling of ψ(ϑ) within the range ϑ ∈ Θ = [ϑmin, ϑmax]. We start with an initial uniform partition into µ¯ subintervals with endpoints µ (c) ω θ = ϑ + (ϑ − ϑ ), µ =0,..., µ,¯ (23) µ min µ¯ max min and we apply the univariate passivity check by computing the ϑ

SHH eigenvalues at each θµ, as discussed in Section V-A. ψ(θµ+1) > 0 θµ+1 An adaptive reﬁnement process is then constructed, based on ν(θµ+1) = 0 the following observations. Considering a single subinterval ψ(θµ) > 0 θµ Θµ = [θµ,θµ+1], ν(θµ) = 0 • if ψ(ϑ)=0, ∀ϑ ∈ Θµ, the model is not passive in this subinterval due to the presence if imaginary SHH eigenvalues. For each ϑ, such eigenvalues delimit at least (d) ω one frequency band Ωi(ϑ) where at least one singular value of the model H(jω; ϑ) exceeds one, with a local maximum σ¯i(ϑ) occurring at frequency ω¯i(ϑ). ϑ • if ψ(ϑ) > 0, ∀ϑ ∈ Θµ, the model is uniformly passive ψ(θµ+1) > 0 θµ+1 in Θµ. ν(θµ+1) = 0 Assuming now that only information at the edges {θµ,θµ+1} of Θµ is available, we have the following subcases (Fig. 2): ψ(θµ) > 0 θµ ν(θµ) = 0 • if ψ(θµ) = ψ(θµ+1) = 0 and ν(θµ) = ν(θµ+1), we can infer that there exist at least one region Γµ,i of the frequency-parameter plane Γ ⊆ [0, +∞) × Θ µ ω where the model is uniformly non-passive (see Figure 2a). (e) For each region , we know the local singular value Γµ,i Fig. 2. Adaptive parameter sampling, see main text for details. maxima σ¯i(θµ), σ¯i(θµ+1), and the corresponding localization frequencies ω¯i(θµ), ω¯i(θµ+1). In addition, we 6

know all the edges of Γµ,i for θµ and θµ+1 from the Algorithm 1 Passivity check of parameterized models sets χ(θµ), χ(θµ+1). Since local passivity violations have Require: frequency basis ϕn for n =0,..., n¯; ¯ been identified, there is no need to refine subinterval Θµ. Require: parameter basis ξℓ for ℓ =1,..., ℓ; • if ψ(θµ) = ψ(θµ+1) = 0 and ν(θµ) 6= ν(θµ+1), Require: model coefficients Rn,ℓ, rn,ℓ in (2) and (5); the number of frequency bands possibly changes when Require: control parameters ϑmin, ϑmax, γ, κ, M; ¯ sweeping ϑ ∈ Θµ (see Figure 2b). In order to obtain a 1: set m =0 and number of initial samples µ¯0 = κ ℓ; 0 precise characterization, we refine Θµ by adding the new 2: set initial samples S0 = {θµ, µ =0,..., µ¯0} as in (23); point 3: repeat 1 4: for do θµ+1/2 = /2(θµ + θµ+1). (24) µ =1,..., µ¯m m 5: construct SHH pencil (M(θµ ), K); • if ψ(θ )=0 and ψ(θ ) > 0, or conversely ψ(θ ) > 0 m µ µ+1 µ 6: find imaginary SHH eigenvalues ωi(θµ ); and ψ(θ )=0, we have a transition from a passive m m µ+1 7: extract local singular value maxima (¯ωµ,i, σ¯µ,i); to a non-passive model while sweeping ϑ ∈ Θµ (see 8: end for Figure 2c). Therefore, we need to refine Θµ through (24) 9: m ← m +1 in order to track the particular ϑ∗ where the onset of a m 10: determine new samples Sm = {θµ , µ =1,..., µ¯m}; passivity violation occurs. 11: until Sm = ∅ or m = M • if both ψ(θ ) > 0 and ψ(θ ) > 0, we can have two m m µ µ+1 12: return passivity violations V = ∪m{(¯ωµ,i, σ¯µ,i), ∀µ,i}. subcases:

– ψ(ϑ) > 0 throughout Θµ: model is uniformly passive in Θµ and no refinement is necessary (Figure 2d); Figure 3 illustrates the proposed passivity check on a – ψ(ϑ) vanishes in some subinterval Θ∗ ⊆ Θµ, denot- practical example (discussed in more detail in Section VII-A). ing a passivity violation that is not visible from the Two different models of a PCB interconnect link are processed edges of Θµ (Figure 2e). We need to refine Θµ. by the proposed passivity check algorithm. Figure 3(a) plots These two cases are here discriminated by computing the function ψ(ϑ) for the two models. The model no. 1 is ψ(θµ+1/2), and by constructing the first-order interpola- not passive for any value of ϑ, hence ψ(θµ) = 0 for all tion error computed µ (the red dots). Correspondingly, Figure 3(b) shows that for all θµ samples, there is at least one imaginary SHH 1 2 1 2 1 2 (25) εµ+ / = ψ(θµ+ / ) − / [ψ(θµ)+ ψ(θµ+1)] . eigenvalue (yellow dots) that defines a frequency band where Refinement is applied if at least one singular value of the model is larger than one (red lines). Local singular value maxima are close to DC (black εµ+1/2 >γ ψ(θµ+1/2) , (26) squares). Note that, since the number of detected imaginary SHH eigenvalues is constant for all θ , no adaptive refinement equivalently, when the estimate of the midpoint through µ is necessary. The blue curve in Figure 3(a) shows instead linear interpolation is not sufficient to infer that ψ(ϑ) is ψ(ϑ) for model no. 2, which is locally not passive only in a uniformly positive within Θµ. The parameter γ can be restricted range of ϑ ∈ Θ∗ ≈ [437, 454] µm. Correspondingly, used to tune the selectivity of the refinement test, here imaginary SHH eigenvalues are only found in this range: see we use the conservative value γ =0.2. Figure 3(c), where adaptive refinement is clearly visible near We perform a total number M of refinement passes. At each the transitions between ψ(ϑ)=0 and ψ(ϑ) > 0, which define 1 pass, the new points θµ+ /2 are added to the previous subset the endpoints of Θ∗. Finally, Figure 3(d) shows the frequency- {θµ, µ =0,..., µ¯}, which is resorted in ascending order and dependent singular values of model no. 1 for ϑ = 400 µm, reindexed, by suitably redefining µ¯. Algorithm 1 summarizes where the frequency of the unique imaginary SHH eigenvalue the proposed multivariate passivity check in pseudocode form is highlighted by a yellow dot. (step 10 embeds the determination of new parameter samples θm at each refinement pass m, according to the above adaptive µ VI. ENFORCING PASSIVITY OF PARAMETERIZED MODELS process). Throughout this work, we use M = 10. A final remark is in order about the initial sampling (23). Based on the multivariate passivity characterization dis- Here, we need to make sure that we do not miss important cussed in Section V-B, we are now ready to formulate our information due to an excessively coarse sampling. On the proposed passivity enforcement algorithm. A perturbed model other hand, we do not want to spend unnecesary computations based on (9) is defined as if the number of samples is too large. The number of initial H(s; ϑ)= H(s; ϑ) + ∆H(s; ϑ) (27) subbands µ¯ depends in fact on the model variations throughout the parameter space, which in turn is directly related to the with b type and number of adopted basis functions ξℓ(ϑ) for model ¯ ∆N(s, ϑ) n¯ ℓ ∆R ξ (ϑ) ϕ (s) parameterization in (5). In this work, we assume entire-domain ∆H(s; ϑ)= = n=0 ℓ=1 n,ℓ ℓ n smooth basis functions (polynomials, orthogonal polynomials, D(s, ϑ) n¯ ℓ¯ P n=0P ℓ=1 rn,ℓ ξℓ(ϑ) ϕn(s) or trigonometric polynomials). Due to the smooth parameter- (28) ization, it is sufficient to consider the heuristic rule µ¯ = κ ℓ¯, The coefficient perturbationsP∆Rn,ℓPwill be computed such with κ> 1. For all documented examples, we set κ =4. that the singular values of the perturbed model that are larger 7

(a) Normalized SHH spectral distance from imaginary axis A. Building Algebraic Passivity Constraints ×10-5 3 Iteration 1 Let us consider a single local singular value maximum Iteration 2 occurring at frequency , 2 σ¯µ,i =σ ¯i(θµ) > 1 ω¯µ,i =ω ¯i(θµ) as resulting from the passivity check of Section V-B. Model

1 perturbation leads to a perturbation of this singular value, which under ﬁrst-order approximation reads [49] 0 H 400 420 440 460 480 500 520 540 560 580 600 σµ,i ≈ σ¯µ,i + Re uµ,i ∆H(j¯ωµ,i; θµ) vµ,i , (31) ϑ (µm)

where uµ,i, vµ,i are the left and right singular vectors of (b) Localization of non-passive bands, Iteration 1 b 600 H(j¯ωµ,i; θµ) associated to σ¯µ,i. Forcing this singular value to be less than one leads to the inequality constraint 550 H m)

µ 500 Re u ∆H(j¯ωµ,i; θµ) vµ,i ≤ 1 − σ¯µ,i. (32)

( µ,i ϑ 450 Deﬁning now

400 T 0 0.5 1 1.5 2 2.5 aµ,i = aµ,i;0,1, ··· ,aµ,i;n,ℓ, ··· ,aµ,i;¯n,ℓ¯ (33) Frequency (Hz) ×108 (c) Localization of non-passive bands, Iteration 2 with the same ordering as in (30), where 460 ξ (θ ) ϕ (j¯ω ) a = ℓ µ n µ,i , (34) 450 µ,i;n,ℓ D(j¯ωµ,i,θµ) m) µ ( ϑ 440 we can cast (32) in algebraic form pT x (35) 430 µ,i ≤ 1 − σ¯µ,i, 0 0.5 1 1.5 2 2.5 3 Frequency (Hz) ×104 where T T H T ϑ (d) Singular values at Iteration 1, =400 µm pµ,i = Re (vµ,i ⊗ uµ,i) ⊗ aµ,i . (36) 1 The constraint (32) operates on an individual singular value maximum σ¯µ,i, attempting to reduce its value to be less than 0.998 one (since based on a ﬁrst-order singular value perturbation, this enforcement is not exact but only approximate). If mul- 0.996 tiple singular value maxima are perturbed concurrently, it is 0 1 2 3 4 5 6 7 8 9 10 sufﬁcient to add a constraint (32) for each µ,i. Frequency (Hz) ×108

Fig. 3. Checking uniform passivity of two PCB link macromodels param- B. Preserving Model Accuracy eterized by the via antipad radius ϑ = r, see Section VII-A). The two models correspond to two different iterations of the passivity enforcement Enforcing constraint (32) does not guarantee that the per- loop, discussed in Section VI. Panel (a): adaptively computed samples of turbed model remains accurate. Hence the need of constructing ψ(θµ) for the two models. Panels (b) and (c): for all computed θµ, each non-passive frequency band is highligthed with a red line; the yellow dots a cost function that casts in algebraic form a suitable norm of highlight the frequencies of imaginary SHH eigenvalues; the black squares the model perturbation. We define such cost function as are the local singular value maxima; the solid black line is an approximation of the contour line for σ = 1 of the singular value trajectories σ(jω; ϑ) (not P 2 2 required by the algorithm and depicted only for illustration purpose). Panel E = Ei,j , (37) (d): singular value plot of model at Iteration 1. i,j=1 X where than one will be displaced below the unit threshold. Such k¯ m¯ 2 2 2 coefficients will be collected in a global vector of decision Ei,j = wi,j;k,m |∆Hi,j (j2πfk, ϑm)| (38) varibles k=1 m=1 X X T T T T denotes the (i, j)-th entry of the squared model perturbation x = x1,1, ··· , xi,j , ··· , xP,P , (29) at frequency fk and parameter ϑm, weighted by the possibly frequency-, parameter-, and entry-dependent weight w . where i,j;k,m Defining now T xi,j = (∆R0,1)i,j , ··· , (∆Rn,ℓ)i,j , ··· , (∆Rn,¯ ℓ¯)i,j ξℓ(ϑm) ϕn(j2πfk) bk,m;n,ℓ = (39) (30) D(j2πfk, ϑm) R collects the (i, j) entries of all matrices ∆ n,ℓ with a suitable and (¯n+1)ℓ¯ Q ordering. We have xi,j ∈ R and x ∈ R , with Q = 2 ¯ T P (¯n + 1)ℓ. bk,m = bk,m;0,1, ··· ,bk,m;n,ℓ, ··· ,bk,m;¯n,ℓ¯ (40) 8 we can cast the elementwise cost function (38) as Algorithm 2 Passivity enforcement of parameterized models Require: frequency basis ϕn for n =0,..., n¯; 2 2 ¯ Ei,j = kFi,j xi,j k2 , (41) Require: parameter basis ξℓ for ℓ =1,..., ℓ; Require: model coefficients Rn,ℓ, rn,ℓ in (2) and (5); where Require: control parameters ϑmin, ϑmax, γ, κ, M; T 1: find passivity violations V via Algorithm 1; w b i,j;1,1 1,1 2: while V 6= ∅ do .  .  3: build constraint (35) for each element in V; Re F . i,j T 4: compute matrix Ψ in (45); Fi,j = , Fi,j = wi,j;k,m bk,m (42) Im nFi,j o  .  5: solve convex optimization problem (46); e  .   .  6: update model coefficients Rn,ℓ ← Rn,ℓ + ∆Rn,ℓ;  n o e  T  e wi,j;k,¯ m¯ b¯  7: find passivity violations V via Algorithm 1;  k,m¯    8: end while 2k¯m¯ ×(¯n+1)ℓ¯ Note that Fi,j ∈ R collects as many rows as 9: return passive model H(s; ϑ). available frequency and parameter samples. The row size can thus be large, since usually 2k¯m¯ ≫ (¯n + 1)ℓ¯. An equivalent b compressed form of (41) reads A. A Printed Circuit Board Interconnect

2 2 Ei,j = kΨi,j xi,j k2 , (43) The first example we consider is a high-speed signal link (see [51] for a detailed description) routed on the inner layers (¯n+1)ℓ¯×(¯n+1)ℓ¯ where Ψi,j ∈ R is obtained through an of two Printed Circuit Boards hinged by a connector. Vertical “economy-size” QR factorization of Fi,j = Qi,j Ψi,j , where interconnection at the feeding ports and at the connector T Qi,j Qi,j = I. Finally (37) can be cast as ports is provided by four through vias. The 2 × 2 scattering responses of the link are parameterized by the via antipad 2 2 E = kΨ xk2 , (44) radius ϑ = r within the range Θ = [400, 600] µm. The raw frequency-domain scattering responses (Courtesy of Prof. where Christian Schuster and Dr. Jan Preibisch, Technische Uni- P versität Hamburg-Harburg, Hamburg, Germany) are obtained Ψ = blkdiag{Ψi,j } (45) i,j=1 through a combination of a full-wave field solver (for the connector), lossy transmission-line models for the stripline C. Iterative Passivity Enforcement segments, and a field model for the vias based on [52]. A total of k¯ = 500 frequency samples up to 10 GHz, combined A first-order singular value perturbation with minimum with m¯ = 9 parameter samples were available for model induced model error is achieved by minimizing (44) while identification. enforcing (35), resulting in the following constrained mini- The parameterized GSK iteration of Section II was applied mization problem with n¯ = 44 basis poles, and using orthogonal (Cheby- chev) polynomials ξℓ(ϑ) to represent parameter variations Ψ 2 T minx k xk2 subject to pµ,i x ≤ 1 − σ¯µ,i, ∀µ, i. (46) through (5). The number of basis functions was determined by trial and error as ℓ¯ = 3, corresponding to quadratic This problem is convex thus straightforward to solve, e.g., polynomials. Only one half of the parameter samples (indices through a Primal-Dual Interior Point method [50]. Since based m =1, 3, 5, 7, 9) were used to fit the model, leaving the even- on a first-order approximation, the above optimization may numbered m =2, 4, 6, 8 for a-posteriori model validation pur- need to be iterated until all passivity violations are removed. poses. The worst-case absolute RMS error among all scattering Algorithm 2 illustrates the resulting passivity enforcement matrix elements at both fitting and validation points resulted scheme in pseudocode form. We see that this algorithm is a − − 8.788 × 10 4, whereas the relative error was 3.367 × 10 3. straightforward extension to the multivariate case of the singu- This model corresponds to the model at Iteration 1 depicted in lar value perturbation scheme [26], which is only applicable Fig. 3. We see from panels (a), (b), and (d) that this model is to univariate models. The key for the proposed multivariate not passive, due to localized low-frequency passivity violations extension is the availability of a reliable process for detecting throughout the parameter range. and extracting the parameter-dependent passivity violations of The proposed passivity enforcement algorithm required the model, as discussed in Section V. three iterations and a runtime of 20 seconds. The final model, as evident from Fig. 4, is uniformly passive since ψ(ϑ) > 0 VII. EXAMPLES for all ϑ ∈ Θ. Figure 5 shows a comparison between the model responses and the raw scattering samples for all fitting The performance of proposed passivity enforcement scheme and validation points. We see that the accuracy is excellent, is now demonstrated on three examples. A laptop with Intel as confirmed by the worst-case RMS errors 8.792 × 10−4 Core i7 CPU running at 2.6 GHz with 16 GB RAM was used (absolute) and 3.368 × 10−3 (relative). These errors are only in all numerical simulations. marginally worse than the errors of the original non-passive 9

Normalized SHH spectral distance from imaginary axis Localization of non-passive bands ×10-4 2.5 1.52

1.515 2 1.51

1.5 1.505 Sidelength (mm)

1.5 1 0 2 4 6 8 10 12 14 16 18 400 420 440 460 480 500 520 540 560 580 600 9 Frequency (Hz) ×10 ϑ (µm) Localization of non-passive bands

Fig. 4. Samples of ψ(ϑ) for the PCB link macromodel after passivity 1.05 enforcement. 1.04

1.03 Sidelength (mm) Scattering response S , magnitude 11 1.02 0.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 S(1,1), data ϑ Frequency (Hz) ×1010 S(1,1), model 0.4 Fig. 7. Localization of non-passive bands (red lines) through imaginary SHH eigenvalues (yellow dots) of original non-passive inductor model. Top and 0.2 bottom panels zoom on two different regions of the frequency-parameter plane. 0 0 1 2 3 4 5 6 7 8 9 10 Frequency (Hz) × 9 Normalized SHH spectral distance from imaginary axis 10 0.04 Scattering response S , magnitude 12 1 0.03

0.9 0.02

0.01 Iteration 1 0.8 ϑ Iteration 6 S(1,2), data 0 S(1,2), model 1 1.1 1.2 1.3 1.4 1.5 0.7 Sidelength (mm) 0 1 2 3 4 5 6 7 8 9 10 Frequency (Hz) × 9 10 Fig. 8. Normalized SHH spectral distance from imaginary axis for the integrated inductor example, before (Iteration 1) and after (Iteration 6) Fig. 5. Scattering responses Sij (jω; ϑ) of the PCB link macromodel after passivity enforcement. passivity enforcement (dashed red lines), compared to the raw data used for model identification (solid blue lines). Both fitting and validation points are displayed; the arrows denote the increasing direction for via antipad radius ϑ. B. An integrated inductor We consider here a 2-port integrated inductor (courtesy of Prof. Madhavan Swaminathan, Georga Institute of Technology, model, thanks to the adopted cost function (44) that is mini- Atlanta, USA). The inductor has a square outline with 1.5 turns mized through passivity enforcement. This level of accuracy routed on two different layers of the substrate. The scattering can be achieved only when the original passivity violations responses were obtained through a full-wave field solver for are very small (the worst-case passivity violation corresponded m¯ = 11 different values of the sidelength L = ϑ ranging to a maximum singular value σmax = 1.000346, which was from 1.02 to 1.52 mm, with k¯ = 477 frequency samples up to larger than one by a very small amount). Finally, we depict the 12 GHz. Model identification via GSK iteration was performed resulting implicitly-parameterized macromodel poles in Fig. 6, using only odd-indexed parameter values, leaving the even- computed by instantiating the model at the original parameter numbered responses for model validation purposes, with n¯ =8 samples ϑm. poles and ¯l =4, 3 Chebychev polynomial basis functions for numerator and denominator, respectively. The initial model was characterized by various passivity violations, illustrated in Fig. 7. These violations required a ×108 Parameterized model poles -3 total of 5 iterations (runtime 15 seconds) to be removed, re- -3.5 sulting in a uniformly passive model throughout the parameter -4 range. Removal of passivity violations is confirmed by Fig. 8, -4.5 which depicts ψ(ϑ) before and after passivity enforcement.

Real part (Hz) -5 -5.5 The RMS errors of original and passive models with respect 0 1 2 3 4 5 6 7 8 9 10 Imag part (Hz) ×109 to the original scattering responses are reported in Table I, showing that both models are very accurate. Figure 9 conﬁrms Fig. 6. Parameterized poles (zoomed view) of the PCB link macromodel, this statement by comparing passive model responses to raw computed and superimposed for all original parameter samples ϑm. data for all ﬁtting and validation points. 10

S , magnitude 12 Localization of non-passive bands, Iteration 1 1 ϑ 2.25 0.9 2.2 0.8

2.15 0.7 S(1,2), data

S(1,2), model Stub length (mm) 0.6 2.1 0 2 4 6 8 10 12 0 0.5 1 1.5 2 2.5 3 Frequency (Hz) ×109 Frequency (Hz) ×1010 S , phase (degrees) 12 Localization of non-passive bands, Iteration 2 200 S(1,2), data 2.25 100 S(1,2), model ϑ 0 2.2

-100 2.15 Stub length (mm) -200 2.1 0 2 4 6 8 10 12 Frequency (Hz) ×109 0 0.5 1 1.5 2 2.5 3 Frequency (Hz) ×1010

Fig. 9. Scattering response S12(jω; ϑ) of the integrated inductor macromodel after passivity enforcement (dashed red lines), compared to the raw data used Fig. 11. Localization of non-passive bands (red lines) through imaginary SHH for model identification (solid blue lines). Both fitting and validation points eigenvalues (yellow dots) of filter model at iteration 1 (top) and 2 (bottom) are displayed; the arrows denote the increasing direction for sidelength ϑ. of the passivity enforcement loop.

Scattering responses, magnitude (dB) TABLE I 0 H ABSOLUTE AND RELATIVE RMS ERRORS OF NON-PASSIVE ( ) AND ϑ Hb PASSIVE ( ) INDUCTOR MODELS AT FITTING AND VALIDATION POINTS. -20

Error Validation points Fitting points -40 abs rel abs rel S(2,1), data −3 −3 −4 −3 S(2,1), model H11 1.84 × 10 4.64 × 10 7.09 × 10 1.90 × 10 -60 −3 −3 −3 −3 0.5 1 1.5 2 H12 2.60 × 10 2.96 × 10 1.74 × 10 1.95 × 10 −3 −3 −4 −3 Frequency (Hz) ×1010 H22 2.04 × 10 5.13 × 10 8.91 × 10 2.24 × 10 Scattering responses, phase (degrees) b −3 −3 −4 −3 H11 1.80 × 10 4.54 × 10 8.58 × 10 2.12 × 10 200 S(2,1), data b −3 −3 −3 −3 H12 2.56 × 10 2.92 × 10 1.72 × 10 1.93 × 10 100 S(2,1), model b −3 −3 −3 −3 ϑ H22 1.94 × 10 4.87 × 10 1.10 × 10 2.72 × 10 0

-100

-200 C. A ﬁlter 0.5 1 1.5 2 Frequency (Hz) ×1010 The last example we propose is a double-folded microstrip

filter (see [53] for a more detailed description), which can Fig. 12. Scattering responses S21(jω; ϑ) of the filter macromodel after be tuned by changing the length ϑ = L of a microstrip stub passivity enforcement (dashed red lines), compared to the raw data used for within the range 2.08–2.28 mm. Scattering responses for model identification (solid blue lines). Only validation points are displayed; m¯ = the arrows denote the increasing direction for stub length ϑ. 21 linearly-spaced stub length values were computed through a field solver at k¯ = 300 samples spanning the frequency band [5, 20] GHz. Model identification based on odd-indexed The localized passivity violations of the initial model are ¯ parameter samples required n¯ = 10 poles and l =3 (quadratic) depicted in the top panel of Fig. 11. Passivity enforcement polynomial basis functions to capture parameter variations. required 5 iterations (runtime 31 seconds) to achieve uniform The resulting model was not passive, as Fig. 10 shows passivity, as confirmed by ψ(ϑ) > 0 in Fig. 10. The passivity by depicting ψ(ϑ)=0 at the first passivity iteration loop. violations at the second iteration are depicted in the bottom panel of Fig. 11, where we can note that the extent of such violations both in the frequency and parameter directions is ×10-3 Normalized SHH spectral distance from imaginary axis 8 reduced. Note also that the presence of multiple singular values Iteration 1 Iteration 6 exceeding one (Fig. 11, Iteration 1, localized at 22–24 GHz 6 and approximately ϑ ∈ [2.08, 2.13] mm), does not pose 4 particular problems, since multiple independent constraints can

2 be enforced while solving (46). Finally, Fig. 12 compares the model responses to the origi- 0 2.08 2.1 2.12 2.14 2.16 2.18 2.2 2.22 2.24 2.26 2.28 nal scattering responses at the validation points (even-indexed Stub length (mm) parameter samples). Also for this example the accuracy is Fig. 10. Normalized SHH spectral distance from imaginary axis for the excellent, with a worst-case RMS error amont all parameter, double-folded microstrip ﬁlter example, before (Iteration 1) and after (Iteration frequency samples and scattering matrix elements equal to 6) passivity enforcement. 3.16 × 10−3 (absolute) and 4.64 × 10−3 (relative). 11

VIII. CONCLUSION [14] P. Triverio, S. Grivet-Talocia, M.S. Nakhla, F. Canavero, R. Achar, “Stability, causality, and passivity in electrical interconnect models”, This paper presented an efficient and robust algorithm for IEEE Trans. on Advanced Packaging, vol. 30, no. 4, pp. 795-808, 2007. checking and enforcing the passivity of behavioral macro- [15] S. Grivet-Talocia, “On driving non-passive macromodels to instability,” models of LTI systems, whose scattering matrix depends International Journal of Circuit Theory And Applications, vol. 37, pp. 863886, Oct 2009. both on frequency and on one additional external parameter. [16] S. Boyd, V. Balakrishnan, P. Kabamba, “A bisection method for com- Thanks to a specialized adaptive sampling process in the puting the H∞ norm of a transfer matrix and related problems”, Math. parameter space, based on the spectral properties of some Control Signals Systems, Vol. 2, 1989, pp. 207–219. [17] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear matrix Skew-Hamiltonian/Hamiltonian matrix pencil associated to the inequalities in system and control theory, SIAM studies in applied model, we are able to detect and localize all frequency- and mathematics, SIAM, Philadelphia, 1994. parameter-dependent passivity violations of the model. An [18] N. Wong, C.-K. Chu, “A fast passivity test for stable descriptor systems via skew-Hamiltonian/Hamiltonian matrix pencil transformations.” IEEE iterative constrained optimization loop is able to remove such Transactions on Circuits and Systems I: Regular Papers vol. 55, no. 2, violations, while retaining model accuracy. To be best of 2008, pp. 635-643. Author’s knowledge, this is the first documented algorithm [19] Z. Zhang and N. Wong, “Passivity Test of Immittance Descriptor Sys- tems Based on Generalized Hamiltonian Methods,” in IEEE Transactions that is able to enforce the uniform passivity of a parameterized on Circuits and Systems II: Express Briefs, vol. 57, no. 1, pp. 61-65, macromodel. Jan. 2010. Future research will be devoted to the extension of proposed [20] Z. Zhang and N. Wong, “An Efficient Projector-Based Passivity Test for Descriptor Systems,” in IEEE Transactions on Computer-Aided Design approach to higher dimensions in the parameter space, with of Integrated Circuits and Systems, vol. 29, no. 8, pp. 1203-1214, Aug. special emphasis on avoiding issues due to the curse of 2010. dimensionality. [21] Z. Zhang and N. Wong, “Passivity Check of S -Parameter Descriptor Systems via S-Parameter Generalized Hamiltonian Methods,” in IEEE Transactions on Advanced Packaging, vol. 33, no. 4, pp. 1034-1042, IX. ACKNOWLEDGEMENTS Nov. 2010. [22] Y. Wang, Z. Zhang, C. K. Koh, G. Shi, G. K. H. Pang and N. The Author is grateful to Prof. Madhavan Swaminathan Wong, “Passivity Enforcement for Descriptor Systems Via Matrix Pencil Perturbation,” in IEEE Transactions on Computer-Aided Design of (Georgia Institute of Technology, Atlanta, USA) for sharing Integrated Circuits and Systems, vol. 31, no. 4, pp. 532-545, April 2012. the integrated inductor data, to Prof. Christian Schuster and [23] S. Grivet-Talocia, “Passivity Enforcement via Perturbation of Hamilto- Dr. Jan Preibisch (Technische Universität Hamburg-Harburg, nian Matrices” , in IEEE Trans. Circuits and Systems I: Fundamental Theory and Applications, pp. 1755-1769, vol. 51, n. 9, September, 2004. Hamburg, Germany) for sharing the PCB interconnect link [24] D. Saraswat, R. Achar and M. Nakhla, “Global Passivity Enforcement data, and to Prof. Piero Triverio (Univ. 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