Passive Reduced Order Multiport Modeling: The Padé-Laguerre, Krylov-Arnoldi-SVD Connection

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1 Passive Reduced Order Multiport Modeling: The Padé-Laguerre, Krylov-Arnoldi-SVD Connection Luc Knockaert and Daniel De Zutter 1 Abstract A reduced order multiport modeling algorithm based on the decomposition of the system transfer matrix into orthogonal scaled Laguerre functions is proposed. The link with Padé approximation, the block Arnoldi method and singular value decomposition leads to a simple and stable implementation of the algorithm. Keywords Reduced order modeling, Laguerre functions, Padé approximation, singular value decomposition. 1 INTRODUCTION Circuit simulation tasks, such as the accurate prediction of the behavior of large RLGC interconnects, generally requires the solution of very large linear networks. Since the main point, from a communications and throughput point of view, is the behavior of the interconnect structure at user-defined ports over a given frequency range, it is of utmost importance to dispose of a reduced but accurate black-box model of the network as seen from the chosen ports. In recent years this has led to the development of reduced order modeling technologies such as asymptotic waveform evaluation (AWE) [1], matrix Padé via Lanczos (MPVL) [2]-[3], symmetric Padé via Lanczos (SyMPVL) [4], block Arnoldi [5] and passive reduced-order interconnect macromodeling (PRIMA) [6]. Though quite different in implementation and numerical stability, most of these algorithms tend to obtain a low order Padé approximant [7] of the system transfer matrix via Krylov subspace modeling. Some of these techniques, such as SyMPVL and PRIMA, are provably passive, partly because the model reduction scheme can be interpreted in terms of congruence transformations 1 INTEC-IMEC, St. Pietersnieuwstraat 41, B-9000 Gent, Belgium. Tel: , Fax: knokaert@intec.rug.ac.be

2 2 [8]. In this paper we propose an algorithm based on the decomposition of the system transfer matrix into orthogonal scaled Laguerre functions [9]. The link with Padé approximation, the block Arnoldi algorithm and the singular value decomposition (SVD) [10] permits a simple and stable implementation of the algorithm. As in PRIMA and SyMPVL, the method is provably passive. 2 THE PADE-LAGUERRE CONNECTION Using any circuit-equation formulation method such as modified nodal analysis (MNA) [11], sparse tableau, etc. [12], a lumped, linear, time-invariant strictly passive [13] multiport circuit of order N can be described by the following system of first-order differential equations: Cẋ = Gx + Bu (1) y = L T x (2) Here, the vector x represents the circuit variables, ẋ represents the time derivative of x, the N N matrix G represents the contribution of memoryless elements, such as resistors, the N N matrix C represents the contribution from memory elements, such as capacitors and inductors, the vector y is the output of interest and the vector u represents the excitations at the ports. Note that L T stands for the transpose of L. Since we consider a multiport formulation with p ports, where in general p N, the rectangular matrices L and B are of dimension N p. Moreover, as explained in [6], without loss of generality we can take L = B. However, for the sake of generality, we will maintain the separate B and L notation throughout this paper. With unit impulse excitations at the ports and zero initial conditions, the Laplace transform of the circuit equations (1) (2) yields the p p port transfer matrix H(s) = L T (G + sc) 1 B (3) and the corresponding p p port impulse response matrix h(t) = L 1 H(s). (4) In most of the reduced-order modeling literature, model reduction strategies are based on Padé approximations of the transfer matrix by means of moment matching. The approach we advocate

3 3 in this paper is the expansion of the impulse response matrix h(t) in scaled Laguerre functions [9], defined as φ α n(t) = 2αe αt l n (2αt) n = 0, 1,... (5) where α is a positive scaling parameter and l n (t) is the Laguerre polynomial l n (t) = et n! d n ( dt n e t t n). (6) It is known [14] that the sequence {φ α n(t)} forms a uniformly bounded orthonormal basis for the Hilbert space L 2 (R + ). Hence the impulse response matrix h(t) admits the Fourier-Laguerre expansion h(t) = F n φ α n(t). (7) n=0 Since the Hardy space H 2 [9] consisting of all analytic and square-integrable functions in the open right halfplane Rs > 0 is the Laplace transform of L 2 (R + ), the sequence of Laplace transforms of the scaled Laguerre functions Φ α n(s) = Lφ α n(t) = 2α s + α ( ) s α n n = 0, 1,... (8) s + α forms a uniformly bounded orthonormal basis for the Hilbert space H 2 equipped with the inner product and norm ν(f) = f f. f g = 1 f(iω)g (iω)dω (9) 2π Considering that the multiport circuit under scrutiny is strictly passive and hence asymptotically stable, and since the transfer matrix is strictly proper, i. e. lim H ij(s) = 0 i, j = 1,...,p, (10) s all the entries H ij (s) of H(s) belong to H 2 and the transfer matrix can be expanded into the orthonormal basis {Φ α n(s)} as This can be rewritten as H(s) = L T (G + sc) 1 B = F n Φ α n(s). (11) n=0 L T (G + sc) 1 B = 2α s + α n=0 ( ) s α n F n. (12) s + α

4 4 Equation (12) has the very simple physical interpretation that any transfer matrix in H 2 can be represented as the product of a simple low-pass filter 2α/(s + α) and a weighted infinite sum of all-pass filters of the type [(s α)/(s + α)] n. Moreover, the bilinear transformation u = s α s + α (13) maps the s domain Laguerre expansion (12) into the u domain power expansion L T ([αc + G] + u[αc G]) 1 B = 1 2α n=0 F n u n. (14) From this we infer that an m th order Padé approximation of the modified transfer matrix Ĥ(u) = L T ([αc + G] + u[αc G]) 1 B (15) in the u domain is equivalent to a an m th order Laguerrre approximation in the s domain, meaning that H(s) can be optimally approximated in the H 2 norm sense by the truncated Fourier- Laguerre expansion m H(s) F n Φ α n(s) = H m (s). (16) n=0 Note that the bilinear transformation (13) maps the open right halfplane Rs > 0 onto the open unit disk u < 1. Although we know that H m (s) converges to H(s) for m when Rs > 0 for a strictly passive system, it is necessary, in order to be able to determine an adequate value for the Laguerre parameter α, to have more information about the convergence rate. We start with the partial fraction expansion H(s) = N k=1 h k s + v k (17) which is valid for simple poles v k with Rv k > 0. From the easily proved identity we obtain n=0 Φ α n(s)φ α n(v) = 1 s + v F n = for ( ) ( ) s α v α < 1 (18) s + α v + α N h k Φ α n(v k ). (19) k=1 Hence, with respect to any matrix norm, we have 2α H(iω) H m (iω) ω 2 + α 2 F k (20) k=m+1

5 5 and in the light of equations (19) and (8) H(iω) H m (iω) 2α N h k ω 2 + α 2 v k=1 k + α v k α v k α m+1 v k + α. (21) For a strictly passive system, this proves the pointwise convergence H m (iω) H(iω) as m approaches infinity. It is seen that the overall convergence rate is dictated by the largest coefficient (v k α)/(v k +α) and hence the optimal α may be found as the solution to the minimax problem α = arg min α>0 max 1 k N v k α v k + α. (22) Of course, the solution to the minimax problem (22) is rather intricate and requires knowledge of all the poles. To set our sights lower, we suppose that the negative poles v k are situated anywhere in the rectangular region Θ = [d 1, d 2 ] [ iy 0, iy 0 ] of the right halfplane. The corresponding minimax problem α = arg min max α>0 v Θ v α v + α (23) can be solved by taking advantage of the maximum modulus principle, which asserts that a function, analytic in a bounded region, attains its maximum modulus on the boundary of this region. The explicit solution for α is α = α = d y0 2 for d 2 d y2 0 (24) d 1 d 1 d 2 y0 2 for d 2 d y2 0. (25) d 1 Since y 0 and the system bandwidth are approximately related through y 0 = 2πf max [2] and since the small time constant τ 2 = 1/d 2 in equation (25) expresses no direct physical nor computational meaning, we may simply take α = d y 2 0 y 0 = 2πf max. (26) Note that the simple low-pass bound expression (20) corroborates this. 3 THE KRYLOV-ARNOLDI-SVD CONNECTION Defining the matrices A = G 1 C and R = G 1 B, (27)

6 6 it is shown in [6] that the column-orthogonal matrix X associated with the block Arnoldi process [5] in order to orthogonalize the columns of the N pq Krylov matrix K q = [ ] R,AR,A 2 R,,A q 1 R (28) yields a reduced order system Cẋ = Gx + Bu (29) y = L T x (30) with C = X T CX B = X T B G = X T GX L = X T L (31) such that the reduced order transfer matrix H(s) = L T ( G + s C) 1 B (32) is a passive Padé approximant of order q 1 for the original transfer matrix. In virtue of the preceding section, the above reasoning remains valid, in the sense of Laguerre approximation, if we define the modified system matrices  = (αc + G) 1 (αc G) and ˆR = (αc + G) 1 B. (33) In other words, we assert that the column-orthogonal matrix ˆX associated with the block Arnoldi process as applied to the N pq modified Krylov matrix [ ˆK q = ˆR,  ˆR,Â2 ˆR,,  ˆR] q 1 (34) yields a reduced order system described by C = ˆX T C ˆX B = ˆX T B such that the reduced order transfer matrix G = ˆXT G ˆX L = ˆXT L (35) H(s) = L T ( G + s C) 1 B (36) is a passive Laguerre approximant of order q 1 for the original transfer matrix.

7 7 The block Arnoldi algorithm [5] can be utilized to generate the column-orthogonal matrix ˆX. The process normally used is a block so-called thin QR factorization based on modified Gram- Schmidt orthogonalization [10]. Numerical experience [5], [15] has shown that some steps in the block Arnoldi algorithm have to be repeated in order to ensure orthogonality to the precision of the computer. To avoid this, we opt for an SVD based technique, which is a numerically more stable algorithm than the Gram-Schmidt process [16]. The idea behind the SVD approach is the following. Putting r = pq < N, the dimension of the Krylov matrix ˆK q is N r. Since the block Arnoldi algorithm is equivalent to modified Gram-Schmidt, we compare the SVD and the thin QR factorization of ˆK q. We have ˆK q = ˆXR t = UΣV T, (37) where ˆX and U are N r column-orthogonal matrices, R t is an r r upper triangular matrix, Σ is an r r diagonal matrix containing the singular values of the Krylov matrix and V is an r r orthogonal matrix. Since ˆX T ˆX = U T U = I r, it is easily seen that ˆX can be written as ˆX = UQ, where Q is an r r orthogonal matrix. From equations (35) and (36) we infer that the reduced order model transfer matrix can be written as ( H(s) = L T ˆX ˆX) 1 ˆXT (G + sc) ˆXT B (38) ( 1 = L T UQ Q T U T (G + sc)uq) Q T U T B (39) ( 1 = L T U U T (G + sc)u) U T B. (40) Note that the mere requirement that Q is nonsingular is sufficient to obtain result (40). The derivations (38)-(40) prove that it is judicious to use the left SVD column-orthogonal factor U instead of ˆX in the reduced order modeling scheme. The complete SVD-Laguerre based algorithm is constructed as follows: Select the values for α and q Solve (G + αc) ˆR 0 = B For k = 1,...,q 1 solve (G + αc) ˆR k = (G αc) ˆR k 1 UΣV T = SVD [ ˆR 0, ˆR 1,, ˆR q 1 ] C = U T CU G = U T GU B = U T B L = U T L

8 8 It is also important to evaluate H(s) or H(s) explicitly as a sum of partial fractions. Taking s 0 such that G + s 0 C is nonsingular, we have H(s) = L T (G + s 0 C + (s s 0 )C) 1 B = L T (I + (s s 0 )E) 1 B 0 (41) where E = (G + s 0 C) 1 C B 0 = (G + s 0 C) 1 B. (42) Supposing E nondefective i.e. diagonalizable, it admits the eigendecomposition E = PΛP 1, (43) where Λ is a diagonal matrix, yielding the partial fraction expression H(s) = L T P (I + (s s 0 )Λ) 1 P 1 B 0. (44) To avoid supplementary LU-decomposition overhead, it can be naturally recommended to choose the parameter s 0 equal to α. 4 NUMERICAL SIMULATIONS 4.1 LOSSY TRANSMISSION LINE Consider a lossy transmission line modeled by M lumped RLGC sections. The circuit equations for the corresponding (p = 2) twoport are C n dv n dt L n di n dt = G n v n + i n i n+1 n = 1,...,M (45) = R n i n + v n 1 v n n = 1,...,M + 1. (46) The input variables are u 1 = v 0, u 2 = v M+1 and the output variables are y 1 = i 1, y 2 = i M+1. The state variables are the entries of the N = 2M + 1 dimensional vector x = (v 1,...,v M, i 1,...,i M+1 ) T. (47)

9 9 The standard C,G,B,L MNA format is easily derived. For example, when M = 2 we have C G C G L ẋ = 1 0 R x u L R (48) L R } {{ } } {{ } } {{ } C G B and of course L = B. A choice of M = 40 equal RLGC sections as in [6] with total parameters G tot = 0, R tot = 10 Ω, L tot = 5 nh, C tot = 15 pf yields a system description of order N = 81. Reduced order Laguerre models of dimensions r = 2q = 8 and r = 2q = 16 are constructed using s 0 = α = 2π10 9. Figures 1 3 respectively show H 11 (f), H 12 (f) and H 22 (f) versus their reduced order counterparts. It is seen that the q = 4, q = 8 Laguerre reduced order models are indistinguishable from the unreduced model up to respectively 2 GHz and 6 GHz. For H 22 (f) the q = 8 reduced order model is even indistinguishable from the unreduced model up to 8 GHz. 4.2 COUPLED LOSSY TRANSMISSION LINES Coupled lossy multiconductor transmission lines can easily be modeled by a multiport generalization of (45)-(46). The circuit equations for M sections, obtained from the discretization of the coupled matrix telegrapher equations [17], can be written as C n dv n dt L n di n dt = G n v n + i n i n+1 n = 1,...,M (49) = R n i n + v n 1 v n n = 1,...,M + 1, (50) where v n and i n are c 1 column vectors and C n, L n, G n and R n are c c square matrices. The standard MNA format is then simply a block matrix generalization of equation (48), i. e.

10 10 for M = 2 we have C C L L L 3 } {{ } C G 1 0 I c I c 0 0 G 2 0 I c I c ẋ = I c 0 R x + I c I c 0 R I c 0 0 R 3 } {{ } G I c I c } {{ } B u (51) where I c is the c c identity matrix. It is seen that the MNA system order is N = (2M + 1)c and the number of columns in B = L is p = 2c. The discretized example of a 5 cm long four-port (c=2) multiconductor transmission line consisting of M = 40 equal sections with total matrices given by (extrapolated from the data in [17]) G tot = S R tot = Ω (52) and L tot = nh C tot = pf (53) yields a system description of order N = 162. Reduced order Laguerre models of dimensions r = 2cq = 16 and r = 2cq = 32 are constructed using s 0 = α = 2π10 9. We observed that the symmetric transfer matrix H(f) is Toeplitz, except for H 12 (f) H 23 (f), and block-symmetric i.e H 14 (f) = H 23 (f). Figures 4 6 respectively show H 11 (f), H 12 (f) and H 13 (f) versus their reduced order counterparts up to 4 GHz. It is seen that the reduced order models closely approximate the original model over the given frequency range. 4.3 A PEEC CIRCUIT As a third example we take the lumped-element equivalent circuit for a three-dimensional electromagnetic problem modeled via PEEC [18] (partial element equivalent circuit) as documented in [2]. The twoport (p = 2) circuit consists of 2100 capacitors, 172 inductors, 6990 inductive couplings, resulting in a MNA system of order N = 306. Reduced order Laguerre models of dimensions r = 2q = 60 and r = 2q = 90 are constructed using s 0 = α = 10π10 9. Figures 7 9 respectively show H 11 (f), H 12 (f) and H 22 (f) versus their reduced order counterparts. It is seen that the q = 30, q = 45 Laguerre reduced order models are very close to the unreduced

11 11 model up to respectively 2 GHz and 4 GHz. We also simulated a reduced order Laguerre model of dimension r = 2q = 120 and found it indistinguishable from the unreduced model over the 5 GHz frequency range. References [1] L. T. Pillage and R. A. Rohrer, Asymptotic waveform evaluation for timing analysis, IEEE Trans. Computer-Aided Design, vol. 9, pp , Apr [2] P. Feldmann and R. W. Freund, Efficient linear circuit analysis by Padé approximation via the Lanczos process, IEEE Trans. Computer-Aided Design, vol. 14, no. 5, pp , May [3] J. I. Aliaga, D. L. Boley, R. W. Freund, and V. Hernández, A Lanczos-type method for multiple starting vectors, Numer. Anal. Manuscript, , Bell Lab.,Murray Hill, NJ, Sep [4] R. W. Freund and P. Feldmann, The SyMPVL algorithm and its applications to interconnect simulation, Numer. Anal. Manuscript, , Bell Lab.,Murray Hill, NJ, June [5] D. L. Boley, Krylov space methods on state-space control models, Circuits Syst. Signal Processing, vol. 13, no. 6, pp , [6] A. Odabasioglu, M. Celik, and L. T. Pileggi, PRIMA: Passive reduced-order interconnect macromodeling algorithm, IEEE Trans. Computer-Aided Design, vol. 17, no. 8, pp , Aug [7] A. Bultheel and M. Van Barel, Padé techniques for model reduction in linear system theory: a survey, J. Comput. Appl. Math., vol 14, pp , [8] K. J. Kerns and A. T. Yang, Preservation of passivity during RLC network reduction via split congruence transformations, IEEE Trans. Computer-Aided Design, vol. 17, no. 7, pp , July [9] Z. Zang, A. Cantoni, and K. L. Teo, Continuous-time envelope-constrained filter design via Laguerre filters and H optimization methods, IEEE Trans. Signal Processing, vol. 46, no. 10, pp , Oct [10] G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore: Johns Hopkins, [11] C-W. Ho, A. E. Ruehli, and P. A. Brennan, The modified nodal approach to network analysis, IEEE Trans. Circuits Syst., vol. CAS-22, no. 6, pp , June [12] J. Vlach and K.Singhal, Computer Methods for Circuit Analysis and Design. New York: Van Nostrand Reinholt, 1983.

12 12 [13] R. A. Rohrer and H. Nosrati, Passivity considerations in stability studies of numerical integration algorithms, IEEE Trans. Circuits Syst., vol. CAS-28, pp , Sept [14] G. Szegö, Orthogonal Polynomials, in Amer. Math. Soc. Coll. Publ., vol. 23, [15] Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., vol. 7, no. 3, pp , July [16] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, Cambridge: University Press, [17] A. R. Djordjević and T. K. Sarkar, Analysis of time response of lossy multiconductor transmission line networks, IEEE Trans. Microwave Theory Tech., vol. MTT-35, no. 10, pp , Oct [18] A. E. Ruehli, Equivalent circuit models for three-dimensional multiconductor systems, IEEE Trans. Microwave Theory Tech., vol. MTT-22, pp , Mar

13 unreduced reduced q=8 reduced q= H 11 (f) f GHz Figure 1: H 11 (f) for the lossy transmission line.

14 unreduced reduced q=8 reduced q= H 12 (f) f GHz Figure 2: H 12 (f) for the lossy transmission line.

15 unreduced reduced q=8 reduced q= H 22 (f) f GHz Figure 3: H 22 (f) for the lossy transmission line.

16 unreduced reduced q=8 reduced q=4 0.5 H 11 (f) f GHz Figure 4: H 11 (f) for the coupled lossy transmission lines.

17 unreduced reduced q=8 reduced q= H 12 (f) f GHz Figure 5: H 12 (f) for the coupled lossy transmission lines.

18 unreduced reduced q=8 reduced q=4 0.5 H 13 (f) f GHz Figure 6: H 13 (f) for the coupled lossy transmission lines.

19 unreduced reduced q=45 reduced q= H 11 (f) f GHz Figure 7: H 11 (f) for the PEEC circuit.

20 unreduced reduced q=45 reduced q= H 12 (f) f GHz Figure 8: H 12 (f) for the PEEC circuit.

21 unreduced reduced q=45 reduced q=30 H 22 (f) f GHz Figure 9: H 22 (f) for the PEEC circuit.