Application of Taylor Models to the Worst-Case Analysis of Stripline Interconnects

Transcription

1 Application of Taylor Models to the Worst-Case Analysis of Stripline Interconnects Paolo Manfredi, Riccardo Trinchero, Igor Stievano, Flavio Canavero 1

2 Motivation q Ever-growing interest in the inclusion of parameters uncertainties in circuit simulations q Variability due to unknown components, manufacturing tolerances, uncertain design parameters q Key resource in the design phase parameters! #! $! " U(s) input CIRCUIT / SYSTEM Y(s,x) random output 2

3 Worst Case q Determining the worst case configuration is fundamental in a number of critical applications (transportation, industrial, medical, etc ) q Standard methods require a large number of deterministic simulations to guarantee an accurate estimation (the worst case often lies in regions of low probability!) Deterministic Simulations samples In principle an infinite number of samples is needed to explore all the possible combinations of the parameters 3

4 State-of-the-art (features) Monte Carlo Polynomial Chaos Interval methods Interval Arithmetic Affine Arithmetic Taylor Models Easy to implement Slow convergence (> samples) Underestimation (i.e., approx. of the bound from inside ) Blind analysis (no parametric model) Nice theoretical framework Improved efficiency No info on error: under/over estimation?? Inherently worst-case Promising alternative that worth to be carefully investigated 4

5 Problem statement q Numerical techniques that propagate bounded variables by redefining the standard operations producing guaranteed enclosures q E.g., the analysis of a circuit defined by components with bounded values requires to handle operations over bounded variables in a conservative way: MNA: 1 ' # ()) +, ()) = / + 1) )4 5# 7()) 0 linear operations addition/subtraction, product,.. nonlinear operations division, exponential, powers, etc complex operations matrix operations Addition/subtraction, product, inversion In order to produce a guaranteed enclosure of the output responses, each operation must provide a conservative bound of its result 5

6 Interval Methods [1] Ø Interval Arithmetic (IA) Random parameters = bounded interval e.g., subtraction: 9, ; <, = = 9 =, ; < 2,3 2,3 = 2 3,3 2 = [ 1,1] 0! [9, ;] 9 ;! No correlation among variables Overestimation, i.e. TOO PESSIMISTIC Ø Affine Arithmetic (AA) Random parameters = linear model via new symbol?!@ =! A +! #? # +! "? " ; 9 Accounts for correlation among parameters 1 1 ℇ Extension to nonlinear operations Non-trivial (even for the product) [1] T. Ding, R. Trinchero, P. Manfredi, I. S. Stievano and F. G. Canavero, "How Affine Arithmetic Helps Beat Uncertainties in Electrical Systems," in IEEE Circuits and Syst. Mag., vol. 15, no. 4, pp ,

7 Taylor model q Hybrid representation of a nonlinear function of an interval variable in terms of a Taylor expansion plus an interval remainder [2]. q Given! = [9, ;] J! N O! = P O! + + O [K, L] a Lagrange Remainder [2] M. Berz and K. Makino, Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor models, Rel. Computing, vol. 4, no. 4, pp , ! I + 3! " b! P(x) gives parametric model of f(x) Lagrange remainder provides conservative error estimation

8 Worst-case bounds J(!) Q J! = [min J!, max J(!)] N O = P O! + + O Q(N O ) = Q(P O! ) ++ O True worst case Q(P O! ) ++ O a b Estimated bound: guaranteed conservative!! 8

9 Polynomial bound: Bernstein q Polynomial bound calculation is non-trivial for multivariate polynomials with order > 2 q Effective solution: conversion of P O into Bernstain polynomials [3] P O (!) X ; Y Q Y (!) q Bernstein polynomials have a notable property: min ; Y P O (!) max ; Y [3] R. Trinchero, P. Manfredi, T. Ding, and I. S. Stievano, Combined parametric and worst-case circuit analysis via Taylor models," accepted for publication in IEEE Trans. Circuits and Syst. (2016). 9

10 Taylor model algebra (basic rules) Sums, subtractions (straightforward)! + [1,2] ! + [ 1,1] = 3 + 3! + 0,3 Multiplication e.g., order 1! + [1,2] \ 3 + 2! + [ 1,1] sum of poly computed via IA e.g., order 1 = 3! + 2! " +! 1,1 + 1, ! + 1,1 1,2 B(2! " ) B(!) B(3 + 2!) computed via IA = 3! + [ 2,15] 10

11 Nonlinear operators q The functional Taylor expansion allows replacing nonlinear operators (e.g., ] N O = 1 N O ) with a mere series of products and sums Sum of scalar coefficients and products b ] N O = P^! + +^ = X ] _ < O `! _ca available!! _ N O < O + Q(/ b (d)) Taylor expansion of ] around Taylor model center < O = P O (0) Lagrange Remainder / b d = ] bg# d h + 1! N O < O bg# with d Q(N O ) [4] [4] M.A. Abramowitz and I.A. Stegun, Handbook of mathematical functions. New York, NY, USA: Dover Publications,

12 Complex and Matrix Operations q Complex calculations are reduced to real-valued operations on Re/Im E.g. 9 r + 19 s ; r + 1; s = 9 r ; r 9 s ; s + 1(9 r ; s + 9 s ; r ) q A Taylor model matrix is a matrix where all the entries are Taylor models j k = N k,## N k,#" N k,"# N k,"" N k,## = P k,##! + + k,## q Linear matrix operations: operate element-wise with scalar operations Matrix Sum/Subtraction j k ± j p Yq = N k,yq ± N p,yq scalar addition Matrix Product j k j p Yq = X N k,y_ N p,_q _ scalar multiplication 12

13 Matrix Inversion q How to invert a matrix TM and still obtain a conservative remainder? ØIdea: Sherman-Morrison formula (SM) t + u 5# = t 5# tr t 5# u (t5# ut 5# ) ü exact solution (NO approximation) ü only requires inversion of t q Requirements t must be (easily) invertible u must be rank-1 suitable iterative extension available 13

14 Solution [3]: Matrix Inversion (cont d) j O = 7 + Q constant matrix (center of the expansion) TM matrix (Poly + remainder) (j x ) 5# = t 5# tr t 5# u (t5# ut 5# ) standard matrix inversion available TM operations poly+remainder handled as it is NO INV NEEDED Rigorous solution for the inversion of an interval matrix!! [3] R. Trinchero, P. Manfredi, T. Ding, and I. S. Stievano, Combined parametric and worst-case circuit analysis via Taylor models," accepted for publication in IEEE Trans. Circuits and Syst. (2016). 14

15 Example (from [4]) R 1 R 6 R 11 R 16 R 21 In R 2 R 3 C 1 R 7 R 8 C 3 R 12 R 13 C 5 R 17 R 18 C 7 C 9 Out Modified Nodal Analysis y + z{ = } R 4 R 9 R 14 R 19 R 5 C 2 R 1 =5.4779kΩ R 6 =4.44kΩ R 15 = kΩ C 1 =12nF R 10 R 2 =2.0076kΩ R 7 =5.9999kΩ R 16 = kΩ C 3 =6.8nF C 4 Solve the system j = j y + zj { 5# } and obtain the WC R 15 C 6 R 3,4,8,9,13,14,18,19 =3.3kΩ R 10 =4.2573kΩ R 17 =1.0301kΩ C 5 =4.7nF R 20 R 11 =3.2201kΩ R 20 = kΩ C 7 =6.8nF C 8 R 5 =4.5898kΩ R 12 = kΩ R 21 =1.2201kΩ C 2,4,6,8,9 =10nF TM Matrices y j y { j { Defining the bounded parameters 2 " = 2 ",A ( ! # ) 2 = 2,A ! # 2 Ä = 2 Ä,A ( ! " ) 2 Å = 2 Å,A ! " 2 Ç = 2 Ç,A ! I where,! Y [ 1,1] [4] T.-A. Pham, E. Gad, M. S. Nakhla, and R. Achar, Decoupled polynomial chaos and its applications to statistical analysis of high-speed interconnects, IEEE Trans. Compon. Packag. Manuf. Techol., 4(10) (2014). 15

16 Application of Taylor Model to the Worst-Case Analysis of Stripline Interconnects SPI2016 Torino, May 8-11 Results (see [3]) q Worst-Case Analysis q Parametric Analysis 1.5 Pass band Ripple MC TM ( or de r 5) Frequency [Hz] MonteCarlo (10000 runs) Very Good accuracy Bernstein polys yield tight bounds TM offers a parameterized description! [3] R. Trinchero, P. Manfredi, T. Ding, and I. S. Stievano, Combined parametric and worst-case circuit analysis via Taylor models," accepted for publication in IEEE Trans. Circuits and Syst. (2016). 16

17 Application example: Coupled Stripline q First application of TMs to the WC analysis of a transmission line q Coupled stripline [4] in a homogeneous dielectric medium with relative permittivity ε r 0.6 mil 5 mil 5 mil 5 mil ε r =4 20 mil q Defining the following bounded parameters 50Ω 10 cm 1pF? É =? É,A ± Δ ÖÜ = 4 ± à = 2 à,a ± Δ âä = 1 ± 0.05 pf 50Ω 1pF [4] T.-A. Pham, E. Gad, M. S. Nakhla, and R. Achar, Decoupled polynomial chaos and its applications to statistical analysis of high-speed interconnects, IEEE Trans. Compon. Packag. Manuf. Techol., 4(10) (2014). 17

18 Application example: Coupled Stripline (ii) q The far- and near-end voltage and current variables are computed through the chain-parameter matrix: å ## = cos ïê ñ å #" = 1ó A sin ïê ò å "# = 1ô A sin ïê { å "" = cos(ïê) ñ ã 0 = å "# ç é + è, å #" å "" è, å ## ç 5# é \ (å "# è, å ## ) 0 = é ç é ã 0 ê = å ## é + å #" å ## ç é ã 0 ë ê = è, (ê) q The WC analysis requires to handle in the TM framework: Ø matrix inversion available Ø sin/cos operations details in SPI paper! 18

19 Application example: Coupled Stripline (iii) q Worst-Case Analysis q Parametric model for ö õ Far-end voltage [db] Far-end crosstalk [db] SPICE (MC) Taylor model Frequency [GHz] Far-end voltage [db] Far-end crosstalk [db] SPICE Taylor model Frequency [GHz] ε r ε r SPICE MonteCarlo (10000 runs): 680 s TM (order 3): 35.6 s 19

20 Conclusions q Promising solution for the direct worst-case analysis of the frequency-domain response of circuits and interconnects (key tool in critical applications) qtm-based approach: Ø represents any variable/response as a polynomial + interval remainder (for approximation and round-off errors) Ø Elegant and robust mathematical framework Ø Extended to handle matrix operations (inversion!!) Ø allows to jointly obtain a parameterized response and WC bounds ØTighter bounds w.r.t. other interval approaches Ø promising results in terms of both accuracy and efficiency 20

21 Thank you for your attention Q&A 21