Model reduction of nonlinear circuit equations

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1 Model reduction of nonlinear circuit equations Tatjana Stykel Technische Universität Berlin Joint work with T. Reis and A. Steinbrecher BIRS Workshop, Banff, Canada, October 25-29, 2010

2 T. Stykel. Model reduction of nonlinear circuit equations p.1 Outline Differential-algebraic equations in circuit simulation Model order reduction problem PAssivity-preserving Balanced Truncation method for Electrical Circuits (PABTEC) Decoupling of linear and nonlinear parts Model reduction of linear equations Recoupling Numerical examples Conclusion

3 Modified Nodal Analysis (MNA) T. Stykel. Model reduction of nonlinear circuit equations p.2 R 6 f 1 C 4 L d 2 e 3 A = [A R, A C, A L, A V, A I ] V 1 R 2 C 3 a b c 0 0 Kirchhoff s current law: Aj = 0, j = [ j T R, jt C, j T L, j T V, jt I Kirchhoff s voltage law: A T η = v, v = [ v T R, vt C, v T L, v T V, vt I Branch constitutive relations: resistors: j R = g(v R ), G(v R ) = g(v R ) v R capacitors: j C = dq(v C), C(v C ) = q(v C) dt v C inductors: v L = dφ(j L), L(j L ) = φ(j L) dt j L ] T ] T

4 MNA circuit equations T. Stykel. Model reduction of nonlinear circuit equations p.3 Consider a linear DAE system E(x) ẋ = A x + f(x) + B u y = B T x with A C(AT C C η)at C A L A V E(x)= 0 L(j L ) 0, A= A T A 0 I L 0 0, B= 0 0, A T V I A R g(a T R η) [ f(x) = 0, u = 0 j I v V ] η, x = j L j V, y = [ ] v I, j V η node potentials, j L, j V, j I currents through inductors, voltage and current sources, v V, v I voltages at voltage and current sources, A R,A C,A L,A V,A I incidence matrices of resistors, capacitors, inductors, voltage and current sources

5 T. Stykel. Model reduction of nonlinear circuit equations p.4 Index Assumptions A V has full column rank (= no V-loops) [A C, A L, A R, A V ] has full row rank (= no I-cutsets) C, L, G are symmetric, positive definite Index characterization [ Estévez Schwarz/Tischendorf 00 ] Index = 0 no voltage sources and every node has a capacitive path to a reference node Index = 1 no CV-loops except for C-loops and no LI-cutsets Index = 2, otherwise

6 T. Stykel. Model reduction of nonlinear circuit equations p.5 Model reduction problem Given a large-scale system E(x) ẋ = A x + f(x) + B u y = C x with x R n and u, y R m, find a reduced-order system Ẽ( x) x = Ã x + f( x) + B u ỹ = C x with x R r, u, ỹ R m, r n. preservation of passivity and stability small approximation error ỹ y tol u for all u U need for computable error bounds numerically stable and efficient methods

7 T. Stykel. Model reduction of nonlinear circuit equations p.6 Model reduction techniques Linear circuit equations Krylov subspace methods ( moment matching ) SyPVL for RC, RL, LC circuits [ Freund et al. 96, 97 ] PRIMA, SPRIM for RLC circuits [ Odabasioglu et al. 96, 97; Freund 04, 05 ] Positive real interpolation [ Antoulas 05, Sorensen 05, Ionutiu et al. 08 ] Balancing-related model reduction methods LyaPABTEC for RC, RL circuits [ Reis/S. 10 ] PABTEC for RLC circuits [ Reis/S. 09 ] Nonlinear circuit equations Proper orthogonal decomposition (POD) [ Verhoeven 08 ] Trajectory piece-wise linear approach (TPWL) [ Rewieński 03 ] (Quadratic) bilinearization + balanced truncation [ Benner/Breiten 10 ]

8 T. Stykel. Model reduction of nonlinear circuit equations p.7 PABTEC Tool [Er,Ar,Br,Cr,... ] = PABTEC(Incidence matrices, element matrices,... ) nonlinear linear Decoupling of inear subcircuits [Erl,Arl,Brl,Crl,... ] = PABTECL(Incidence matrices, element matrices,... ) (no dynamics) no L or no C no L and no C Topology no CVI loops no LVI cutsets no R otherwise (unreducible) Lyapunov Riccati Lur e Preprocessing (Projectors) Preprocessing (Projectors) Preprocessing (Projectors) Solving Lyapunov equations (ADI, Krylov methods) Model reduction Solving Riccati equations (Newton s method) Model reduction Solving Lur e equations (Newton s method) Model reduction Postprocessing Postprocessing Postprocessing linear nonlinear Recoupling of subsystems

9 Decoupling T. Stykel. Model reduction of nonlinear circuit equations p.8 + u in u out Large linear RLC circuits arise in modelling transmission lines and pin packages modelling circuits elements by Maxwell s equations via partial element equivalent circuits (PEEC) Assume that C(A T C η) = A C = [ A Cl, A Cn ], AL = [ A Ll, A Ln ], AR = [ A R l, A R n ], [ C l 0 0 C n (A T C n η) ], L(j L ) = [ Ll 0 0 L n (j Ln ) ] [ ] Gl, g(a T R η) = A T R l η. g n (A T R n η)

10 Replacement of nonlinear elements T. Stykel. Model reduction of nonlinear circuit equations p.9 :-( LI-cutsets may arise :-) :-( LI- or I-cutsets may arise

11 Replacement of nonlinear elements T. Stykel. Model reduction of nonlinear circuit equations p.10 :- additional variables :-) :-( CV- or V-loops may arise

12 Replacement of nonlinear elements T. Stykel. Model reduction of nonlinear circuit equations p.11 :- additional variables :-) :- additional variables

13 Decoupled system T. Stykel. Model reduction of nonlinear circuit equations p.12 Linear RLC equations: E ẋ l = Ax l + B u l y l = B T x l with A C CA T C 0 0 A R GA T R A L A V E = 0 L 0, A= A T L 0 0 A C = [ A Cl ] [, A L = A Ll 0 [ ] G A V A Cn l 0 0 A V =, G = 0 G 1 0, G 2 A T V 0 0 ] [ ] [ A R, A R = l A R n,1 A R n,2, A I = 0 I I A I 0, B= I A I A R n,2 A Ln 0 I 0 [ ] x T l = η T ηz T j T Ll jv T jt, Cn [ ] u T l = j T I jz T j T Ln u T V ut ; Cn Nonlinear equations: C n (v Cn ) d dt u C n = j Cn, L n (j Ln ) d dt j L n = A T L n η, j z = (G 1 + G 2 )G 1 1 g n(a T R n η) G 2 A T R n η, ],

14 T. Stykel. Model reduction of nonlinear circuit equations p.13 Balanced truncation System G = ( E, A, B, C ) is balanced if the controllability and observability Gramians X and Y satisfy X = Y = diag(σ 1,..., σ n ). Idea: balance the system, i.e., find an equivalence transformation ( Ê, Â, ˆB, Ĉ ) = (Wb ET b, W b AT b, W b B, CT b ) ( [ ] [ ] [ ] E = 11 E 12 A, 11 A 12 B, 1, [C 1, C 2 ] E 21 E 22 A 21 A 22 B 2 such that ˆX = Ŷ = diag(σ 1,..., σ n ) and truncate the states corresponding to small σ j G = (E 11, A 11, B 1, C 1 ). ) DAEs: G(s) = C(sE A) 1 B = G sp (s) + P(s) G(s) = G sp (s) + P(s)

15 T. Stykel. Model reduction of nonlinear circuit equations p.14 Projected Lur e equations If G=(E, A, B, C ) is passive, then there exist matrices X = X T 0, J c, K c and Y = Y T 0, J o, K o that satisfy the projected Lur e equations (A BC) XE T + EX(A BC) T + 2P l BB T P T l = 2K c K T c, X = P r XP T r, E XC T P l BM T 0 = K cj T c, I M 0 M T 0 = J cj T c, (A BC) T Y E + E T Y (A BC) + 2Pr T C T CP r = 2Ko T K o, B T Y E + M0 TCP r = Jo T K o, I M0 TM 0 = JT o J o, Y = P T l Y P l, where M 0 = I 2 lim s C(sE A + BC) 1 B, P r and P l are the spectral projectors onto the left and right deflating subspaces of the pencil λe A + BC corresponding to the finite eigenvalues. 0 X min X X max, 0 Y min Y Y max X min controllability Gramian, Y min observability Gramian

16 Passivity-preserving BT method T. Stykel. Model reduction of nonlinear circuit equations p.15 Given a passive system G = (E, A, B, C ). 1. Compute P r, P l, M Compute X min = RR T, Y min = LL T (= solve the Lur e equations). ] 3. Compute the SVD L T Π1 ER = [U 1, U 2 ][ [V 1, V 2 ] T. Π 2 4. Compute the reduced-order model [ ] [ I 0 2 W T AT 2 W T B C Ẽ =, Ã = B CT 2I B C [ ] W B T B = B / [, C = CT C / 2 ] 2 ], with I M 0 = C B, W = LU 1 Π 1/2 1 and T = RV 1 Π 1/2 1.

17 Properties T. Stykel. Model reduction of nonlinear circuit equations p.16 G = (Ẽ, Ã, B, C ) is passive G = (Ẽ, Ã, B, C ) is reciprocal ( G(s) = Σ GT (s)σ ) Error bounds: G H := sup G(iω) 2 ω R If 2 I + G H (π lf π nf ) < 1, then G G H 2 I + G 2 H (π lf π nf ). If 2 I + G H (π lf π nf ) < 1, then G G H 2 I + G 2 H (π lf π nf ).

18 Application to circuit equations T. Stykel. Model reduction of nonlinear circuit equations p.17 A CAT C C 0 0 A GA T R R AIAT I A L A V E = 0 L 0, A BC = A T A I 0 L 0 0, B= 0 0 =C T A T V 0 I 0 I Compute P r and P l using the canonical projectors technique [März 96] P r = H 5 (H 4 H 2 I) H 5 H 4 A L H H 6 0 A T V (H 4H 2 I) A T V H 4A L H 6 0 with H 1 = Z T CRIV A L L 1 A T L Z CRIV, H 2 =..., H 3 = Z T C H 2Z C, H 4 = Z C H 1 3 ZT C, H 5 = Z CRIV H 1 1 ZT CRIV A L L 1 A T L I, H 6 = I L 1 A T L Z CRIV H 1 1 ZT CRIV A L, Z C and Z CRIV are basis matrices for ker A T C and ker[a C, A R, A I, A V ] T. P l = S P T r S T with S = diag(i nη, I nl, I nv )

19 Application to circuit equations T. Stykel. Model reduction of nonlinear circuit equations p.18 A CAT C C 0 0 A GA T R R AIAT I A L A V E = 0 L 0, A BC = A T A I 0 L 0 0, B= 0 0 =C T A T V 0 I 0 I Compute P r and P l = S P T r S T with S = diag(i nr, I nl, I nv ). Compute M 0 = I 2 lim s C(sE A + BC) 1 B M 0 = [ I 2A T I ZH 1 0 ZT A I 2A T I ZH 1 0 ZT A V 2A T V ZH 1 0 ZT A I I + 2A T V ZH 1 0 ZT A V ] where H 0 = Z T (A R GA T R +A I AT I +A V AT V )Z, Z = Z CZ RIV C, Z RIV C is a basis matrix for ker [A R, A I, A V ] T Z C and [Z RIV C, Z RIV C ] is nonsingular.

20 Application to circuit equations T. Stykel. Model reduction of nonlinear circuit equations p.19 Compute M 0, P r, P l = S P T r S T with S = diag(i nr, I nl, I nv ). Compute X min =RR T, Y min =FF T (solve the projected Lur e equations) If D 0 = I M 0 M0 T is nonsingular, then the projected Lur e equations are equivalent to the projected Riccati equation (A BC)XE T + EX(A BC) T + 2P l BB T P T l + 2(EXC T P l BM T 0 )D 1 0 (EXCT P l BM T 0 ) = 0, X = P rxp T r compute a low-rank approximation X min R R T, R R n,k, k n, using the generalized low-rank Newton method [ Benner/St. 10 ] Y min = S X min S T S R R T S T = F F T D 0 is nonsingular, if the circuit contains neither CVI-loops except for C-loops nor LIV-cutsets except for L-cutsets

21 Application to circuit equations T. Stykel. Model reduction of nonlinear circuit equations p.20 Compute M 0, P r, P l = S P T r S T with S = diag(i nr, I nl, I nv ). Compute X min R R T, Y min = S X min S T S R R T S T = F F T. Compute the SVD of F T E R F T E R = R T SE R is symmetric compute the EVD R T SE R Λ1 = [U 1, U 2 ][ instead of the SVD Λ 2 ] [U 1, U 2 ] T W = S RU 1 Λ 1 1/2 and T = RU 1 Λ 1 1/2 sign(λ 1 ) with Λ 1 = diag( λ 1,..., λ lf ), sign(λ 1 ) = diag(sign(λ 1 ),..., sign(λ lf ))

22 T. Stykel. Model reduction of nonlinear circuit equations p.21 Application to circuit equations Compute M 0, P r, P l = S P T r S T with S = diag(i nr, I nl, I nv ). Compute X min R R T. Compute the EVD R T SE R = [U 1, U 2 ][ Λ1 Λ 2 ] [U 1, U 2 ] T and W = S RU 1 Λ 1 1/2, T = RU 1 Λ 1 1/2 sign(λ 1 ). Compute B and C such that C B = I M 0 (I M 0 )Σ with Σ = diag(i ni, I nv ) is symmetric compute the EVD (I M 0 )Σ = U 0 Λ 0 U T 0 B = sign(λ 0 ) Λ 0 1/2 U T 0 Σ and C = U 0 Λ 0 1/2

23 PABTECL algorithm T. Stykel. Model reduction of nonlinear circuit equations p.22 Compute M 0, P r, P l = S P T r S T with S = diag(i nr, I nl, I nv ). Compute X min R R T. Compute the EVD R T SE R = [U 1, U 2 ][ Λ1 Λ 2 ] [U 1, U 2 ] T and W = S RU 1 Λ 1 1/2, T = RU 1 Λ 1 1/2 sign(λ 1 ). Compute the EVD (I M 0 )Σ = U 0 Λ 0 U T 0 with Σ = diag(i ni, I nv ) and B = sign(λ 0 ) Λ 0 1/2 U T 0 Σ, C = U 0 Λ 0 1/2. Compute the reduced-order model [ ] I 0 Ẽ =, Ã = 1 [ 2 W T AT 2 W T BC B CT 2I B C [ W B = T B B / [ ], C = CT, C 2 / ] 2. ],

24 Recoupling T. Stykel. Model reduction of nonlinear circuit equations p.23 Linear reduced-order model: Ẽ x [ l = Ã x l + B1, B 2, B 3, B 4, B ] 5 u l, C 1 A T I η C 2 A T R n η + G 1 1 g n(a T R n η) ỹ l = C 3 x l y l = A T L n η C 4 j V C 5 j Cn Nonlinear equations: C n (v Cn ) d dt u C n = j Cn, L n (j Ln ) d dt j L n = A T L n η, j z = (G 1 + G 2 )G 1 1 g n(a T R n η) G 2 A T R n η

25 Reduced-order nonlinear system T. Stykel. Model reduction of nonlinear circuit equations p.24 We obtain the reduced-order system Ẽ( x) x = Ã x + f( x) + B u ỹ = C x with Ẽ Ã + B 2 (G 1 + G 2 ) C 2 B3 B5 B2 G 1 0 L Ẽ( x) = n ( j Ln ) C n (ũ Cn ) 0, Ã= C C , G 1 C2 0 0 G 1 B 1 B4 0 x l 0 0 B= 0 0, 0 f( x) = 0, x = j Ln ũ Cn. 0 0 g n (ũ Cn ) Remark: If n Cn = 0 and n Ln = 0, then passivity is preserved and under some additional topological conditions we have the error bound ỹ y 2 c(π lf π nf )( u 2 + ỹ 2 ). [ Heinkenschloss/Reis 09 ] ũ R n

26 Example: linear RLC circuit T. Stykel. Model reduction of nonlinear circuit equations p.25 n = , m = resistors inductors capacitors Bounded real characteristic values X min R R T, R R n,84 Reduced model: r = j Frequency responses Full order PABTEC Absolute error and error bound Error Error bound Magnitude 10 5 Magnitude Frequency w Frequency w

27 Example: nonlinear circuit T. Stykel. Model reduction of nonlinear circuit equations p.26 1 Input: voltage source 2000 linear capacitors 1990 linear resistors 991 linear inductors 10 nonlinear inductors 10 diodes 1 voltage source original reduced system system Dimension Simulation time u i V x 10 3 t Output: negative current of the voltage source x 10 5 t Error in the output orig. system red. system 4 Model reduction time 822 Error in the output 4.4e-05 i V t

28 T. Stykel. Model reduction of nonlinear circuit equations p.27 Conclusions and future work Model reduction of nonlinear circuit equations topology based partitioning balancing-related model reduction of linear subsystems with preservation of passivity and computable error bounds Exploiting the structure of MNA matrices E, A, B, C use graph algorithms for computing the basis matrices use modern numerical linear algebra algorithms for solving large-scale projected Riccati/Lyapunov equations MATLAB Toolbox PABTEC Preservation of passivity and error bounds for general circuits Numerical solution of large-scale Lur e equations

Model order reduction of electrical circuits with nonlinear elements

Model order reduction of electrical circuits with nonlinear elements Andreas Steinbrecher and Tatjana Stykel 1 Introduction The efficient and robust numerical simulation of electrical circuits plays a