Passivity Preserving Model Reduction for Large-Scale Systems. Peter Benner.

Transcription

1 Passivity Preserving Model Reduction for Large-Scale Systems Peter Benner Mathematik in Industrie und Technik Fakultät für Mathematik Sonderforschungsbereich 393 S N benner@mathematik.tu-chemnitz.de SIMULATION joint work with Heike Faßbender, Enrique S. Quintana-Ortí Model Order Reduction, Coupled Problems and Optimization Lorentz Center, Leiden, September 19 23, 2005

2 Outline Outline Linear systems in circuit simulation Model reduction Positive-real balanced truncation Implementation of PRBT Newton s method for algebraic Riccati equations Implementation based on matrix sign function Implementation based on ADI method Passive reduced-order models by interpolating spectral zeros Conclusions SIMULATIONSN 2

3 Linear systems in circuit simulation Linear Systems Linear time-invariant systems in generalized state-space form: Eẋ(t) = Ax(t) + Bu(t), t > 0, x(0) = x 0, y(t) = Cx(t) + Du(t), n generalized states, i.e., x(t) R n (n is the order of the system); m inputs, i.e., u(t) R m ; m outputs, i.e., y(t) R m ; A λe stable, i.e., λ (A, E) C { } system is stable, Corresponding transfer function: G(s) = C(sE A) 1 B + D. In frequency domain, y(s) = G(s)u(s). SIMULATIONSN 3

4 Linear systems in circuit simulation Linear Systems in Circuit Simulation In circuit simulation, linear systems arise from a modified nodal analysis (MNA) using Kirchhoff s laws for linear RLC circuits, resulting from, e.g., decoupling large sub-circuits of a given layout/network, modeling interconnect (transmission lines), modeling the pin package of VLSI circuits; linearization of nonlinear circuits around a DC operating point (e.g.,in small-signal analysis). SIMULATIONSN 4

5 Model reduction Model Reduction Often, order n is too large to allow simulation in an adequate time or to even tackle the model using available solvers. Idea: replace order-n original system Eẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t), by reduced-order system Ẽ x(t) = Ã x(t) + Bu(t), ỹ(t) = C x(t) + Du(t), of order l n with ỹ(t) R p such that the output error is small. y ỹ = Gu Gu G G u SIMULATIONSN 5

6 Model reduction Passive Systems Important property of circuits to be preserved in reduced-order model: passivity. Definition: A linear system is passive if t u(τ)t y(τ) dτ 0 t R, u L 2 (R, R m ). The system cannot generate energy. system is passive its transfer function is positive real Definition: A real, rational matrix-valued function G : C C m m is positive real if 1. G is analytic in C + := {s C Re(s) > 0}, 2. G(s) + G T ( s) 0 for all s C +. SIMULATIONSN 6

7 Model reduction Goal For passive linear system, compute passive reduced-order system with computable global error bound. Padé-type methods in general do not preserve passivity, post-processing necessary [Bai/(Feldmann)/Freund 98, 01]. PRIMA [Odabasioglu et al. 96, 97] preserves passivity for interconnect models, basically Arnoldi process. SyPVL preserves passivity for RLC circuits [Feldmann/Freund 96, 97]. LR-ADI/dominant subspace approximation can preserve passivity [Li/White 01]. No computable error bounds available for Krylov-type methods. Here: alternative approach for general passive systems based on positive-real balancing. SIMULATIONSN 7

8 Positive-real balanced truncation Positive-Real Balancing Set R := D + D T (positive real R 0, here assume R > 0), Â := A BR 1 C, and consider the two dual positive-real algebraic Riccati equations (PAREs) 0 = ÂP ET + EP ÂT + EP C T R 1 CP E T + BR 1 B T, 0 = ÂT QE + E T QÂ + EQBR 1 B T QE + C T R 1 C. Let P min, Q min > 0 be the minimal solutions, then the system is positive real balanced iff P min = E T Q min E = diag(σ 1,..., σ n ). P min, E T Q min E are called positive-real Gramians. Note: P min, Q min > 0 are the stabilizing solutions of the PAREs. SIMULATIONSN 8

9 Positive-real balanced truncation Positive-Real Balanced Truncation I 1. Find positive-real balancing equivalence transformation (E, A, B, C, D) (T ES 1, T AS 1, T B, CS 1, D) =: (Ẽ, Ã, B, C, D), [ ] P min = E T Σ1 Σ Q min E =, 1 = diag(σ 1,..., σ r ), Σ 2 = diag(σ r+1,..., σ n ), Σ 2 σ 1 σ 2... σ r > σ r+1 σ r+2... σ n > Truncate the states x r+1,..., x n of the balanced system ([ ] [ ] [ ] (Ẽ, Ã, B, C, D) E11 E = 12 A11 A, 12 B1,, [ ) ] C E 21 E 22 A 21 A 22 B 1 C 2, D, 2 i.e., the reduced-order model and transfer function are (E r, A r, B r, C r, D r ) := (E 11, A 11, B 11, C 11, D) G r (s) = C r (se r A r ) 1 B r + D r SIMULATIONSN 9

10 Positive-real balanced truncation Positive-Real Balanced Truncation II Properties: Reduced-order model is passive ( stable), relative error bound if G H D 2 : G G r H G H G G r H G + D T H 2 R 2 2 G r + D T H n k=r+1 σ k Computation: Analogous to balanced truncation: let P min = S T S, E T Q min E = R T R, transformation matrices and reduced-order model are computed from SVD of SR T. Often, P min, Q min have low numerical rank = use (numerical) full-rank factors rather than Cholesky factors = cheap SVD, cost O(n) instead of O(n 3 ) = need method to compute factored solutions of PAREs. SIMULATIONSN 10

11 Implementation Newton s Method for AREs Newton s method for AREs Consider algebraic Riccati equation (ARE) 0 = R(Q) = C T C + A T Q + QA QBB T Q. Frechét derivative of R at Q: R Q : Z (A BB T Q) T Z + Z(A BB T Q) Newton-Kantorovich method: Q j+1 = Q j ( R Q j ) 1 R(Qj ), j = 0, 1, 2,... SIMULATIONSN 11

12 Implementation Newton s method for AREs = Newton s method for AREs [Kleinman 68, Mehrmann 91, Lancaster/Rodman 95] FOR j = 0, 1, 2,... A j A BB T Q j =: A BK j. Solve Lyapunov equation A T j N j + N j A j = R(Q j ). Q j+1 Q j + N j. END FOR j Convergence: A j is stable j 0. 0 Q... Q j+1 Q j... Q 1. lim j R(Q j ) F = 0, lim j Q j = Q 0 (quadratically), acceleration of (initially slow) convergence possible using line searches. Need efficient Lyapunov solver, depending on data structures and computing full-numerical-rank factors. SIMULATIONSN 12

13 Implementation Factored Newton Iteration Newton s method for AREs Rewrite Newton s method for AREs [Kleinman 68] A T j N j + N j A j = R(Q j ) A T j (Q j + N j ) + (Q }{{} j + N j ) A }{{} j = C T C Q j BB T Q }{{} j =Q j+1 =Q j+1 =: W j W T j Let Q j = Y j Y T j for rank (Y j ) n: A T j ( Yj+1 Yj+1 T ) ( + Yj+1 Yj+1) T Aj = W j Wj T SIMULATIONSN 13

14 Implementation Method based on Matrix Sign Function matrix sign function implementation Want method for solving Lyapunov equations which computes full-rank factor Y j+1 directly (without ever forming Q j+1 ). Consider F T X + XF + E = 0. Newton s method for the matrix sign function yields [Roberts 71]: F 0 F, E 0 E, for j = 0, 1, 2,... F k+1 1 ( Fk + c 2 2c kf 1 ) k, k E k+1 1 ( Ek + c 2 2c kf T k k = X = 1 2 lim j E k E k F 1 ) k. Here: E = B T B or C T C, F = A T or A, want factor R of solution. SIMULATIONSN 14

15 Implementation Solving Lyapunov Equations for Full-Rank Factor matrix sign function implementation Consider now A T X + XA + C T C = 0. For E 0 = C T 0 C 0 := C T C, C R p n obtain E k+1 = 1 ( Ek + c 2 2c ka T k k E ka 1 ) 1 k = 2c k [ C k c k C k A 1 k ] T [ C k c k C k A 1 k ]. = Re-write E k iteration: [ C 0 := C, C k+1 := 1 2ck C k c k C k A 1 k ], = 1 2 lim k C k = R Problem: C k R p k n = C k+1 R 2p k n Cure: limit work space by computing rank-revealing QR factorization in each step. SIMULATIONSN 15

16 Implementation Application to Positive-Real Balancing I matrix sign function implementation Factored Newton method not directly applicable to PAREs! Need modification: right-hand side of Lyapunov equation W j W T j. with R > 0. RHS = C T R 1 C+Q j BR 1 B T Q j =: C C T + B j BT j Lyapunov equation is non-singular linear system of equations = write A T j Q j+1 + Q j+1 A j = C T C + BT j Bj as A T j (Q j+1 Q j+1 ) + (Q j+1 Q j+1 )A j = W j Wj T ( W j Wj T = Solve two Lyapunov equations per step with equal Lyapunov operator. ). SIMULATIONSN 16

17 Implementation Application to Positive-Real Balancing II matrix sign function implementation To get factored Newton iterates need factor of Q := Q jmax Q jmax = Z jmax Z T j max Z jmax Z T j max 0. Solution: similar to stochastic balanced truncation [Varga/Fasol 93, Varga 00] Get full-rank factor from stable, nonnegative Lyapunov equation A T (Z T Z) + (Z T Z)A + C T C = 0 where C = R 1 2 C R 1 2 B [Zjmax, Z jmax ] [ Z T jmax Z T j max ]. SIMULATIONSN 17

18 Implementation Need method for solving Lyapunov equations Sparse Implementation sparse implementation A T j ( Yj+1 Yj+1 T ) ( + Yj+1 Yj+1 T ) Aj = W j Wj T, where W j = [C T, Y j (Yj T B)], which computes Y j+1 directly (without ever forming X j+1 ), and uses the structure of A j, A j = A BK j = A B (B T Y j ) Y T j, = sparse m Note: as m n, we can efficiently apply Sherman-Morrison-Woodbury formula (A BK j ) 1 = (I n + A 1 B(I m K j A 1 B) }{{} K j )A 1 m m = (I n + ˆB(I m K j ˆB) 1 K j )A 1 SIMULATIONSN 18

19 Implementation ADI Method for Lyapunov Equations sparse implementation For A R n n stable, W R n w (w n), consider Lyapunov equation A T X + XA = W W T. ADI Iteration: [Wachspress 88] (A T + p k I)X (j 1)/2 = W W T X k 1 (A p k I) (A T T + p k I)X k = W W T X (j 1)/2 (A p k I) with parameters p k C and p k+1 = p k if p k R. For X 0 = 0 and proper choice of p k : lim X k = X superlinear. k Re-formulation using X k = Z k Z T k yields iteration for Z k... SIMULATIONSN 19

20 Implementation Set X k = Z k Z T k Factored ADI Iteration [Penzl 97, Li/Wang/White 99, B./Li/Penzl], some algebraic manipulations = sparse implementation V 1 p 2Re (p 1 )(A T + p 1 I) 1 W, Z 1 V 1 FOR j = 2, 3,... r V k `I (pk + p k 1 )(A T + p k I) 1 V k 1, Z k ˆ Z k 1 V k where and Re (p k ) Re (p k 1 ) Z kmax = [ V 1... V kmax ] V k = C n w Z kmax Z T k max X Note: Implementation in real arithmetic possible by combining two steps. SIMULATIONSN 20

21 Implementation Newton-ADI for ARE [B./Li/Penzl] Solve Lyapunov equation (A BK j ) T Y j+1 Y T j+1 + Y j+1y T j+1 (A BK j) = W j W T j with factored ADI iteration. sparse implementation Obtain low-rank approximations Z 0, Z 1,..., Z kmax to Lyapunov solution. Newton s method with factored iterates Q j+1 = Y j+1 Y T j+1 = Z k max Z T k max. Factored solution of ARE: Q Y jmax Y T j max. SIMULATIONSN 21

22 Numerical examples Numerical Examples 1. Transmission line based on RLC loops used as interconnect model [Infineon 2002] Partitioning into nseg segments = n = 3nseg inputs, 2 outputs. E nonsingular, but ill-conditioned. 2. RLC ladder network, also used for interconnect modeling [Gugercin/Antoulas 2003] Cascadic interconnection of nsec sections, each section consisting of an RLC loop in parallel with an additional resistance = n = 2nsec. 1 input, 1 output. E = I n. Hardware: Xeon cluster with 30 nodes (2.4GHz, 1GByte RAM each) 3,800 Mflops for matrix product. Interconnection network uses Myrinet switch 10 µsec latency, 1.9 Gbit/sec bandwidth. SIMULATIONSN 22

23 Numerical examples Example 1: Accuracy Here, n = 199, m = p = 2, r = 20. Difficulty: catch falling edge of output signal around 100 Hz. Bode Diagram From: In(1) 0 From: In(2) 50 To: Out(1) Magnitude (db) To: Out(2) Frequency (rad/sec) SIMULATIONSN 23

24 Numerical examples Example 2: Accuracy n = 1000, m = p = H error for r = 8 Numerical ranks of Gramians are 90, 75. Reduced-order selection: r = max{j σ j /σ } = r = 8. G(jω) G 8 (jω) Newton iterations, 9 sign iterations each Frequency ω SIMULATIONSN 24

25 Numerical examples Parallel Performance Mflop rate of the numerical kernels Execution times max n 10, 000 n = 2002 (Ex. 1), n = 2000 (Ex. 2) 3000 Random stable system; n/sqrt(n p ) = 2000; m=p=1 psb03odc pdgeclnw pdgecrny 3 x 104 Passive model reduction of RLC Examples Example 1 Example Mflops per node 1500 Execution time (sec.) Number of processors Number of processors Speed-up #Nodes (n p ) Ex. 1 Ex SIMULATIONSN 25

26 Interpolating spectral zeros Passive Reduced-Order Models by Interpolating Spectral Zeros Definition: Spectral factorization of a positive real transfer function: G(s) + G T ( s) = W (s)w T ( s), W (s) stable, rational. Spectral zeros of G: S G := {λ C det W (λ) = 0}. Theorem: (Antoulas 2002/05) Let S G be the spectral zeros of a reduced-order model. If S G S G, G(λ) = G(λ) λ S G G is a minimal degree rational interpolant of the values of G on the set S G, then the reduced-order model corresponding to G is both stable and passive. SIMULATIONSN 26

27 Interpolating spectral zeros Computing an Interpolatory Reduced-Order Model I Choose 2l distinct points s 1,..., s 2l S G. Let Ṽ = [(s 1 I A) 1 B... (s k I A) 1 B] W = [(s k+1 I A T ) 1 C T... (s 2k I A T ) 1 C T ]. Assume det W T Ṽ 0, let V = Ṽ W = W (Ṽ T W ) 1 and compute the projected system à = W T AV, B = W T B, C = CV, D = D. Then G(s i ) = G(s i ), i = 1, 2,..., 2l. and the projected system is stable and passive. [Antoulas 2002/05] SIMULATIONSN 27

28 Interpolating spectral zeros Computing an Interpolatory Reduced-Order Model II Theorem: (Antoulas 2002/05) The (finite) spectal zeros are the (finite) eigenvalues of M λl = A B A T C T C B T D + D T λ I I 0. Method: (Sorensen 2003/05) Compute partial Schur reduction MQ = LQT where Re(λ) > 0 for all λ λ (T ) using a Cayley transformation (µl M) 1 (µl + M), µ R. Let Q T = [X T, Y T, Z T ]. and compute SVD X T Y = Q x S 2 Q T y. Let V = XQ x S 1 and W = Y Q y S 1. Then à = W T AV, B = W T B, C = CV is stable and passive. SIMULATIONSN 28

29 Interpolating spectral zeros Computing an Interpolatory Reduced-Order Model III For R := D +D T > 0, the finite spectral zeros (eigenvalues of M λl) are the eigenvalues of the Hamiltonian matrix [ ] A BR 1 C BR 1 B T C T R 1 C (A BR 1 C) T. Instead of partial Schur decomposition, compute using the Hamiltonian Lanczos process. Advantages: HQ = QT where Re(λ) > 0 λ λ (T ) Hamiltonian spectral symmetry is preserved. Faster convergence if complex shifts are used. Slightly cheaper iterations. SIMULATIONSN 29

30 Interpolating spectral zeros Numerical Example Example 2, again: (RLC ladder network [Gugercin/Antoulas 2003], here n = 400, r = Spectral Zeros Right Half Plane Spectral Zeros and Interpolation Points Imaginary Part Imaginary Part Cayley approach Hamiltonian approach spectral values Real Part Real Part SIMULATIONSN 30

31 Interpolating spectral zeros Numerical Example: Accuracy Bode Diagram Cayley approach Magnitude (db) original system reduced system Frequency (rad/sec) Bode Diagram Hamiltonian approach Magnitude (db) original system reduced system Frequency (rad/sec) SIMULATIONSN 31

32 Conclusions Conclusions Guaranteed passive reduced-order models. PRBT reduced-order models more accurate than models computed via moment matching/pvl for same order. Global (though conservative) error bound. PRBT applicable to fairly large models using parallelization. New variant of method based on interpolation of spectral zeros using structure-preserving method. Descriptor case for sparse systems not treatable yet. Parallel implementation based on sign function for software library PLiCMR available. SIMULATIONSN 32

33 Conclusions Ad(é) Thank you for your attention! SIMULATIONSN 33