Order Reduction of (Truly) Large-Scale Linear Dynamical Systems - PowerPoint PPT Presentation

Order Reduction of (Truly) Large-Scale Linear Dynamical Systems Roland W. Freund Department of Mathematics University of California, Davis, USA http://www.math.ucdavis.edu/ ˜ freund/ Supported in part by NSF

Motivation • Need for order reduction in VLSI circuit simulation • Corollary to Moore’s Law • RCL networks: Electric networks consisting of only resistors ( R ’s), capacitors ( C ’s), and inductors ( L ’s) • These networks are (truly) large

Moore’s law

VLSI chip scaling

VLSI interconnect • Wires are not ideal: Resistance Capacitance Inductance • Consequences: Timing behavior Noise Energy consumption Power distribution

Lumped-circuit paradigm • Replace ‘pieces’ of the interconnect by RCL networks • Up to O (10 6 ) circuit elements per network • Up to O (10 6 ) networks

Need for order reduction

Outline • The order reduction problem • Projection + Krylov = Pad´ e-type reduction • SPRIM for general RCL networks • SPRIM–SVD • Pad´ e-type approximation properties of SPRIM • Concluding remarks

Outline • The order reduction problem • Projection + Krylov = Pad´ e-type reduction • SPRIM for general RCL networks • SPRIM–SVD • Pad´ e-type approximation properties of SPRIM • Concluding remarks

RCL networks as descriptor systems • System of linear time-invariant DAEs of the form C d dt x ( t ) + G x ( t ) = B u ( t ) y ( t ) = B T x ( t ) where C , G ∈ R N × N and B ∈ R N × m • x ( t ) ∈ R N is the unknown vector of state variables • m inputs, m outputs • s C + G is nonsingular except for finitely many values of s ∈ C

Reduced-order models • System of DAEs of the same form: d dt z ( t ) + G n z ( t ) = B n u ( t ) C n y ( t ) = B T n z ( t ) � • But now: C n , G n ∈ R n × n B n ∈ R n × m and where n ≪ N

Transfer functions • Original descriptor system: H ( s ) = B T ( s C + G ) − 1 B • Reduced-order model: n ( s C n + G n ) − 1 B n H n ( s ) = B T • ‘Good’ reduced-order model ⇐ ⇒ ‘Good’ approximation H n ≈ H

Problem of structure preservation • Any RCL network is stable, passive, . . . • Reduced-order model should be stable, passive, . . . • More difficult problem: Reduced-order model of an RCL network should be synthesizable as an RCL network

Preservation of RCL structure

General RCL network equations • System of linear time-invariant DAEs of the form C d dt x ( t ) + G x ( t ) = B u ( t ) y ( t ) = B T x ( t ) where       G 1 G 2 G 3 C 1 0 0 B 1 0       − G T       C =  , G =  , B = 0 C 2 0 0 0 0 0   2   − G T 0 0 0 0 B 2 0 0 3 • Moreover: G + G T � 0 C � 0 and (This implies passivity!)

Outline • The order reduction problem • Projection + Krylov = Pad´ e-type reduction • SPRIM for general RCL networks • SPRIM–SVD • Pad´ e-type approximation properties of SPRIM • Concluding remarks

Projection-based reduction • Let V n ∈ R N × n be any matrix with full column rank n • Use V n to explicitly project the data matrices of C d dt x ( t ) + G x ( t ) = B u ( t ) y ( t ) = B T x ( t ) onto the subspace spanned by the columns of V n

Projection-based reduction, continued • Resulting reduced-order model d dt z ( t ) + G n z ( t ) = B n u ( t ) C n y ( t ) = B T n z ( t ) � where C n = V T G n = V T B n = V T n C V n , n G V n , n B • Passivity is preserved: C � 0 , G + G T � 0 C n � 0 , G n + G T ⇒ n � 0

Projection-based order reduction • PRIMA Passive Reduced Interconnect Macromodeling Algorithm (Odabasioglu, ’96; Odabasioglu, Celik, and Pileggi, ’97) • Split-congruence transformations (Kerns, Yang, ’97) • SPRIM Structure-Preserving Reduced Interconnect Macromodeling (F., ’04 and ’07)

PRIMA reduced-order models • Let V n be any matrix whose columns span the n -th Krylov subspace K n ( A , R ) where � � − 1 C � � − 1 B A := s 0 C + G and R := s 0 C + G and s 0 ∈ R is a suitably chosen expansion point • Projection + Krylov subspace = Pad´ e-type approximant : H n ( s ) = H ( s ) + O (( s − s 0 ) q ) , where q ≥ ⌊ n/m ⌋

Structure is not preserved • Structure of the data matrices:       G 1 G 2 G 3 C 1 0 0 B 1 0       − G T       C = 0 C 2 0  , G = 0 0  , B = 0 0   2   − G T 0 0 0 0 B 2 0 0 3 • Structure of PRIMA reduced-order matrices: C n = , G n = , B n =

Outline • The order reduction problem • Projection + Krylov = Pad´ e-type reduction • SPRIM for general RCL networks • SPRIM–SVD • Pad´ e-type approximation properties of SPRIM • Concluding remarks

SPRIM • As in PRIMA, let V n be any matrix such that K n ( A , R ) = colspan V n • Key insight that is exploited in SPRIM: In order to have a Pad´ e-type property as in PRIMA, we can project with any matrix ˜ V n such that K n ( A , R ) ⊆ colspan ˜ V n • ... ; Odabasioglu, ’96; Grimme, ’97; Odabasioglu, Celik, and Pileggi, ’97; ...

SPRIM, continued • Recall:       G 1 G 2 G 3 C 1 0 0 B 1 0       − G T       C = G = B = 0 C 2 0  , 0 0  , 0 0     2 − G T 0 0 0 0 B 2 0 0 3 • Partition V n accordingly:   V (1) n     V (2) V n =   n   V (3) n

SPRIM, continued • Set   V (1) 0 0 n     V (2) ˜ V n =  0 0  n   V (3) 0 0 n K n ( A , R ) = colspan V n ⊆ colspan ˜ • Then: V n • This guarantees a Pad´ e-type property!

SPRIM models • Recall:       G 1 G 2 G 3 C 1 0 0 B 1 0       − G T       C =  , G =  , B = 0 C 2 0 0 0 0 0   2   − G T 0 0 0 0 B 2 0 0 3 and   V (1) 0 0 n     V (2) ˜ V n =  0 0  n   V (3) 0 0 n

SPRIM models, continued • The projection now preserves this structure:       ˜ ˜ ˜ ˜ ˜ G 1 G 2 G 3 C 1 0 0 B 1 0       G T ˜ − ˜       C n =  , G n =  , B n = 0 0 0 0 0 C 2 0     2 ˜ − ˜ G T 0 0 0 0 B 2 0 0 3 • Pad´ e-type property: H n ( s ) = H ( s ) + O (( s − s 0 ) q ) with q ≥ ⌊ n/m ⌋

An RCL circuit with mostly C’s and L’s 0 10 Exact PRIMA model SPRIM model −1 10 −2 10 −3 10 abs(Z ( 2,1)) −4 10 −5 10 −6 10 −7 10 −8 10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Frequency (Hz) 9 x 10 Exact and models corresponding to n = 120

A package example 0 10 Exact PRIMA model SPRIM model −1 10 V1int/V1ext −2 10 −3 10 −4 10 8 9 10 10 10 10 Frequency (Hz) Exact and models corresponding to n = 80

Package example, high frequencies 0 10 Exact PRIMA model SPRIM model −1 10 V1int/V1ext −2 10 10 10 Frequency (Hz) Exact and models corresponding to size n = 80

A finite-element model of a shaft 0 10 −1 10 −2 10 −3 10 abs(Z) −4 10 −5 10 −6 10 −7 10 Exact PRIMA model SPRIM model −8 10 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz) Exact and models corresponding to n = 15

SPRIM vs. PRIMA • Pros: Same computational work SPRIM preserves block structure and reciprocity Higher accuracy • Cons: SPRIM models are two or three times as large as corresponding PRIMA models

Outline • The order reduction problem • Projection + Krylov = Pad´ e-type reduction • SPRIM for general RCL networks • SPRIM–SVD • Pad´ e-type approximation properties of SPRIM • Concluding remarks

SPRIM–SVD • Columns of V n span K n ( A , R ) • SPRIM projection:     V (1) V (1) 0 0 n n         V (2) V (2) ˜ V n = = ⇒ V n =     0 0 n n     V (3) V (3) 0 0 n n • But: # of rows of V (3) ≪ # of rows of V (1) and V (2) n n n

The RCL circuit with mostly C’s and L’s 0 10 Exact PRIMA model SPRIM model −1 10 −2 10 −3 10 abs(Z ( 2,1)) −4 10 −5 10 −6 10 −7 10 −8 10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Frequency (Hz) 9 x 10 Exact and models corresponding to n = 120

Singular values of projection subblocks 1.4 (1) V n (2) V n (3) 1.2 V n 1 Singular values 0.8 0.6 0.4 0.2 0 0 20 40 60 80 100 120

SPRIM–SVD, continued • For l = 1 , 2 , 3, replace V ( l ) by the matrix U ( l ) containing the n n left singular vectors corresponding to the ’non-zero’ singular values • SPRIM–SVD projection:     V (1) U (1) 0 0 0 0 n n         V (2) U (2) ˜ ˆ V n = = ⇒ V n =     0 0 0 0 n n     V (3) U (3) 0 0 0 0 n n • For the example: 3 n = 360 = ⇒ 74 + 72 + 1 = 147