1. Model Order Reduction Slides adopted from Luca Daniel
University of California, Berkeley
Massachusetts Institute of Technology
with contributions from:
Joel Phillips, Cadence Berkeley Labs
Jacob White, Massachusetts Instit. of Technology

2. Overview Reduction via moment matching: (Projection Framework)
general Krylov Subspace methods
case 1: Arnoldi
case 2: PVL
case 3: multipoint moment matching
Importance of preserving passivity
PRIMA

3. Projection Framework: Change of variables
reduced state
Note: q << N
original state

4. Projection Framework
Original System
Substitute
Note: now few variables (q<<N) in the state, but still thousands of equations (N)

5. Projection Framework (cont.)
Reduction of number of equations: test multiplying by VqT
If V and U biorthogonal

6. Projection Framework (cont.)
qxn
qxq
nxn
nxq

7. Projection Framework
Change of variables
Equation Testing

8. Approaches for picking V and U
Use Eigenvectors
Use Time Series Data
Compute
Use the SVD to pick q < k important vectors
Use Frequency Domain Data
Use Singular Vectors of System Grammians?
Use Krylov Subspace Vectors?

9. Overview Reduction via moment matching: (Projection Framework)
general Krylov Subspace methods
case 1: Arnoldi
case 2: PVL
case 3: multipoint moment matching
Importance of preserving passivity
PRIMA

10. Intuitive view of Krylov subspace choice for change of base projection matrix
Taylor series expansion:
change base and use only the first few vectors of the Taylor series expansion: equivalent to match first derivatives around expansion point U

11. Combine point and moment matching: multipoint moment matching
Multipole expansion points give larger band
Moment (derivates) matching gives more
accurate behavior in between expansion points

12. Compare Pade’ Approximations and Krylov Subspace Projection Framework
moment matching at
single DC point
numerically very
ill-conditioned!!!
Krylov Subspace Projection Framework:
multipoint moment
matching
numerically very
stable!!!

13. Aside on Krylov Subspaces - Definition
The order k Krylov subspace generated from matrix A and vector b is defined as

14. Projection Framework: Moment Matching Theorem (E. Grimme 97)If
and
Then

15. Overview Reduction via moment matching: (Projection Framework)
general Krylov Subspace methods
case 1: Arnoldi
case 2: PVL
case 3: multipoint moment matching
Importance of preserving passivity
PRIMA

16. Special simple case #1: expansion at s=0,V=U, orthonormal UTU=I
If U and V are such that:
Then the first q moments (derivatives) of the
reduced system match

17. Algebraic proof of case #1: expansion at s=0, V=U, orthonormal UTU=I
apply k times lemma in next slide

18. Lemma:
Note in general:
BUT...
Substitute: Iq
U is orthonormal

19. Need for Orthonormalization of U
Vectors will line up with dominant eigenspace!

20. Need for Orthonormalization of U (cont.)
In "change of base matrix" U transforming to the new reduced state space, we can use ANY columns that span the reduced state space
In particular we can ORTHONORMALIZE the Krylov subspace vectors

21. Orthonormalization of U: The Arnoldi Algorithm
For i = 1 to k
Generates k+1 vectors!
Orthogonalize new vector
For j = 1 to i
Normalize new vector

22. Overview Reduction via moment matching: (Projection Framework)
general Krylov Subspace methods
case 1: Arnoldi
case 2: PVL
case 3: multipoint moment matching
Importance of preserving passivity
PRIMA

23. Special case #2: expansion at s=0, biorthogonal VTU=I
If U and V are such that:
Then the first 2q moments of reduced system match

24. Proof of special case #2: expansion at s=0, biorthogonal VTU=UTV=Iq (cont.)
apply k times the lemma in next slide

25. Lemma: . Substitute: Substitute: Iq biorthonormality Iq

26. PVL: Pade Via Lanczos [P. Feldmann, R. W. Freund TCAD95]
PVL is an implementation of the biorthogonal case 2:
Use Lanczos process to biorthonormalize the columns of U and V: gives very good numerical stability

27. Overview Reduction via moment matching: (Projection Framework)
general Krylov Subspace methods
case 1: Arnoldi
case 2: PVL
case 3: multipoint moment matching
Importance of preserving passivity
PRIMA

28. Case #3: Intuitive view of subspace choice for general expansion points
In stead of expanding around only s=0 we can expand around another points
For each expansion point the problem can then be put again in the standard form

29. Case #3: Intuitive view of Krylov subspace choice for general expansion points (cont.)
Hence choosing Krylov subspace
s1=0 s1 s2 s3
matches first kj of transfer function around each expansion point sj

30. Overview Reduction via moment matching: (Projection Framework)
general Krylov Subspace methods
case 1: Arnoldi
case 2: PVL
case 3: multipoint moment matching
Importance of preserving passivity
PRIMA

31. Interconnected Systems
In reality, reduced models are only useful when connected together with other models and circuit elements in a composite simulation
Consider a state-space model connected to external circuitry (possibly with feedback!)
ROM
Can we assure that the simulation of the composite system will be well-behaved? At least preclude non-physical behavior of the reduced model?

32. Passivity
Passive systems do not generate energy. We cannot extract out more energy than is stored. A passive system does not provide energy that is not in its storage elements.
If the reduced model is not passive it can generate energy from nothingness and the simulation will explode

33. Interconnecting Passive Systems
The interconnection of stable models is not necessarily stable
BUT the interconnection of passive models is a passive model: Q D C - + Q D C - + Q D C - + Q D C - +

34. Positive Real Functions
A positive real function is a function internally stable with non-negative real part
(no unstable poles)
(real response)
(no negative resistors)
Hermittian=conjugate and transposed
It means its real part is a positive semidefinite matrix at all frequencies

35. Positive Realness & Passivity
For systems with immittance (impedance or admittance) matrix representation, positive-realness of the transfer function is equivalent to passivity
ROM + + - -

36. Necessary conditions for passivity for Poles/Zeros
The positive-real condition on the matrix rational function implies that:
If H(s) is positive-real also its inverse is positive real
If H(s) is positive-real it has no poles in the RHP, and hence also no zeros there.
Occasional misconception : “if the system function has no poles and no zeros in the RHP the system is passive”.
It is necessary that a positive-real function have no poles or zeros in the RHP, but not sufficient.

37. Sufficient conditions for passivity
Note that these are NOT necessary conditions (common misconception)

38. Congruence Transformations Preserve Positive Semidefinitness
Def. congruence transformation
same matrix
Note: case #1 in the projection framework V=U produces congruence transformations
Property: a congruence transformation preserves the positive semidefiniteness of the matrix
Proof. Just rename
Note:

39. Overview Reduction via moment matching: (Projection Framework)
general Krylov Subspace methods
case 1: Arnoldi
case 2: PVL
case 3: multipoint moment matching
Importance of preserving passivity
PRIMA

40. PRIMA (for preserving passivity) (Odabasioglu, Celik, Pileggi TCAD98)
A different implementation of case #1:
V=U, UTU=I, Arnoldi Krylov Projection Framework:
Use Arnoldi: Numerically very stable

41. PRIMA preserves passivity
The main difference between and case #1 and PRIMA:
case #1 applies the projection framework to
PRIMA applies the projection framework to
PRIMA preserves passivity because
uses Arnoldi so that U=V and the projection becomes a congruence transformation
E and A produced by electromagnetic analysis are typically positive semidefinite while may not be.
input matrix must be equal to output matrix

42. Algebraic proof of moment matching for PRIMA expansion at s=0, V=U, orthonormal UTU=I
Used Lemma: If U is orthonormal (UTU=I) and b is a vector such that

43. Proof of lemma
Proof:

44. Compare methods number of moments matched by model of order q
preserving passivity
case #1 (Arnoldi, V=U, UTU=I on
sA-1Ex=x+Bu) q no
PRIMA (Arnoldi,
V=U, UTU=I on
sEx=Ax+Bu)
yes
necessary when model is used in a time domain simulator
case #2 (PVL, Lanczos,V≠U, VTU=I on sA-1Ex=x+Bu) 2q
more efficient
(good only if model is used in frequency domain)

45. Overview Reduction via moment matching: (Projection Framework)
general Krylov Subspace methods
case 1: Arnoldi
case 2: PVL
case 3: multipoint moment matching
Importance of preserving passivity
PRIMA
Summary and Conclusions

46. Summary: Conventional Design Flow
Funct. Spec
Logic Synth.
Gate-level Net.
RTL
Layout
Floorplanning
Place & Route
Front-end
Back-end
Behav. Simul.
Gate-Lev. Sim.
Stat. Wire Model
Parasitic Extrac.

47. Summary: Parasitic Extraction
thousands of wires
e.g. critical path
e.g. gnd/vdd grid
Parasitic
Extraction
identify some ports
produce equivalent circuit that models response of wires at those ports
tens of circuit
elements for
gate level spice
simulation

48. Summary: Model Order Reduction (the second step of parasitic extraction)
Electromagnetic
Analysis
(Tuesday)
thin volume filaments
with constant current
small surface panels
with constant charge
million of elements
Model Order
Reduction
(Today)
tens of elements

49. Conclusions
Reduction via moment matching: Krylov Subspace Projection Framework
allows multipoint expansion moment matching (wider frequency band)
numerically very robust
use PVL is mode efficient for model in frequency domain
use PRIMA to preserve passivity if model is for time domain simulator