Model Order Reduction

1
Model Order Reduction
Bo Hu Mixed Signal CAD Electrical Engineering
Department University of Washington
2
Outline

• Overview of the problem
• Linear Model Order Reduction
• Non-linear Model Order reduction
• Reference

3
The Problem
Slow to simulate
x(t)
u(t)
y(t)
N is Large
reduce
u(t)
y(t)
z(t)
qltltN
Fast to simulate
4
Model Order Reduction

• Model Order Reduction Construct a simplified
  system to approximate the original system with
  reasonable accuracy.

5
Linear Model Order Reduction
Reduction Methods are mature for Linear System

• Typical application RC, RL,LC, and RLC circuits.
• Speed-up 10 to 100 or more, depends on problems

6
Linear Model Order Reduction con't

• Basic Idea Construct a reduced order system
  whose transfer function Hr(s) is a pade
  approximation to the transfer function H(s) of
  the original system.
• Why Pade ?

7
Approximation methods

• Taylor Series
• Pade Approximations
• Lagrange Polynomials
• Spline
• ... Many other approximations
• The choice depends on specific problem

8
Pade Method con't

• The Dynamic Systems transfer function has the
  following structure
• H(s) A(s)/B(s),
• for such kind of function, Pade approximation
  is simple and often better !

9
Pade Approximation Example
10
Pade Approximation Example cont
11
Pade Based Algorithm

• Moment Matching
• Construct Pade Function Hq(s) to approximate
  H(s), such that their first q moments are the same

12
Moment matching methods

• Asymptotic Waveform Evaluation(AWE) (Explicit
  moment matching)
• Arnoldi Algorithm(Implicit)
• Lanczos algorithm(Implicit)

13
AWE Method--Explicit moment-matching algorithm

• Developed by Pillage, Rohrer in 1990.
• Basic Idea compute the first 2q moments of the
  transfer function H(s), and then find the pade
  approximation function Hr(s) to match those 2q
  moments .
• using Hr(s), we can do frequency domain analysis
  and time domain analysis of the system.
• Advantages easy to understand and implement,
  when q is small, AWE gives good results.

14
AWE Process for SISO

• Compute the first 2q moment of H(s)

15
AWE Process for SISO cont

• Solve the coefficients of Hq(s) based on the 2q-1
  moments of H(s)

16
AWE for SISO cont
17
AWE Process for SISO cont
18
AWE for MIMO

• AWE for MIMO system(m-input,p-output)
• get pade approximation separately for each pair
  of inputs and outputs
• then apply the superposition property of linear
  networks, group them into one matrix Hq(s)
• However the computation cost increase quadraticly
  with the number of portsO(m x p).

19
Numerical problem in AWE Process

1. AWE gives good result when qlt10
2. Beyond that, AWE has numerical instability
   problem.
3. The reason is that when compute the coefficients
   ai and bi of Hr(s), the AWE method encounter
   ill-conditioned matrix M.

20
Arnoldi Algorithm

• Developed by Silveira, Kamon, White, Elfadel,
  Ling etc. in 1990.
• Basic Idea Perform variable substitution xVz,
  such that the reduced system has a transfer
  function Hq(s) Pade-Approximate to original
  transfer function H(s).
• The construction of V in Arnoldi algorithm is a
  modified Gram-Schmidt Process

21
Arnoldi Algorithm for SISO
22
SISO Arnoldi Algorithm con't
23
Block Arnoldi Algorithm Outline
24
Notes about Arnoldi method

• Arnoldi algorithm is a modified Gram-Schmidt
  process on Krylov subspace
• Hq(s) from Arnoldi method matches up to the qth
  moments.
• In Arnoldi algorithm, A -inv(G)C When compute V
  AR, the practical implementation is

25
PRIMA

• Passive Reduced Order Interconnect Macromodeling
  Algorithm combination of moment matching with
  congruence transformation.
• The advantage of PRIMA be able to preserve the
  passivity during the reduction process and in the
  same time, numerically stable.
• Compared to Lanczos algorithm, PRIMA trades part
  of the accuracy for Passivity, Lanczos algorithm
  is more efficient, but could lose Passivity.

26
PRIMA and Passivity

• A circuit is passive if none of its elements
  generates energy.
• A system is passive iff

27
PRIMA

1. Use Arnoldi Process to obtain V, and perform
   variable substitution x Vz
2. In the same time, perform congruence
   transformations as follows

28
PRIMA

• PRIMA is useful, especially when the system has
  both linear and nonlinear part
• The linear part can be reduced and keep
  passivity, which is very important for the
  stability of the dynamic system.

Nonlinear
Nonlinear
reduce
Linear
Linear
29
Pade Via Lanczos(PVL) --Another implicit moment
matching algorithm

• Basic Idea Implicitly Match first q moment of
  H(s) by Lanczos Process.
• Developed by Feldmann and Freund in1995
• Advantage Numerically more stable and
  computationally efficient.
• Disadvantage for RLC system, it does not keep
  passivity.

30
Lanczos algorithm overview

• Originally Developed by Lanczos at 1950s to solve
  eigenvalue problems.
• Basic Idea Given matrix A(N by N), and starting
  with given nonzero vectors r,l (N by 1), run the
  lanczos process for n steps to obtain matrix Tn(n
  x n, typically nltltN), Tn is often a very good
  approximation to matrix A, and Tns eigenvalue is
  close to As eigenvalue.

31
Lanczos Algorithm Application

• Applications of Lanczos algorithm
• Compute the approximate eigenvalue of matrix
  A
• Solve large systems of linear equations Ax
  b
• Used in PVL Algorithm since 1990's.

32
Lanczos Process
33
Lanczos Process
34
Lanczos Algorithm for SISO

• For SISO system
• Run Lanczos Process, we obtain Tq, and
• the qth Pade-Approximation to Hq(s) is
  obtained as follows(e1 is the first unit vector
  of N by 1)

35
MPVL Algorithm Outline--with deflation and
look-ahead technique

• MPVL multi input and multi output PVL.
• Basic idea after variable transformation xVz,
  the reduced systems transfer matirx Hr(s) is
  pade-approximation to original systems transfer
  matrix H(s).

36
MPVL Algorithm Outline cont--with deflation and
look-ahead technique

• Practical implementation of MPVL is similar to
  SISO PVL,
• But MPVL requires deflation and look-ahead
  techniques

37
MPVL Algorithm Outline--with deflation and
look-ahead technique

• Deflation procedure detect and delete linearly
  dependent or almost linearly dependent vectors in
  the block Krylov subspace.
• For example if the kth vector in KrylovA,r can
  be represented by the former k-1 vectors, then
  the kth vector should be deleted from
  KrylovA,rthis procedure is called deflation.
• After deflation, the size of KrylovA,r and
  KrylovA,lmay be different, the MPVL process
  terminates when either Krylov subspace is
  exhausted.

38
MPVL Algorithm Outline cont--with deflation and
look-ahead technique

• Break-down in lanczos process could happen when
  vi and wi are orthogonal or almost orthogonal to
  each other.
• Look-ahead technique must be taken to remedy
  break-down in lanczos process.

39
MPVL Algorithm Outline cont--with deflation and
look-ahead technique
40
Comparison of Three methods

• AWE is more straight forward to understand and
  implement, but it is numerically unstable.
• Lanczos algorithm is numerically stable, and
  efficient but can lose passivity.
• Improved Arnoldi method(PRIMA) is stable, and
  keep the passivity.

41
Computational cost

• The computation cost for the three methods are
  all under O(N3), depends on how sparse those
  coefficient matrices are
• Better than solve it directly which requires
• O(N4) in general

42
Nonlinear model order reduction

• The problem(m inputs and p outputs)

43
Available Approaches

• Linearization method
• Quadratic method
• Piece-wise-linear method
• Balancing technique

44
Linearization method

• Expand f(x) to first order, and convert the
  non-linear problem as linear problem
• Disadvantage it strongly depends on how f(x) is
  similar to a linear function.

45
Quadratic method

• Basic idea expand f(x) to second
  order
• Disadvantages depends on how closely f(x) is
  similar to quadratic function.

46
Piece-Wise-Linear method

• Basic idea represent the non-linear system with
  a piecewise-linear system and then reduce each of
  the pieces with linear model reduction methods.
• Procedure supply a training input, trace the
  trajectory of the non-linear system, and in the
  same time generate a set of piecewise linear
  systems as an approximation to the original
  non-linear system.

x(t)
the exact trajectory ---- the pwl
approximation
t
The response to a training input
47
Piece-Wise-Linear method cont

• Works better than linear-reduction and quadratic
  reduction method.
• Disadvantages the resultant piece-wise-linear
  systems accuracy depends on the training input
  not qualified as a general approach.

48
Reference

• 1   A Trajectory Piecewise linear
  Approach to model order reduction and fast
  simulation of nonlinear circuits and
  micromachined devices, Rewienski White,J 2  
  A quadratic method for nonlinear model order
  reduction, chen,Y white,J 3   Model Order
  Reduction for Nonlinear System, Y.Chen  4  
  PRIMA Passive reduced order interconnect
  macromodeling algorithm, Odabasioglu 5  
  Reduced-order modeling of large linear passive
  multi-terminal circuits using matrix-Pade
  approximation, Freund 6    Asymptotic waveform
  evaluation for timing analysis ,Pillage, L.T.
  Rohrer, R.A 7    Feldmann, P. and Freund, R.
  W., Efficient Linear Circuit Analysis by Pade
  Approximation via the Lanczos Process 8   
  Pade approximants / George A. Baker, Jr., Peter
  Graves-Morris 9  Reduced-order modeling of
  large linear subcircuits via a block lanczos
  algorithm, Feldman, Freund 10   A
  lanczos-type method for multiple starting
  vectors, Freund,hernandez,boley,aliaga 11  
  Reduced-Order Modeling Techniques Based on Krylov
  Subspaces and Their Use in Circuit Simulation,
  Freund 12   A Block Rational Arnoldi
  Algorithm for Multipoint Passive Model-Order
  Reduction of Multiport RLC Networks, Elfadel,
  Ling