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Model Order Reduction. Bo Hu Mixed Signal CAD Electrical Engineering Department University of Washington. Outline. Overview of the problem Linear Model Order Reduction Non-linear Model Order reduction Reference. The Problem. Slow to simulate. x(t). u(t). y(t). N is Large. reduce.

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model order reduction
Model Order Reduction

Bo Hu

Mixed Signal CAD

Electrical Engineering Department

University of Washington

outline
Outline
  • Overview of the problem
  • Linear Model Order Reduction
  • Non-linear Model Order reduction
  • Reference
the problem
The Problem

Slow to simulate

x(t)

u(t)

y(t)

N is Large

reduce

u(t)

y(t)

z(t)

q<<N

Fast to simulate

model order reduction4
Model Order Reduction
  • Model Order Reduction: Construct a simplified system to approximate the original system with reasonable accuracy.
linear model order reduction
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
linear model order reduction con t
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 ?
approximation methods
Approximation methods
  • Taylor Series
  • Pade Approximations
  • Lagrange Polynomials
  • Spline
  • ... Many other approximations
  • The choice depends on specific problem
pade method con t
Pade Method con\'t
  • The Dynamic System’s transfer function has the following structure:

H(s) = A(s)/B(s),

for such kind of function, Pade approximation is simple and often better !

pade based algorithm
Pade Based Algorithm
  • Moment Matching:
  • Construct Pade Function Hq(s) to approximate H(s), such that their first q moments are the same
moment matching methods
Moment matching methods
  • Asymptotic Waveform Evaluation(AWE) (Explicit moment matching)
  • Arnoldi Algorithm(Implicit)
  • Lanczos algorithm(Implicit)
awe method explicit moment matching algorithm
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.
awe process for siso
AWE Process for SISO
  • Compute the first 2q moment of H(s)
awe process for siso con t
AWE Process for SISO con’t

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

awe for mimo
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 ports:O(m x p).
numerical problem in awe process
Numerical problem in AWE Process
  • AWE gives good result when q<=10
  • Beyond that, AWE has numerical instability problem.
  • The reason is that: when compute the coefficients ai and bi of Hr(s), the AWE method encounter ill-conditioned matrix M.
arnoldi algorithm
Arnoldi Algorithm
  • Developed by Silveira, Kamon, White, Elfadel, Ling etc. in 1990.
  • Basic Idea: Perform variable substitution x=Vz, 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
notes about arnoldi method
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:
prima
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.
prima and passivity
PRIMA and Passivity
  • A circuit is passive if none of its elements generates energy.
  • A system is passive iff
prima27
PRIMA
  • Use Arnoldi Process to obtain V, and perform variable substitution: x = Vz
  • In the same time, perform congruence transformations as follows:
prima28
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

pade via lanczos pvl another implicit moment matching algorithm
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.
lanczos algorithm overview
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 n<<N), Tn is often a very good approximation to matrix A, and Tn’s eigenvalue is close to A’s eigenvalue.
lanczos algorithm application
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.
lanczos algorithm for siso
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):

mpvl algorithm outline with deflation and look ahead technique
MPVL Algorithm Outline--with deflation and look-ahead technique
  • MPVL: multi input and multi output PVL.
  • Basic idea: after variable transformation x=Vz, the reduced system’s transfer matirx Hr(s) is pade-approximation to original system’s transfer matrix H(s).
mpvl algorithm outline con t with deflation and look ahead technique
MPVL Algorithm Outline con’t--with deflation and look-ahead technique
  • Practical implementation of MPVL is similar to SISO PVL,
  • But MPVL requires deflation and look-ahead techniques
mpvl algorithm outline with deflation and look ahead technique37
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 Krylov{A,r} can be represented by the former k-1 vectors, then the kth vector should be deleted from Krylov{A,r}—this procedure is called deflation.
  • After deflation, the size of Krylov{A,r} and Krylov{A’,l’}may be different, the MPVL process terminates when either Krylov subspace is exhausted.
mpvl algorithm outline con t with deflation and look ahead technique38
MPVL Algorithm Outline con’t--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.
comparison of three methods
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.
computational cost
Computational cost
  • The computation cost for the three methods are all under O(N^3), depends on how sparse those coefficient matrices are
  • Better than solve it directly which requires

O(N^4) in general

nonlinear model order reduction
Nonlinear model order reduction
  • The problem(m inputs and p outputs)
available approaches
Available Approaches
  • Linearization method
  • Quadratic method
  • Piece-wise-linear method
  • Balancing technique
linearization method
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.
quadratic method
Quadratic method
  • Basic idea: expand f(x) to second order:
  • Disadvantages: depends on how closely f(x) is similar to quadratic function.
piece wise linear method
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”

piece wise linear method con t
Piece-Wise-Linear method con’t
  • Works better than linear-reduction and quadratic reduction method.
  • Disadvantages: the resultant piece-wise-linear system’s accuracy depends on the training input; not qualified as a general approach.
reference
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

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