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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 l.jpg
Model Order Reduction

Bo Hu

Mixed Signal CAD

Electrical Engineering Department

University of Washington

Outline l.jpg

  • Overview of the problem

  • Linear Model Order Reduction

  • Non-linear Model Order reduction

  • Reference

The problem l.jpg
The Problem

Slow to simulate




N is Large






Fast to simulate

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Model Order Reduction

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

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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

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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 ?

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Approximation methods

  • Taylor Series

  • Pade Approximations

  • Lagrange Polynomials

  • Spline

  • ... Many other approximations

  • The choice depends on specific problem

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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 !

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Pade Based Algorithm

  • Moment Matching:

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

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Moment matching methods

  • Asymptotic Waveform Evaluation(AWE) (Explicit moment matching)

  • Arnoldi Algorithm(Implicit)

  • Lanczos algorithm(Implicit)

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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.

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AWE Process for SISO

  • Compute the first 2q moment of H(s)

Awe process for siso con t l.jpg
AWE Process for SISO con’t

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

Awe for mimo l.jpg

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).

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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.

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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

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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 l.jpg

  • 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.

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PRIMA and Passivity

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

  • A system is passive iff

Prima27 l.jpg

  • Use Arnoldi Process to obtain V, and perform variable substitution: x = Vz

  • In the same time, perform congruence transformations as follows:

Prima28 l.jpg

  • 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.






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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.

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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.

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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.

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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 l.jpg
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 l.jpg
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

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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 l.jpg
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.

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MPVL Algorithm Outline con’t--with deflation and look-ahead technique

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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.

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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

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Nonlinear model order reduction

  • The problem(m inputs and p outputs)

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Available Approaches

  • Linearization method

  • Quadratic method

  • Piece-wise-linear method

  • Balancing technique

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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.

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Quadratic method

  • Basic idea: expand f(x) to second order:

  • Disadvantages: depends on how closely f(x) is similar to quadratic function.

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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.


the exact trajectory

---- the pwl approximation


The response to a “training input”

Piece wise linear method con t l.jpg
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 l.jpg

[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