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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

Model Order Reduction

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

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

- 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

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

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

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

Moment matching methods

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

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

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

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 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

- 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

- 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

- 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

- 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

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

PRIMA

- Use Arnoldi Process to obtain V, and perform variable substitution: x = Vz
- In the same time, perform congruence transformations as follows:

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

- 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

- 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

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

- 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: 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

- 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 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 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.

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

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

- 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

- The problem(m inputs and p outputs)

Available Approaches

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

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

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

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

- 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

[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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