Model Order Reduction

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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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Presentation Transcript
Model Order Reduction

Bo Hu

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

u(t)

y(t)

z(t)

q<<N

Fast to simulate

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.
Approximation methods
• Taylor Series
• Lagrange Polynomials
• Spline
• ... Many other approximations
• The choice depends on specific problem
• 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 !

• 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
• 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

• 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: 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
• 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.
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
• 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.
• 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