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PPT – A more reliable reduction algorithm for behavioral model extraction PowerPoint presentation | free to download - id: 10a6e8-ZDc1Z

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A more reliable reduction algorithm for

behavioral model extraction

- Dmitry Vasilyev, Jacob White
- Massachusetts Institute of Technology

Outline

- Background
- Projection framework for model reduction
- Balanced Truncation algorithm and approximations
- AISIAD algorithm
- Description of the proposed algorithm
- Modified AISIAD and a low-rank square root

algorithm - Efficiency and accuracy
- Conclusions

Model reduction problem

inputs

outputs

Many (gt 104) internal states

inputs

outputs

few (lt100) internal states

- Reduction should be automatic
- Must preserve input-output properties

Differential Equation Model

- state

A stable, n x n (large) E SPD, n x n

- vector of inputs

- vector of outputs

- Model can represent
- Finite-difference spatial discretization of PDEs
- Circuits with linear elements

Model reduction problem

n large (thousands)!

q small (tens)

Need the reduction to be automatic and preserve

input-output properties (transfer function)

Approximation error

- Wide-band applications model should have small

worst-case error

gt maximal difference over all frequencies

?

Projection framework for model reduction

- Pick biorthogonal projection matrices W and V
- Projection basis are columns of V and W

x

Vxr x

x

n

V

q

xr

Ax

WTAVxr

Most reduction methods are based on projection

Projection should preserve important modes

u

y

LTI SYSTEM

t

t

input

output

P (controllability) Which modes are easier to

reach?

Q (observability) Which modes produce more output?

X (state)

- Reduced model retains most controllable and most

observable modes - Mode must be both very controllable and very

observable

Balanced truncation reduction (TBR)

Compute controllability and observability

gramians P and Q (n3) AP PAT BBT

0 ATQ QA CTC 0 Reduced model keeps

the dominant eigenspaces of PQ

(n3) PQvi ?ivi wiPQ ?iwi

Reduced system (WTAV, WTB, CV, D)

Very expensive. P and Q are dense even for

sparse models

Most reduction algorithms effectively separately

approximate dominant eigenspaces of P and Q

- Arnoldi Grimme 97 V colspA-1B, A-2B, ,

WVT , approx. Pdom only - Padé via Lanczos Feldman and Freund

95 colsp(V) A-1B, A-2B, , - approx. Pdom

colsp(W) A-TCT, (A-T )2CT, , - approx. Qdom - Frequency domain POD Willcox 02, Poor Mans

TBR Phillips 04

colsp(V) (j?1I-A)-1B, (j?2I-A)-1B, , -

approx. Pdom colsp(W) (j?1I-A)-TCT,

(j?2I-A)-TCT, , - approx. Qdom

However, what matters is the product PQ

RC line (symmetric circuit)

V(t) input i(t) - output

- Symmetric, PQ all controllable states are

observable and vice versa

RLC line (nonsymmetric circuit)

Vector of states

- P and Q are no longer equal!
- By keeping only mostly controllable and/or only

mostly observable states, we may not find

dominant eigenvectors of PQ

Lightly damped RLC circuit

R 0.008, L 10-5 C 10-6 N100

- Exact low-rank approximations of P and Q of

order lt 50 leads to PQ 0!!

Lightly damped RLC circuit

Top 5 eigenvectors of Q

Top 5 eigenvectors of P

Union of eigenspaces of P and Q does not

necessarily approximate dominant eigenspace of

PQ .

AISIAD model reduction algorithm

Idea of AISIAD approximation

Approximate eigenvectors using power

iterations Vi converges to dominant

eigenvectors of PQ Need to find the product

(PQ)Vi

How?

Approximation of the product Vi1 qr(PQVi),

AISIAD algorithm

Wi qr(QVi)

Vi1 qr(PWi)

Approximate using solution of Sylvester equation

Approximate using solution of Sylvester equation

More detailed view of AISIAD approximation

Right-multiply by Wi

(original AISIAD)

X

X

H, qxq

M, nxq

Modified AISIAD approximation

Right-multiply by Vi

X

X

H, qxq

Approximate!

M, nxq

Modified AISIAD approximation

Right-multiply by Vi

X

X

H, qxq

Approximate!

M, nxq

We can take advantage of numerous methods, which

approximate P and Q!

Specialized Sylvester equation

-M

X

X

A

H

q x q

n x q

n x n

Need only column span of X

Solving Sylvester equation

Schur decomposition of H

-M

X

X

A

Solve for columns of X

X

Solving Sylvester equation

Schur decomposition of H

- Applicable to any stable A
- Requires solving q times

Solution can be accelerated via fast MVP

Another methods exists, based on IRA, needs Agt0

Zhou 02

Solving Sylvester equation

Schur decomposition of H

- Applicable to any stable A
- Requires solving q times

For SISO systems and P0 equivalent to matching

at frequency points ?(WTAW)

Modified AISIAD algorithm

LR-sqrt

- Obtain low-rank approximations of P and Q
- Solve AXi XiH M 0, gt Xi PWi where

HWiTATWi, M P(I - WiWiT)ATWi BBTWi - Perform QR decomposition of Xi ViR
- Solve ATYi YiF N 0, gt Yi QVi where

FViTAVi, N Q(I - ViViT)AV CTCVi - Perform QR decomposition of Yi Wi1 R to get new

iterate. - Go to step 2 and iterate.
- Bi-orthogonalize W and V and construct reduced

model

(WTAV, WTB, CV, D)

For systems in the descriptor form

Generalized Lyapunov equations

Lead to similar approximate power iterations

mAISIAD and low-rank square root

Low-rank gramians

(cost varies)

mAISIAD

LR-square root

(inexpensive step)

(more expensive)

For the majority of non-symmetric cases, mAISIAD

works better than low-rank square root

RLC line example results

H-infinity norm of reduction error (worst-case

discrepancy over all frequencies)

N 1000, 1 input 2 outputs

Steel rail coolling profile benchmark

Taken from Oberwolfach benchmark collection,

N1357 7 inputs, 6 outputs

mAISIAD is useless for symmetric models

For symmetric systems (A AT, B CT) PQ,

therefore mAISIAD is equivalent to LRSQRT for

P,Q of order q

RC line example

Cost of the algorithm

- Cost of the algorithm is directly proportional to

the cost of solving a linear system (where

sjj is a complex number) - Cost does not depend on the number of inputs and

outputs

(non-descriptor case)

(descriptor case)

Conclusions

- The algorithm has a superior accuracy and

extended applicability with respect to the

original AISIAD method - Very promising low-cost approximation to TBR
- Applicable to any dynamical system, will work

(though, usually worse) even without low-rank

gramians - Passivity and stability preservation possible via

post-processing - Not beneficial if the model is symmetric