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### Introduction to Model Order Reduction

### III.3 - Reduction of Strongly NonLinear Systems via Trajectory PieceWise Linear Method

### III.4 Reduction of Strongly NonLinear Systemsvia Trajectory-PieceWise Linear (TPWL) + Truncated Balance Realization (TBR)

Luca Daniel

Massachusetts Institute of Technology

luca@mit.edu

http://onigo.mit.edu/~dluca/2006PisaMOR

www.rle.mit.edu/cpg

Numerical Simulation

Quick intro to PDE Solvers

Quick intro to ODE Solvers

Model Order reduction

Linear systems

Common engineering practice

Optimal techniques in terms of model accuracy

Efficient techniques in terms of time and memory

Non-Linear Systems

Parameterized Model Order Reduction

Linear Systems

Non-Linear Systems

Today

Model Order Reduction of Non-Linear Systems

- Introduction
- Reduction of weakly nonlinear systems (Volterra Series)
- Reduction of strongly nonlinear systems Trajectory PieceWise Linear (TPWL) and Polynomial (PWP)
- with moment matching
- with Truncated Balance Realization

- Inlet density disturbance d:

DISCRETIZE

Reduction of NonLinear Systems

NonLinear Reduction

PDE Field Solvers or Circuit Simulators

Non-Linear analog components

e.g. MEMs, VCO, LNA

for nonlinear dynamic systems

in

- Substitute:

- Reduced system:

- A problem:

q=10

n=104

n=104

q=10

- Using VTF(Ux) is too expensive!

Nonlinear Problem is MUCH Harder

- In what basis should we project?
- No simple frequency domain insight
- No eigenmodes
- No Krylov subspace
- How do you represent the projection?
- New problem for nonlinear

How to get the change of variable matrix

- Analysis of linearized models [Ma88]
- Sampling of time-simulation data + principal components analysis [Sirovich87]
- Nonlinear balancing [Scherpen93],
- Time-derivatives [Gunupudi99], …

for NonLinear Systems

- Representation of F(x) using a polynomial (e.g. Taylor’s expansions, Volterra Series) [Phillips00]*****

- Representation of F(x) using several linearizations (Trajectory Piece-Wise Linear TPWL) [Rewienski01]

- Representation of F(x) with several polynomials (PWP PieceWise Polynomial) [Dong03]

Introduction to Model Order Reduction

III.2 – Reduction of Weakly Non Linear Systems

Luca Daniel

Thanks to Joel Phillips, Cadence Berkeley Labs

Representation of F(x) based on Taylor’s expansions

- Approximate nonreduced model:

MOR for nonlinear dynamic systems

(J. Phillips, Y. Chen, F. Wang):

- Linear, quadratic reduced order models:

Expand nonlinearity in multi-dimensional polynomial series

Differential equation becomes

To match first few terms in functional series expansion, only need first few polynomial terms

Each term is a -dimensional

tensor, represented as an matrix

Polynomial Approximations[note : not static models; frequency effects retained]

where

etc.

Draw from reduced space as

Identity for Kronecker products

Project tensors

Gives reduced model

Projection of polynomial termsetc.

Projection procedure produces reduced model in same polynomial form

Key tensor components are compressed to lower dimensionality

Kronecker forms provide convenient general notation

How to get projection spaces V?

Reduced polynomial modelsIntroduce variational parameter

Consider parametrized system

Expand response in powers of

Substitute into differential equation

Compare and collect terms for each power of

Variational AnalysisCoupled set of differential equations

Each set computes response at one order

Each set is linear in its own state variable

First set is just linearized model

Variational Analysisetc.

First order equation set driven by input of low rank

Krylov subspace for projection

Second order equation set driven by input

Key insight : recall is well-approximated by thus

Model ReductionSecond order equation set is driven by

Generate second Krylov space using

Compute models by projection

each equation order set separately, or

project original polynomial system using UNION of

number of matched moments of order-i given by size of appropriate Krylov subspace

Model ReductionSize of Krylov spaces grows quickly with functional series order

potentially large models

intrinsic in polynomial descriptions (e.g. Volterra series)

Practical for weak nonlinearities needing only few terms in functional series

example: RF datapath

strong nonlinearities included in bias conditions

Many local optimizations, extensions are possible

two-sided projectors

multiple reduction steps

Computational ConsiderationsProjection + polynomial series + variational analysis = rigorous nonlinear reduction procedure

Compact Kronecker notation is very useful

Is there a way to reduce number of terms in higher-order Krylov space?

Smaller models faster

Diagnose when models based on first order system are failing?

Follow-on reduction?

Summary weakly nonlinear MORfor NonLinear Systems

- Representation of F(x) using a polynomial (e.g. Taylor’s expansions, Volterra Series) [Phillips00]*****

- Representation of F(x) using several linearizations (Trajectory Piece-Wise Linear TPWL) [Rewienski01]

- Representation of F(x) with several polynomials (PWP PieceWise Polynomial) [Dong03]

Luca Daniel

Massachusetts Institute of Technology

Thanks to Michał Rewieński, Jacob White and Brad Bond

www.rle.mit.edu/cpg

Piecewise-linear representation

[M. Rewienski, J. White ICCAD01]

- Linearizations around xi , i=0,…, s-1

- Weighted combination of the models:

- Project linearized models:

Background – TPWL [Reiwenski01]

x8

x1

x7

x6

x5

x2

x4

x14

x10

xa

xb

x0

x11

x9

x13

x3

x15

Model i only valid near xi

x2

#linearizations =#samplesn

n = 104

#samples = 100

10010000 = LARGE

Use collection of linear models

x1

Background – TPWL: Picking Linearization Points

Use training trajectories to pick linearization points

y(t)

x2

State Space

Time Domain Simulation

t

x1

Linearization at current state xi

Quasi-piecewise-linear MOR – computing weights

- For i=0,…,(s-1) compute:
- Compute
- For i=0,…,(s-1) compute:
- Normalize wi.

Model from linearization 1

Model from linearization k

Model from linearization 2

K2

A1

K1

A2

Ak

Kk

Background – TPWLReduction of the Linearized SystemsBackground – TPWLConstructing the projection matrix V

Use moments from EACH linear model to construct V

Background – TPWL: Weighting / Simulation[Riewinski01,Tiwary05,Dong05]

Linearization 3

x2

Use weighting functions to combine linear models during simulation

also well

approximated

C – poorly

approximated

Well approximated

Current state

Linearization 2

x1

Linearization 1

Example of a microswitch

Generate the reduced linear model

Use the reduced linear system

for simulation

x2

x1

x0

A

x3

Fast approximate simulation

x2

x1

Computational results - model order reduction

*Matlab implementation

Computational results – OpAmp example

OpAmp:

- 70 MOSFETs
- 13 resistors
- 9 capacitors
- 51 nodes

training input

testing

input

Computational results – shock propagation in jet engine inlet

- Density disturbance d:

Computational results – shock propagation in 1D

- Burgers equation:

- How should we pick training inputs?
- How “often” should we compute linearized models along the training trajectory?
- What are the errors of representing a nonlinear systems as a trajectory piecewise linear system?
- What are “the best” reduced order bases for the trajectory PWL model?

- TPWL macromodels can be accurate for a number of strongly nonlinear dynamical systems, outperforming reduced models based on single-state polynomial expansions.
- TPWL models use simple, cost-efficient representation of system’s nonlinearity, resulting in very short simulation times.
- TPWL models may be extracted very efficiently by using a fast approximate simulator.

Model Order Reduction of Non-Linear Systems

- Introduction
- Reduction of weakly nonlinear systems (Volterra Series)
- Reduction of strongly nonlinear systems (TPWL) Trajectory Pieace Wise Linear and Polynomial
- with moment matching
- with Truncated Balance Realization

Luca Daniel

Massachusetts Institute of Technology

Thanks to Dmitry Vasilyev, Michał Rewieński, Jacob White

Krylov-subspace methods

Fast

Don’t guarantee accuracy

Balanced-truncation methods

Expensive (~n3)

Guarantee accuracy

Obtaining projection basisFor example,

V=W=

colspan(A-1B, A-2B, … , A-q B)

Let’s now use this!

Our Approach:

- Use TPWL to handle nonlinearity
- Before we used Krylov-subspace linear reduction (less accurate)
- Here we use TBR for projection matrices W and V

x0

x2

x1

…

xn

TBR reduction

u

y

LTI SYSTEM

t

t

Past

input

Future

output

X (state)

P (controllability)

Which states are easier to reach?

Q (observability)

Which states produces more output?

TBR algorithm includes into projection basis

most controllable and most observable states

Numerical results –RLC transmission line

TBR-based TPWL beat

Krylov-based

4-th order TBR TPWL reaches the limit of TPWL representation

Error in transient

||yr – y||2

Order of the reduced model

Reducing cost of TBR reduction - Combined Krylov-TBR algorithm

Krylov reduction (Wi , Vi):

Ai = WiTAVi

Bi = WiTB

Ci = CVi

Initial Model:

(A B C), n

Intermediate Model:

(Ai Bi Ci), ni

TBR reduction (Wt , Vt):

Ar = WtTAVt

Br = WtTB

Cr = CVt

Reduced Model:

(Ar Br Cr), q

Performance of Krylov- TBR TPWL MOR extraction procedures*

Cost of Krylov-TBR almost equals Krylov

*Matlab implementation

Comparing accuracies of Krylov TPWL method and TBR-based TPWL algorithm

5x reduction in order – 125x improvement in efficiency

*Testing input equal to training input

Future work

- Are TBR-based TPWL models valid for unstable linearizations?
- What about systems having the following form (i.e. circuits with nonlinear capacitors):

Conclusions

- TBR-based linear reduction procedure to generate TPWL reduced models
- Order reduced 5 times while maintaining comparable accuracy with Krylov TPWL method (efficiency improved 125 times!)
- Combined Krylov-TBR reduction allows to extract TPWL models at low cost

Numerical Simulation

Quick intro to PDE Solvers

Quick intro to ODE Solvers

Model Order reduction

Linear systems

Common engineering practice

Optimal techniques in terms of model accuracy

Efficient techniques in terms of time and memory

Non-Linear Systems

Parameterized Model Order Reduction

Linear Systems

Non-Linear Systems

Today

Tomorrow

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