Parameterized Model Order Reduction for Nonlinear Dynamical Systems

1
Parameterized Model Order Reduction for Nonlinear
Dynamical Systems

• Brad Bond, MIT
• Luca Daniel, MIT

This work was supported by the MARCO Gigascale
Systems Research Center, by the Semiconductor
Research Corporation, and by the National Science
Foundation
www.rle.mit.edu
2
Outline

• Introduction
• Background
• NonLinear Parameterized Model Order Reduction
  (NLPMOR)
• Examples
• Results
• Future Work

3
Introduction Motivation
Possible parameters
Nonlinear components
System geometry
Material properties
Preserve parameter dependence
Capture nonlinear effects

• Optimization and design
• May want to simulate a model many times at
  different parameter values

4
Outline

• Introduction
• Background
• Moment Matching
• Parameterized Model Order Reduction (PMOR)
• Trajectory PieceWise Linear (TPWL) Approach
• NonLinear Parameterized Model Order Reduction
  (NLPMOR)
• Examples
• Results
• Future Work

5
Background Moment Matching
Want
Taylor Series Expansion
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Background Moment Matching Grimme97
Project each system matrix into subspace
The columns of V define the new subspace
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Parameterized Model Order Reduction.Previous work

• Moment Matching approaches - (Pullela97,
  Gunupudi00,
  Prudhomme02, Daniel02, Li05)
• Can handle nonlinear dependence on parameter
• Can handle extremely large systems

Truncated Balance Realizations - (Heydari01,
Phillips04) - Does not handle extremely large
systems

• Optimization based approaches - (Sou05)
• Good for fitting data from measurements
• Cannot construct large order reduced models

• Statistical Data Mining - (Liu02)
• Can handle nonlinear parameter dependence and
  nonlinear systems
• Does not work well for extremely large systems

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Background PMOR Daniel02
Obtain linear dependence on some new parameters
P-variable Taylor series expansion
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Previous work on Non-Parameterized MOR for
nonlinear systems

• Representation of F(x) using a polynomial (e.g.
  Taylors 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

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Background Nonlinear MOR
Still requires large function evaluation
Take linear approximation of F(x) at xi
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Background TPWL Reiwenski01
Model i only valid near xi
x2
linearizations samplesn
n 104
samples 100
10010000 LARGE
Use collection of linear models
x1
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Background TPWL Picking Linearization Points
Use training trajectories to pick linearization
points
x2
y(t)
State Space
x1
t
Time Domain Simulation
Linearization at current state xi
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Background TPWL Weighting / SimulationRiewins
ki01,Tiwary05,Dong05
also well approximated
Linearization 3
Use weighting functions to combine linear models
during simulation
x2
Well approximated
C poorly approximated
Current state
Linearization 2
x1
Linearization 1
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Background TPWL Constructing V
Use moments from EACH linear model to construct V
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Outline

• Introduction
• Background
• NonLinear Parameterized Model Order Reduction
  (NLPMOR)
• Constructing the system
• How to pick linearization points
• How to construct V
• Examples
• Results
• Conclusions
• Future Work

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NLPMOR Constructing the System
Start with system possessing nonlinear dependence
on state x and parameters sj
By linear approximation, polynomial fitting to
data points
Obtain Linear dependence on new parameters
Single linear model
Linearize to get weighted sum of linear models
Weighting basis functions
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NLPMOR Picking Linearization Points 2 Options
Use training trajectories to pick linearization
points
y(t)
x2
time
s2
x1
State Space
Could train again at different points in
parameter space
Populate state space with linear models
Parameter Space
s1
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NLPMOR Constructing V 2 Options
Option 1
Single variable Taylor series expansion of x for
each model, as in TPWL
MOR V
OR
Option 2
p-variable Taylor series expansion of x as in
PMOR, but for each model
PMOR V
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NLPMOR Constructing V
For a model with p parameters and k linear
models, total number of vectors for V is O(kpm)
to match m moments
Keeping all vectors would results in a huge
reduced model
Hence we perform an SVD on V and keep only the q
most important vectors
O(kpm)
q
SVD
N
N
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NLPMOR 4 Algorithms
Original Model
Linearize in parameters sj
MOR
Train to pick linearization points xi
PMOR
Train at single p-space point
Train at multiple p-space points
MOR V
PMOR V
MOR V
PMOR V
Parameter Space
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Outline

• Introduction
• Background
• NLPMOR
• Examples
• Circuit
• Beam
• Results
• Conclusions
• Future Work

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Analog Circuit Example
Nonlinear terms
Picture by Michal Rewienski
Analog circuit with nonlinear components
distributed throughout
Using constitutive relations for each element,
apply KCL at each node to obtain state-space model
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Analog Circuit Example State Space System
Capacitor values
Saturation current
Turn on voltage
Resistor values
Single equation at node j
State-Space System
Possible parameters
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Analog Circuit Example One possible
parameterized model
Selected parameter a
Obtained linear dependence on parameters
Single point training
Multiple point training
MOR V
PMOR V
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Micromachined Switch Example
Governing Equations
Discretize
Picture by Michal Rewienski
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Micromachined Switch ExampleState Space System
Possible Parameters
Material Properties
State-Space System
Beam height
Beam width
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Micromachined Switch ExampleOne possible
parameterized model
System parameterized in
Single training
Multiple training
MOR V
PMOR V
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Outline

• Introduction
• Background
• NLPMOR
• Examples
• Results
• Algorithm Comparison
• Algorithm Cost
• Conclusions
• Future Work

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Analog Circuit Example Algorithm 1
Single training
Multiple training
Full Model ALG1 ROM
MOR V
PMOR V
1.4
1
0.9
- Training points
- Evaluation points
1e-10
1.5e-10
0.5e-10
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Results Benefit of PMOR V
Analog Circuit
Analog Circuit
TPWL ROM ALG1 ROM
Single p-space point training
Multiple p-space point training
MOR V
PMOR V
Id0
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Analog Circuit Example
Single training
Multiple training
This model is parameterized in Id and 1/R, and
was created using ALG2
MOR V
Full Model ALG2 ROM
PMOR V
2
1.5
1
0.5
- Training points
- Evaluation points
1e-10
1.5e-10
0.5e-10
2e-10
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Results Benefit of multiple training points in
p-space
MEMs Switch
Analog Circuit
TPWL ROM ALG2 ROM
Error
time (s)
Single p-space point training
Multiple p-space point training
MOR V
PMOR V
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Results Algorithm Comparisons
Each algorithm covers a different region of
p-space
S2
s2B
s2A
All algorithms produce model with same reduced
order and same number of linear pieces
S1
s1A
s1B
Parameter Space Validity of Models
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Results Algorithm Comparisons
Each algorithm has a different cost
Cheap
Majority of cost lies in making vectors for V
Majority of cost lies in training
p - of parameters k - of linear models per
trajectory r - of parameter values used m -
of moments matched per parameter
Expensive in both training and creating V
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Conclusions

• NLPMOR
• Captures nonlinear effects
• Preserves parameter dependence
• Algorithm choice is situation/system dependent
• TPWL, ALG1 can provide high accuracy locally in
  parameter space
• ALG2, ALG3 can provide global accuracy in
  parameter space
• ALG1, ALG3 expensive to create V
• ALG2, ALG3 expensive to train

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Future Work

• Ensure stability and passivity of reduced order
  models
• Quantify parameter space coverage