Symbolic Model Reduction for Linear and Nonlinear DAEs 2 Introduction to symbolic analysis and approximation motivation of symbolic methods principles of symbolic simplification approximation

Symbolic Model Reductionfor Linear and Nonlinear DAEs

Symposium on Recent Advances in MORTU Eindhoven, The NetherlandsNovember 23rd, 2007

Thomas [email protected]

Slide 2

Introduction to symbolic analysis and approximationmotivation of symbolic methodsprinciples of symbolic simplificationapproximation methods for linear systemsnonlinear symbolic model generation

EDA tool Analog Insydes

Overview

platform overview

Methodology transfermechatronics

Slide 3

Symbolic Methods

M1

M3

IB

M4 M6

M5 M7

M2

M8VDD

VSS

vin Vout

CC

CL

M9 numerical simulation

symbolic analysis

Slide 4

A=1A=1Loop filter

F(s)Loop filter

F(s)

VCOVCO

ff ii

ff oo

Ctrl. logicCtrl. logic

Phasecomparator

Phasecomparator

Design Flow

system level

block level

transistor level

• error analysis• gaining system understanding• specification transfer between levels• automated behavioral model generation• multi-physics modeling and system simulation

validation / redesign(on all levels)

++

--

VDDVDD

VSSVSS

ININOUTOUT

VDD

VSS

OUTIN+

IN-

Application fields of symbolic methods:

Slide 5

Folded-Cascode OpAmp

ProblemWhat causes resonanceat 10 MHz?

Classic approachparameter variations and numerical simulationfailed because of hugenumber of parameters

Symbolic analysiscalculation of the transferfunction and derivation ofa symbolic formula for theresonance frequency

Complexity problem

Symbolic Analysis

exact symbolic transfer function consists of more than 5·1019 termsprinted: paper stack with a height of 15·109 milesdifferential order: 19

Slide 6

Folded-Cascode OpAmp

ProblemWhat causes resonanceat 10 MHz?

Classic approachparameter variations and numerical simulationfailed because of hugenumber of parameters

Symbolic Analysis

Symbolic analysiscalculation of the transferfunction and derivation ofa symbolic formula for theresonance frequency

Complexity problem ⇒ symbolic approximation

Slide 7

Numerical Validation

first design

design aftersymbolic analysis

Slide 8

Text book:

Complexity Problem

Exact solution using computer algebra

132 Terms

132 Terms

= −R

Cout

E

RV

Slide 9

Symbolic analysis:exact transfer function

Netlist

Symbolic Approximation: Basic Idea

+ + + +2 4

1 2 1 3 1 4 2 3 2 4

R RR R R R R R R R R R

R1 = R2 = 10 ΩR3 = R4 = 1000 Ω

+ + + +

4

2 4 4 4 410

10 10 10 10 10Numerical evaluation

+ +2 4

1 2 3 4( )( )R R

R R R RTransfer function aftersymbolic approximation

relative error:0.0025

Slide 10

Symbolic analysis:exact transfer function

Netlist

Symbolic Approximation: Basic Idea

+ + + +2 4

1 2 1 3 1 4 2 3 2 4

R RR R R R R R R R R R

R1 = R2 = 10 ΩR3 = R4 = 1000 Ω

+ + + +

4

2 4 4 4 410

10 10 10 10 10Numerical evaluation

+ +2 4

1 2 3 4( )( )R R

R R R RTransfer function aftersymbolic approximation

relative error:0.0025Problem:

Exact symbolic transfer function can not be calculated

for real world problems.

Slide 11

Symbolic Approximation: Ranking

equations + −⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥− + + ⎣ ⎦⎣ ⎦⎣ ⎦

1 2 2 1

2 2 3 4 2

10

R R R iR R R R i

netlist

numeric result R1 = R2 = 10 ΩR3 = R4 = 1000 Ω ⇒ i2 = 2.493· 10-4 A

relative error

positionsymbol

1.000E+0(1, 1)R2

1.000E+0(1, 1)R1

1.000E+0(2, 1)-R2

9.950E-1(2, 2)R4

9.950E-1(2, 2)R3

5.013E-3(2, 2)R2

2.488E-3(1, 2)-R2

Slide 12


equations + −⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥− + + ⎣ ⎦⎣ ⎦⎣ ⎦

1 2 2 1

2 2 3 4 2

10

R R R iR R R R i

netlist


relative error

positionsymbol

1.000E+0(1, 1)R2

1.000E+0(1, 1)R1

1.000E+0(2, 1)-R2

9.950E-1(2, 2)R4

9.950E-1(2, 2)R3

5.013E-3(2, 2)R2

2.488E-3(1, 2)-R2

cumulative error 0.002488

Slide 13


equations + −⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥− + + ⎣ ⎦⎣ ⎦⎣ ⎦

1 2 2 1

2 2 3 4 2

10

R R R iR R R R i

netlist


relative error

positionsymbol

1.000E+0(1, 1)R2

1.000E+0(1, 1)R1

1.000E+0(2, 1)-R2

9.950E-1(2, 2)R4

9.950E-1(2, 2)R3

5.013E-3(2, 2)R2

2.488E-3(1, 2)-R2


Slide 14


equations + −⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥− + + ⎣ ⎦⎣ ⎦⎣ ⎦

1 2 2 1

2 2 3 4 2

10

R R R iR R R R i

netlist


relative error

positionsymbol

1.000E+0(1, 1)R2

1.000E+0(1, 1)R1

1.000E+0(2, 1)-R2

9.950E-1(2, 2)R4

9.950E-1(2, 2)R3

5.013E-3(2, 2)R2

2.488E-3(1, 2)-R2


STOP

Slide 15


equations + −⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥− + + ⎣ ⎦⎣ ⎦⎣ ⎦

1 2 2 1

2 2 3 4 2

10

R R R iR R R R i

netlist


relative error

positionsymbol

1.000E+0(1, 1)R2

1.000E+0(1, 1)R1

1.000E+0(2, 1)-R2

9.950E-1(2, 2)R4

9.950E-1(2, 2)R3

5.013E-3(2, 2)R2

2.488E-3(1, 2)-R2


Slide 16


netlist

symbolic approximatedtransfer function

relative error

positionsymbol

1.000E+0(1, 1)R2

1.000E+0(1, 1)R1

1.000E+0(2, 1)-R2

9.950E-1(2, 2)R4

9.950E-1(2, 2)R3

5.013E-3(2, 2)R2

2.488E-3(1, 2)-R2

=+ +

2 42

1 2 3 4( )( )R Ri

R R R R

equations + −⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥− + + ⎣ ⎦⎣ ⎦⎣ ⎦

1 2 2 1

2 2 3 4 2

10

R R R iR R R R i

Slide 17

Overview Symbolic Simplification Methods

Simplified equationsTransfer functions

Simplified equationsautomaticerror control

Nonlinear equationsLinear equations

Insights ABM generationInsights

Design task

AC, P/Z, Noise AC, DC, Transient, TEMP

Slide 18

Linear Symbolic Analysis

• Starting from netlist descriptionSymbolic methods

• Standard analysis techniques (MNA, STA) forsetting up analytic system equations

Slide 19

Linear Symbolic Analysis

• SBG (Simplification Before Generation)matrix approximation based on Sherman-Morrison formula for error prediction

Simplification methods

= −R

Cout

E

RV

• SAG (Simplification After Generation)approximation of transfer function in canonical SOPform to equal coefficient error

• Pole/Zero techniques

Slide 20

Mathematical Formulation

Small-signal behavior can mathematically be described by alinear system of equations:

( ) ˆ; ( ; )

( ) lij l

A s p x b s p

a a p s

⋅ =

= ⋅∑

formulated inLaplace frequency s

coefficients of A

1

ˆ( , , )ˆ

n

N

k

xp p py x

∈=

=

LR

With:internal variables

parameters

output component

Slide 21

Linear Symbolic Analysis: Benefits and Limits

• symbolic analysis is a tradeoff between accuracyand complexity

• symbolic extraction of poles and zeros possible fortransfer functions of order ≤ 3 only

Limits

• support for symbolic computation oftransfer functions, etc.

• support for symbolic pole/zero analyses• application of industrial-sized circuits (up to

100 transistors) in interactive computation time• advanced research area: efficient and sophisticated

algorithms available

Benefits

Slide 22

Nonlinear Symbolic Analysis

AHDL model

netlistdescription

originalDAE system

simplifiedDAE system

index controlerror control

reference simulation

originalsimplified

Slide 23

Mathematical Formulation

System behavior can mathematically be described by aset of differential algebraic equations (DAE system):

( )( ) 0);(),(),(

0);(),(),(),(=

=ptutytxg

ptutytxtxf &

),( gfF

differential part

algebraic part

1

1

1

1

( , , )( , , )( , , )( , , )

m

n

l

N

u u ux x xy y yp p p

==

=

=

LLLL

With:inputs

internal variables

outputs

parameters

Slide 24

Numerical Analyses

Transient analysisdynamic behaviornonlinear DAE system

DC/DT analysisstatic behaviornonlinear equation system

AC analysislinear behaviorlinear equation systemin frequency domain

2×10 6 4×10 6 6×10 6 8×10 6 0.00001t

2

1

1

2

2 1 1 2Vin

0.2

0.4

0.6

0.8

1

1.2

vout

1.0E0 1.0E2 1.0E4 1.0E6 1.0E8Frequency

50

40

30

20

10

0

ed

ut

in

ga

M�Bd� 1.0E0 1.0E2 1.0E4 1.0E6 1.0E8

1.0E0 1.0E2 1.0E4 1.0E6 1.0E8Frequency

80

60

40

20

0

es

ah

P�ged�

1.0E0 1.0E2 1.0E4 1.0E6 1.0E8

Slide 25

Symbolic Model Reduction

Specificationsinputs u, outputs v

numerical analyses A

Error controlerror function Eerror bound ε

),A( uv FF =

DAEF

DAEG

),A( uv GG =ε<),E( GF vv

Goalfind DAE system G with reduced complexity and defined accuracy

Slide 26

Symbolic Model Reduction

),A( uv FF =

DAEF

DAEG

),A( uv GG =ε<),E( GF vv

1R kR2R

Specificationsinputs u, outputs v

numerical analyses A

Error controlerror function Eerror bound ε

Goalfind DAE system G with reduced complexity and defined accuracy

Simplification processiterative application of reduction techniques R

Slide 27

Symbolic Reduction Techniques (1)

Algebraic manipulationelimination of variables

removal of independent blocks of equations

),(0)(yxg

yfx== ( )yyfg ),(0 =

Slide 28


Branch reductiondetection and removal of unused branches of piecewise-defined functions

=)(xf)(1 xf ax <)(2 xf bxa ≤≤)(3 xf bx >

)()( 2 xfxf = für alle x

Slide 29


0)(:1

=∑=

N

iij xtF 0)~(:

1=∑

≠=

N

kii

ij xtG

Term reductionreplace terms of equations with zero

applicable to nested expressions

Slide 30


0)(:1

=∑=

N

iij xtF 0)~(:

1=+∑

≠=

κN

kii

ij xtG

Term substitutionreplace terms of equations with constant value

applicable to nested expressions

Slide 31

Algorithm

perform reductions

error ok?yes


DAE system

simplified DAE system

reject error analysis

no

Slide 32

Ranking

arbitrary order: 59 reductions optimized order: 64 reductions

DefinitionRanking is the estimation of the influence of each reduction on the output behavior and a suitable ordering.Advantages

increased number of reductionsdecreased computation time

Implementationaccurate estimationsimple and efficient computationtradeoff between accuracy and efficiency

rejected

accepted

Slide 33

Clustering

no clustering: 65 iterations, 26.2 s with clustering: 5 iterations, 3.0 s

DefinitionClustering is the bundling of reduction steps with similar influence on the output behavior. Advantages

many simultaneous reduction stepsreduced effort for error analyseshighly decreased computation time

Implementationranking data can be used for clustering

Slide 34

Index Monitor

DefinitionIndex k of a DAE system is a measure for its distance to a regular ODE system.For k >1 the numerical solving is an ill-posed problem.Advantages

monitoring of the index during model reductionassuring numerical stability of reduced system

Implementationmany index concepts not appropriate: algebraic/symbolic due to complexity, structural/graph-based due to modified topologynumerical computation of tractability index and strangeness index

Slide 35

Original Algorithm

perform reductions

error ok?yes


DAE system


reject error analysis

no

Slide 36

Improved Algorithm

perform reductions

yes


DAE system


reject error analysis, index analysis

no

(re)compute clustering

compute ranking

error ok?index ok?

Slide 37

Example Application: Operational Amplifier

VB

Vsig

1k

RF1

I4

VCC

VDD

I1

I3

Q2N2907AQ4

Q2N2222

Q5

Q2N2907A

Q3

Q2N2907A

Q7 Q8Q2N2907A

Q2

Q2N2222

Q1

Q2N2222

Q6

Q2N2907A

R1 20

R2 20

10k

RF2

RL 100k

Comp

30p

0

0

0

0

0

VDB

8

10

7

3

in

413

11

6

2

2

12

1

5

9

InputOutput 8 bipolar transistors

7 sources

5 resistors

1 capacitor

Taskcomputation of a behavioral model

input: Vsig, output: V9

Slide 38


DT specificationerror bound

error function

Transient specificationerror bound

error function∞

⋅==

DT

DT

E25.0 Vε

∞⋅=

=

Tran

Tran

E0.1 Vε

Slide 39


Original system73 equations

350 terms

94 parameters

Slide 40


Symbolic reduction73 → 6 equations

350 → 24 terms

94 → 21 parameters

Model reduction time792 s

Slide 41


Symbolic reduction73 → 6 equations

350 → 24 terms

94 → 21 parameters

Behavioral modeloriginal: 166.0 s

model: 2.3 s

Slide 42

Infineon Design Problem: Folded-cascode OpAmp

cfcamp equations top-levelterms parameters factor

originalmodel 1120 100 % 16306 100 % 1977 100 % 1

simplified model 22 2 % 54 0.3 % 94 5 % 137

Applicationautomatic generation of behavioral models (e.g. VHDL-AMS, Verilog-A)⇒ system simulation, optimization, system understanding

Slide 43

Nonlinear Symbolic Analysis: Benefits and Limits

• support for automated generation of behavioralmodels for different behavioral modeling languagesand simulators

• automatic and error-controlled complexity reduction • supported analysis modes: DC-Transfer, AC, Transient

Benefits

Limits • limited circuit size (currently up to 20 transistors)• algorithms under development• explicit symbolic results for static systems

in general not possible

• due to the general mathematical conceptthe methods are also applicable to other domains(e.g. system simulation of mechatronical systems)

Synergies

Slide 44

Tool for analysis, modeling,and optimization of analogcircuits

Commercially distributedsince 1998

Current version: 2.1.1

Mathematica Add-on

Evaluation version available at:www.analog-insydes.de

EDA Tool Analog Insydes

Slide 45

EDA Tool Analog Insydes

Mathematica Kernel

Interfaces: PSpice, Eldo, Saber, Spectre, Titan

High-performanceMathLink Binaries

SymbolicModel Library

Online HelpMathematica

Frontend

Slide 46

MOR Techniques – Current Activities

bottom-upbehavioral modeling

model optimization

device model compilation

Analog Insydes

CircuitSchematic

Setup ofCircuit Equations

Model Approximation

Analytic ModelAHDL

Netlist Import

Device Model Library

PerformanceOptimization

Device ModelSPICE / CMI

Model Export

AHDL Import

Behavioral ModelAHDL

Simulation

coupling of

numeric/symbolic MOR

Involved in publicproject initiatives: • VeronA• SymTecO• SyreNe

netlist-basedMOR

Slide 47

Transfer of Methodology

Hydraulicsmodeling ofgas pipeline networks

Enzyme kineticsinterface between Analog Insydes andPathwayLab (simulatorfor enzyme kinetics)

Energy distribution networksmodel reduction for high-voltageenergy distribution networkswith dispersed power generation

Slide 48

Transfer of Methodology: Mechatronical Systems

analogies electronics mechanics

through variables current force, torque

across variables voltage rotation, displacement

equation setup Kirchhoff lawsd‘Alemberts

principle

Usage of analogiesfor methodology transfer

• Decompose mechanical system in finite elements• Implement elements in terms of symbolic component

models• Support for vector-valued variables• Automated mapping to an equivalent netlist with

scalar-type nodes and branches

Adding support formechanics

Slide 49

Application Example: Symbolic Analysis of Mechanical System

density ρelasticity E

shear modulus Glength A1 lA1

length A2 lA2

height A1,A2 hA

length B lBheight B hB

width A1, A2, B w

Parameters: System consists of silicon beams:

lA1= 223.5 μm

lA2= 12 μm

lB= 250 μm

Goal:Find symbolic formulafor displacement as afunction of force

Slide 50

Application Example: Symbolic Analysis of Mechanical System

Goal:Find symbolic formulafor displacement as afunction of force

Step 1: Analyze the system numerically in order to find

out a suitable discretisation. Therefore we com-pare the dynamics with different discretisations.

Step 2: Derive the symbolic model equations for

the static model from dynamic model.

Step 3: Apply symbolic model reduction in order to

reduce the static model and solve the reduced symbolic model equations for displacement in C3.

Step 4: Validate the derived solution against the numerical solution of the full model.

Slide 51

31 1 23

1 2

22

A A A

A A A

l l lu FEh w l l

+=

+

Result Validation

Compare reduced against full model:

Slide 52

Result Validation

31 1 23

1 2

22

A A A

A A A

l l lu FEh w l l

+=

+


Slide 53

Result Validation

31 1 23

1 2

22

A A A

A A A

l l lu FEh w l l

+=

+

Bh

Bl


Slide 54

Application Example: Acceleration Sensor

• Three metal plates formingseries connection of twocapacities

• Movable center plate withmass mdyn and Hooks lawIn case of acceleration

→ Plate moves→ Change of capacities→ Voltage drop at Vout

Simple mechatronical system• Acceleration sensor with

electronic circuit attachedto a point mass

• Force F accelerates the system

Acceleration sensor component

Slide 55

Application Example: Model Reduction

Analog Insydes command CancelTerms approximates the DAE system:

Slide 56

Application Example: Model Reduction

Analog Insydes command CancelTerms approximates the DAE system

Slide 57

Application Example: Reduced Model Equations

Post processing with Analog Insydes command CompressDAE:

Reduced equation system5 equations

Comparison of simulationresults

original modelreduced model

Slide 58

Summary and Outlook

• Application of symbolic analysis only possible usingapproximation techniques coupled with numericsimulation

• Linear symbolic analysis used successfully for industrialcircuit analysis tasks

• Nonlinear symbolic analysis under development withpromising results

Summary

Outlook • Extension of nonlinear simplification methods

• Extension towards support for multi-physics systems

• Integration of symbolic analysis into industrial designflows

Documents

Symbolic Model Reduction for Linear and Nonlinear DAEs 2 Introduction to symbolic analysis and approximation motivation of symbolic methods principles of symbolic simplification approximation