Control Engineering Laboratory BOND GRAPHS - Physical ... · PDF fileControl Engineering...

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7/11/2003 Summerschool Bertinoro7-11 July 2003

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Control Engineering Laboratory

BOND GRAPHS -Physical systems modeling 1:

Fundamental concepts

BOND GRAPHS -Physical systems modeling 1:

Fundamental concepts

Cornelis J. Drebbel Institute for Mechatronicsand Control Engineering Laboratory, Electrical Engineering Department

University of Twente, [email protected]

Peter BreedveldPeter Breedveld

© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (2)

AimAim•• System theoretic approach to physical system dynamics System theoretic approach to physical system dynamics

based onbased on–– classification of phenomena in terms of energyclassification of phenomena in terms of energy–– fundamental principles of thermodynamicsfundamental principles of thermodynamics

•• shown how, as a result:shown how, as a result:–– variables and relations describing physical systems may be variables and relations describing physical systems may be

classifiedclassified–– models may be organized as (‘portmodels may be organized as (‘port--based approach’)based approach’)

•• multiport elementsmultiport elements•• interconnected in an interconnection structure corresponding to interconnected in an interconnection structure corresponding to a a

generalized networkgeneralized network–– multiport elements describe basic behaviors with respect to enermultiport elements describe basic behaviors with respect to energy gy

and entropyand entropy•• special attention:special attention:

–– role of analogiesrole of analogies–– analogue behavioranalogue behavior

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IntroductionIntroduction

• introduction & some philosophy:• Why physical systems modeling?• What is physical systems modeling?• Context of explanation (focus of engineering

physics) versus justification (focus of math)• elements and components, theory building• synthesis between classical approaches

• choice of variables:–– mechanical versus thermodynamic mechanical versus thermodynamic

frameworkframework


Three-world meta-modelThree-world meta-model

•• Real worldReal world (assuming existence of (assuming existence of ‘‘objectiveobjective’’environmentenvironment’’))

•• Conceptual worldConceptual world (in our brain)(in our brain)•• ‘‘PaperPaper’’ worldworld, including electronically stored , including electronically stored

datadata

–– Only in/via the paper world:Only in/via the paper world:•• CommunicationCommunication•• SupportSupport•• SystematizationSystematization

–– exchangeable abstractions/concepts >>>exchangeable abstractions/concepts >>>–– importance of importance of symbolssymbols & & notation notation (representation)(representation)

3


Are physical concepts ‘real’ ?Are physical concepts ‘real’ ?

•• For example:For example:–– energy, time, momentum, causality,…energy, time, momentum, causality,…

•• In a context of In a context of justificationjustification: : NO!NO!•• In a context of In a context of discoverydiscovery//explanationexplanation: :

YES, YES, but rather ‘really useful’ than but rather ‘really useful’ than just ‘real’just ‘real’


The useless ‘quest for truth’The useless ‘quest for truth’

•• A model is necessarily A model is necessarily incomplete:incomplete:‘‘all models are wrongall models are wrong’’, but , but ‘‘truthtruth’’ is is notnot

what countswhat counts•• The issue is whether a model is The issue is whether a model is

competentcompetent (to solve a problem in a (to solve a problem in a given given problem contextproblem context in the most in the most generic sense of the word)generic sense of the word)

4


What is a model?What is a model?

Some sort of abstraction Some sort of abstraction (in the (in the ‘‘paperpaper’’ world)world) that enablesthat enables•• insight insight in in the real world counterpartthe real world counterpart•• communicationcommunication about about the real world counterpartthe real world counterpart•• observationobservation of of the real world counterpartthe real world counterpart•• troubleshootingtroubleshooting of of the real world counterpartthe real world counterpart•• designdesign of new aspects related to the real world counterpartof new aspects related to the real world counterpart•• modificationsmodifications of of the real world counterpartthe real world counterpart•• ‘‘explanationexplanation’’ of of functionalityfunctionality of the real world counterpartof the real world counterpart•• measurementmeasurement of of the real world counterpartthe real world counterpart

but, most importantly, that is :but, most importantly, that is :•• competentcompetent to solve a given problem and make decisions to solve a given problem and make decisions

related to the real worldrelated to the real world


ModelingModeling

• Given a specific problem context, the decision process to obtain a competent model to solve this problem

• Approaches between two extremes:–– a priori knowledgea priori knowledge–– ‘‘black boxblack box’’ (also an axiomatic concept with a priori (also an axiomatic concept with a priori

assumptions!!!)assumptions!!!)•• concept concept ‘‘inputinput’’ implicitly contains model of imposing some implicitly contains model of imposing some

action with negligible back effect (action with negligible back effect (‘‘high input impedancehigh input impedance’’))•• concept concept ‘‘outputoutput’’ implicitly contains model of measurement implicitly contains model of measurement

with negligible effect on the system being observed (with negligible effect on the system being observed (‘‘low low output impedanceoutput impedance’’))

•• after a competent inputafter a competent input--output relation is found, it is not output relation is found, it is not open for modifications or physical interpretationopen for modifications or physical interpretation

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Model representationModel representation

•• model model representationrepresentation::–– symbols used to representsymbols used to represent

•• the the conceptsconcepts being usedbeing used•• (the structure of) their (the structure of) their relationsrelations (interconnection)(interconnection)

•• model model manipulationmanipulation (as opposed to modeling!) (as opposed to modeling!) ::–– transformation to transformation to differentdifferent representations (including representations (including

the the ‘‘solutionsolution’’) to) to•• increase insightincrease insight•• draw conclusions, etc.draw conclusions, etc.


Role of representationsRole of representations

Process timeProcess timeversusversus

processprocessinging timetime

A

A

A

A

A

T

T

T

T

T

a

b

c

d

e

x1

x2

x3

x4

x5

Sequential:Sequential:

A

T

T

A

x

t

x1 x5x4

x3x2

Simultaneous:Simultaneous:

6


Physical systems modelingPhysical systems modeling

•• all concepts used in the model are, or have a direct relation all concepts used in the model are, or have a direct relation to, to, physically relevant conceptsphysically relevant concepts (use of a priori knowledge!)(use of a priori knowledge!)

•• physicalphysical relationsrelations are maintained as much as possibleare maintained as much as possible•• herein constrained to (for sake of herein constrained to (for sake of ‘‘simplicitysimplicity’’):):

–– deterministicdeterministic mathematical models of mathematical models of macroscopicmacroscopic systems systems thatthat

•• obey obey basic principlesbasic principles of macroscopic physics:of macroscopic physics:–– energyenergy conservationconservation–– positive positive entropyentropy productionproduction

•• describe the describe the behaviorbehavior in time of the common physical properties:in time of the common physical properties:–– mechanical (incl. hydraulic and pneumatic)mechanical (incl. hydraulic and pneumatic)–– electricalelectrical–– magneticmagnetic–– chemicalchemical–– materialmaterial


time & uncertaintytime & uncertainty•• timetime::

–– derivedderived measure for ‘measure for ‘regularityregularity’ based on ’ based on counting ‘ticks’ of a ‘timecounting ‘ticks’ of a ‘time--base’ based on base’ based on repetitive behaviorrepetitive behavior requiringrequiring statestate and and change change (in (in order to be able to count)order to be able to count)

–– within the smallest unit used: necessarily within the smallest unit used: necessarily uncertainty uncertainty

•• cf. Heisenberg u.r. for displacement (= elastic state, cf. Heisenberg u.r. for displacement (= elastic state, kinetic state of change) and momentum (= elastic state kinetic state of change) and momentum (= elastic state of change, kinetic state)of change, kinetic state)

–– dialectic concepts!dialectic concepts!

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concepts of‘state’ & ‘change’

concepts of‘state’ & ‘change’

•• dialecticdialectic fundamental conceptsfundamental concepts•• basis for any basis for any dynamicdynamic model (more model (more

than time!)than time!)•• within a within a context of discoverycontext of discovery a shift to a shift to

spacespace--time is understandable and time is understandable and useful, like the shift from positionuseful, like the shift from position--momentum in Hamiltonian mechanics to momentum in Hamiltonian mechanics to positionposition--velocity in Lagrangian velocity in Lagrangian mechanicsmechanics


Position: a ‘special’ statePosition: a ‘special’ state

PositionPosition•• has a dual nature:has a dual nature:

–– EnergyEnergy state (related to a state (related to a conservation or symmetry principle conservation or symmetry principle like all other states)like all other states)

–– ConfigurationConfiguration statestate•• does does notnot transform like a tensortransform like a tensor

end of introductionend of introduction

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Contents of the sequelContents of the sequel

•• Modeling pitfallsModeling pitfalls•• PortPort--based modeling based modeling •• Basic Concepts (ports, bonds)Basic Concepts (ports, bonds)•• Dynamic conjugation (effort, flow)Dynamic conjugation (effort, flow)•• Multidomain modeling and the role of Multidomain modeling and the role of

energyenergy•• (Computational) Causality(Computational) Causality


Modeling pitfallsModeling pitfalls

•• ‘‘Every model is wrong’Every model is wrong’

•• Model depends on problem contextModel depends on problem context

•• Competent modelsCompetent models

•• Analogies are not identitiesAnalogies are not identities

•• Avoid implicit assumptionsAvoid implicit assumptions

•• Avoid model extrapolationAvoid model extrapolation

•• Confusion of components with elementsConfusion of components with elements

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Physical components versus conceptual elements

Physical components versus conceptual elements

dominant behavior:dominant behavior:••when falling: when falling: ideal massideal mass••when pulling load: when pulling load: ideal springideal spring••for vibration isolation: for vibration isolation: ideal resistorideal resistor••etc.etc.

physical component:physical component:piece of rubber hosepiece of rubber hose


Parasitic elementsParasitic elements

•• Next to dominant behavior:Next to dominant behavior:

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In engineering models:In engineering models:

•• Avoid implicit assumptions, e.g. aboutAvoid implicit assumptions, e.g. about–– problem contextproblem context–– referencereference

–– orientationorientation–– coordinatescoordinates

–– metricmetric

–– ‘negligible’ phenomena, etc.‘negligible’ phenomena, etc.

•• Avoid model extrapolationAvoid model extrapolation–– danger of ignoring earlier assumptionsdanger of ignoring earlier assumptions

•• Focus at competence, not ‘truth’Focus at competence, not ‘truth’


Port-based modelingPort-based modeling•• multidomainmultidomain approach:approach:

–– ‘mechatronics’ (and beyond)‘mechatronics’ (and beyond)•• multiple viewmultiple view approach: other graphical approach: other graphical

representations: iconic diagrams, (linear representations: iconic diagrams, (linear graphs), block diagrams, graphs), block diagrams, bond graphsbond graphs, etc. and , etc. and equationsequations

•• domain independentdomain independent notation using ports:notation using ports:–– bond graphs (& some other benefits…)bond graphs (& some other benefits…)

•• portport--basedbased approach:approach:–– underlying structure of underlying structure of 2020--simsim, ideal tool for , ideal tool for

demonstrationdemonstration•• what are what are portsports and what are and what are bond graphsbond graphs??

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ideal ideal motormotor

J

ideal ideal inertiainertia

Intuitive introduction of the ‘port’ concept

Intuitive introduction of the ‘port’ concept

ideal ideal transmissiontransmission

ideal current ideal current sourcesource

P

potentiometerpotentiometer

Dominant behaviorDominant behavior

(not necessarily competent in each context)(not necessarily competent in each context)


Simple modelSimple model

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Addition of relevant parasitic behavior

Addition of relevant parasitic behavior

Polymorphic Polymorphic modelingmodeling

Depending on Depending on problem problem context:context:


MECH

J

ii∆∆uu11

ii ii

∆∆uu22 ∆∆uu33∆∆TT22 ∆∆TT33

ωω ωω

PP ==∆∆uu⋅⋅ ii(power)(power)

PP ==∆∆TT⋅⋅ωω(power)(power)

ii∆∆uu44

∆∆TT11

ωω

electrical portselectrical ports

mechanical portsmechanical ports

What are ports?What are ports?

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Comparison with familiar model views

Comparison with familiar model views

Iconic diagramsIconic diagrams(‘ideal physical models’):(‘ideal physical models’):

R L

C

R

K=1/CFext

Usource

i v

F m

electricalelectrical mechanicalmechanical


Iconic diagram symbolsIconic diagram symbols

Port-based, but domain dependentPortPort--based, but based, but domain dependentdomain dependent

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Iconic diagram symbolsIconic diagram symbols

C-type storage

I -type storage

(M)R (dissipation,

irreversible transduction)

Se (effort source)

Sf (flow source)

TF (transformer)


Iconic diagramsIconic diagrams

•• Icons do not (always) represent Icons do not (always) represent physical structure:physical structure:

TF (transformer)TF (transformer)

1

i

2020--simsim

icon:icon:

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Common block diagram models


R L

C

R

K=1/CFext

Usource

i v

F m




Note: not all signals Note: not all signals are physically are physically

meaningful variablesmeaningful variables

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Ports in iconic diagramsPorts in iconic diagrams

R L

C

R

K=1/CFext

Usource

i v

F m


Ports in iconic diagramsPorts in iconic diagrams

R L

C

R

K=1/C

Fext

Usource

i

v

F

m

17


Ports in block diagramsPorts in block diagrams

Note: each signal is a physically Note: each signal is a physically meaningful variablemeaningful variable


Structure as multiportStructure as multiport

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Different formsDifferent forms


Different formsDifferent forms

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Compact & domain independentCompact & domain independent

Source

Coil

Capacitor

Resistor

Source

Mass

Spring

Damper

Domain dependent


Compact & domain independentCompact & domain independent

I

C

R

Se

I

C

R

1Se

Domain independent

1

Structure explicitlyStructure explicitly

representedrepresented

as multiport:as multiport:

JUNCTION!JUNCTION!

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Ports in bond graph viewPorts in bond graph view

C

R

Se

I

1


Bond graphsBond graphs

Inventor (MIT, 1959): Prof. Henry M. Paynter (1923-2002)

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BondsBonds

Energy exchange = powerEnergy exchange = power

Storage of kinetic energyStorage of kinetic energyStorage of elastic energyStorage of elastic energy

C ICompact notation:Compact notation:

••Terminology and notation induced by chemical bonds:Terminology and notation induced by chemical bonds:


Ports and power bondsPorts and power bonds

•• (power) bond connecting two elements (power) bond connecting two elements via (power) ports (Harold Wheeler, via (power) ports (Harold Wheeler, 1949)1949)

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Dynamic conjugationDynamic conjugation

•• Between signals of bilateral signal Between signals of bilateral signal flow of relation:flow of relation:–– rate of change: ‘flow’rate of change: ‘flow’

(zero in equilibrium)(zero in equilibrium)•• e.g. molar rate during diffusione.g. molar rate during diffusion

–– equilibrium determining variable: ‘effort’equilibrium determining variable: ‘effort’•• e.g. concentratione.g. concentration


Power conjugationPower conjugation

= special case of dynamic conjugation := special case of dynamic conjugation :–– ‘effort’ and ‘flow’ relate to ‘effort’ and ‘flow’ relate to powerpower–– functional relation is commonly a functional relation is commonly a productproduct

–– sumsum in case of scattering variablesin case of scattering variables

∂ ∂= = + = +

∂ ∂

∂= =

∂∂

= =∂

∑ ∑ ∑ ∑ddd

d d d

d 'effort'

d

d 'flow'

d

jii i j j

i ji j i j

ji j

i

ii j

j

pqE E EP e f e f

t q t p t

pEe e

q t

q Ef f

t p

23


Conjugate variables & corresponding statesConjugate variables & corresponding states


mass-spring systemmass-spring system

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Basic dynamic behaviors& mnemonic codes

Basic dynamic behaviors& mnemonic codes

•• StorageStorage (reversible)(reversible)–– CC, , I (qI (q--type and ptype and p--type storage) type storage)

•• Irreversible transformationIrreversible transformation (‘dissipation’):(‘dissipation’):–– (M)(M)RR(S)(S)

•• DistributionDistribution–– 00--junction, junction, 11--junctionjunction

•• Supply and demandSupply and demand::–– (M)(M)SeSe, (, (M)M)SfSf

•• Reversible transformationReversible transformation–– (M)(M)TFTF, (M), (M)GYGY

•• >> >> 9 basic elements9 basic elements


Orientation conventionsOrientation conventions

•• 11--ports: into most elements (R, C, I), except ports: into most elements (R, C, I), except sources (Se, sources (Se, SfSf))

•• 22--port transducers: 1 in, 1 outport transducers: 1 in, 1 out•• junction structure elements: arbitraryjunction structure elements: arbitrary•• multiport generalizations: same as simple formmultiport generalizations: same as simple form•• MOST IMPORTANT: obeying grammar rules MOST IMPORTANT: obeying grammar rules

minimizes sign errors!minimizes sign errors!

25


Basic one-port elementsBasic one-port elements


Basic two- and multiport elements

Basic two- and multiport elements

26


Constitutive relations not necessarily linear!

Constitutive relations not necessarily linear!

e.g. zener diode:e.g. zener diode:dominant behavior:dominant behavior:

irreversible irreversible transduction transduction (resistor)(resistor)

with nonlinear with nonlinear constitutive relationconstitutive relation

R


Other form of non-linearity:Modulation

Other form of non-linearity:Modulation

iconic diagram (IPM)iconic diagram (IPM)

capstancapstan

e.g. modulation of a transducer:e.g. modulation of a transducer:

MR

27


Bilateral signal flow (computational causality)

Bilateral signal flow (computational causality)

2 possibilities:2 possibilities:


NotationNotation

Causal stroke:Causal stroke:

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Back to the exampleBack to the example

C

R

Se

I

1


Back to the exampleBack to the example

C

R

Se

I

1

2020--sim demosim demo

29


Causal port propertiesCausal port properties

•• FixedFixed causalitycausality

•• PreferredPreferred (integral) causality(integral) causality


Causal constraintsCausal constraints

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Arbitrary causalityArbitrary causality


Causality assignmentCausality assignment

Algorithmic (SCAP):Algorithmic (SCAP):–– 1) fixed causal ports with propagation via 1) fixed causal ports with propagation via

constraintsconstraintscausal ‘conflict’: causal ‘conflict’: ‘ill‘ill--posednessposedness’’

–– 2) preferred causal ports with 2) preferred causal ports with propagation via constraintspropagation via constraintscausal ‘conflict’: causal ‘conflict’: dependent state(s)dependent state(s)

–– 3) choice of arbitrary causal port with 3) choice of arbitrary causal port with propagation via constraintspropagation via constraintsmeans existence of means existence of algebraic algebraic loop(sloop(s))

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ExampleExample

Se GY

II

R

1

R

1

2nd order loop2nd order loop

1st order loop1st order loop1st order loop1st order loop

••Important modeling feedbackImportant modeling feedback

••Automatic in 20Automatic in 20--simsim

••Visible in bond graph causalityVisible in bond graph causality

••‘Hidden’ in case of iconic diagrams‘Hidden’ in case of iconic diagrams

J

u

i u

i

u

i

u

i

u

i

T

ω

T

ω


Positive orientationPositive orientation

orientationorientation NOT THE SAME AS NOT THE SAME AS direction!direction!

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Causality does NOT affect orientation!Causality does NOT affect orientation!


Mechanical versus Thermodynamic framework

Mechanical versus Thermodynamic framework

Split domains (Split domains (thermtherm.) and couple by SGY .) and couple by SGY (mech.):(mech.):

CC--SGYSGY--CC

Only relaxation Only relaxation behavior: Cbehavior: C--RR

Oscillatory behavior Oscillatory behavior (damped): C(damped): C--I(I(--R)R)

OneOne type of storagetype of storageTwoTwo types of storagetypes of storageThermodynamics:Thermodynamics:Mechanics:Mechanics:

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Symplectic gyratorSymplectic gyrator

•• Unit gyrator:Unit gyrator:

•• SGY:SGY:

•• Multiport:Multiport:

r = 1GY0 0C C

SGY0 0C C

0 -1+1 0

C11 C12C12 C22 SGYC 0

2020--sim demosim demo


Symplectic GYratorSymplectic GYrator

•• Mechanical (Mechanical (xx,,pp))

–– Only in Only in inertialinertial frames: (Newton's 2nd law)frames: (Newton's 2nd law)

•• Electrical network (Electrical network (qq,,λλ):):

–– Only Only quasiquasi--stationarystationary (non(non--radiating):radiating):

SGY0 0C C

SGY0 0C C

∂=

∂potEex

dd

=potxft

∂=

∂kinEep

dd

=kinpft

dd

= −pFt

=potf v

=pote F=kine v

dd

= −pFt

λ∂

=∂magEe

ddλ

=magft

ddλ

− = ut

=mage i

dd

=elecqft

=elecf i

∂=

∂elecEeq=elece u

ddλ

− = ut

=mage i

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Unit gyrator (SGY) as ‘dualizer’Unit gyrator (SGY) as ‘dualizer’Port equivalent:Port equivalent: Port equivalent:Port equivalent:Original:Original: Original:Original:

C1 SGY 1 I

R1 SGY 1 R

1 SGY TF SGY 0 1 TF 0

SGY GY SGY 1 1 GY 1

SGY1

1 TF 0SGY1

1 GY 1

0GY

TF 1

1 0SGY SGY1

0

SGY

1

1

0

1 0

0

SGY SGY0

1

SGY

1 0

1

1

0

0

Se SGY 1

0Sf SGY

Sf 1

Se 0

I 0 C0 SGY


Mechanical framework of variables

Mechanical framework of variables

35


Generalized thermodynamic framework of variables

Generalized thermodynamic framework of variables

f flow

e effort

q f t= ∫ d

generalized state

electric i current

u voltage

q i t= ∫ d charge

magnetic u voltage

i current

λ = ∫u td magnetic flux linkage

elastic/potential translation

v velocity

F force

x v t= ∫ d displacement

kinetic translation F force

v velocity

p F t= ∫ d momentum

elastic/potential rotation

ω angular velocity

T torque

θ = ∫ ωdt angular displacement

kinetic rotation T torque

ω angular velocity

b T t= ∫ d angular momentum

elastic hydraulic ϕ volume flow

p pressure

V t= ∫ ϕd volume

kinetic hydraulic p pressure

ϕ volume flow

Γ = ∫ p td momentum of a flow tube

thermal T temperature

fS entropy flow

∫= tSS f d

entropy

chemical µ

chemical potential

fN molar flow

∫= tNN f d

number of moles


Co-Energy & Legendre Transformations

A function is homogeneous of order n if

Define

then or is homogeneous of order (n-1).

For a homogeneous function Euler’s theorem holds:

By definition:

but also:

Hence:

F x x x xk( ) , , with = 1 …F x nF x( ) ( )α α=

yi xFxi

( ) =∂

∂

yi xF x

xi

n F xxi

nyi x( )

( ) ( )( )α

∂ α

∂α

α

α

∂

∂α= = ⋅ =

−1yi x( )

∂

∂

Fxii

kxi n x x

nyi

i

kxi n

yT x=∑ ⋅ = ⋅ =

=∑ ⋅ = ⋅ ⋅

1

1

1

1 F or F( ) ( )

dF Fxi

dxi yidxi yT dxi

k

i

k= ⋅ = = ⋅

=∑

=∑

∂∂ 11

dF dn

yT xn

yT dxn

dy T x= ⋅

= ⋅ + ⋅

1 1 1( )

( )( )dy T x n yT dx⋅ = − ⋅1

for n : ( )= ⋅ =1 0dy T x

for n : dF =-

≠ ⋅11

1ndy T x( )

36


HomogeneousEnergy Functions

The energy of a system with k state variables q is:

If qi is an “extensive” state variable, this means that:

Hence E(q) is first order (n = 1) homogeneous, so

is zeroth order (n – 1 = 0) homogeneous, which means that ei(q) is an “intensive” variable, i.e.

This means also that in case n=1 and k=1 e(q) is constant, i.e.which changes the behavior of this element into that of a source.in order to enable storage: storage elements = multiports (k>1)‘1-port storage element’ = n-port storage element with flows of n-1 ports kept zerocorresponding n-1 states constant and not recognized as states.Such a state is often considered a parameter:if then E’(q1) not necessarily first-order homogeneous in q1

E q E q qk( ) ( , , )= 1 …

E q E q E q( ) ( ) ( )α α α= = 1

ei qEqi

( ) =∂

∂

e q e q e qi i i( ) ( ) ( )α α= =0

∂

∂

eq

dedq

= = 0

E q q q E qn dq ii( , ,..., ) ( ),1 2 0 1 1= ∀ ≠

= ′


HomogeneousEnergy Functions

In bond graph form:

For n=1 and k independent extensities:only k–1 independent intensitiesbecause for n=1 we find Gibbs' fundamental relation:

by definition:

from the above equations follows the Gibbs-Duhem relation:

E q eT q( ) = ⋅

dE eT dq= ⋅

( )de T q⋅ = 0

37


Legendre Transformations Legendre transform of F(x) with respect to xi is by definition:

with

and the total Legendre transform of F(x) is:Note that for n = 1: L=0

Now

or

which means that xi is replaced by yi as independent variable or “coordinate”!Hence

L F x xiLxi

F x yi xi{ ( )} ( )= = − ⋅

yiFxi

=∂∂

L F x L F x yii

kxi{ ( )} ( )= = −

=∑

1

dLxidF d yixi dF yidxi xidyi y jdx j xidyi

j i= − = − − = −

≠∑( )

LxiLxi

x xi yi xi xk= − +( , , , , , , )1 1 1… …

L L(y xidyi dy T xi

k= =− =− ⋅

=∑); ( ) dL

1


Co-Energy FunctionsThe co-energy of E(q) with respect to qi is by definition:

Hence

The total co-energy E*(e) of E(q) is: E* = -L, hence

For n=1:confirming earlier conclusion that there are only k-1 independent ei

For n=2:

For n=3:

Eqi∗

EqiLqi

Eqiq qi ei qi qk

∗ =− = ∗− +( , , , , , , )1 1 1… …

E q Eqiei ei qi( ) ( , , )+ ∗ = ⋅… …

E q E e eT q( ) ( )+ ∗ = ⋅

E e∗ =( ) 0

E q E e eT q( ) ( )= ∗ = ⋅12

E q eT q( ) = ⋅13

E e eT q∗ = ⋅( )23

38


Relations forCo-Energy Functions

dEqidei qi e j dq j

j∗ = ⋅ − ⋅

≠∑

1

dE deii

kqi de T q n eT dq n dE∗ =

=∑ ⋅ = ⋅ = − ⋅ = −

11 1( ) ( ) ( )

E n En

neT q

neT q∗ = − =

−⋅ ⋅ = −

⋅( )1

11

1


Legendre transforms in simple thermodynamics

In thermodynamic systems with– internal energy U– entropy S– temperature T– volume V– pressure p– total mole number N– total material potential µtot

– mole number per species i Ni

– chemical potential µi:Free energy F:

Enthalpy H:

F LS U TS pV iNitot N

i

m= = − = − + + ⋅

=

−∑ µ µ

1

1

dF SdT pdV idNitot dN

i

m= − − + + ⋅

=

−∑ µ µ

1

1 F T,V N N N f T,v c( , , ) . ( , ) ( )=

H LV U pV U pV= = − − = +( )

dH TdS Vdp i dNitot dN

i

m= + + ⋅ + ⋅

=

−∑ µ µ

1

1H S p N N N h s p c( , , , ) . ( , , ) ( )=

39


Legendre transforms in simple thermodynamics

Gibbs free enthalpy G:

For m=1:

G LS V U TS pV tot N iNii

m= = − − − = ⋅ +

=

−∑, ( ) µ µ

1

1

dG SdT Vdp idNitot dN

i

m= − + + + ⋅

=

−∑ µ µ

1

1

G T, p N N N g T, p c( , , ) . ( , ) ( )=

g tot T, p= µ ( )


Legendre Transforms and Causality

• If an effort is “forced” on a port of a C element (“derivative causality” or “flow causality”), this means that the roles of e and q are interchanged in the set of independent variables, which means that the energy has to be Legendre transformed.

• This is particularly useful when the effort e is constant (e.g. an electrical capacitor in an isothermal environment with T=Tconst):

withwith

uq. s e

TS

. : T const

dF udq SdT udq= − =

Cuq.

P u q dFdt

= ⋅ =

40


Constitutive Relations

• The function ei(q) is called constitutive relation, also called constitutive equation, constitutive law, state equation, characteristic equation, etc.

• If ei(q) is linear, i.e. first order homogeneous, then E(q) is second order homogenous, i.e. E(q) is quadratic. In this case, and only in this case:

E q E q

E q eT q

de T q eT dq dE

( ) ( )

( )

( )

α α=

= ⋅

⋅ = ⋅ =

2

12


Maxwell Reciprocity

From the principle of energy conservation (First law) can be derived that:

i.e. the Jacobian matrix of the constitutive relations is symmetric.

This is called Maxwell reciprocity or Maxwell symmetry.

∂∂ ∂

∂∂ ∂

2 2Eqi q j

Eq j qi

=

∂

∂

∂

∂

e jqi

eiq j

=

41


Intrinsic Stability

Intrinsic stability requires that this Jacobian is positive-definite:

and that the diagonal elements of the Jacobian are positive:

det ∂∂

eq

> 0

∂

∂

eiqi

i> ∀0


Legendre Transforms in Mechanics In mechanical systems with

– kinetic energy T potential energy V– displacements x momenta p– velocities v forces F:

• Hamiltonian H:

• Lagrangian L:

• co-Hamiltonian :

• co-Lagrangian or Hertzian :

E q H x p T V( ) ( , )= = +

Hp Lp vT p H T T T V T V L(x v∗ =− = ⋅ − = + ∗ − + = ∗ − =( ) ( ) , )

with v Hp

=∂∂

Hx p,* * T TH v p F q H

x,p

* *(T T ) (V V ) (T V) T V H (F, v)

= ⋅ + ⋅ − =

∗ ∗ ∗= + + + − + = + =

with F Hx

=∂∂

Hx∗ Hx FT q H V V T V V T L F p∗ = ⋅ − = + ∗ − + = ∗ − =−( ) ( ) *( , )

42


Legendre Transforms in Electrical Circuits

In electrical circuits with– capacitor charges q and voltages u– coil flux linkages Φ and currents i:

E q Ec q EL( , ) ( ) ( )Φ Φ= +

E u i uT q iT E E∗ = ⋅ + ⋅ − =↑

( , ) Φ

only in linear case!


Example nonlinear 3-port CExample nonlinear 3-port C

•• energy storage in GAS:energy storage in GAS:–– IdealIdeal

–– VanVan--derder--WaalsWaals

( )0

00

, ev v

R s sc cvT s v T

v

− −

=

0

00

( , ) v v

R s sc cv bT v s T e

v b

− − −

= −

pv RT=

( )2ap v b RTv

+ − =

43


Regular causal form in thermodynamics:

Regular causal form in thermodynamics:

thermalport

mechan-icalport

materialport

d- -dV Vt

= d- -dS St

=

d- -dN Nt

=( , )p Tµ µ=

( , )p p v T= T

p(T,V,N)S(T,V,N)µ(T,V,N)

C

differential causality!


Desired integral causal form:Desired integral causal form:

thermalport

mechan-icalport

materialport

d- -dV Vt

= d- -dS St

=

d- -dN Nt

=( , )p Tµ µ=

( , )p p v T= T

p(S,V,N) T(S,V,N)µ(S,V,N)

C

integral causality!

44


∫

=ddS St

d/dt

p T p(v,T)

s(v,T)

∫

µ (p,T)

µ =ddN Nt

− = −ddV Vt

v

v sS

N

- V

Differentialcausality

Compute all conjugate variables:Compute all conjugate variables:


∫

=ddS St

p T p(v,T)

T(v,s)

∫

µ (p,T)

µ =ddN Nt

− = −ddV Vt

v

v sS

N

∫ -

V

Integralcausality

Change to integral causality (T(v,s)?):Change to integral causality (T(v,s)?):

45


∫

=ddS St

p T p(v,T)

T(v,s)

∫

µ (p,T)

µ =ddN Nt

− = −ddV Vt

v

v sS

N

∫ -

????

T0

S0

V

Find S0 from T0 (?):Find S0 from T0 (?):


Form requires T0, not S0Form requires T0, not S0

( )0

00

d 0

1, exp dv

v v vv

s s pT s v T vc c T

∂∂ =

− = − ∫

46


∫

=ddS St

p T p(v,T)

T(v,s)

∫

µ (p,T)

µ =ddN Nt

− = −ddV Vt

v

v sS

N

∫ -

T0

V


Concluding remarksConcluding remarks

PortPort--based modeling:based modeling:–– no a priori decision about input and output of a no a priori decision about input and output of a

bilateral relationbilateral relation–– two variables of bilateral relation (port):two variables of bilateral relation (port):

•• dynamically conjugateddynamically conjugated•• power conjugated in case of energy conservation:power conjugated in case of energy conservation:

–– effort and floweffort and flow

–– behavior with respect to energy is domain behavior with respect to energy is domain independent (C,I; R; TF,GY; independent (C,I; R; TF,GY; Se,SfSe,Sf; 0,1); 0,1)

–– causality assignment gives feedback on causality assignment gives feedback on modeling decisionsmodeling decisions

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