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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
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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)
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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’
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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)
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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
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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.
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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:
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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
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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
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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
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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
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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’
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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)
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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:
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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
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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)
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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
Common block diagram models
R L
C
R
K=1/CFext
Usource
i v
F m
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Common block diagram models
Common block diagram models
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
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Ports in iconic diagramsPorts in iconic diagrams
R L
C
R
K=1/C
Fext
Usource
i
v
F
m
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Ports in block diagramsPorts in block diagrams
Note: each signal is a physically Note: each signal is a physically meaningful variablemeaningful variable
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Structure as multiportStructure as multiport
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Different formsDifferent forms
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Different formsDifferent forms
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Compact & domain independentCompact & domain independent
Source
Coil
Capacitor
Resistor
Source
Mass
Spring
Damper
Domain dependent
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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
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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:
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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
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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
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Conjugate variables & corresponding statesConjugate variables & corresponding states
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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
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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!
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Basic one-port elementsBasic one-port elements
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Basic two- and multiport elements
Basic two- and multiport elements
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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
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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
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Bilateral signal flow (computational causality)
Bilateral signal flow (computational causality)
2 possibilities:2 possibilities:
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NotationNotation
Causal stroke:Causal stroke:
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Back to the exampleBack to the example
C
R
Se
I
1
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Back to the exampleBack to the example
C
R
Se
I
1
2020--sim demosim demo
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Causal port propertiesCausal port properties
•• FixedFixed causalitycausality
•• PreferredPreferred (integral) causality(integral) causality
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Causal constraintsCausal constraints
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Arbitrary causalityArbitrary causality
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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
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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!
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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
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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
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Mechanical framework of variables
Mechanical framework of variables
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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
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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
© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (72)
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= ∀ ≠
= ′
© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (73)
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
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© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (74)
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
© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (75)
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
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© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (76)
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
© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (77)
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
© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (78)
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 ( )
© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (79)
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
© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (80)
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
© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (81)
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
=
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© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (82)
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
© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (83)
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∗ = ⋅ − = + ∗ − + = ∗ − =−( ) ( ) *( , )
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© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (84)
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!
© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (85)
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
© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (86)
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!
© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (87)
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
© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (88)
∫
=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:
© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (89)
∫
=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
© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (90)
∫
=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 (?):
© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (91)
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
© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (92)
∫
=ddS St
p T p(v,T)
T(v,s)
∫
µ (p,T)
µ =ddN Nt
− = −ddV Vt
v
v sS
N
∫ -
T0
V
© Peter BreedveldSummerschool Bertinoro, 7-11 July 2003 (93)
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