Transcript
Control Engineering Laboratory
PORT-BASED MODELING OF DYNAMIC SYSTEMS: fundamental concepts and bond graphs Peter Breedveld Control Engineering Laboratory, Faculty of Electrical Engineering, Mathematics and Computer Science University of Twente, Netherlands
[email protected] 7/18/2005
Summerschool Bertinoro 17-23 July 2005
© P.C. Breedveld 1
Some remarks • Role of concepts (‘intellect’, multiple views) • Ports from a historical, energy point of view (energy states) • Focus on linear motion and fixed axis rotation, simple configuration constraints: no need for abstract geometrical concepts (like in the rest of the course) • Separation between configuration state and energy state: structures influence of kinematic constraints
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General Outline • Background and history • Introduction to ports, bonds and physical domains • Introduction to bond graph modeling: causality, domain-independence, etc.
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Assumed prerequisites • Some experience in: – – – –
Block diagram models Numerical simulation (methods) Linear and nonlinear analysis Basic physics (electrical circuits, simple rigid body mechanisms, principles of thermodynamics, etc.
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Introduction – Modular modeling (prevailing trend in modeling and simulation of complex physical systems):
• represented as a network interconnection of ideal elements
– Advantages
• flexibility • re-usability of model parts • physical insight • support for automated modeling • fundamental to systems and control theory
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Introduction – Disadvantages
• equations of motion obtained directly from network modeling are often – complicated – without apparent structure – will easily contain algebraic constraints arising from the interconnection of the sub-systems
• no problem for digital simulation, but • may not be very well-suited to analysis and control purposes, (nonlinear systems!)
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Course objectives – particular type of network modeling, viz. port-based modeling:
• sub-systems are interacting through power exchange represented by pairs of conjugated variables • unified way to deal with systems from different physical domains (e.g., mechanical and electrical) • resulting systems of equations possess an underlying generalized Hamiltonian structure, leading to the geometric notion of a port-Hamiltonian system
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Course objectives • exploiting the underlying Hamiltonian structure offers many possibilities for – – –
analysis simulation control
of complex physical systems
• illustrated in variety of application areas • possible extension to classes of distributed-parameter systems
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Appetizer: Newton cradle example • Experiment
• 20-sim demo • Animation
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Dual role example: Falling and bouncing object • generally considered as conceptually complex:
– switching models reinitialization, timing – energy bookkeeping – prone to sign errors
• port-based approach with explicit structure, allows conceptually simple solution Summerschool Bertinoro, 17-23 July 2005 (10)
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Aim • System theoretic approach to physical system dynamics based on – classification of phenomena in terms of energy – fundamental principles of thermodynamics
• shown how, as a result: – variables and relations describing physical systems may be classified – models may be organized as (‘port-based approach’) • multiport elements • interconnected in an interconnection structure corresponding to a generalized network
– multiport elements describe basic behaviors with respect to energy and entropy
• special attention: – role of analogies – analogue behavior Summerschool Bertinoro, 17-23 July 2005 (11)
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Fundamental issues • 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 framework Summerschool Bertinoro, 17-23 July 2005 (12)
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‘Why?:’ Three-world meta-model • Real world (assuming existence of ‘objective’ environment’) • Conceptual world (in our brain) • ‘Paper’ world, including electronically stored data – Only in/via the paper world: • • •
Communication Support Systematization
– exchangeable abstractions/concepts >>> – importance of symbols & notation (representation) Summerschool Bertinoro, 17-23 July 2005 (13)
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‘Why?’: Are physical concepts ‘real’ ? • For example:
– energy, time, momentum, causality,…
• In a context of justification: NO! • In a context of discovery/explanation: YES, but rather ‘really useful’ than just ‘real’
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‘Why?:’ The useless ‘quest for truth’ • A model is necessarily incomplete: ‘all models are wrong’, but ‘truth’ is not what counts • The issue is whether a model is competent (to solve a problem in a given problem context in the most generic sense of the word)
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What is a model? Some sort of abstraction (in the ‘paper’ world) that enables • • • • • • • •
insight in the real world counterpart communication about the real world counterpart observation of the real world counterpart troubleshooting of the real world counterpart design of new aspects related to the real world counterpart modifications of the real world counterpart ‘explanation’ of functionality of the real world counterpart measurement of the real world counterpart
but, most importantly, that is :
• competent to solve a given problem and make decisions related to the real world
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Modeling • Given a specific problem context, the decision process to obtain a competent model to solve this problem • Approaches between two extremes: – a priori knowledge – ‘black box’ (also an axiomatic concept with a priori
assumptions!!!) • concept ‘input’ implicitly contains model of imposing some action with negligible back effect (‘high input impedance’) • concept ‘output’ implicitly contains model of measurement with negligible effect on the system being observed (‘low output impedance’) • after a competent input-output relation is found, it is not open for modifications or physical interpretation
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‘Special’ states Position/displacement • has a dual nature:
– energy state (related to a conservation or symmetry principle like all other states) – configuration state
• does not transform as a tensor Matter • convects all matter-bound properties (not volume!) • conjugate intensity depends on other intensities • boundary criterion Volume • boundary criterion Entropy • can be ‘locally’ produced (only ‘locally’ conserved)
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Model representation • model representation:
– symbols used to represent • the concepts being used • (the structure of) their relations (interconnection)
• model manipulation (as opposed to modeling!) :
– transformation to different representations (including the ‘solution’) to • increase insight • draw conclusions, etc.
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Role of representations Sequential:
x1
a
T
A
x2
b c
T x3
d
A T x4
A
AT x5
e
AT t
Simultaneous: T
Process time versus processing time Summerschool Bertinoro, 17-23 July 2005 (20)
A
A
x1
T
x2
x
x3
x5 x4
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What are we trying to describe? • (dynamic) behavior • engineering systems
• herein constrained to (for sake of ‘simplicity’):
– deterministic models of systems that obey basic principles of macroscopic physics: • energy conservation (‘first law’) • positive entropy production (‘second law’)
• that describe the temporal trajectory of the common physical properties:
– mechanical (incl. hydraulic and pneumatic), electrical, magnetic, chemical, material, thermal, etc. >>> multidomain – configuration
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Physical systems modeling • all concepts used in the model are, or have a direct relation to, physically relevant concepts (use of a priori knowledge!) • physical relations are maintained as much as possible • herein constrained to (for sake of ‘simplicity’): – deterministic mathematical models of macroscopic systems that • obey basic principles of macroscopic physics: – energy conservation – positive entropy production
• describe the behavior in time of the common physical properties: – – – – –
mechanical (incl. hydraulic and pneumatic) electrical magnetic chemical material
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time & uncertainty • time:
– derived measure for ‘regularity’ based on counting ‘ticks’ of a ‘time-base’ based on repetitive behavior requiring state and change (in order to be able to count) – within the smallest unit used: necessarily
uncertainty
• cf. Heisenberg u.r. for displacement (= elastic state, kinetic state of change) and momentum (= elastic state of change, kinetic state)
– dialectic concepts!
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concepts of ‘state’ & ‘change’ • dialectic fundamental concepts • basis for any dynamic model (more than time!) • within a context of discovery a shift to space-time is understandable and useful, like the shift from positionmomentum in Hamiltonian mechanics to position-velocity in Lagrangian mechanics Summerschool Bertinoro, 17-23 July 2005 (24)
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Contents of the sequel • • • • •
Modeling pitfalls Port-based modeling Basic Concepts (ports, bonds) Dynamic conjugation (effort, flow) Multidomain modeling and the role of energy • (Computational) Causality
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Modeling pitfalls • ‘Every model is wrong’ • Model depends on problem context • Competent models • Analogies are not identities • Avoid implicit assumptions • Avoid model extrapolation • Confusion of components with elements
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Port-based modeling • multidomain approach:
– ‘mechatronics’ (and beyond)
• multiple view approach: other graphical representations: iconic diagrams, (linear graphs), block diagrams, bond graphs, etc. and equations • domain independent notation using ports: – bond graphs (& some other benefits…)
• port-based approach:
– underlying structure of 20-sim, ideal tool for demonstration
• what are ports and what are bond graphs? Summerschool Bertinoro, 17-23 July 2005 (27)
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Physical components versus ideal elements
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Physical component versus ideal elements
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Physical component versus ideal elements
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Component mounter
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Physical components versus conceptual elements physical component: piece of rubber hose dominant behavior: •when falling: ideal mass •when pulling load: ideal spring •for vibration isolation: ideal resistor •etc. Summerschool Bertinoro, 17-23 July 2005 (32)
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‘Parasitic’ elements • Next to dominant behavior:
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In engineering models: • Avoid implicit assumptions, e.g. about – problem context – reference – orientation – coordinates – metric – ‘negligible’ phenomena, etc.
• Avoid model extrapolation – danger of ignoring earlier assumptions
• Focus at competence, not ‘truth’ Summerschool Bertinoro, 17-23 July 2005 (34)
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Intuitive introduction of the ‘port’ concept
J
ideal current source
ideal motor
Dominant behavior
ideal inertia
ideal transmission
P
potentiometer
(not necessarily competent in each context) Summerschool Bertinoro, 17-23 July 2005 (35)
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Simple model
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Addition of relevant parasitic behavior Depending on problem context:
Polymorphic modeling Summerschool Bertinoro, 17-23 July 2005 (37)
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What are ports? ∆ u2
∆u1
i
P =∆u⋅ i
(power)
i
∆ u3
i
i ∆ u4
∆T1
ω
electrical ports
∆T2
∆T3 J
ω
ω
MECH
P =∆T⋅ω (power)
mechanical ports
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Comparison with familiar model views Iconic diagrams (‘ideal physical models’): R
L
i
Usource
v C
F
R
m
Fext K=1/C
electrical
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mechanical
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Iconic diagram symbols
Port -based, but Port-based, domain dependent
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Iconic diagram symbols C-type storage I -type storage (M)R (dissipation, irreversible transduction) Se (effort source) Sf (flow source) TF (transformer) Summerschool Bertinoro, 17-23 July 2005 (41)
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Iconic diagrams • Icons do not (always) represent physical structure:
20-sim icon: 1 i
TF (transformer) Summerschool Bertinoro, 17-23 July 2005 (42)
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Common block diagram models R
L
Usource
i
v C
F
R
m
Fext K=1/C
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Common block diagram models
Note: not all signals are physically meaningful variables Summerschool Bertinoro, 17-23 July 2005 (44)
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Ports in iconic diagrams
R
L
Usource
i
v C
F
R
m
Fext K=1/C
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Ports in iconic diagrams
R
m
L
F
i Usource
R Fext C
v K=1/C
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Ports in block diagrams
Note: each signal is a physically meaningful variable
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Structure as multiport
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Different forms
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Different forms
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Compact & domain independent Domain dependent
Coil
Mass
Source
Spring
Source
Capacitor
Resistor
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Damper
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Compact & domain independent Domain independent
I
I
Se
1
Structure explicitly
R
C
Se
1
C
R
represented as multiport: JUNCTION!
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Ports in bond graph view I
Se
1
C
R
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Bond graphs
Inventor (MIT, 1959): Prof. Henry M. Paynter (1923-2002) Summerschool Bertinoro, 17-23 July 2005 (54)
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Analog simulation with OP-AMPS
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Bonds Storage of kinetic energy
Storage of elastic energy
Compact notation:
C
I Energy exchange = power
•Terminology and notation induced by chemical bonds:
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Ports and power bonds • (power) bond connecting two elements via (power) ports (Harold Wheeler, 1949)
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mass-spring system
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Dynamic conjugation • Between signals of bilateral signal flow of relation: – rate of change: ‘flow’ (zero in equilibrium) • e.g. molar rate during diffusion
– equilibrium determining variable: ‘effort’ • e.g. concentration
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Power conjugation = special case of dynamic conjugation : – –
‘effort’ and ‘flow’ relate to power functional relation is commonly a product – sum in case of scattering variables
dE ∂E dqi ∂E dp j P= =∑ +∑ = ∑ ei fi + ∑ e j f j dt d d t t i ∂qi j ∂p j i j dp j ∂E 'effort' ei = ej = dt ∂qi fi =
dqi dt
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fj =
∂E ∂p j
'flow'
© Peter Breedveld
Conjugate variables & corresponding states
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Basic dynamic behaviors & mnemonic codes • Storage (reversible) – C, I (q-type and p-type storage) • Irreversible transformation (‘dissipation’): – (M)R(S) • Distribution – 0-junction, 1-junction • Supply and demand: – (M)Se, (M)Sf • Reversible transformation – (M)TF, (M)GY • >> 9 basic elements Summerschool Bertinoro, 17-23 July 2005 (62)
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Orientation conventions • 1-ports: into most elements (R, C, I), except sources (Se, Sf) • 2-port transducers: 1 in, 1 out • junction structure elements: arbitrary • multiport generalizations: same as simple form • MOST IMPORTANT: obeying grammar rules minimizes sign errors!
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Convention on effort and flow positions
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Basic one-port elements
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Basic two- and multiport elements
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Constitutive relations not necessarily linear! e.g. zener diode:
dominant behavior: irreversible transduction (resistor)
with nonlinear constitutive relation
R Summerschool Bertinoro, 17-23 July 2005 (67)
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Other form of non-linearity: Modulation e.g. modulation of a transducer:
iconic diagram (IPM)
capstan MR Summerschool Bertinoro, 17-23 July 2005 (68)
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Bilateral signal flow (computational causality) 2 possibilities:
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Notation Causal stroke:
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Back to the example I
Se
1
C
R
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Back to the example I
Se
1
C
R
20-sim demo Summerschool Bertinoro, 17-23 July 2005 (72)
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Causal port properties • Fixed causality
• Preferred (integral) causality
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Causal constraints
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Arbitrary causality
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Causality assignment Algorithmic (SCAP):
– 1) fixed causal ports with propagation via constraints causal ‘conflict’: ‘ill-posedness’
– 2) preferred causal ports with propagation via constraints
causal ‘conflict’: dependent state(s)
– 3) choice of arbitrary causal port with propagation via constraints means existence of algebraic loop(s)
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Example u
u
u T
i
i
u
u
i
J ω
i
i
I
1
I
GY
ω •Important modeling feedback
2nd order loop
Se
T
•Automatic in 20-sim •Visible in bond graph causality •‘Hidden’ in case of iconic diagrams
1
1st order loop
1st order loop
R Summerschool Bertinoro, 17-23 July 2005 (77)
R © Peter Breedveld
Positive orientation orientation NOT THE SAME AS direction!
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Causality does NOT affect orientation!
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Basic form of physical models interfaces and structure have to satisfy basic laws in degenerate way storage = 0, production = 0
Basic ‘laws’
boundary addition
structure constraint relations distribution, reversible transformation transport
first ‘law’ conservation ‘laws’ storage
second ‘law’ irreversible transformation
constitutive relations determine particular occurrence (instantiation) Summerschool Bertinoro, 17-23 July 2005 (80)
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Thermodynamic approach System boundary Energy balance
GJS GJS
for the total system
Irreversible thermodynamics of ‘forces’ and ‘fluxes’ (irreversible part)
Gibbs’ relation (Energy ‘balance’ of the reversible part)
Power continuous junction structure
• Difference is used to find the entropy production: Structure is conceptual ! • Power continuous structure not made explicit, except for instantaneous Carnot engines (1-junction for entropy flow) Summerschool Bertinoro, 17-23 July 2005 (81)
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Mechanical versus Thermodynamic framework Mechanics:
Thermodynamics:
Two types of storage One type of storage Oscillatory behavior Only relaxation behavior: C-R (damped): C-I(-R) Split domains (therm.) and couple by SGY (mech.): C-SGY-C
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Symplectic gyrator • Unit gyrator:
C
0
GY
0
C
0
C
r=1
• SGY: • Multiport:
C
C11 C12 C12 C22
0
C
SGY
0
SGY
0 -1 +1 0
20-sim demo
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Symplectic GYrator • Mechanical (x,p)
ekin =
∂E ∂p
C
ekin = v
0 f kin =
dp dt
SGY F =−
e pot =
e pot = F
dp dt
0 f pot =
f pot = v
∂E ∂x dx dt
C
dp
– Only in inertial frames: F = − (Newton's 2nd law) dt
• Electrical network (q,λ): emag =
C f mag =
∂E ∂λ dλ dt
∂E eelec = u eelec = ∂q
emag = i
0 −
dλ =u dt
SGY
– Only quasi-stationary (non-radiating): Summerschool Bertinoro, 17-23 July 2005 (84)
−
0 f elec = i
dq felec = dt
dλ =u dt
emag = i
C
© Peter Breedveld
Unit gyrator (SGY) as ‘dualizer’ Original:
Port equivalent:
Original:
Port equivalent:
0
SGY
I
0
C
Se
SGY
1
Sf
1
1
SGY
C
1
I
Sf
SGY
0
Se
0
1
SGY
R
1
R 1
SGY
1 1
SGY
TF
SGY
0
1
TF
SGY
0
1
0
0
0 SGY 0
SGY
1
GY
SGY
1
1
GY
1 0
1
SGY
TF
1
1
GY
1 1
1
SGY
GY
0
1
TF
SGY
0
SGY
0
1
1
0 SGY
1
0 1
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0
Basic model structure (generalized bond graph): Generalized Anti Junction -reciprocal Structure (power part (generalized Boundary conditions continuous)gyrator) (sources)
Irreversible Energy storage Junction Weighted transformation Structure (reciprocal) (entropy production) Summerschool Bertinoro, 17-23 July 2005 (86)
Weighted part (generalized transformer)
Simple Junction Structure (generalized Kirchhoff laws) © Peter Breedveld
Mechanical framework of variables
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Generalized thermodynamic framework of variables electric
magnetic
thermal
chemical
f flow
e effort
i current
u voltage
u voltage
i current
T temperat ure
fS entropy flow
µ
fN molar flow
chemical potential
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∫
q = f dt
generalized state
q = ∫ idt
charge
λ = ∫ udt
magnetic flux linkage
S=
∫ fS dt
entropy
N=
∫ fN dt
number of moles
© Peter Breedveld
Generalized thermodynamic framework of variables f flow
e effort
elastic/potential translation
v velocity
F force
kinetic translation
F force
v velocity
elastic/potential rotation
ω angular velocity
T torque
kinetic rotation
T torque
ω angular velocity
ϕ volume flow
p pressur e
p pressure
ϕ volume flow
elastic hydraulic
kinetic hydraulic
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∫
q = f dt
generalized state x = ∫ v dt
displacement p = ∫ Fdt
momentum θ = ∫ ωdt
angular displacement b = ∫ Tdt
angular momentum V = ∫ ϕdt
volume Γ=
∫ pdt
momentum of a flow tube
© Peter Breedveld
Energy as starting point • Energy
( )
– E = E q = E ( q1 ,..., qi ,...qn ) – Homogeneous function of set of state variables
• In (‘generalized’) mechanics : Hamiltonian
( )
– E = H q , p = H ( q1 ,..., qi ,...qk , p1 ,..., pi ,... pk )
• In thermodynamics of ‘simple’ systems: U = U ( q ) = U ( S , q ,..., q ,...q ) internal energy
( )
2
S = S (U , q2 ,..., qi ,...qn ) – E = U q = U (V , S , N ) – more species: E = U q = U (V , S , N ) =
i
n
( )
= U (V , S , N1 ,..., N i ,..., N m ) = = U (V , S , N1 ,..., N i ,..., N m −1 , N ) Summerschool Bertinoro, 17-23 July 2005 (90)
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Configuration influence on energetic structure • can become an additional energy state: geometric parameter in storage relation results in a force
F ( q, x ) =
∂E ( q, x ) ∂x
– examples: LVDT, electret microphone, relay, (ideal) gas etc. (MP C) – cycles allow transduction – changes of causality correspond to Legendre transforms!! (relation to dissipation)
• can modulate an energy relation
– examples: crank-slider mechanism, etc. (MTF)
• can switch a contact (behavior) Summerschool Bertinoro, 17-23 July 2005 (91)
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Modulated capacitor?
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System (boundary) definition • Open (thermodynamic) systems • dN = 0: ‘Lagrangian coordinates’: N not a state • dV = 0: ‘Eulerian coordinates’: V not a state • Almost always mixed boundaries: N and V remain states! • Network: globally Eulerian, locally Lagrangian (e.g. network of tubes) Summerschool Bertinoro, 17-23 July 2005 (93)
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‘Classical network case’ • Both N and V :
– constant – NOT CONSIDERED STATES, but parameters
• As a result:
– energy function not necessarily first order homogenous – no dependency between storage port efforts (no generalized Gibbs-Duhem equation) – no clear distinction between intensive and extensive description
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If not a network: • Globally Lagrangian and locally Eulerian possible! • Example: balloon
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Part of a tube network daxialV=0 (V not a convected property) daxial N≠0
dradialN=0 dradial V≠0 (flexible tube, local changes)
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System boundary definition • Mechanical system of masses, springs, dampers, etc. (network): – globally Eulerian coordinates
• because the network (topology) of spatial elements ‘cells’ is constant
– locally Lagrangian coordinates
• e.g. the constant mass with its motion restricted to the element cell: ‘one element passing a neighboring element destroys the topology’
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Summary •
System theoretic approach to physical system dynamics based on – classification of phenomena in terms of energy – fundamental principles of thermodynamics
•
as a result: – variables and relations describing physical systems may be classified – models may be organized as • multiport elements • interconnected in an interconnection structure corresponding to a generalized network
– multiport elements describe basic behaviors with respect to energy and entropy
•
Explicit conceptual separation between two roles of displacement variable: – energy state (stored potential or elastic energy) – configuration state (‘position in space’)
•
strongly reduces conceptual complexity even though numerical complexity may become temporarily higher
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