bond graphs: physical systems modeling 2 ... - unibo.it · physical systems modeling 2: physical...

1

7/11/2003 Summerschool Bertinoro7-11 July 2003

1

Control Laboratory

BOND GRAPHS:Physical systems modeling 2Physical systems modeling 2:

Applications

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

University of Twente, [email protected]

Peter BreedveldPeter Breedveld

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

AimAim

•• Impression of practical power ofImpression of practical power of–– port conceptport concept–– polymorphic modelingpolymorphic modeling–– hierarchical modelinghierarchical modeling–– simultaneoussimultaneous representationrepresentation–– domain independent representationdomain independent representation–– explicit structureexplicit structure

•• Immediate feedback on modeling Immediate feedback on modeling decisionsdecisions

2


ContentsContents•• Relation between iconic diagram view, bond Relation between iconic diagram view, bond

graph view and conventional linear system graph view and conventional linear system viewsviews

•• Causal conflictsCausal conflicts•• Feedback on (structural) modelling Feedback on (structural) modelling

decisionsdecisions–– case: case: aquarium pumpaquarium pump

•• Computational versus conceptual Computational versus conceptual complexitycomplexity–– case: case: bouncing ballbouncing ball


Multidomain modeling and the role of energy

Multidomain modeling and the role of energy

•• energy: ‘glue’ between domainsenergy: ‘glue’ between domains•• properties with respect to energy properties with respect to energy

domaindomain--independentindependent•• energy conservation (proper energy energy conservation (proper energy

book keeping) part of the grammar of book keeping) part of the grammar of the bond graph languagethe bond graph language

3


Word bond graph ofmulti-domain systemWord bond graph ofmulti-domain system

source motor pump

nozzle


Submodels allowhierarchical approach

Submodels allowhierarchical approach

drill

drill headelectric motor

gearbox

gearboxgears

bearings

axes

4


ComponentsComponents

motormotor loadloadflexibleflexibletransmissiontransmission

currentcurrentamplifieramplifier potentiometerpotentiometersetpointsetpoint


TT

ω ω

TT

ω ω

uu

ii

Word bond graphWord bond graph



5


TT

ω ω

TT

ω ω

uu

ii

Word bond graphWord bond graph




Stationary transduction Stationary transduction

uu

iiTT

ω ω MSf GY TF TF q

TT

ω ω

FF

v v

6


Stationary transduction Stationary transduction

MSf GY TF TF q


Dynamics of inertiasDynamics of inertias

MSf GY TF TF

I

1

I

1 q

7


Adding frictionAdding friction

MSf GY TF TF

I

R

1

I

1

R

q

Does not solve dependenceDoes not solve dependence


Adding belt complianceAdding belt compliance

C

MSf GY TF TF

I

1

I

10 q

Solves dependenceSolves dependence

3x preferred integral:

3 independent initial conditions

8


Second orderSecond order

C

MSf GY TF TF

I

1

I

10 q

Not preferred when differentialNot preferred when differential

Note: immediate feedback on

modeling decisions!


Third orderThird order

C

MSf GY TF TF

I

R

1

I

1

R

0 q

J

motormotor

ME

transmissiontransmission

PJ

loadload

9


Closing the loopClosing the loopC

MSf GY TF TF

I

R

1

I

1

R

0 qPDSP

MV s

C

MSf GY TF TF

I

R

1

I

1

R

0d/dt

PISP

MV


CollocatedCollocatedC

GY TF TF

I

R

1

I

1

R

0

C

TF TF

I

R

1

I

1

R

0

IR

C

TF TF

I

R

I

1

R

0

R

C

C

MSf GY TF TF

I

R

1

I

1

R

00

MSf 0

∫

PDSP

MV s

K

∫K

0 GYMSf

MSe 1 1

C

MSf GY TF TF

I

R

1

I

1

R

00

K

K ∫

Differentiating:

virtual damper

Proportional:

virtual spring

10


Equation viewEquation view

•• 11 bonds 11 bonds 22 variables, 22 ports, 22 22 variables, 22 ports, 22 equations (3 states equations (3 states 19 algebraic)19 algebraic)

C

MSf GY TF TF

I

R

1

I

1

R

0 q


Constitutive equationsConstitutive equationsω

ωω ω

ω

= =

= =

= =

= =

1 2

set point

m m

1 2

1 2

1 1 1 11 1

(Sf) (R)

(GY)d d1 1 (I) (I)d d

m mmR R

motor motor motor

J J

TF TF

i i T Ru K T K i

T Tt J t J

v r T r Fω

ω ω ω ω ω ω

= =

= = − −

= = =

= = =

1

2 2 2 22 2

12 1

1 1 11

12 1 2 1 2

(TF) (TF)

d (C) ;d

; ; (1)

- ; ;

m

m

TF TF

CmotorJ RTF

motor RTF

C C

v r T r FF Kv T T T Tt

v v v F F F Fω ω ω ω

ω ϕ ω

= − = =

= = ∫2 load

load load

2 22 2

2load load

(0) ; ; (1)

(R) ( d ; sensor)loadJ R RTF TF

R R

T T T

T R t

11


Eliminate algebraic equations

Eliminate algebraic equations

( )

( )

( )

ωω

ωω

ω ω

ϕω

= − −

= −

= −

=

1m sp 1 1

1

22 2load

2

1 1 2 2

load2

d 1dd 1dddd

d

mC

C

C

K i r F Rt J

r F Rt J

F K r rt

t3rd order open loop behavior3rd order open loop behavior

4th order (open integrator > closed loop)4th order (open integrator > closed loop)


Block Diagram View

12


Block Diagram ExpansionBlock Diagram Expansion

Expand bonds:Expand bonds:

Expand elements:Expand elements:


Signs & orientations in junction expansions

Signs & orientations in junction expansions

13


Add signs:Add signs:




Contract oneContract one--ports:ports:

Choose forward path and combine gains:Choose forward path and combine gains:

14


Case 1: Aquarium air pumpCase 1: Aquarium air pump

LinePowerSupply

LinearMotor

Lever

Bellows

CheckValve1CheckValve2

AmbientPressure

0

0


Fixed causality of check valveFixed causality of check valve21

2 dp C vρ∆ =2

12 dp C

Aϕρ ∆ =

22dCpA

ρϕ ϕ∆ =

Title

-0.8 -0.4-0.2 0 0.2 0.4 0.6 0.8p.e

p.f

-0.5

0

0.5

1

Title

-0.8 -0.4-0.2 0 0.2 0.4 0.6 0.8p.e

p.f

-1

-0.5

0

0.5

1

( )22

sgnd

Ap p

Cϕ

ρ= ∆ ∆

2if 0 then else 0 end;

dp A p

Cϕ

ρ= ∆ > ∆

port1 port20 1 0

R

Title

-0.8 -0.4-0.2 0 0.2 0.4 0.6 0.8p.f

p.e

-1

-0.5

0

0.5

1

15


Fixed causality of check valveFixed causality of check valve21

2 dp C vρ∆ =2

12 dp C

Aϕρ ∆ =

22dCpA

ρϕ ϕ∆ =

Title

-0.8 -0.4-0.2 0 0.2 0.4 0.6 0.8p.e

p.f

-0.5

0

0.5

1

Title

-0.8 -0.4-0.2 0 0.2 0.4 0.6 0.8p.e

p.f

-1

-0.5

0

0.5

1

( )22

sgnd

Ap p

Cϕ

ρ= ∆ ∆

2if 0 then else 0 end;

dp A p

Cϕ

ρ= ∆ > ∆

port1 port20 1 0

R

Title

-0.8 -0.4-0.2 0 0.2 0.4 0.6 0.8p.f

p.e

-1

-0.5

0

0.5

1


Iconic diagram modelingIconic diagram modeling

16


Causality information:Causality information:


Problem with processingProblem with processing

17


Problem with processingProblem with processing


Simulation possibleno dynamics

Simulation possibleno dynamics

18


Adding dynamicsAdding dynamics


Adding dynamicsAdding dynamics

19


ResonanceResonance


ResonanceResonance

20


ResonanceResonance


ResonanceResonance

21


ResonanceResonance


Optimal outputOptimal output

22


Optimal ouputOptimal ouput



23




Submodels still emptySubmodels still empty

24


Bond causality:immediate modeling feedback


LinePowerSupply

LinearMotor

Lever

Bellows

CheckValve1CheckValve2

AmbientPressure

0

0




LinePowerSupply

LinearMotor

Lever

Bellows

CheckValve1

AmbientPressure

0

0

CheckValve2

C

need for air compressibility

25




resonance?: magnet massLinePowerSupply

LinearMotor

Lever

Bellows

CheckValve1

AmbientPressure

0

0

CheckValve2

C


LinePowerSupply

LinearMotor

Lever

Bellows

CheckValve1

AmbientPressure

0

0

CheckValve2

C

I

1



resonance?: magnet mass

26




No resonance!: add internal resistance

LinePowerSupply

LinearMotor

Lever

Bellows

CheckValve1

AmbientPressure

0

0

CheckValve2

C

I

1


LinePowerSupply

LinearMotor

Lever

Bellows

CheckValve1

AmbientPressure

0

0

CheckValve2

C

I

1

R

1



No resonance!: add internal resistance

27


LinePowerSupply

LinearMotor

Lever

Bellows

CheckValve1

AmbientPressure

0

0

CheckValve2

C

I

1

R

1

C

1



resonance: add bellows compliance


In regular bond graph:In regular bond graph:

GY TFMSe TF

0

R

1 R

28


GY TFMSe TF

0

R

1 R

C



GY TFMSe TF

0

R

1 R

C

I

1


29


GY TFMSe TF

0

R

1 R

C

I

1

R

1



GY TFMSe TF

0

R

1 R

C

I

1

R

1

C

1


30


GY TFMSe TF

0

R

1 R

I

1

R

1

C

1



GY TFMSe TF

0

R

1 R

C

I

1

R

1

C

1


31


SimulationSimulation

Title

0 0.1 0.2 0.3 0.4 0.5time {s}

p.ph

i {m

3/s}

p.f {

m3/

s}

outp

ut

-0.001

-0.0008

-0.0006

-0.0004

-0.0002

0

0.0002

-0.0005

-0.0003

-0.0001

0.0001

0.0003

0.0005

0.0007

5e-006

2.5e-005

4.5e-005

6.5e-005

8.5e-005

0.000105

0.000125


Forward pathForward path

GY TFMSe TF

0

R

1 R

C

I

1

R

1

C

1

1s

1s

2forward gain As

=

32


LoopsLoops

GY TFMSe TF

0

R

1 R

C

I

1

R

1

C

1

1mKL

Rms= − 2

2 2bellowsK nLms

= −

2 2

3 2airK n ALms

= −

4air

in

KLR s

= −5

air

out

KLR s

= −

Non-touching pairs: L1.L4, L2.L4, L1.L5, L2.L5

Denominator shape:

Shape of H:

2 31 B C Ds s s

+ + +

2

3 22 31

AAssH B C D s Bs Cs D

s s s

= =+ + ++ + +


Linear analysisLinear analysis

33


Poles & zerosPoles & zeros


OptimizationOptimization

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Aquarium pump: conclusions

Aquarium pump: conclusions

•• iconic diagram solutioniconic diagram solution•• systematical integration of expertise systematical integration of expertise

from various domainsfrom various domains•• feedback on modeling decisions via feedback on modeling decisions via

bond causalitybond causality•• causal assignment suggest nature of causal assignment suggest nature of

changes towards competent modelchanges towards competent model•• graph predicts structure of linear graph predicts structure of linear

analysis resultanalysis result


Case 2: Falling and bouncing object

Case 2: Falling and bouncing object

•• generally considered as conceptually generally considered as conceptually complex:complex:–– switching models switching models reinitializationreinitialization, timing, timing–– energy bookkeepingenergy bookkeeping–– prone to sign errorsprone to sign errors

•• portport--based approach with explicit based approach with explicit structure, allows conceptually simple structure, allows conceptually simple solutionsolution

35


Falling and bouncing objectFalling and bouncing object

1 ISe


1

1

ISe

Sf


Explicit reference

36


1 ISe

C

R



1

1

1

ISe

C

RSf

0


Explicit reference

37


1

1

ISe

Sf



X0e

1

1

1

ISe

C

RSf


38


Energy states versus configuration states

Energy states versus configuration states

•• Special role of the mechanical domainSpecial role of the mechanical domain–– ConfigurationConfiguration statestate: integrators used to : integrators used to

determine contactdetermine contact–– Energy stateEnergy state: spring displacement: spring displacement


energy state at bond graph level

Switched junctionSwitched junction

X0e

1

1

1

ISe

C

RSf

configuration states at signal level

39



X0e

1

1

1

ISe

C

RSf



X0e

1

1

1

ISe

C

RSf

Title

17.898 17.9 17.902 17.904 17.906 17.908 17.91 17.912 17.914 17.916time {s}

boun

ce

outp

ut

-0.3975

-0.39745

-0.3974

-0.39735

-0.3973

-0.39725

-0.000598046

-0.000347718

-9.73898e-005

0.000152938

0.000403266

0.000653594without event

40



X0e

1

1

1

ISe

C

RSf

requires fixed causality



X0e

1

1

1

ISe

C

RSf

41



X0e

1

1

1

ISe

C

RSf



X0e

1

1

1

ISe

C

RSf

42


Title

0 5 10 15 20time {s}

boun

ce

outp

ut

0

0.5

1

1.5

-2

3

8

13


X0e

1

1

1

ISe

C

RSf


Title

3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8time {s}

boun

ce

outp

ut

0

0.2

0.4

0.6

0.8

1

-2

0

2

4

6

8


X0e

1

1

1

ISe

C

RSf

43


Title

3.75 3.8 3.85 3.9 3.95time {s}

boun

ce

outp

ut

0

0.2

0.4

0.6

0.8

1

-2

0

2

4

6

8


X0e

1

1

1

ISe

C

RSf


Title

3.82 3.825 3.83 3.835 3.84 3.845 3.85time {s}

boun

ce

outp

ut

0

0.2

0.4

0.6

0.8

1

-2

0

2

4

6

8


X0e

1

1

1

ISe

C

RSf

44


Bouncing object: conclusions

Bouncing object: conclusions

•• Conceptual complexity 'low':Conceptual complexity 'low':–– Close to physicsClose to physics–– No sign or bookkeeping problemsNo sign or bookkeeping problems–– Simple implementationSimple implementation–– No model switchingNo model switching

•• Numerical complexity 'higher’ (compared to Numerical complexity 'higher’ (compared to switching momenta: no higher modes)switching momenta: no higher modes)–– No problem with variable step method or No problem with variable step method or

modern hardwaremodern hardware


General ConclusionsGeneral Conclusions

1.1. distinction between distinction between systemsystem to be modeled and to be modeled and “properties” of “properties” of modelmodel (hybrid, cont., (hybrid, cont., discrdiscr. lumped, etc.). lumped, etc.)

2.2. portsports versus inputversus input--output relations (output relations (simultaneoussimultaneous vsvssequential)sequential)

3.3. generalized mechanical (‘Hamiltonian’) and thermodynamic generalized mechanical (‘Hamiltonian’) and thermodynamic frameworks can be united (SGY)frameworks can be united (SGY)

4.4. importance of domain independence and importance of domain independence and explicitexplicit structure structure representation (junction structure)representation (junction structure)

5.5. explicit SGY demonstrates that common electrical explicit SGY demonstrates that common electrical network models and mechanical system models are special network models and mechanical system models are special cases (quasicases (quasi--stationary and inertial frame respectively) stationary and inertial frame respectively)

6.6. network and open thermodynamic systems can be modelednetwork and open thermodynamic systems can be modeled

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