Modeling & Simulation 2019Lecture 8. Bond graphs

Claudio AltafiniAutomatic Control, ISYLinköping University, Sweden

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Summary of lecture 7

• General modeling principles• Physical modeling: dimension, dimensionless quantities, scaling• Models from physical laws across different domains• Analogies among physical domains

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Lecture 8. Bond graphs

Summary of today

• Analogies among physical domains• Bond graphs• Causality

In the book: Chapter 5 & 6

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Basic physics laws: a survey

Electrical circuits

Mechanics – translational

Mechanics – rotational

Hydraulics

Thermal systems

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Electrical circuits

Basic quantities:

• Current i(t) (ampere)• Voltage u(t) (volt)

• Power P (t) = u(t) · i(t)

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Electrical circuits

Basic laws relating i(t) and u(t)

• inductance

Ld

dti(t) = u(t) ⇐⇒ i(t) =

1

L

∫ t

0

u(s)ds

• capacitance

Cd

dtu(t) = i(t) ⇐⇒ u(t) =

1

C

∫ t

0

i(s)ds

• resistance (linear case)u(t) = Ri(t)

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Electrical circuits

Energy storage laws for i(t) and u(t)

• electromagnetic energy

T (t) =1

2Li2(t)

• electric field energy

T (t) =1

2Cu2(t)

• energy loss in a resistance

T (t) =

∫ t

0

P (s)ds =

∫ t

0

u(s)i(s)ds = R

∫ t

0

i2(s)ds =1

R

∫ t

0

u2(s)ds

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Electrical circuits

Interconnection laws for i(t) and u(t)

• Kirchhoff law for voltagesOn a loop:

∑k

σkuk(t) = 0, σk =

{+1, σk aligned with loop direction−1, σk against loop direction

• Kirchhoff law for currentsOn a node:

∑k

σkik(t) = 0, σk =

{+1, σk inward−1, σk outward

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Electrical circuits

Transformations laws for u(t) and i(t)

• transformer

u1 = ru2

i1 =1

ri2

u1i1 = u2i2 ⇒ no power loss

• gyrator

u1 = ri2

i1 =1

ru2

u1i1 = u2i2 ⇒ no power loss

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Electrical circuits

Example

State space model:

d

dti =

1

L(us −Ri− uC)

d

dtuC =

1

Ci

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Mechanical – translational

Basic quantities:

• Velocity v(t) (meters per second)• Force F (t) (newton)

• Power P (t) = F (t) · v(t)

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Mechanical – translational

Basic laws relating F (t) and v(t)

• Newton second law

md

dtv(t) = F (t) ⇐⇒ v(t) =

1

m

∫ t

0

F (s)ds

• Hook’s law (elastic bodies, e.g. spring)

1

k

d

dtF (t) = v(t) ⇐⇒ F (t) = k

∫ t

0

v(s)ds

• viscosity (e.g. dry friction)

F (t) = bv(t)

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Mechanical – translational

Energy storage laws for F (t) and v(t)

• kinetic energy

T (t) =1

2mv2(t)

• potential energy

T (t) =1

2kF 2(t)

• energy loss due to friction

T (t) =

∫ t

0

P (s)ds =

∫ t

0

F (s)v(s)ds = b

∫ t

0

v2(s)ds =1

b

∫ t

0

F 2(s)ds

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Mechanical – translationalInterconnection laws for F (t) and v(t)• sum of forces; same velocity (series connection)

F = F1 + F2

v1 = v2

• sum of velocities; same force (parallel connection)

F1 = F2

v = v1 + v2

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Mechanical – translational

Transformations laws for F (t) and v(t)

• levers

F1 = −`2`1F2

v1 = −`1`2v2

• pulleys

F1 =1

2F2

v1 = −1

2v2

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Mechanical – rotational

Basic quantities:

• Angular Velocity ω(t) (radians per second)• Torque M(t) (newton · meter)

• Power P (t) =M(t) · ω(t)

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Mechanical – rotational

Basic laws relating M(t) and ω(t)

• Newton second law

Jd

dtω(t) =M(t) ⇐⇒ ω(t) =

1

J

∫ t

0

M(s)ds

• torsion of a body

1

k

d

dtM(t) = ω(t) ⇐⇒ M(t) = k

∫ t

0

ω(s)ds

• rotational friction (typically nonlinear)

M(t) = h(ω(t))

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Mechanical – rotational

Energy storage laws for M(t) and ω(t)

• rotational energy

T (t) =1

2Jω2(t)

• torsional energy

T (t) =1

2kM2(t)

• energy loss due to rotational friction

T (t) =

∫ t

0

P (s)ds =

∫ t

0

M(s)ω(s)ds

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Mechanical – rotationalInterconnection laws for M(t) and ω(t)

• sum of torques; same angular velocity (series connection)

M =M1 +M2

ω1 = ω2

• sum of velocities; same force (parallel connection)

M1 =M2

ω = ω1 + ω2

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Mechanical – rotational

Transformations laws for M(t) and ω(t)

• gears

M1 = rM2

ω1 = −1

rω2

• “gyrator”

My = rωz

ωy = −1

rMz

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Flow system

Basic quantities:

• Flow Q(t) (cubic meters per second)• Pressure p(t) (newton per square meter)

• Power P (t) = p(t) ·Q(t)

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Flow systemBasic laws relating p(t) and Q(t)

• Newton second law

ρ`

A︸︷︷︸Lf=inertance

d

dtQ(t) = p(t) ⇐⇒ Q(t) =

1

Lf

∫ t

0

p(s)ds

• pressure of a liquid column

A

ρg︸︷︷︸Cf=capacitance

d

dtp(t) = Q(t) ⇐⇒ p(t) =

1

Cf

∫ t

0

Q(s)ds

• flow resistance (laminar flow)

p(t) = Rf︸︷︷︸flow resistance

Q(t)

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Flow system

Energy storage laws for p(t) and Q(t)

• kinetic energy (fluid in a tube)

T (t) =1

2LfQ

2(t)

• potential energy (fluid in a tank)

T (t) =1

2Cfp

2(t)

• energy loss due to flow resistance

T (t) =

∫ t

0

P (s)ds =

∫ t

0

p(s)Q(s)ds

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Flow systemInterconnection laws for p(t) and Q(t)

• series connection

p = p1 + p2

Q1 = Q2

• parallel connection

p1 = p2

Q = Q1 +Q2

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Flow system

Transformations laws for p(t) and Q(t)

• flow transformer

p1 = rp2

Q1 =1

rQ2

where r = A2/A1

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Analogies among physical domains

Electrical Mechanical Mechanical Hydraulic Thermaltranslational rotational

flow current velocity angular volume heat flowvelocity flow

effort voltage force torque pressure temperaturepower power power power power power ·

temperatureinductive inductor inertia moment of inertia of −−element inertia fluid

capacitive capacitor spring torsional tank heatelement spring storageresistive resistor friction friction friction thermalelement resistance

transformer transformer lever gears transducer −−gyrator gyrator −− gyro −− −−

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Conversions among domains

• Mechanical (translational) – Hydraulics

transformer:

p =1

AF

Q = A v

• Mechanical (rotational) – Electrical

gyrator:

M = k i

ω =1

ku

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Analogies among physical domains

• two "power" variables1. flow

f ={i, v, ω, Q, q

}2. effort

e ={u, F, M, p, T

}• their product: power

e · f ={u · i, F · v, M · ω, p ·Q, T · q

}

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Analogies, cont’d

• three energy "exchange" relationships1. inductance =⇒ effort storage

df

dt=

1

αe =⇒ f(t) =

1

α

∫ t

0

e(s)ds

2. capacitance =⇒ flow storage

de

dt=

1

βf =⇒ e(t) =

1

β

∫ t

0

f(s)ds

3. resistance =⇒ loss of energy

e = γf (possibly nonlinear: e = h(f))

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Bond graphs

Bond graph theory: an exchange of energy is a bond

e−−−⇀f

• line = connection between parts of the system• above the line (harpoon side): effort• below the line: flow• direction of half arrow = direction of energy flow(i.e., direction in which power p = e · f is positive)

u−−⇀i

F−−−⇀v

M−−−⇀ω

p−−−⇀Q

T−−⇀q

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Bond graphs for energy exchanges

• bond graph for effort storage: I-type element

f(t) =1

α

∫ t

0

e(s)ds ⇐⇒ e−−−⇀f

I : α

• bond graph for flow storage: C-type element

e(t) =1

β

∫ t

0

f(s)ds ⇐⇒ e−−−⇀f

C : β

• bond graph for resistive elements: R-type element

e(t) = γ f(t) ⇐⇒ e−−−⇀f

R : γ

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Bond graphs for sources

• bond graph for effort source

See−−−−⇀f

[system ]

• bond graph for flow source

Sfe−−−−⇀f

[system ]

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Bond graphs for junctions

bond graph for series connection: s - junction

..........................e1

f1s..........................

e2 f2

..........................

en fn

f1 = f2 = . . . = fn =⇒ common flow

e1 + e2 + . . .+ en = 0

outgoing harpoon at ej : change sign to ej in the summation

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Bond graphs for junctions

bond graph for parallel connection: p - junction

............................e1

f1p............................

e2 f2

............................

en fn

e1 = e2 = . . . = en =⇒ common effort

f1 + f2 + . . .+ fn = 0

outgoing harpoon at fj : change sign to fj in the summation

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Bond graphs for transformer and gyrators

• bond graph for transformer: TF - junction

e1−−−−⇀f1

TF e2−−−−⇀f2

e2 = ne1, f2 =1

nf1

• bond graph for gyrator: GY - junction

e1−−−−⇀f1

GY e2−−−−⇀f2

e2 = rf1, f2 =1

re1

Both conserve power

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Bond graphs and ports

Edges (bonds) of the bond graph are connecting points (or ports) of thesystem. The number of ports can be• one port: Se, Sf , C, I, R• two ports: TF , GY• multiport: s-junction, p-junction

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Example: electrical circuit / hydraulic system

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Example: electrical circuit

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Example: electrical circuit

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Example: an hydraulic system

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Example: an hydraulic system

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State space description

Is it possible to obtain a state space description out of a bond graph?

Information flow in a state space model

x = f(x, u)

• for given x och u it is possible to compute x?

• in a bond graph: causality marking

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Information flow

Information flow for bond graphs of C , I, and Se, Sf types:

︸ ︷︷ ︸integral causality

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CausalityCausal strokes• “A sets e and B sets f ” (i.e., A is of C and Se types)

• “B sets e and A sets f ” (i.e., A is of I and Sf types)

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Causality, cont’d

• Bond graphs with fixed causality strokes:1. sources

2. C and I type

3. transformers

4. gyrators

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Causality, cont’d• Causality at junctions

1. s-junctions

2. p-junctions

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Causality, cont’d

• Elements with free causality1. R-type

– e can be computed from a given f– f can be computed from a given e– certain nonlinearities may constitute an exception

e = φ(f) f = φ−1(e)

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Bond graphs

Meaning of bond graphs:• describes various physical domains in the same way• expresses simple addition rules for effort and flow variables• illustrates the structure of a system from its components• translates physical laws into graphical interactions• formalism prone to object-oriented programming

Claudio Altafini

www.liu.se