Bond Graph Theory

Interconnection systems Elements Modulated elements Causality

Lecture 1: Bond graph Theory

Arnau D`oria-Cerezo

Institut d’Organitzaci´oi Control de Sistemes Industrials Universitat Polit`ecnica de Catalunya

Posgrado en Automatizaci´on Modelado y Simulaci´on de Sistemas de Control Junio 2006. Universidad de Cuenca, Ecuador

Modelling and Simulation of Control Systems Lecture 1: Bond graph Theory Interconnection systems Elements Modulated elements Causality Outline

1 Interconnection systems

2 Elements 1-port elements Energy 2-port elements 3-port elements

3 Modulated elements

4 Causality Fixed causality elements Constrained causality elements Preferred causality elements Indiﬀerent causality elements

Modelling and Simulation of Control Systems Lecture 1: Bond graph Theory Interconnection systems Interconnection systems Elements Engineering multiports Modulated elements Ports and bonds Causality Interconnection systems

Port Interconnected System

fA fB System e e System System System A A B B A B

A port is an interface of an element with other. A port contains two variables, e (eﬀort) and f (ﬂow). Usually, a port exchanges power.

+ + + + iC iC iL iL vL vC vL vC L C L C - - - -

Interconnection rules

vL = vC , iL = −iC .

eﬀort variables (vC , vL), ﬂow variables (iC , iL).

Some engineering subsystems:

P Q i

v ω τ ω τ Hydraulic pump Electric motor

Shaft

τ τ ω2 ω1

Interconnection: Electrical motor + Shaft + Hydraulic pump

e A B f

A and B are related by a power bond. The half arrow represents the positive power flow convention. Pair of signals: flow f and effort e, called power variables. Flow: current, speed... Effort: voltage, torque...

The Bond Graph Theory can be, roughly speaking, summarized in:

Energy Sources Power Se: effort Bonds Sf: flow Power preserving elements Storage Transformation Elements Junctions Elements I: kinetic 0: common effort relates e with e (or f-f) common flow TF: C: potential 1: n-port elements GY: relates f with e Dissipation 2-port elements Elements R 1-port elements

Modelling and Simulation of Control Systems Lecture 1: Bond graph Theory Interconnection systems 1-port elements Elements Energy Modulated elements 2-port elements Causality 3-port elements Basic elements of the BG theory

Each element is represented by a multiport. Port are connected by a bond.

1-port elements: energy storing, dissipation or supplying elements. C-elements and I-elements. R-elements. Se-elements and Sf-elements. 2-ports elements: Transformers and gyrators. 3-ports elements: 0 and 1 junctions.

2-port and 3-port elements describes a network interconnection. 2-port and 3-port elements are power preserving.

Constitutive relation e q˙ = f C: ΦC f −1 where q is called displacement, and e =ΦC (q).

Linear case Integral relation from f to e For an ideal capacitor:

t −1 q q −1 ΦC (q)= v = e(t)=ΦC q(t0)+ f(τ)dτ C C Zt0 1 t q(t) v(t)= i(τ)dτ | {z } C Zt0

Constitutive relation e p˙ = e I: ΦI f −1 where p is called momentum, and f =ΦI (p).

Linear case e f Integral relation from to For an ideal inductor:

t −1 p p −1 ΦI (p)= i = f(t)=ΦI p(t0)+ e(τ)dτ L L Zt0 1 t p(t) i(t)= v(τ)dτ | {z } L Zt0

Constitutive relation e e =Φ (f) R: ΦR R f −1 or f =ΦR (e), algebraic relation.

Represents a power dissipation.

Linear case For an ideal resistor: ΦR(f)= Rf v(t)= Ri(t)

e Constitutive relation E : Se f e = E(t).

The eﬀort e, does not depend on f!!!. Output power ﬂow.

Examples Electrical voltage source. Torque source.

e Constitutive relation F : Sf f f = F (t).

Flux f, does not depend on e!!!. Output power ﬂow.

Examples Electrical current source. Speed source.

The R, C and I-type elements can be placed along the tetrahedron of state:

d dt ΦC

Φr p q

ΦI d dt f

The energy variation for an element is describe by:

t H(t) − H(t0)= e(τ)f(τ)dτ. Zt0

C-element −1 The constitutive relation is: e =ΦC (q), q˙ = f. −1 q For a linear case, ΦC (q)= C , 1 H(q) − H(q )= (q2 − q2). 0 2C 0

The energy variation for an element is describe by:

t H(t) − H(t0)= e(τ)f(τ)dτ. Zt0

I-element −1 The constitutive relation is: f =ΦI (p), q˙ = e. −1 p For a linear case, ΦI (p)= I , 1 H(p) − H(p )= (p2 − p2). 0 2I 0

The energy variation for an element is describe by:

t H(t) − H(t0)= e(τ)f(τ)dτ. Zt0

R-element −1 The constitutive relation is: f =ΦR (e) or e =ΦR(f). Then, t − −1 H(t) H(t0)= e(τ)ΦR (e(τ))dτ. Zt0

e1 e2 2-port f1 f2

2-port elements are power preserving. With the sign convention of the figure, then power conservation means e1f1 = e2f2. Any power flowing into one side of the 2-port is simultaneously flowing out of the other side.

2-port elements Transformers. Gyrators.

e1 e2 TF f1 : f2 τ

One way to satisfy the power conservation is by means of

e1 = τe2 τf1 = f2.

This deﬁne a 2-port element known as transformer, and denoted by TF. The parameter τ is called the transformer modulus. Examples Ideal rigid level, gear pair, electrical transformer, hydraulic ram...

e1 e2 GY f1 : f2 γ

A second way to satisfy the power conservation equation is

e1 = γf2 γf1 = e2. This deﬁne a 2-port element known as gyrator, and denoted by GY. The parameter γ is called the gyrator modulus.

Gyrators are somehow more mysterious than transformers. In the early days of bond graph theory, before the importance of the gyrator was recognized, systems which were in fact gyrators were approximated using transformers. Gyrators are needed, at fundamental level, to model the Hall eﬀect and some electromagnetic phenomena at microwave frequencies, among others.

e3 f3

e1 e2 3-port f1 f2

3-port elements are power preserving. Following the ﬁgure power convention,

e1f1 + e2f2 + e3f3 = 0.

They are often called junctions. They are, in fact, n-port elements.

0-junction or, common eﬀort junction. e3 f3

e1 e2 e1 = e2 = e3. 0 f1 f2

With the power sign convention shown, power conservation is

e1f1 + e2f2 + e3f3 = 0.

Then using the common eﬀort relation, we get

f1 + f2 + f3 = 0

which justiﬁes the ﬂow junction name.

1-junction or, common ﬂow junction. e3 f3

e1 e2 f1 = f2 = f3. 1 f1 f2

With the power sign convention shown and using the common ﬂow relation, we get

e1 + e2 + e3 = 0

which justiﬁes the eﬀort junction name.

Modelling and Simulation of Control Systems Lecture 1: Bond graph Theory Interconnection systems Elements Modulated elements Causality Modulated elements

Some elements can be modulated. Some parameters (or constitutive relations) depends on an external signal, which does not has energy contribution. In Bond graph Theory, is represented by an activated bond.

Example: a modulated transformer

τ e1 = τ(t)e2

e1 e2 τ(t)f1 = f2 MTF f1 f2 In this case the transformer modulus is given by a variable τ(t).

Modelling and Simulation of Control Systems Lecture 1: Bond graph Theory Interconnection systems Elements Modulated elements Causality Other examples

A variable resistor A DC motor e ki (t) MR R(t) f f

The resistance value is high e1 e2 MGY dependent on the temperature, f1 f2 or in a piezoelectric systems on the position... Equations: They are also MSe, MSf, MI, v = kif (t)ω τ = kif (t)ia. MC...

Modelling and Simulation of Control Systems Lecture 1: Bond graph Theory Interconnection systems Fixed causality elements Elements Constrained causality elements Modulated elements Preferred causality elements Causality Indiﬀerent causality elements Causality

A bond relates two elements, one of them states the eﬀort and the other one the ﬂow. Example: electric drill

Electric drill ﬁx the eﬀort (torque), and it turns depending on the material.

The causal stroke determines which element ﬁx the eﬀort.

A B A B

A ﬁx e and B returns f. B ﬁx e and A returns f.

Causality does not depend on the power direction. Causality is required to obtain the dynamical equations. Causality assignation depends on the interconnected elements.

This occurs at sources: Eﬀort source, Se e Se f

Flow source, Sf e Sf f

This occurs at: Transformer, TF Gyrator, GY e1 e2 e1 e2 TF GY

f1 f2 f1 f2

or or

e1 e2 e1 e2 TF GY

f1 f2 f1 f2

This occurs at: 0-junctions 1-junctions

e1 = e2 = . . . = en f1 = f2 = . . . = fn

3 3 e f e3 f3 en en fn f 1 2 n e e e1 e2 0 1 f1 f2 f1 f2

The storage elements have a preferred causality: integral causality.

C I

Integration is a process which can be realised physically. Numerical diﬀerentiation is not physically realisable, since information at future points is needed.

Example When a voltage v is imposed on an electrical capacitor, the current is: dv i = C Diﬀerentiation!!! dt

When the current is imposed v = v0 + idt.

Modelling and Simulation of Control Systems LectureR 1: Bond graph Theory Interconnection systems Fixed causality elements Elements Constrained causality elements Modulated elements Preferred causality elements Causality Indiﬀerent causality elements Blocs diagram equivalence

C-element I-element

C I

C-element I-element

f q = fdt e p = edt R R

q p

−1 −1 e e = φC (q) f f = φI (p)

When there are no causal constraints. For a linear R-element, it does not matter which of the port variables is the output.

R or R

1 e Rf f = e = R

Modelling and Simulation of Control Systems Lecture 1: Bond graph Theory