Bond Graph Modeling

Bond-Graph Modeling

In the 19th century, Lord Kelvin and James ClerkMaxwell both observed that a wide range of phenome-na give rise to similar forms of equations, findinganalogies between heat flow and electric force andbetween lines of force and fluid streamlines. In the

1940s and 1950s, H.M. (Hank) Paynter of MIT worked oninterdisciplinary engineering projects including hydroelec-tric plants, analog and digital computing, nonlineardynamics, and control [1]. Through this experience, he

observed that similar forms of equations are generated bydynamic systems in a wide variety of domain (for exampleelectrical, fluid, and mechanical); in other words, such sys-tems are analogous. Paynter incorporated the notion of anenergy port into his methodology, and thus bond graphswere invented. Since that time, his group and many othershave developed the basic concepts of bond-graph model-ing into a mature methodology.

The bond-graph method is a graphical approach tomodeling in which component energy ports are connect-ed by bonds that specify the transfer of energy betweensystem components. Power, the rate of energy transportbetween components, is the universal currency of physi-cal systems.

The graphical nature of bond graphs separates thesystem structure from the equations, making bond

graphs ideal for visualizing the essential characteris-tics of a system. Indeed, by creating bond graphs,

designing and analyzing the structure of a sys-tem—perhaps the most important part of themodeling task—can often be undertaken usingonly a pencil and paper. Modelers can thusfocus on the relationships among componentsand subsystems rather than the implementa-tion details of their particular modeling soft-ware. Even before a computer is used, bondgraphs can provide an engineer with infor-mation about constrained states, algebraicloops, and the benefits and consequences ofpotential approximations and simplifications.

Many computer-based modeling tools areavailable for generating and processing bondgraphs, see “Further Reading.” These toolsgenerally have capabilities that extend far

beyond those of traditional block-diagramsoftware, including generation of symbolic rep-

resentations, model inversion, and parametricidentification as well as the ability to producesimulations, frequency responses, and other

design aids. Bond-graph models can therefore beused by engineers not only to perform straightfor-

ward numerical analysis but also, more importantly,to gain qualitative insight.

PETER J. GAWTHROP and GERAINT P. BEVAN

A TUTORIAL INTRODUCTION FOR CONTROL ENGINEERS

24 IEEE CONTROL SYSTEMS MAGAZINE » APRIL 2007 1066-033X/07/$25.00©2007IEEE

© DIGITALSTOCK

Authorized licensed use limited to: INSTITUTO TECNOLOGICO DE AERONAUTICA. Downloaded on January 19, 2010 at 11:39 from IEEE Xplore. Restrictions apply.

This article has two purposes. First, we provide a tuto-rial introduction to bond graphs using examples that arefamiliar to control engineers. Second, the article is intend-ed to motivate readers to apply the bond-graph approachthemselves and to read more on the topic.

This article includes two case studies. The first casestudy is described in “Case Study 1: Laboratory Experi-ment,” while the second is described in the section“Case Study 2: An Aircraft Fuel System.” The early sec-tions of this tutorial cover enough of the bond-graphmethod to model the system in Case Study 1. The latersections introduce more advanced topics required forCase Study 2.

Italic text (like this) highlights material that is directlyapplicable to the sidebar “Case Study 1.”

ANALOGIESBond-graph modeling is based on three types of analogies,namely, signal analogies, component analogies, and con-nection analogies.

Signal AnalogiesTable 1 shows four signal categories with examples fromfour engineering domains:

» effort signals, with the generic symbol e, includingelectrical voltage and mechanical force

» flow signals, with the generic symbol f , includingelectrical current and mechanical velocity

» integrated effort signals, with the generic symbol p,including electrical lines of flux and mechanical(translational and angular) momentum

» integrated flow signals, with the generic symbol q,including electrical charge and mechanical displacement.

Some entries in the table are less familiar but neverthe-less provide a guide for the systematic choice of signals forsystem modeling. Additional domains, including magneticand thermal, can also be incorporated in this scheme. Akey insight is that the product of the effort and flow sig-nals in each domain is power, that is,

effort × flow = power. (1)

For this reason, effort and flow signal pairs are deemedto be carried by the single power bond of Figure 1(a). Thedirection of the half-arrow indicates the positive directionof energy transport; in Figure 1(a), energy transport fromleft to right is regarded as positive.

The system shown in Figure S1 of Case Study 1 is driven byelectrical power (voltage × current), which is converted to rotation-al mechanical power (torque × angular velocity) by a dc motor.

It should be noted that the half arrow does not denotean input or output in the same way as an arrow on a blockdiagram. Inputs and outputs are assigned by causalstrokes, which are introduced later in the section “Causalityand Block Diagrams.”

Figure 1(b) shows the active bond, which carries eithereffort or flow. The active bond thus corresponds to a block-diagram signal and can therefore act as an interfacebetween a system modeled as a bond graph and anothersystem modeled as a block diagram.

Component AnalogiesTable 2 lists one-port bond-graph components with analo-gous examples from four engineering domains. For exam-ple, the generic

» Se component, which can correspond to an idealvoltage source or an applied force, is a source ofeffort

» Sf component, which can correspond to an ideal cur-rent source or an applied velocity, is a source of flow

» De component, which can correspond to a voltmeteror a force sensor, is a detector of effort

» Df component, which can correspond to an ammeteror a tachometer, is a detector of flow

» R component, which can correspond to an electricalresistor or a mechanical damper, dissipates energy

» C component, which can correspond to an electricalcapacitor or a mechanical spring (or compliance),stores energy

» I component, which can correspond to an electricalinductor or a mechanical mass, stores energy

» SS components (not shown in table) model colocat-ed sensor-actuator pairs: Se-Df or Sf-De. Thesecomponents also represent energy ports of com-pound components.

In the linear case, the corresponding equations for theR, C, and I components in terms of the generic variables ofTable 1 are, respectively,

Further Reading» Books: A. Mukherjee, R. Karmakar and A.K. Samantaray

[23], D.C. Karnopp, D.L. Margolis, and R.C. Rosenberg

[3], P.J. Gawthrop and L.P.S. Smith [24], P.C. Breedveld

and G. Dauphin-Tanguy [25], F.E. Cellier [26].

» Conference proceedings: [27]–[33].

» Journal special issues: [34]–[36].

» Web sites:

• Bond Graph Compendium: www.ece.arizona.edu/∼cellier/bg.html

• Software reviews: www.bondgraphs.com/software.

html

» Software:

• Model Transformation Tools: mtt.sf.net

• CAMP-G: www.bondgraph.com

• 20-Sim: www.20sim.com

• Dymola: www.dynasim.se

• Symbols 2000: www.symbols2000.com

APRIL 2007 « IEEE CONTROL SYSTEMS MAGAZINE 25


26 IEEE CONTROL SYSTEMS MAGAZINE » APRIL 2007

F igure S1(a) shows a laboratory experiment comprising a flexi-

ble beam driven by a dc motor through a gearbox. As with any

physical system, it is up to the engineer to decide which effects to

include and which to ignore. In this case, following the experimen-

tal manual, the main approximations are:

» The armature inductance of the dc motor is ignored.

» The beam is approximated by a rotational mass-spring

system analogous to that of Figure 4(a).

» The gearbox is assumed to be rigid and free of

backlash.

FIGURE S1 Case Study 1: a laboratory experiment. (a) The Quanser flexible beam experiment. The dc motor drives the flexible beamthrough a gearbox, a strain gauge measures the beam deflection, and a potentiometer measures the motor angular position. (b) Sys-tem schematic diagram. For simplicity, the system is approximated as indicated in this mixed electrical/mechanical schematic dia-gram; in particular, the armature inductance is ignored, and the beam is given a lumped approximation. (c) The system bond graph.R :ra, GY:k, I :mm and R :rm model the motor; TF:n, I :mg, and R :rg model the gearbox; and C :cb, I :mb, and R :rb model the beam.Sf:y and Se :yb measure b and τb, respectively. (d) A simplified bond graph. The drive components are combined into the fourequivalent components I :me, R :re, R :r0, and Se :u0.

Strain Gage

Gears

dc Motor

Flexible Beam

τ,Ω

(a)

Ω Ω

τm

Ωm Ωm

ττg τ

Ωb

0V = u

ia

Va

ia

R:rb

1

I:mb

0

R:rg

1

I:mg C:cb

TF:n1

I:mm

R:rm

Df:yGY:km1

R:ra

Se:u

De:yb

Ωb

(c) (d)

Ω Ωτ

Ωb

τ

Ωb

0

Ω0 = υ0

τ0τ0

R:rb

1

I:mb

01

I:me C:cb

0

R:reR:r0

Sf:u0 Df:y

De:yb

u = V Va

cb

mmia mbn,mg

Ω Ωb

τ ττm

Ωm

(b)

Case Study 1: A Laboratory Experiment

R e = rf, (2)

C

e = qc , (3)

q = f, (4)

I

f = pm , (5)

p = e, (6)

where r, c, and m are constants describing the correspond-ing physical system. In the electrical domain, (2) corre-sponds to Ohm’s law and (3) to Coulomb’s law; in themechanical domain, (3) corresponds to Hooke’s law, while(5) corresponds to Newton’s second law.

Because the same type of component usually occursmore than once in a given system, the colon “:” notation isused to distinguish between multiple instances of each

component type. In particular, the symbol preceding thecolon refers to the component type, while the symbol fol-lowing the colon labels the particular instance. Thus C :c1

refers to a C component labeled c1, which is equivalent toplacing the label c1 adjacent to the symbol for a capacitorin an electrical circuit diagram.

Connection AnalogiesTwo components can be connected by a power bondthus giving them the same effort and flow. Figure 2(a)shows two mechanical components, while Figure 2(b)shows two analogous electrical components. Each ofthese physical systems can be represented by the bondgraph of Figure 2(c).


Figure S1(b) shows the corresponding schematic. The

left-hand part shows the conventional electrical model of the

motor armature comprising the armature resistance ra driven

by the applied voltage V and the back EMF Va. Since V is the

system input, it is labeled by the conventional symbol u. The

connection between the electrical and mechanical systems is

given by the usual dc motor equations

τm = kmia,

Va = kmm.

Figure S1(c) gives the bond graph corresponding to this

apparatus.

Using the simplification rules of Figure 9, Figure S1(c)

can be simplified to give Figure S1(d), where the GY, TF,

and one of the I components have been removed. In terms

of the original components, the components of the simpli-

fied system are

r0 = k2m

n2ra,

re = rg + rm

n2,

me = mg + mm

n2,

u0 = nkm

u.

Although the simplified bond graph retains the same input-out-

put dynamics as the original system, it is easier to understand.

For example, it is clear that R :r0 acts as a natural derivative

controller as in Figure 11.

Figure S2 gives the frequency response relating the two

outputs y = b and yb = τb to the equivalent system input u0.

The frequency response, which is generated automatically

from the bond graph and the numerical values of Table S1, is

the first step in the control-design process.

FIGURE S2 Frequency response for the Quanser experiment.The numerical values of Table S1 combined with the bond graphof Figure S1 yield the frequency response of y = b andyb = τb, measured in rad/s and N-m, respectively, to the equiva-lent system input u0, measured in rad/s; the resonant peak cor-responds to the flexure natural frequency of about 20 rad/s.

yb = τby = Ωb

20

0

−20

−40

−60

|g(jω

)| (

dB)

1 10 100

ω (rad/s)

TABLE S1 Numerical values. These numerical valuescorresponding to each bond-graph component in Figure S1 are taken from the Quanser manual [14].

Component Value Unitskm 0.00767 N-m/Ara 2.6

mm 3.87e−7 Kg-m2

rm 0.0 N-m-s/radn 1

70mg 2.5200e−5 Kg-m2

rg 0.0 N-m-s/radcb 2 rad/N-mmb 0.0012368 Kg-m2

TABLE 1 Analogous signals. Systematic modeling, including the bond-graph approach, uses the conceptof analogous signals to bring together different physical domains. One such analogy is the effort/flow analogy displayed here,where each row contains analogous signals and each column corresponds to a domain. In each case, effort × flow= power.

Bond Graph Translation Rotation Electrical Hydraulic

Effort Force Torque Voltage Pressuree F N τ N-m V V P Pa

Flow Velocity Angular velocity Current Flowf v m/s rad/s I A Q m3/s

Integrated effort Momentum Angular momentum Lines of flux Momentum per unit areap = ∫

edt p kg-m/s h kg-m2/s λ V-s p kg-m/s

Integrated flow Position Angle Charge Volumeq = ∫

f dt x m θ rad q C V m3



Because there are only two components, the electricalcomponents of Figure 2(b) share both voltage and current.But, more generally, electrical connections are either paral-lel [Figure 3(a)] or series [Figure 3(b)].

Figure S1 of Case Study 1 includes examples of both mechan-ical (damper) and electrical (resistor) R components.

The parallel connection obeys Kirchhoff’s voltage law,whereas the series connection obeys Kirchhoff’s current law.The bond-graph approach uses a 0 junction to model a paral-lel electrical connection [Figure 3(c)] and a 1 junction [Figure3(d)] to model a series electrical connection. However, theseries/parallel analogy can be misleading in the mechanicaland other non-electrical domains; a more useful abstractionis to view an electrical parallel connection as a common-effort or 0 connection and an electrical series connection as a

common-flow or 1 connection. The effort on each bondimpinging on a 0 junction is equal, while the flows on thesebonds sum to zero. A 0 junction with m bonds in and nbonds out is therefore described by the equation pair

0

ein1 =· · ·= ein

m = eout1 =· · ·= eout

n , (7)

∑mi=1 f in

i − ∑nj=1 f out

j = 0, (8)

where the efforts e in1 , . . . , e in

m and the flows f in1 , . . . , f in

nare carried on bonds pointing into the junction, while theefforts e out

1 , . . . , e outn and the flows f out

1 , . . . , f outn are car-

ried on bonds pointing out of the junction. Likewise, theefforts on a 1 junction sum to zero while the flows are allequal, so a dual junction is described by the equation pair

1

f in1 = · · · = f in

m = f out1 = · · · = f out

n , (9)

∑mi=1 e in

i − ∑nj=1 e out

j = 0. (10)

The system shown in Figure S1 of Case Study 1 contains anelectrical 1 junction carrying armature current as well as severalmechanical 1 junctions carrying velocity.

The symbols 0 and 1 are chosen to be neutral withrespect to the physical domain.

Simple RCI SystemFigure 4(a)–(c) shows analogous systems drawn from threephysical domains. Since the systems are analogous, theyshare the bond graph of Figure 4(d). As can be seen from Fig-ure 4(a) and (c), the concepts of parallel and series connectioncan be misleading, whereas the concepts of common effortand flow junctions provide a domain-neutral formulation.

Power Conversion with Transformers and GyratorsThe effort and flow variables within each physical domainof Table 1 have different units and therefore cannot be


FIGURE 1 Types of bond. (a) A power bond. The effort and flow sig-nal pair of Table 1 are carried by a single power bond. The half arrowindicates the direction of positive power transport. The signals atopposite ends are equal, that is, e2 = e1 and f2 = f1. Note thateffort × flow = e1 f1 = e2 f2 = power. (b) An active bond that carrieseither effort or flow. The active bond, which corresponds to a block-diagram signal, can act as an interface between a system modeledas a bond graph and another system modeled as a block diagram.

f2f1

e1 e2

e

f

(a)

(b)

TABLE 2 Analogous components with one energy port. The analogous signals of Table 1 lead to the analogouscomponents of this table; the first column gives the generic bond-graph component, while the remaining columnsgive the domain-specific analogues.

Bond Graph Translation Rotation Electrical Hydraulic

ExternalSe Applied force Applied torque Applied voltage Applied pressureDe Force sensor Torque sensor Voltmeter Pressure sensor

F N T N/m V V P PaSf Applied velocity Applied rotation Applied current Applied flowDf Speedometer Tachometer Ammeter Flow meter

v m/s ω rad/s i A Q m3/s

1 portC Spring Torsional spring Capacitor Accumulator

K N/m K N-m/rad C F K Pa/m3

I Mass Moment of inertia Inductor Flow inertiam kg J kg-m2 L H I kg/m4

R Damper Rot. Damper Resistor Restrictord N-s/m d N-m-s/rad R K Pa-s/m3


directly connected. However, since power is the universalcurrency of physical systems, the power-converting bond-graph components TF (a generic transformer) and GY (ageneric gyrator) of Figure 5(c) and (d) provide a means forconverting power and thus connecting different domains.The TF component generalizes an electrical transformer,which has the property that theratio of voltages (efforts) at thetwo terminals is the inverse of theratio of current, which is consis-tent with the fact that power isconserved in the sense that theinstantaneous power at the inputport equals the instantaneouspower at the output port at eachinstant of time. Figure 5(a)shows a physical system that, inidealized form, corresponds tothe TF component of Figure 5(c).Additional examples of physicalcomponents with an ideal TFrepresentation include a piston formechanical-to-hydraulic powerconversion and a rack-and-piniongear converting translational torotational power.

The GY component is thesame as the TF component inso-far as power is conserved; the dif-ference is that flow at one portdepends on effort at the other,and vice versa. Figure 5(b) showsa physical system that, in ideal-ized form, corresponds to the GYcomponent of Figure 5(d). Thename gyrator arises from theproperty of a gyroscope thatangular velocity (flow) is convert-ed into torque (effort).

In the linear case, the TF andGY components have the equations

TF

e2 = ne2, (11)

f1 = nf2, (12)

GY

e2 = kf1, (13)

e1 = kf2, (14)

where n and k are nondimension-al constants describing the corre-sponding physical system. Thepairs (11)–(12) and (13)–(14) bothdescribe energy-conserving com-ponents since, in both cases, theinput and output power is thesame, that is, e2 f2 = e1 f1.

The system shown in Figure S1 of Case Study 1 has a GYcomponent representing conversion of electrical to mechanicalpower within a dc motor as well as a TF component (a gearratio) representing conversion of mechanical power within agearbox. In each case, losses are accounted for using R compo-nents, which dissipate power.


FIGURE 3 Connecting three components. (a) A parallel connection. This connection obeys Kirch-hoff’s voltage law; the voltage (effort) is common. (b) A series connection. This connection obeysKirchhoff’s current law; the current (flow) is common (c) Bond-graph generalization of diagram(a). In a 0, or common-effort, junction, the efforts are equal and the flows sum to zero, that is,e1 = e2 = e3, f1 − f2 − f3 = 0. (d) Bond-graph generalization of diagram (b). In a 1, or com-mon-flow, junction, the flows are equal and the efforts sum to zero, that is, f1 = f2 = f3 ,e1 − e2 − e3 = 0.

V1 = e1 V2 = e2

i2 = f2 i3 = f3

i1 = f1

V3 = e3

f3

e3

f1

e1

e2 f2

0

V2 = e2 V3 = e3

i1 = f1

i2 = f2 i3 = f3V1 = e1

f3

e3

f1

e1

e2 f2

1

(a) (b)

(d)(c)

FIGURE 2 Connecting two components. (a) A mechanical mass m and spring with compliance care connected together; the mass and spring share the same velocity (flow) v2 = v1 and thesame action and reaction (effort) F1 = F2. (b) An electrical inductor with inductance m and capac-itor with capacitance c are connected together; the components share the same current (flow)i 2 = i 1 and voltage (effort) V1 = V2. (c) The bond graph I:m and C:c components are connectedtogether using the power bond of Figure 1(a); these components share a flow f2 = f1 and efforte1 = e2. The colon notation I:m and C:c associates the label m with the I component and thelabel c with the C component. The color coding is used to help interpretation; it is not part of thebond-graph method.

f2f1

e1 e2C:c I:m

c

mmc

f1 = i1 f2 = i2

e1 = V1 e2 = V2

(a) (b)

(c)


CAUSALITY ANDBLOCK DIAGRAMSAlthough block diagrams are famil-iar to control engineers, the sidebar“Why Bond Graphs Are Better thanBlock Diagrams” explains thatblock diagrams have an unfortunatedrawback, namely, they representassignment statements rather thanequations. In other words, a blockdiagram cannot be drawn until theinputs and outputs of each compo-nent are specified. For example, theupper right-hand part of Figure 6(a)shows an electrical resistor corre-sponding to the equation V = ri.Two possible block diagrams areshown below this component,where one has voltage (effort) out-put and corresponds to the assign-ment statement V := ri, while theother has current (flow) output andcorresponds to the assignmentstatement i := (1/r)V . In contrast,the bond-graph representation inthe upper left-hand part of Figure6(a) is acausal and represents anequation. The addition of a causalstroke in each of the two lowerbond graphs assigns the input andoutput of each R component. Thiscausal assignment is not part of theinitial modeling but is added later.This approach has the importantadvantage that bond-graph compo-nents are reusable within different


FIGURE 5 Transformers and gyrators. (a) A simple gearbox. Rotational mechanical power is converted between two shafts with gear ration > 1; the device corresponds to the transformer (TF) bond graph of (c). (b) a dc motor with power conversion between electrical terminalsand the mechanical shaft, corresponding to the gyrator (GY) bond graph of (d). (c) The TF component provides power conversion such thate1 = ne2 and f1 = nf2. (d) The GY component provides power conversion such that e2 = kf1 and e1 = kf2, where k is the back EMF con-stant of the motor.

f2 = Ω2

e2 = τ2e1 = V1

f1 = i1GY:k

τ1, Ω1

τ2, Ω2

V1, i1

τ2, Ω2

TF:ne2 = τ2

f2 = Ω2f1 = Ω1

e1 = τ1

(a)

(c) (d)

(b)

FIGURE 4 Analogous second-order systems. (a) Mechanical system comprising a trolley labeledm1, a spring labeled k1,and a damper labeled r1. (b) Analogous electrical system comprising anelectrical inductor labeled m1, a capacitor labeled c1, and a resistor labeled r1. (c) Analogoushydraulic system, comprising a tank with an outlet pipe that has both resistive and inertial charac-teristics, corresponding to a flow-dependent pressure drop and the fluid momentum. (d) Bondgraph representing the mechanical, electrical, and hydraulic systems. Each system has two ener-gy ports, one at the left and one at the right, each corresponding to the domain-specific effort/flowpair of Table 1. Analogous components are the same color, the blue italic text indicating the sig-nals in each case. Component C :c1 models extension of the spring, accumulation of charge in thecapacitor, or storage of fluid in the tank. It would typically be parameterized by the spring stiffnessk in the mechanical domain, by capacitance C in the electrical domain, and by tank cross-sectionalarea A and fluid density ρ in the hydraulic domain. Component I :m1 models the momentum of thetrolley, the lines of the flux in the inductor, or the momentum of fluid in the pipe. It would typicallybe parameterized by the trolley mass m, electrical inductance L , or by the pipe length l and densi-ty ρ of the fluid within it. Component R :r1 models the friction of the damper, the resistance of theresistor, or the friction within the pipe. It is typically parameterized by a damping coefficient b, by anelectrical resistance R, or by a hydraulic loss coefficient CD and orifice cross-sectional area A.

e1 = F1

f1 = v1

e0 = F0

f0 = v0

r1

k1

m1

f0 = i0

c1e0 = V0

r1 f1 = i1m1

f0 = Q0 f1 = Q1

e1 = P1e0 = P0

e0

f0 f1

e0

em f1

f1

f1

e1

e0

er

fc

R:r1

0

C:c1 I:m1

1

(a)

(c) (d)

(b)


causal contexts, whereas block-diagram components are not.While the bond-graph approachthus has one model for a resistor,the block-diagram approach hastwo. In bond-graph terminology,the middle R imposes effort andhas flow imposed on it, whereasthe lower R imposes flow and haseffort imposed on it.

Because the C component ofFigure 6(b) is dynamic, the distinc-tion between the two forms ofcausality is more significant. Inparticular, the middle diagramcorresponds to integral causality,while the lower corresponds toderivative causality. The former ispreferred if we wish to have astate-space system representation.The I component is similar to theC component of Figure 6(b) butwith e and f reversed.

Figure 7(a) is the causally com-plete equivalent of Figure 4(d),Figure 7(b) is the correspondingblock diagram, and the followingcomments explain the details:

» The C and I componentsare in preferred integralcausality; for the C compo-nent, this relation implieseffort out and flow in,whereas, for the I compo-nent, this relation impliesflow out and effort in.

» The R component has effortoutput since the corre-sponding flow is caused bythe I component.

» The 0 junction has exactlyone bond imposing efforton it, whereas the 1 junc-tion has exactly one bondimposing flow on it.

» The 0 junction of Figure 7(a)corresponds to the first sum-mation block of Figure 7(b)and the connection to thesecond summation block.

» The 1 junction of Figure 7(a)corresponds to the secondsummation block of Figure7(b) and the various con-nections involving f1.

FIGURE 6 Component causality. (a) R. The topmost bond is acausal and represents an equation,for which the corresponding electrical component is shown. The middle bond has a causal stroke(perpendicular end-bar) indicating that e is the output, while the lower bond has a causal strokeindicating that f is the output. Because of the causal stroke, these two bonds represent assign-ment statements rather than equations. The corresponding block diagrams for these assignmentstatements are shown. (b) C. This component is similar to the R component except that the equa-tion is differential, not algebraic. The middle diagram shows integral causality, while the lower dia-gram shows derivative causality. (c) I. This component is similar to the C component but with eand f reversed. The middle diagram shows derivative causality, while the lower diagram showsintegral causality.

f

e

f

e

ƒe

f

e

f e

e f

e := Ldfdt

Lf :=

1∫ edt

Ls

1Ls

Lf − ∫ edt = 0

I:L

I:L

I:L

(c)

f

e

f

e

f

e

e := rf

e − rf = 0

f := 1r e

f

e

rf e

1r

e fR:r

R:r

R:r f

e

ƒe e

f

e

e :=1c

∫ fdt

ce − ∫ fdt = 0

f : = cdedt

f

e

1cs

f

cse f

C:c

C:c

C:c

(a) (b)


Why Bond Graphs Are Better Than Block Diagrams» Acausal

• Equation based

• Causality, that is component input and

output, determined after modeling

• Causality issues clear

» Energy conserving

• Bonds convey power

• Automatically obeys laws of physics

» Compact

• Each bond conveys two related signals

• Connections are localized

• Components are localized

• Topology is closer to the physical system

• Graphical depiction of sign convention

» Reusable subsystems

• Subsystem causality adapts in response

to impinging subsystems


Causal AssignmentAbstracting the physical system as an acausal bond graphprovides a complete description of the corresponding model.However, for the purposes of analysis, there are many waysof representing the system as a set of equations, and eachsuch representation has its own particular uses. For example,the control engineer typically uses a state-space representa-tion for system analysis and simulation, whereas themechanical engineer may prefer a Lagrangian representationand the mathematician may prefer a Hamiltonian represen-tation. Each of these representations corresponds to a partic-ular causality and thus each representation can be extractedby imposing a particular pattern of causal strokes on theacausal bond graph [2], a procedure known as completingcausality. This article focuses on generating a state-space rep-resentation of a system.

Figure S1 of Case Study 1 has examples of R components ineach type of causality. For example, the armature resistor R :ra

imposes the armature current (flow), whereas R :rg and R :rbeach impose torque (effort). Figure S1 also has examples of bothintegral (I :mg, C :cb, and I :mb) causality as well as derivative(I :mm) causality.

The assignment of causality to a bond graph can usuallybe accomplished automatically by computer if the causalityis specified at key points on the graph, usually the externalports, and if some general preference for integral or deriva-tive causality [Figure 6(b) and (c)] is expressed by the mod-eler. The best known method for automating causalassignment is the sequential causal assignment procedure(SCAP) [3], which gives a state-space system representa-tion. If, indeed, the system has a state-space representation,the details of the resulting pattern of causal strokes,

although helpful in understanding the inner workings ofthe model, need not be viewed by the modeler. However, ifthe model does not possess a state-space representation,then the pattern of causal strokes clarifies the situation andhelps the modeler reconsider the model in terms of theunderlying physical system.

For proper causal completion, which will result in a setof explicit assignment statements, it is necessary that exact-ly one bond impose a flow on each 1 junction. Similarly,exactly one bond must impose an effort on each 0 junction.The causality of TF and GY components is also subject toconstraints if self-consistent system models are to be gen-erated. In particular, causality is transmitted unalteredthrough TF components, that is, one impinging bondimposes effort (flow), while the other has effort (flow)imposed on it. Causality is reversed through a GY compo-nent so that both impinging bonds impose effort (or flow)and have flow (or effort) imposed on them. Within theseconstraints, causality can be assigned arbitrarily, althoughgeneral guidelines, or preferences, are usually expressed.

After specifying the causality at the external interfaces,it is generally advisable for the modeler to specify thepreferred causality of the system C and I components. Asdiscussed in Figure 6, C and I components may haveeither integral or derivative causality. For simulation orstate-space representations, integral causality is usuallypreferable since it leads to ordinary differential equations(ODEs), which can be computed without recourse tocomputationally intensive differential algebraic equation(DAE) solvers.

Constraints due to 0 and 1 junctions or TF and GY com-ponents may make it impossible to place all of the energy

storage elements in integralcausality. In this case, dependentstates result in DAEs.

Insofar as modeling is the artof approximation, that is, decid-ing which components, features,and behaviors to omit, bondgraphs can help engineers decidewhich approximations are usefulbefore generating the equations.For example, approximate mod-els with derivative causality canbe converted to integral causalityeither by adding greater detail tothe model, in the form of addi-tional states, or by combiningstates to simplify the model. Anexample of this conversion is thedecision to model a shaft con-necting two rotating masses aseither rigid or compliant. Themodeler may wish to considerwhether the additional difficulty

FIGURE 7 System causality. (a) Causal bond graph with causality completed on each compo-nent. As indicated by the causal strokes, the C :c1 component has its preferred causality of flowin and effort out, the I :m1 component has its preferred causality of effort in and flow out, where-as the R :r1 component has, in this system, a causality of effort out and flow in. Annotationssuch as q1 are for clarity and are not part of the bond-graph notation. (b) Block-diagram. The input and output of each component corresponds to the causal stroke on thebond-graph components of (a); the C and I components are in integral causality since they leadto block-diagram integrators. From Table 1, the associated states are the integrated flow q1 andintegrated effort p1. The block diagram and bond graph share the same color coding to indicatethe corresponding elements.

y2 = e0

f0 = u1 f1

e0

em f1

f1

f1 = y1

e1 = u2

e0

er

fc

q1 p1

C:c1 I:m1

R:r1

0 1+ −−+

er

1c1s

1m1s

fc e0 −em

r1

e1

f1 = y1f0 = u1

y2

(a) (b)



of solving DAEs outweighs potential disadvantages associ-ated with an ODE representation, such as numerical stiff-ness or reduced transparency in the meaning of quantitiesrepresented by system states.

Although either causal configuration can be meaning-ful for an R component, the modeler should be aware ofthe possibility of algebraic loops arising between multipleR components. An algebraic loop occurs when the outputof one component is required to determine the input ofanother, and vice versa. The formation of an algebraicloop exists on a bond graph if, between R componentswith opposing causality, there exists a path that includesneither a 1 junction with flow imposed by an I or Sf com-ponent nor a 0 junction with the effort imposed by a C orSe component. In other words, to avoid the formation ofan algebraic loop, at least one system input or storage ele-ment with integral causality must impose a flow on a 1junction or an effort on a 0 junction in each path betweenR components with opposing causality. A slight compli-cation is that GY components effectively reverse thecausality of the two subsystems on either side of themwith respect to each other. Thus R components may haveopposing causality even if the causal stroke is at the sameend of the bond attached to each component due to a anodd number of GY components appearing in the pathbetween them.

In Figure S1 of Case Study 1 the component I :mm has deriv-ative causality. An ordinary differential equation representationcan be obtained by adding further detail, such as motor shaftcompliance, or by simplifying the model.

One of the benefits of bond-graph modeling is that thepresence of algebraic loops is explicitly visible to the mod-eler. Although algebraic loops lead to implicit equations,such loops can be broken in various ways to allow the cre-ation of an explicit model, for example:

» adjacent R components can be combined to formequivalent components

» additional dynamics can be modeled by introducingC or I components between the R components

» algebraic or numerical solvers can be introduced(using the SS component) to resolve the causal con-flict

» the causality of adjacent components can be alteredso that both R components can be assigned the samecausality.

Despite the possibility of completely automating causalassignment, it is generally advisable for the modeler to beinvolved in the process since the causal assignments madewithin a model carry important information about the sys-tem and the suitability of the model for any intended pur-pose. In particular, the information gleaned from causalassignment gives the modeler immediate feedback as tothe consequences of including or deleting a componentfrom the bond-graph model of the physical system withoutthe need to generate equations.

State-Space Equations and Block DiagramsBond graphs are an acausal system representation. By assign-ing a causal stroke to each bond, a causal representation canbe generated. The causally complete model can be convertedinto other causal representations such as state-space equa-tions and block diagrams. This section demonstrates the prin-ciples of this conversion. Most bond-graph software supportsthis conversion and provides an interface to standard controlengineering tools such as Matlab and Octave [4].

The causal strokes on a bond graph provide “signposts”to guide the generation of state-space equations and blockdiagrams. Although software can perform these transfor-mations automatically, state-space equations can be gener-ated by hand.

Both the bond graph and simplified bond graph of Figure S1 ofCase Study 1 correspond to state-space equations with three statessince three components have integral causality in each case. The state-space equations can be derived by hand or by using symbolic software.

The following steps demonstrate this procedure usingFigure 7(a):

» Identify the states. The system states are the inte-grated flows q associated with C components, aswell as integrated efforts p associated with I compo-nents, in integral causality [Figure 6(b)]. In this casethe states are q1, the integrated flow variable associ-ated with C :c1, and p1, the integrated effort variableassociated with I :m1.

» Write state derivatives in terms of states and inputs.By definition, (dq1/dt) = fc . Following the causalstrokes, fc = f0 − f1. In this example, f0 is an inputand, following the causal strokes, f1 = (p1/m1) ,where p1 is a state. Thus, the first state equation is

dq1

dt= f0 − p1

m1. (15)

By definition, (dp1/dt) = em . Following the causalstrokes, we obtain em = e0 − er − e1 . In this exam-ple, e0 = (q1/c1), and e1 is an input. Using the prop-erties of the R component given in Figure 6(a),er = r1 f1 = r1(p1/m1). Thus the second state equation is

dp1

dt= q1

c1− r1

p1

m1− e1. (16)

» Write outputs in terms of states and inputs. Theoutputs f1 and e0 can be written in terms of thestates as f1 = (p1/m1) and e0 = (q1/c1). Defining

x =[

q1p1

], y =

[f1e0

], u =

[f0e1

], (17)

(15) and (16) can be written in state-space form

dxdt

(t) = Ax(t) + Bu(t), (18)

y(t) = Cx(t), (19)



where

A =[

0 − 1m1

1c1

− r1m1

], B =

[1 00 −1

], C =

[ 1c1

0

0 1m1

]. (20)

In a similar fashion the block diagram of Figure 7(b) can bederived from the causally complete bond graph of Figure 7(a).

Case Study 1 uses simplification to reduce the number ofbond-graph components and remove the component I :mm inderivative causality.

Simplification and ApproximationIn the same way that two resistors in series can be com-bined into a single resistor or two rigidly connectedmasses can be combined into a single mass, bond-graphcomponents can be combined to give a simplified bondgraph with the same external behavior as the originalbond graph. Such simplification can be useful in under-standing the behavior of a complex system in terms of itssimplified version.

Figure 8 shows two I components separated by a Ccomponent, a bond graph that can represent two rotatingmasses separated by a compliant shaft. Under steady stateconditions, the masses rotate at the same speed, and thereis no change in the twist of the shaft. This synchronous

rotation is manifested as zero flow through the bond to theC component. Even under transient conditions, the flow isgenerally small. It may therefore be reasonable to approxi-mate the flow to zero, an approximation that can beaccomplished explicitly by replacing C with a flow sourceSf adding zero flow to the junction. Having made thisapproximation, the bond graph can be simplified by elimi-nating the Sf component entirely because bonds addingzero flow to a 0 junction have no effect on the system.

Using some simple rules, further simplifications can easi-ly be made. A junction connecting only two bonds is redun-dant since it merely constrains the effort and flow in eachbond to be the same. An identical effect can be achieved byreplacing both bonds and the junction with a single bond,thereby eliminating the 0 junction. Simplifying further, twojunctions of the same type, which are connected by a singlebond, can be replaced by a single junction of the same type.The result is a greatly simplified bond graph with two I com-ponents connected to a single 1 junction. Finally, the approx-imate system can be simplified as in Figure 9(a) and (b).

In more complex examples, analysis of the causality ofthe original model and the simplified version yields infor-mation that can be used to determine whether the approxi-mation is effective. The original model has three energystorage states to which integral causality can be simultane-

ously assigned. The model there-fore produces a set of three ODEs.The approximated model has onlytwo states and is therefore in somesense simpler. However, followingthe rules of causal assignment,described previously, it is not pos-sible to simultaneously place bothof the I components in integralcausality. One state must beplaced in derivative causality [Fig-ure 6(b)], which (without simplifi-cation) results in a set of DAEs.Whether the replacement of threeODEs with two DAEs is a usefulapproximation or not depends onthe purpose of the model; never-theless, a benefit of bond-graphcausality is that the consequencesof such an approximation areexplicit to the modeler.

FIGURE 8 Approximation of bond graphs. (a) Three energy storage states. This bond graph mightrepresent two rotating masses I :m1 and I :m2 connected by a flexible shaft C:c. (b) If the shafthas very low compliance, the shaft can be explicitly modeled as rigid by replacing the C compo-nent with a zero-flow Sf. (c) Since the addition of zero flow to a 0 junction does not affect the sys-tem dynamics, the bond graph can be simplified by removing the Sf component entirely.

e2

f2

e1

f101 1

I:m1 C:c I:m2

(a)

e2

f2

e1

f1

f=0

01 1

I:m1 I:m2Sf:zero

(b)

e2

f2

e1

f101 1

I:m1 I:m2

(c)


The bond-graph method is an approach to modeling in which

component energy ports are connected by bonds that specify the

transfer of energy between system components.


ADVANTAGES OF BOND GRAPHSOVER BLOCK DIAGRAMSCase Study 1 provides evidence for the assertions in the side-bar “Why Bond Graphs Are Better Than Block Diagrams.”The evidence is presented below. In particular, Figure 10 ismore complex than the bond graph of Figure S1 ofCase Study 1.

The acausal, equation-based nature of a bondgraph is apparent from the way that componentscan be treated identically regardless of thecausality imposed on them. The resistances ra

and rm of Case Study 1 are implemented identi-cally in the bond-graph model of the motor withonly the later addition of a causal stroke, specify-ing the inputs and outputs required to generateassignment statements from the acausal equa-tions. By contrast, the same resistances ra and rm

depicted in block-diagram form in Figure 10(b)are implemented differently, with the reciprocalof the resistance multiplying the input signal inone case, but not the other.

A more striking example can be seen in thehandling of the motor inertia. Whether the inertiamust be modeled with an integrator or differentia-tor changes according to its causality, as can beseen by comparing Figure 10(b) and (c), where achange to the input and output of the componentmodel causes the inertia causality to change. In thefirst configuration, an output torque is generatedin response to motor speed for a given voltage,and the inertia is represented by a differentiator.

The block diagram of Figure 10(a), corresponding tothe bond graph of Figure S1 of Case Study 1 is compli-cated, bearing little relation to the system topology. Acomparison with the bond graph reveals many of the ben-efits that bond graphs provide for modeling the system.

In the second configuration, the motor torqueis an input, and the shaft speed an output, butthe differentiator must then be changed to anintegrator. It is also necessary to reverse thedirection of some arrows and to change the signof the signal connecting the motor torque τm tothe motor mass, which is not necessary in thebond graph because the positive direction ofenergy transport is automatically handled by thesign convention specified by the bond directions.

The disadvantages of requiring two block dia-grams to represent one component depending onits inputs and outputs extend beyond the need tocreate and maintain twice as many models. Thereis also the problem of unit testing. While it is asimple matter to verify the dynamic behavior ofthe block diagram with integral causality in Fig-ure 10(c) by specifying constant inputs, this sim-plicity is not true of the version with a

differentiator in Figure 10(c). Applying a constant shaftspeed does not reveal any of the dynamic characteristics ofthe motor. The need for different forms of verificationmeans that it is necessary to write separate tests for everyparticular instantiation of a model in block-diagram form,


FIGURE 9 Simplification of bond graphs. (a) Connected I and R pairs. This bondgraph might represent two rigidly connected masses, each of which is connect-ed to a damper. (b) Simplified version of diagram (a). The parameters of each Iand R component can combine as re = r1 + r2 and me = m1 + m2 to create asimpler model with equivalent properties. (c) TF connected I and R pairs. Thisbond graph might represent an electromechanical actuator in which electricalpower is provided to a first-order electrical circuit, with inductance m1 and resis-tance r1, after which the power is transformed into mechanical power andapplied to a mass-damper with parameters m2 and r2. (d) Simplified version ofdiagram (c). The system parameters can combine to form equivalent parame-ters re = r1 + n 2r2 and me = m1 + n 2m2 . The interpretation of the inputchanges to ue = ku in accordance with the elimination of the gain, but the out-put remains the same. (e) Subsystem with GY connection. This bond graphcan represent a linear electric pump with inductance m and resistance r . (f)Simplified version of diagram (e). Eliminating the GY:k component, the causali-ty of the system reverses with respect to the output SS:y, requiring the 1 junc-tion to be replaced by a 0 junction, and the I:m component to be replaced withthe C :ce component. The parameters associated with the system componentsalso change to accommodate the elimination of the GY gain, giving ce = m/k2

and re = k2/r . Again, the output remains the same, but the interpretation of theinput changes to ue = ku.

1

I:m1

R:r1

1

I:m2

R:r2

1

I:me

R:re

1Se:u

I:m1

TF:n 1

I:m2

R:r1 R:r2

Df:y 1

R:re

Se:ue Df:y

I:me

(a)

(c)

1Se:u

I:m

R:r

GY:k SS:y 0

C:ce

SS:y

R:re

Sf:ue

(e) (f)

(d)

(b)


whereas it suffices to test a single configuration of a bondgraph with confidence that the same model can be usedregardless of the external causality.

Maintenance of bond-graph models is also made easi-er by the localization of components. It can be seen thatthe block-diagram representation of the gearbox requiresthat the gear ratio n be specified in two places in Figure10(d), whereas the corresponding TF :n component needonly be inserted once for the bond-graph representation(Figure S1). Multiple specification of parameters is usual-ly only a minor inconvenience when initially creatingmodels, but can easily lead to the introduction of hard-to-detect bugs when models are updated to accommo-date future changes.

ADVANCED TOPICSThe bond-graph method has been developed in severalways since its inception, and many of these developmentshave been initiated at the biennial International Confer-ence on Bond Graph Modeling. From these extensions, wediscuss several topics that we believe to be of importanceto control engineers. Sources of information for these top-ics can be found in “Further Reading.”

Physical-Model-Based ControlAs discussed in [5], [6] physical-model-based controlregards feedback controllers as physical systems. Thisapproach has the advantage that both physical intuitionand energy-based stability analysis can be used to design


FIGURE 10 Block-diagram representation of the system described in Figure S1. (a) The top level system comprises three subsystems,namely, a motor, a gearbox, and a flexible beam. (b) The motor calculates the torque output in response to a given input voltage and shaftspeed. Note that the resistances Ra and Rm (purple) are implemented differently; Rm appears as a multiplier, whereas the reciprocal of Ra

multiplies its input. This difference corresponds to the different causalities imposed on each of these resistors. The motor gain km appearstwice in the model; if different numerical values were inadvertently assigned to each instance, the model would no longer obey physicallaws. (c) An alternative motor model shows how diagram (b) would need to be modified to cause the model to output shaft speed inresponse to an applied load, a change that might be required to permit unit testing of the motor submodel. Note that, as well as reversingsome arrow directions, the integrator 1/s corresponding to the motor inertia (red) must be replaced with a differentiator s, and the sign of thesignal from the motor torque to the summing junction must be reversed. (d) The gearbox block diagram includes the gearing, gear inertia,and friction. Note that the gear ratio n (brown) appears twice in the model; once again, different values assigned to each block result in anonphysical system. (e) The flexible-beam block diagram comprises the beam compliance, inertia, and linear friction model. Note that theoutput torque is calculated by the beam compliance (green), which cannot therefore be easily removed.

v

–km

1/Ra ia

Km

+ +

+ ++ 1/Mm

−Rm

+1

s Ωm

τm

(b)

Ωm

τm Ω

Ωτ

τ+ +

+

++

+

+1/Mg

1/Mb

−Rg

−Rb

1/s1/s

1/s

1/Cb

1/n

n

−1

(d)

Ωm

Ωb

τm

v+ +

1/Ra

−Km

ia

Km+ ++

1/Mm

−Rm

1/s

−1

−1

(c)

vv

Motorτm

ΩmGearbox

Ω

τBeam

ΩbΩb

(a)

(e)


stable controllers. One exampleof physical-model-based controlis the impedance control method-ology of [7], which is applied inthe robotics field and, morerecently, in the design of structur-al dynamics experiments [8].

Perhaps the simplest versionof physical-model-based controlis the realization that a PID con-troller is analogous to a mass-spring-damper system and cantherefore (as in Case Study 1) betreated as a physical systemattached to the controlled physi-cal system. Using bond-graphrepresentations, Figure 11 showsthe physical systems correspond-ing to two versions of PID control. In particular, if kp = k,ki = (k/Ti), and kd = kTd, then Figure 11(a) corresponds tothe PID controller

u = k(

1 + 1Tis

+ Tds)

(w − y).

It is well known that such a control, although a conve-nient idealization, is not practical due to the pure deriva-tive action, which is reflected in the fact that the Icomponent in Figure 11(a) must be assigned derivativecausality. Figure 11(b) shows the bond graph correspond-ing to a PID control with filtered derivative. Note that the Icomponent now has integral causality. In particular ifkf = (Tf /kTd), the controller equation becomes

u = k

(1 + 1

Tis+ Td

s1 + Tf s

)(w − y).

A crucial feature of this approach is that the systeminput u and output y must be colocated, that is, the bondat the lower right of each controller of Figure 11 can bedirectly connected to a port of the controlled system. Ifthis is not the case and u and y are associated with differ-ent system ports, one approach is to revert to convention-al control-system design based on the state-spaceapproach or the block-diagram approach. In other words,the bond-graph methodology is used just for systemmodeling, and the bond graph is converted into state-space or transfer function form to be analyzed using con-

ventional software tools such as Matlab or Octave. As analternative, current research investigates physical-model-based control in the noncolocated case [8], [9].

Inversion and BicausalityThe systems of Figure 4 have the acausal bond graph ofFigure 4(d). When natural causality is imposed as in Fig-ure 7, the system input u and output y are

u =[

f0e1

], y =

[e0f1

]. (21)

The corresponding system transfer function is

G =

m1 s+r1c1m1 s2+c1 r1 s+1

1c1m1 s2+c1 r1 s+1

1c1m1 s2+c1 r1 s+1

−c1 sc1m1 s2+c1 r1 s+1

. (22)

This system has two poles corresponding to the compo-nents C :c1 and I :m1 in integral causality.

For some applications, such as actuator sizing [10], [11],it is of interest to compute inputs in terms of outputs,which leads to system inversion. Because of their acausalnature, [such as Figure 4(d)], bond-graph models areamenable to inversion. In Figure 12(a) the causality at theleft-hand port is reversed to change the choice of systeminput and output from (21) to

u =[

e0e1

], y =

[f0f1

]. (23)


FIGURE 11 PID Control. (a) Pure PID. R :kp, C :ki, and I :kd give the proportional, integral, andderivative terms, respectively; I :kd has derivative causality. (b) Filtered PID. R :kf and the associ-ated 1 junction give a lowpass filtering effect on the derivative action; I :kd now has integralcausality.

uyw

0

(a) (b)

SS:[in] SS:[out]

1 C:ki

I:kd

R:kp

uyw

0SS:[in] SS:[out]

1 C:kiR:kp

0 I:kdR:kf

Simple rules govern the transformation of bond graphs into

other system representations.


The transfer function arising from this choice is

G =[ c1m1 s2+c1 r1 s+1

m1 s+r1

−1m1 s+r1

1m1 s+r1

−1m1 s+r1

]. (24)

Not surprisingly, this transfer function is improper; thefact that it has only one pole follows from the fact that onlythe component I :m1 is in integral causality.

Another choice of inputs and outputs is

u =[

e1f1

], y =

[eo

fo

]. (25)

However, the concept of bond-graph causality must beextended in this case. Remember that each bond carriestwo covariables, namely effort and flow. The place-ment of a single causal stroke at one end of the bond, asin Figure 6, indicates which component sets the effortand which sets the flow. This configuration is uni-causal. By contrast, a bicausal model contains bonds in

which one component sets both the effort and the flow.In Figure 12(b), bicausality is signified on a bond graphby separating the causal stroke into two half strokes[12], [13].

The model corresponding to the bond graph of Figure4(d) can now be inverted by changing the causality of thebonds to that depicted in Figure 12(c). The resultingimproper transfer function is

G =[

1 m1s + r1c1s c1m1s2 + c1r1s + 1

]. (26)

The transfer function has no poles because both C :c1 andI :m1 are now in derivative causality [Figure 6(b)].

It is worth emphasizing the qualitative aspects of inver-sion by this method. Model inversion is accomplished bychanging the causality at the model interfaces. Changes incausality propagate through the model automatically andthus require no change to the model itself. Although trans-fer functions are given for completeness, the number of

poles and zeros is deduced bycounting the number of compo-nents in integral causality in thebond graphs of the system andinverse system, respectively.

Hierarchical SystemsSimple bond graphs can be con-structed entirely from the stan-dard basic components listed inTable 2. These models can beused in further bond graphs tocreate models of greater com-plexity, namely, hierarchicalbond-graph models. Hierarchi-cal bond graphs offer many ofthe benefits that are frequentlyassociated with object-orientedprogramming techniques.

When working with bondgraphs, it is natural to con-struct models by performing atop-down decomposition of thesystem of interest. By encap-sulating low-level functionalitywithin self-contained compo-nent models, clutter can beminimized easing visual in-spection, thus helping the mod-eler focus on an appropriatelevel of abstraction at eachstage of the model develop-ment. Such decomposition canalso greatly ease the process ofverifying that the structure of


FIGURE 12 Bicausality and Inversion. (a) Partial inversion. The RCI model of Figure 7 can be partiallyinverted by changing the causality at the left-hand system port. The output f0 of the model is the flowrequired to cause the effort e0 to track a desired signal given a disturbance e1. (b) Bicausality. Usinghalf strokes, each bond can have one of four bicausal configurations. (c) Inversion. The RCI model ofFigure 7 can be inverted by changing the causality at the system interfaces. These changes propagatethrough the model and cause C :c1 and I :m1 to be placed in derivative causality. The output of themodel f0 is the flow required to cause the flow f1 to track a desired signal given a disturbance e1; e0

is the corresponding effort, and e0f0 is the power required.

e0

f0 f1

e0

em f1

f1

f1

e1

e0

er

fc

C:c1 I:m1

R:r1

0 1

e1 := e2

f2 := f1

e1 := e2

f1 := f2

f1 := f2

e2 := e1

f2 := f1

e2 := e1

(a) (b)

e0

f0 f1

e0

em f1

f1

f1

e1

e0

er

fc

C:c1 I:m1

R:r1

0 1

(c)


the model accurately mimics the structure of the systemthat it is intended to represent. Subsystem encapsulationmakes it easy to maintain libraries of components and swapentire classes of components within a model. Therefore, it isappropriate in the early stages of model construction to usevery simple subsystem models. As model developmentprogresses, components can be refined, and subsystems ofgreater complexity can be constructed, tested, and insertedinto the higher-level model.

Hierarchical models are implemented using externalport components, which are equivalent to combinedsource-sensors. These external ports are connected to othersubsystem models using standard component bonds, withcausality assigned in the same way as for any other bond.It is therefore possible for the causality of the bonds at theexternal interfaces to change. This property is particularlyuseful for embedding bond-graph models within system-of-systems models, where the subsystem models adapttheir causal configuration to the context in which they findthemselves. This feature of external ports without fixedcausality is a significant advantage of bond graphs overblock-diagram-based modeling methods.

Hybrid SystemsMany useful engineering systems incorporate switches,that is, components that fundamentally change the glob-

al nature of the system by making or breaking local con-nections. The effects of switches on the global system aregenerally much larger than the local physical effects ofthe switching action itself, and thus it is often useful torepresent switches as an instantaneous change in a sys-tem variable.

When the switch establishes a connection within asystem, any causal configuration can be meaningfulacross the switch. However, when it is used to break aconnection, there is generally a definite causal config-uration associated with it. Opening an electrical con-tact or closing a fluid valve can be represented on abond graph by imposing zero flow on the 1 junctionrepresenting the wire or pipe in which the switch orvalve is placed. Similarly, the imposition of zero efforton a 0 junct ion models the ef fect of breaking amechanical link.

In certain circumstances, it is desirable to ensure thatthe inclusion of a switch does not change the global causal-ity of a model. For example, it is often desired that all Cand I components be assigned integral causality so that noalgebraic equations need to be solved. Inertial switches(ISW) or capacitive switches (CSW), which combine anideal switching element with an I or C component, can beused to ensure that there is no change in causality due tothe action of the switch.


FIGURE 13 Bouncing ball. (a) The bond graph superimposed on the schematic diagram. The CSW component models the ground, while theINTF component integrates the flow (velocity) to give height. (b) In this simulation, the ball is dropped from 10 m. The air resistance reduceseach rebound, while the ground compliance allows negative height.

Ball

Ground

v

−v

R:air_resistance 1 SS:gravity

I:ball_mass

CSW:ground

0

INTF:intf SS:x

(a)

10

8

6

4

2

0

0 1 2 3 4 5 6 7 8 9 10−2

x (m

)

t (s)

(b)

The bond-graph method is a useful modeling tool, particularly

well suited for describing physical systems.


Figure 13 gives an example of using a CSW componentin a mechanical context. When the ball is above ground, theCSW is open and no force is exerted on the ball; when theball is below ground, the CSW acts as an ordinary spring,thus creating a bounce.

Distributed-Parameter SystemsAlthough the bond-graph approach does not explicitly han-dle distributed-parameter systems described by partial dif-

ferential equations, these systems can be approximated usingN discrete lumps. For example, the flexible beam of FigureS1(a), approximated in S1(c) by a single lump, can be betterapproximated using N lumped elements each of the form ofFigure 14(a). Using the Bernoulli-Euler approximation, eachlump has the bond-graph representation of Figure 14(b).

Figure 14(c) shows the frequency response relatingangle and angular velocity at the fixed end of the beam fordifferent N. The final choice of N depends on the band-width over which the approximation is required.

CASE STUDY 2: AN AIRCRAFT FUEL SYSTEMThe second case study is an aircraft fuel system modeledusing bond graphs. The design presented is a hybrid ofthree aircraft.


FIGURE 14 A uniform beam. (a) An infinitesimal lump of width δz. The vertical velocity v is driven by the net force δF , while the torque τ isdriven by the net angular velocity δ. (b) Lumped bond graph. The interaction between angular and linear motion is expressed by the trans-former TF:dz, the linear motion of the lump is expressed by I:dm, and the angular twist by C:dk. The term R:dr expresses structural damp-ing. (c) Frequency responses |g( jω)| for N lumped elements, where N = 5, 20, 40.

δz

υ

F

Ω

τ τ + δτ

Ω + δΩ

υ + δυ

F + δF

F F + δF F + δF

υ + δFυυ

Ωτ + δτ

Ω + δΩτ τ + δτ

ΩSS:[a in]

1 0

I:dm

SS:[y out]SS:[y in]

SS:[a out]1

TF:dz

1 C:dkR:dr

(a)

(c)

(b)

1e–06

1e–05

1e–04

0.001

0.01

0.1

1

10

100

10 100 1,000 10,000

52040

ω (rad/s)

|g(jω

)|(N

-m)/

(rad

/s)

0

Power is the universal currency

of physical systems.


The model that we develop is suitable for preliminaryfuel-system architectural design, typically performed duringthe conceptual design phase, or as a realistic fuel system sub-stitute for integration testing during later design phases.More interestingly perhaps for control engineers, the modelis eminently suitable for use in specifying, developing, and

verifying complex system control software, which is themost expensive part of most modern aircraft development.

For architectural design work, the main requirementsare that the model should be easy to modify, easy toanalyze, and give reasonably accurate results for minimaleffort if the design changes frequently and questions about


FIGURE 15 A simple aircraft fuel model. (a) Fuel tank layout. (b) Bond graph. The refuel system is depicted in green; the feed system in red.The system consists of nine tanks of which there are six types.

F1

F2

F3

F4L F4R

WL2 WR2

WR1WL1

(a)

[Lef

tAftF

eed]

[Eng

ine]

[OuterRefuel]

[CentreRefuel]

[OuterFeed] [Feed]

[LeftWingFeed]

[RightWingFeed]

[Feed] [OuterRefuel]

[CentreRefuel]

[OuterFeed]

[RightRefuel]

[Refuel][For

war

dRef

uel]

[For

war

dFee

d]

[LeftRefuel]

[Lef

tRef

uel]

[Rig

htA

ftFee

d]

[Rig

htR

efue

l]

CentreTank:F2

ForwardTank:F1

Engine:Jet

InnerWingTank:WL2

OuterWingTank:WR1FeedTank:F3

AftTank:F4RAftTank:F4L

OuterWingTank:WL1

Refuel:Ground

InnerWingTank:WR2

(b)


system performance must beanswered quickly.

For integration testing and controlsoftware development, models musttypically be capable of running in realtime, while exhibiting the main fea-tures of the system behavior anddynamically exciting all relevantinterfaces through which the model isconnected to other systems. In thesecases, models are often used by peo-ple not closely associated with thesystem hardware itself and so mustbe reliable and not require specializedknowledge of the system in order forthe model to be operated successfully.

Model DesignThe Fuel model [Figure 15(b)] is ahierarchical bond graph repre-senting a simple aircraft fuel-man-agement system. The systemcomprises nine fuel tanks, ofwhich there are six types, namely,one instance each of the Forward-Tank, CentreTank, and FeedTankfuel tanks, and the InnerWing-Tank [Figure 16(a)], OuterWing-Tank and AftTank types, whichare each instantiated twice due tothe lateral symmetry of the air-craft. For simplicity, the system isathermal, and the pipework isassumed to be coplanar and per-pendicular to the external gravita-tional field. The connectionsbetween the tanks form two majorsubsystems: Refuel (green) andFeed (red) the respective purposesof which are to load fuel into theaircraft and transfer it from thestorage tanks to the feed tank,which supplies the jet engine.These subsystems can also beused to transfer fuel around theaircraft according to the demandsof the flight control system.

Fuel enters the system throughthe ground refueling point (Refu-el:Ground) attached to the centerfuselage tank (CentreTank:F2) andleaves the system through the sin-gle jet engine (Engine:jet). Theengine is supplied with fuel fromthe feed tank (FeedTank:F3).

FIGURE 16 Submodels. (a) Tank. Each tank submodel includes the Valve and Pump compo-nents, which are located within the tank, and a Volume, which contains the fuel and inert gas.(b) Volume. This component represents the storage of the fuel and inert gas, which is con-tained within the tank. The fuel is stored in a C component with an incompressible constitutiverelationship. The gas is stored in a C component with a compressible constitutive relationship.A TF represents the surface separating the two fluid domains. (c) Valve. (d) Pump. Thesecomponents are the electromechanical actuators that control the movement of the fluid withinthe fuel system. Interfaces with the dc electrical system are included in the component models.

0

Volume:volume

Valve:valveCPump:pumpF

Pump:pumpW Valve:valveW

SS:[CentreRefuel]

SS:[OuterRefuel] SS:[OuterFeed]

SS:[Feed]

[1]

[2]

TF:surface

1 C:fuel

SS:[In]

C:gas

SS:Vent

(a) (b)

[mod]

dc Motor Operated Valve

EMTF:controller

1

1

SS:electrical

Se:command

FMR:orifice

SS:[out]

I:position R:resistance

1SS:[in]

ISW:inertia

R:pipe

(c)

[2]

[1]

[mod]

DC Pump

R:rotor

1

Df:flow

SS:[in] 0

De:pressure

1I:inertia R:friction

EMTF:controller

SS:electrical

Se:command

SS:[out]

(d)



Each of the six tank types contains a Volume [Figure16(b)], which contains stored fuel and inert gas, andValve and Pump components [Figure 16(c) and (d)],which allow fuel to enter or leave the tank.

The valves contain resistive losses representing thevalve orifices and the pipes to which they are connected.The frictional losses in the pipes are implemented with asimple resistive component (R:pipe). Losses associatedwith the valve orifice are implemented using a flow-modu-lated resistor (FMR:orifice), the resistance of which ismodulated according to the valve position, which is con-trolled by an electronic actuator. Inertial switches are usedto allow or disallow the flow of fuel through the valves.

The Valve and Pump components contain controllersthat regulate the flow of fuel. The control logic is imple-mented as a simple text file, which actuates these compo-nents according to the quantity of fuel in the tanks (thesystem states). The simple aim of the control logic asimplemented is to keep the feed tank as full as possible atall times so that the engine does not run dry. The resultsof a simulation using this model and the control logic canbe seen in Figure 17, which shows the volume levels offuel in the tanks as fuel is consumed by the engine. A realfuel-management system would have additional require-ments, such as apportioning the remaining fuel betweenthe other tanks to control the center of gravity of the fuelto within appropriate longitudinal and lateral limits forthe flight mode to enhance agility, or maintain stabilitywhile optimizing range.

Benefits of Modelingthe System as a Bond GraphImplementing the fuel systemmodel as a bond graph providesseveral benefits over traditional sig-nal flow models that would typical-ly be produced using tools such asSimulink or EASY5. Perhaps themost important benefit is the adapt-able nature of the interfaces. Afterdevelopment and testing, compo-nent and subsystem models mustbe capable of being embedded indiverse environments and able toadapt to a range of conditions.

Consider the refuel system.Design of pipe geometry requiresthat pressure drops remain withincertain limits when maximum flowis forced through the pipes, so themodel must be able to accept aflow of fuel as an input at the refu-eling end, and the pipes mustdetermine the system pressure loss.However, to verify that the result-

ing design meets performance requirements relating tomaximum permissible refuel time, it is necessary toinstead perform simulations with an appropriate fuel pres-sure applied at the input and fuel flow rates calculatedthrough the pipes and valves. Signal flow models wouldrequire separate models, with a consequent doubling ofthe amount of testing required.

The need for additional models would also mean thatmore new models are used throughout the designprocess, with a risk of new errors being introduced. It isgenerally preferable to use existing tried-and-tested mod-els wherever possible.

Adaptable interfaces are also beneficial when inte-grating system models to produce a virtual integratedaircraft model. If all of the systems are produced as bondgraphs, all of the interfaces are guaranteed to transactpower, which greatly simplifies the task of stitchingmodels together. The task is further simplified by thefact that the causality of the interfaces can be changedwithout producing new models. Thus, an electric fuelpump, which accepts a constant voltage input duringfuel system simulations, can easily be adapted to output

FIGURE 17 Simulation of the hybrid fuel model. The volume of fluid in each tank changes asfuel is consumed by the jet engine. A simple logic controller aims to keep the feed tank (F3)replenished from the transfer tanks, using the valves and pumps. The steps at 0.9 m3 are dueto thresholds within the control logic that prevent valve chatter. The volume of fuel in the trans-fer tanks never quite reaches zero because residual fuel always remains due to tank geometry;this behavior is modeled by low-level thresholds in the control logic. A more sophisticated con-troller would attempt to also maintain the center of gravity within acceptable limits.

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

−0.1

Vol

ume

of F

uel (

m3 )

0 600 1,200 1,800 2,400 3,000 3,600 4,200 4,800 5,400 6,000

Time (s)F1 F2 F3 F4L WL1 WL2

This article is an introduction

to bond graphs.



a voltage when connected to a current-producing electri-cal distribution model.

The compact nature of bond graphs also makes themideal for representing fluid flow, which is just a particularform of energy transport. Unlike signal flow diagrams, it isnot necessary to take any particular care with the sign con-vention for flows at each port of a component model. Sincethe bond-graph sign convention is simply specified by the

direction of bonds impinging on a component, there is nodifficulty with removing components and embeddingthem in other parts of the system model.

It should be noted that producing bond-graph systemmodels does not require that control engineers give up tra-ditional design tools. Several bond-graph software pack-ages can convert bond graphs to formats that can beembedded directly within Matlab and Simulink—as m-files, mex files, and S functions—and within other softwarepackages in their native formats. Using appropriate toolsfor each stage of work generally yields better results thanattempting to use the most readily available tool for allaspects of system design.

We recommend creating models with specialist model-ing tools, running simulations within appropriate simula-tion harnesses, and performing control design withappropriate design tools. Pencil and paper can of course besubstituted for software at any stage of the developmentprocess except real-time simulation.

CONCLUSIONSThis article has presented an introduction to bond graphsfor control engineers. Although the notation can initiallyappear daunting, the bond graph method is firmlygrounded in the familiar concepts of energy and power.The essential element to be grasped is that bonds representpower transactions between components. Engineers canquickly become adept at applying the technique with just alittle practice.

The use of generic components and variables makes it asimple matter to model multidomain systems, allowingengineers to analyze complex problems and interactionsthat might normally be hidden by more traditionalapproaches to subsystem division and analysis.

The graphical nature of bond graphs enables modelersto easily identify potentially troublesome areas of theirsystem representations and to quickly determine the formof remedy that can make the model more appropriate forthe task in hand. The method is particularly beneficial inidentifying where supposedly simplifying assumptionsand approximations might be counterproductive.

Simple rules govern the transformation of bond graphsinto other system representations, and readily availablesoftware exists to perform conversions automatically. Sys-tems modeled as bond graphs can thus be easily integratedwith familiar control engineering toolsets.


Related Paradigms

B ond graphs are related to other modeling approaches. Two

of these of particular interest are the behavioral approach

and energy-based methods, such as dissipative systems and

the port-controlled Hamiltonian approach.

The behavioral approach [15], [16] and the bond-graph

approach provide two of the many ways of describing and

understanding dynamical systems. As discussed in more detail

in [17], the two approaches are similar in that

» the system description does not distinguish inputs and

outputs, but rather is viewed as a constraint on a set of

variables: the manifest variables in behavioral terms and

port variables in bond-graph terms

» systems are connected without assigning the input/output

structure beforehand

» state-variable descriptions are regarded as representa-

tions to be derived from the basic system representation

only when decisions have been made about which vari-

ables are to be regarded as inputs and outputs.

The two approaches are different in that

» the bond-graph approach is graphical, whereas the

behavioral approach is mathematical

» the bond-graph approach is explicitly based on energy

concepts and uses the systematic modeling approach

whereby physical system variables are classified accord-

ing to Table 1

» the behavioral approach handles distributed systems,

described by partial differential equations. The bond-

graph approach does not.

It is becoming recognized that energy-based methods are

relevant to control engineers, see [18]. Bond graphs are inher-

ently energy based and thus are related to other energy-based

methods, including dissipative systems [19]–[21] and, as shown

in [22], port-Hamiltonian systems.

Bond graphs are inherently energy based and thus are related

to other energy-based methods, including dissipative systems

and port-Hamiltonian systems.


The bond-graph approach is not unique in focusing onenergy and in determining causality after modeling. Alter-native approaches with these characteristics are discussedin “Related Paradigms.” However the bond-graphapproach is unique in combining these features with anintuitively appealing graphical modeling and causalityanalysis formulation.

We believe that the bond-graph method is a usefulmodeling tool, particularly well suited for describing phys-ical systems, and can provide a powerful way for engi-neers to analyze and solve the problems that they face.

REFERENCES[1] H. Paynter, “An epistemic prehistory of Bond Graphs,” in P. Breedveldand G. Dauphin-Tanguy, Eds., Bond Graphs for Engineers. Amsterdam:North-Holland, 1992, pp. 3–17.[2] W. Marquis-Favre and S. Scavarda, “Alternative causality assignmentprocedures in bond graph for mechanical systems,” Trans. ASME, J. Dyn.Syst., Meas. Control, vol. 124, no. 3, pp. 457–463, 2002.[3] D. Karnopp, D.L. Margolis, and R.C. Rosenberg, System Dynamics: Model-ing and Simulation of Mechatronic Systems, 3rd ed. New York: Horizon, 2000.[4] Octave, “Octave home page,” 2002 [Online]. Available: http://www.octave.org[5] A. Sharon, N. Hogan, and D.E. Hardt, “Controller design in the physicaldomain,” J. Franklin Inst., vol. 328, no. 5, pp. 697–721, 1991. [6] P.J. Gawthrop, “Physical model-based control: A bond graph approach,”J. Franklin Inst., vol. 332B, no. 3, pp. 285–305, 1995.[7] N. Hogan, “Impedance control: An approach to manipulation. Part I–The-ory,” Trans. ASME, J. Dyn. Syst., Meas. Control, vol. 107, pp. 1–7, Mar. 1985.[8] P. Gawthrop, M. Wallace, and D. Wagg, “Bond-graph based substructur-ing of dynamical systems,” Earthquake Eng. Struc. Dyn., vol. 34, pp. 687–703,May 2005.[9] P.J. Gawthrop, “Bond graph based control using virtual actuators,” J.Syst. Control Eng., vol. 218, pp. 251–268, Sept. 2004.[10] R.F. Ngwompo and P.J. Gawthrop, “Bond graph based simulation ofnonlinear inverse systems using physical performance specifications,” J.Franklin Inst., vol. 336, pp. 1225–1247, Nov. 1999.[11] R.F. Ngwompo, S. Scavarda, and D. Thomasset, “Physical model-basedinversion in control systems design using bond graph representation Part 1:theory,” J. Syst. Control Eng., vol. 215, pp. 95–103, Apr. 2001.[12] P.J. Gawthrop, “Bicausal bond graphs,” in Proc. Int. Conf. Bond GraphModeling Simulation (ICBGM’95), Las Vegas, 1995, vol. 27, pp. 83–88.[13] P.J. Gawthrop, “Physical interpretation of inverse dynamics usingbicausal bond graphs,” J. Franklin Inst., vol. 337, pp. 743–769, 2000.[14] J. Apkarian, A Comprehensive and Modular Laboratory for Control SystemsDesign and Implementation. Quanser Consulting, 1995. [15] J.W. Polderman and J.C. Willems, Introduction to Mathematical SystemTheory: A Behavioral Approach (Applied Mathematics Series 26). New York:Springer-Verlag, 1997.[16] J. Willems, “On interconnections, control, and feedback,” IEEE Trans.Automat. Contr., vol. 42, pp. 326–339, Mar. 1997.[17] P.J. Gawthrop, “Bond graphs in a behavioral context,” in Proc. 13thEuropean Simulation Symp.: Simulation in Industry, Marseille, France, 2001,pp. 745–749.[18] R. Ortega, A.J. van der Schaft, I. Mareels, and B. Maschke, “Puttingenergy back in control,” IEEE Control Syst. Mag., vol. 21, pp. 18–33, Apr. 2001.[19] J.C. Willems, “Dissipative dynamical systems, Part I: general theory,Part II: Linear system with quadratic supply rates,” Arch. Rational MechanicsAnal., vol. 45, no. 5, pp. 321–392, 1972. [20] D.J. Hill and P.J. Moylan, “Dissipative dynamical systems: Basic input-output and state properties,” J. Franklin Inst., vol. 208, no. 5, pp. 327–357, 1980.[21] R. Lozano, B. Brogliato, O. Egelund, and B. Maschke, Dissipative Systems:Analysis and Control. New York: Springer-Verlag, 2000.[22] D. Vink, D. Ballance, and P. Gawthrop, “Bond graphs in model matchingcontrol,” Math. Comput. Model. Dyn. Syst., vol. 12, no. 2-3, pp. 249–261, 2006.[23] A. Mukherjee, R. Karmakar, and A. Samantaray, Bond Graph in Modeling,Simulation and Fault Detection. New Delhi: I.K. International Publishing, 2006.[24] P.J. Gawthrop and L.P.S. Smith, Metamodelling: Bond Graphs and DynamicSystems. Hemel Hempstead, Herts, England: Prentice Hall, 1996.

[25] P. Breedveld and G. Dauphin-Tanguy, Eds. Bond Graphs for Engineers,Amsterdam: North-Holland, 1992.[26] F.E. Cellier, Continuous System Modelling. New York: Springer-Verlag, 1991.[27] F.E. Cellier and J.J. Granda, in Proc. 1995 Int. Conf. Bond Graph ModelingSimulation (ICBGM’95), Las Vegas, 1995, vol. 27. [28] J.J. Granda and F.E. Cellier, in Proc. Int. Conf. Bond Graph Modeling(ICBGM’93), La Jolla, CA, 1993, vol. 25.[29] J.J. Granda and G. Dauphin-Tanguy, in Proc. 1997 Int. Conf. Bond GraphModeling Simulation (ICBGM’97, Phoenix, AZ, 1997, vol. 29.[30] J.J. Granda and F. Cellier, in Proc. 1999 Int. Conf. Bond Graph ModelingSimulation (ICBGM’99), San Francisco, CA, 1999, vol. 31.[31] J.J. Granda and F. Cellier, in Proc. Int. Conf. Bond Graph Modeling Simula-tion (ICBGM’01), Phoenix, AZ, 2001, vol. 33.[32] Proc. Int. Conf. Bond Graph Modeling and Simulation (ICBGM’03), Simula-tion Series, Orlando, Florida, U.S.A.: Society for Computer Simulation, 2003.[33] Proc. Int. Conf. Bond Graph Modeling and Simulation (ICBGM’05), Simula-tion Series, New Orleans, U.S.A.: Society for Computer Simulation, 2005.[34] P.C. Breedveld, “Editorial: Special issue on bond graphs,” J. FranklinInst., vol. 328, no. 5-6, pp. 523–524, 1991.[35] P.J. Gawthrop and S. Scavarda, “Special Issue on Bond Graphs:Editorial,” in Proc. Institution of Mechanical Engineers Pt. I: J. Systems and Con-trol Engineering, Mar. 2002, vol. 216, pp. i–v.[36] W. Borutzky and P. Gawthrop, “Special issue on bond graph model-ling (Editorial),” Math. Comput. Model. Dyn. Syst., vol. 12, pp. 103–105,Apr.-June 2006.

AUTHOR INFORMATIONPeter J. Gawthrop ([email protected]) obtained hisB.A., D.Phil., and M.A. degrees in engineering science fromOxford University in 1973, 1977, and 1979, respectively. Fol-lowing a period as a research assistant with the Departmentof Engineering Science at Oxford University, he becameW.W. Spooner Research Fellow at New College, Oxford. Hethen moved to the University of Sussex as a lecturer, and latera reader in control engineering. In 1987, he took up the WylieChair of Control Engineering in the Department of Mechani-cal Engineering at Glasgow University, where he was afounding member of the Centre for Systems and Control.Since 2002, he has held an honorary research position at Glas-gow. He has also held visiting appointments in Australia atthe universities of New South Wales, Newcastle, Sydney, andthe Royal Melbourne Institute of Technology. In the UnitedKingdom, he has a visiting position at the University of Bris-tol. His research interests include continuous-time model-based predictive control, intermittent control, physiologicalcontrol systems, system identification, and system modelingand control using the bond-graph method. He can be con-tacted at the Centre for Systems and Control, Universityof Glasgow, Glasgow G12 8QQ, U.K.

Geraint P. Bevan graduated from the University ofSalford in 1996 with a bachelor of engineering degree inaeronautical engineering. He then joined British Aero-space and worked on several aircraft projects in theUnited Kingdom and the United States until 2003, whenhe became a research assistant at the University of Glas-gow, where he is also a part-time doctoral student. He iscurrently undertaking research to develop automaticautomotive controllers for active vehicle safety withinthe CEmACS project, part of the European Sixth Frame-work Programme.

