sca_bond_graph_paper.tex 293 2007-06-04 13:31:07Z maehne

Proposal for a Bond Graph Based Model of Computationin SystemC-AMS

Torsten Mähne∗, Alain Vachoux†

Laboratoire de Systèmes Microélectroniques (LSM)Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland

AbstractSystemC-AMS currently offers modelling formalismswith specialised solvers mainly focussing on the electri-cal domain. There is a need to improve its modellingcapabilities concerning conservative continuous timesystems involving the interaction of several physical do-mains and their interaction with nonconservative digitalcontrol components. Bond graphs unify the descriptionof multi-domain systems by modelling the energy flowbetween the electrical and non-electrical components.They integrate well with block diagrams describing thesignal processing part of a system. It is proposed todevelop an extension to the current SystemC-AMS pro-totype, which shall implement the bond graph method-ology as a new Model of Computation (MoC).

1 IntroductionAdvances in processing technologies allow to packan increasing number of features in single or multi-chip components (called System-on-a-Chip (SoC) orSystem-in-a-Package (SiP), respectively). The in-creasing complexity of the implemented functionalities,the diversity of possible implementations (e.g., digi-tal/analogue/RF hardware, software, Micro-Electro-Mechanical System (MEMS)), and the rapid evolutionof the needs (e.g., time to market, product diversity,compliance to standards) ask, on the one hand, forearly partitioning decisions that meet design constraintsand, on the other hand, for easy reuse and retargeting.Design complexity is usually tackled with the help ofmodelling and abstraction concepts, which in turn de-pend on the application domain and are grouped underthe generic term of Model of Computation (MoC).

SystemC [8] is a set of C++ classes and methodsthat provides means to develop behavioural and struc-tural models of digital hardware systems from abstractspecifications to Register Transfer Level (RTL) mod-els. SystemC is currently based on the event-drivenMoC, which can be used to model abstract behaviourssuch as transactions between processing units as wellas logic behaviours in digital hardware. However, byits very nature, SystemC can be extended to supportother MoCs [13] such as Synchronous Data Flow (SDF)or Finite State Machine (FSM).

∗torsten.maehne@epfl.ch†alain.vachoux@epfl.ch

When it comes to supporting Analogue and Mixed-Signal (AMS) systems, several attempts to extend Sys-temC have been done and are reviewed in Section 2.This paper presents a proposal to support bond graphbased models in SystemC-AMS as a way to model physi-cal systems such as MEMS while providing the necessaryabstraction level and simulation speed for system-leveldesign. Section 3 introduces the bond graph method-ology and its most recent theoretical advances. Sec-tion 4 discusses the application of this methodologyto SystemC-AMS, namely the requirements to supportbond graphs in SystemC-AMS.

2 AMS extensions for SystemCVachoux et al. [15] details the current SystemC-AMS prototype, which has been chosen to be de-veloped further under the auspices of the Open Sys-temC Initiative (OSCI) Analog/Mixed-signal WorkingGroup (AMSWG). Two formalisms, or MoCs, arecurrently implemented: a variant of the SynchronousData Flow (SDF) MoC is used to model signal pro-cessing dominated behaviours as well as more generalcontinuous-time behaviours using oversampled modelsand the Linear Network MoC provides a library of linearelectrical primitives for describing linear macro models.First experiments for an extension of SystemC-AMS tomodel conservative elements with non-linear dynamicalgebraic equations are presented in Einwich et al. [6].All these MoCs are synchronised with the discrete-eventSystemC simulation kernel through a synchronisationlayer, thereby allowing mixed-signal, or mixed-MoC,simulation. The applicability of the current prototypefor the modelling of MEMSs, which involve several phys-ical domains under the rule of energy conservation laws,was examined on micromechanical inertial sensors [10].The mechanical part was chosen to be modelled usinga non-conservative block diagram with feedback andsimulated using the SDF MoC instead of mapping themechanical elements like dampers, springs, and masseson their electrical counterparts resistor, capacitor, andinductor. The ∆C-U -converter was modelled using thelinear electrical elements.

In Al-Junaid et al. [1] an extended version of Sys-temC, called SystemC-A, is presented. SystemC-A pro-vides constructs to support user-defined Ordinary Dif-ferential Equations (ODEs) and Differential AlgebraicEquations (DAEs). Conservative systems are analysed

sca_bond_graph_paper.tex 293 2007-06-04 13:31:07Z maehne

using a Modified Nodal Analysis (MNA) by specifyingthe contributions of each conservative terminal to theJacobian and the right hand side of the DAE system.However, the presented solution has the drawback thatit required modifications to the standard SystemC ker-nel itself to couple the analogue solver with the discrete-event solver, instead of providing an abstraction layeron top of SystemC to allow the parallel integration ofvarious continuous time MoCs.

Another effort, called SystemC-WMS, is presentedin Orcioni et al. [12]. It only uses the standard Sys-temC hierarchical channels to implement analogue mod-ules, which communicate by exchanging energy wavesthrough wave channel interfaces. These interfaces avoidthe interconnexion problems commonly found in signal-flow representations of conservative blocks, where theinput or output direction of the to one port associatedacross (e.g., voltage) and through quantities (e.g., cur-rent) has to be decided at the implementation timeof the module. This is too early because the direc-tion of the information flow is determined from theinterconnexion of the modules, which is only known tothe simulator at elaboration time. The wave channelmethodology avoids this problem since incident waveshave always the input role and reflected waves theoutput role. Parallel and series connexion of modulesare achieved through special channels, which dispatchthe waves to the modules they connect together. Thisis similar to the scattering junction of Wave DigitalFilters (WDFs). The response of a module to the inci-dent waves is described through a, b parameters used inthe WDF theory. The methodology can be extended tocircuits with mildly non-linear elements. The observedsimulation performance of SystemC-WMS was howeverlimited because of the scheduling of several discreteevents for each time step preventing the independentexecution of the discrete and continuous parts.

Like in the “classical” AMS-Hardware DescriptionLanguages (HDLs) (e.g., VHDL-AMS and Verilog-AMS), SystemC-AMS and SystemC-A use a generalisednetwork model to represent conservative systems, inwhich modified Kirchoff’s voltage and current laws ac-count for the energy conservation. The generalised net-works represent accurately the physical structure of theconservative system but they have the drawback thatthey don’t visualise the computational structure of thesystem answering the question: Which quantities act asinput and which as outputs? The relation between theacross and through quantities at the terminals of eachcomponent are described using ODEs or DAEs. Theequation setup is quite low-level in SystemC-AMS (over-loaded equation method, which calculates procedurallythe new quantity values from the ones of the previoustime step) and SystemC-A (overloaded method, whichsetups for each time step the Jacobian matrix and righthand side of the DAE to be solved) compared to thedirect entry of DAEs in VHDL-AMS. The algorithmsused to solve the equation system for the whole gen-eralised network and thus the simulation performanceare similar for SystemC-AMS, SystemC-A, and classicAMS-HDLs simulators. One shared problem is that the

solvability of the implemented model is usually onlychecked insufficiently at model elaboration time: e.g.,VHDL-AMS only checks that the number of definedacross and through equations matches, which is a nec-essary but not sufficient condition. In Haase [7] theproblem of the possible definition of syntactically cor-rect simulation problems without a solution is discussedfrom a mathematical point of view to give reasons andestablish rules that may help to avoid these difficulties.

3 The Bond Graph MethodologyOne way to unify the description of a multi-domainsystem is the bond graph methodology [4, 9], which isused in mechanical engineering, mechatronics, controltheory, and to some extend in power electronics [3].Each conservative system can be transformed from itsdomain specific representation (e.g., electrical circuit,mechanical multi-body system, rigid bodies, fluidic net-works, thermal networks) to a bond graph representinggraphically the energy flow through the multi-domainsystem. As an example the transformation of a simpleelectrical circuit to an equivalent bond graph annotatedwith causality and then derived computational structureis shown in Figure 1.

The bonds (half-arrows) represent the energy flowbetween the ports of the elements of the graph, each onehaving associated effort e and flow f variables, whichproduct gives the power:

P (t) = e(t) · f(t) (1)

They are also called power variables for this reason.By definition, the half-arrow points into the direction,in which the power flows for positive e and f . Forthe description of dynamic systems, two other variabletypes are important, which belong to the class of energyvariables: The (generalised) momentum p(t) is definedas the time integral of an effort:

p(t) =∫ t

e(t) dt = p0 +∫ t

t0

e(t) dt (2)

The (generalised) displacement q(t) is defined as thetime integral of a flow:

q(t) =∫ t

f(t) dt = q0 +∫ t

t0

f(t) dt (3)

with p0 and q0 the initial values of p and q, respectively,at the time t0.

The energy E(t) is defined as the time integral of thepower P (t):

E(t) =∫ t

P (t) dt =∫ t

e(t)f(t) dt (4)

The energy can be also expressed as a function of theenergy variables by inserting the differential form of (2)and (3) in (4):

E(q) =∫ q

e(q) dq E(p) =∫ p

f(p) dp (5)

sca_bond_graph_paper.tex 293 2007-06-04 13:31:07Z maehne

R1

C1

L1

R2

C2 U(t)

L2

(a) Simple electrical circuit

01

1

1R : R1

C : C1

I : M1

R : R2 C : C2

I : M2

Se : U(t)10

9

8

5

41

2

3

6 7

(b) Bond graph with annotated causality

U(t) q2 q7 p3 p9

e10 e2 e7 f3 f9

f1 f4 f8

e1 f5

f6

e6

e5

e8 e4

e9 = p9 e3 = p3 f2 = q2 f7 = q7 f10

(c) Dependency graph representing the computationalstructure of the bond graph

Figure 1: Example of the transformation of an electrical circuit to an equivalent bond graph annotated withcausality and then derived computational structure

The definition for the generalised power and energyvariables is independent of a particular physical domain.

Three generalised 1-port elements represent the resis-tive R, inertial I, and capacitive C elements independentof the considered physical domain. Energy sources aremodelled as effort source Se and flow source Sf elements.Quantity transformations (also across domain bound-aries) are represented through the transformer TF andgyrator GY 2-port elements, while still ensuring energyconservation. Multi-port junction elements representexplicitly the connexion of elements, which are exposedto a common effort (0-junction) or a common flow (1-junction). All elements (Table 1) can have non-linearcharacteristic equations.

One main advantage of bond graphs is that they canbe annotated in a systematic way with the causality foreach bond (flow-out or effort-out causality), which vi-sualises the computational structure of the bond graphand allows to sort the element equations in the rightorder for model execution (Figure 1c). The perpendic-ular stroke at one end of a bond states that at thisside the effort variable e is known (it acts as an in-put) and f can be calculated as a function f = Φi(e).Consequently, the flow f is known on the other sideof the bond and acts as an input to a function to cal-culate the effort e: e = Φj(f). Table 1 summarisesthe basic elements of a bond graph with the permittedcausality assignments and their defining equations. Theassigned causalities allow some further formal checks onthe model: the number of states and non-states in thesystem, the presence of algebraic loops during modelexecution, or if it is an ill-posed model.

The causality assignment allows also for a well in-tegration of bond graphs with signal flow graphs. Anexample is given in Figure 2 in form of a car wheelmodel of an electronically controlled suspension systemincorporating a semi-active damper and a fast load-leveller [9]. Both ways of representing the systems aregiven: the “classic” domain-specific way and the wayusing bond graphs for the energy conservation partand block diagrams for the signal processing part. Theblocks in the signal flow graph can take the calculated

power or energy variables as input and can modulatethe sources or element parameters of the bond graph.

There are extensions to classical bond graphs, whichallow, e.g., hierarchical models with more abstractword bond graphs [4] or handling of discrete switch-ing due to external signals by extending the set ofbond graph elements with idealised controlled junc-tions leading to so-called hybrid bond graphs [2, 11].In Karnopp et al. [9] the generalisation of the basicbond graph elements R, I, and C to multiport fields isintroduced and junction structures as an assembly ofpower-conserving 0- and 1-junctions, transformers TF ,and gyrators GY are presented, which allow the effec-tive modelling of complex multiport systems combiningstructural details and clarity with a convenient visu-alisation. The usefulness of these concepts is demon-strated on transducer, nonlinear mechanical systems,distributed parameter systems (e.g., a beam), mag-netic circuits and thermofluid systems examples. Theseproperties of bond graphs make them attractive forthe design and verification of Systems-on-Chips (SoCs)since they unify and thus ease the description of elec-trical and non-electrical system components and theirinteractions. They also offer a higher level of abstrac-tion and thus possible run-time advantages over thementioned classical AMS-HDLs.

Besides specialised bond graph tools such as MTT,20-sim, and Enport (reviews can be found, e.g., onhttp://www.bondgraphs.com/), there are also efforts tointegrate bond graph support in widely used mathemat-ical tools such as MATLAB/Simulink (e.g., BondLab)or Mathematica (e.g., Bond graph tool box for Math-ematica). Another approach is to use the capabilitiesof an AMS-HDL to represent bond graphs. In Cellierand McBride [5] the implementation of causal/a-causalbond graphs of the publicly available BondLib for Mod-elica/Dymola is presented. Its usage is demonstrated ona position control system involving a hydraulic motor.However, Modelica’s capabilities are weak on the dis-crete event side, which is one reason why it is not usedin the microelectronics community and thus not suitablefor mixed-signal SoC design. Pêcheux et al. [14] showsthat bond graphs can be represented in VHDL-AMS.

sca_bond_graph_paper.tex 293 2007-06-04 13:31:07Z maehne

Table 1: Basic elements of a bond graph with their legal causality assignments

Defining relation Examples from electrical, mech-

Class Name Symbol General case Linear case anical, and hydraulic domains

1-port effort source See

fe(t) given, f(t) arbitrary voltage, force, pressure sources

flow source Sfe

ff(t) given, e(t) arbitrary current, velocity, volume flow rate

sources

(generalised)resistor

e

fR

e

fR

e = ΦR(f)

f = Φ−1R (e)

e = Rf

f = 1Re

electrical resistor, translationaldamper, rotational damper, hy-draulic throttle

(generalised)capacitor

e

f = qC

e

f = qC

q = ΦC(e)

e = Φ−1C (q)

q = Ce

e = 1C q

electrical capacitor, spring, tor-sion bars, gravity tank, accumula-tor

(generalised)inductor

e = p

fI

e = p

fI

p = ΦI(f)

f = Φ−1I (p)

p = If

f = 1I p

electric inductor, mass, rotatingdisk with moment of inertia, sec-tion of a fluid-filled pipe with fluidinertia

2-port (generalised)transformer

e1

f1TF

e2

f2e1

f1TF

e2

f2

e1 = me2, f2 = mf1

e2 = 1me1, f1 = 1

mf2

electrical transformer, ideal rigidlever, gear pair, hydraulic ram

(generalised)gyrator

e1

f1GY

e2

f2e1

f1GY

e2

f2

e1 = rf2, e2 = rf1

f2 = 1r e1, f1 = 1

r e2

electrical gyrator, gyroscope,voice coil transducer

n-port flow junction,0-junction,common effortjunction

0e1

f1

ei fi

ej

fj

en fne1 = . . . = ei = . . . = en

fi = −(∑n

j=1,j 6=i fj

) parallel connexion of electricalconductors; situation involving asingle force and n velocities sum-ming up to zero; parallel connex-ion of hydraulic passages

effort junction,1-junction,common flowjunction

1e1

f1

ei fi

ej

fj

en fn f1 = . . . = fi = . . . = fn

ei = −(∑n

j=1,j 6=i ej

) series connexion of electrical con-ductors; dynamic equilibrium ofn forces associated with a singlevelocity; series connexion of hy-draulic passages

They are used in the models of a Pb/Fe battery and acomplex airbag SoC including a MEMS accelerometer,a digital control, a thermal network, a laser diode, andthe chemical reaction to inflate the cushion. The imple-mentation of bond graphs is not very efficient becausethe support of VHDL-AMS for continuous systems islimited to non-conservative signal flow description basedon free quantities and energy conserving generalisednetworks. Thus, from a bond graph point of view, effortand flow variables of a port have to be referred fromthe across and through quantities defined between twoterminals. Satisfying Kirchhoff’s voltage and currentlaws inside the generalised networks leads, in the caseof bond graphs where these laws are already ensuredthrough the 0- and 1-junctions, to duplicated equations.Causality cannot be assigned to the bonds and thus not

be benefited from. All three aspects have a negativeimpact on the simulation performance of the model.

4 Application of the Bond GraphMethodology to SystemC-AMS

The goal of this work is to improve the modelling andsimulation capabilities of SystemC-AMS regarding con-servative continuous time components and their inter-action with discrete time (digital) control components.For a high simulation performance, the models shouldavoid the setup of global DAE systems, which require acomplex implicit solver. But this is the case for the cur-rent available MoCs for linear and nonlinear elementsin energy conserving generalised networks. For non-

sca_bond_graph_paper.tex 293 2007-06-04 13:31:07Z maehne

F0(t)

Fc

vc

vb, pb

vw, pw

kb,c

kw,0xw,0

xb,w

Fc = Bxb,w −BAvb

vc = −hvb − g (xc − xb,w)

xc(t)

xb,w

vb

Sf

v0(t)

Body

Wheel mw

mb

db,w F

U

xU

ControllerSoC

xU

. . .

(a) “Classic” domain-specific system representation

Sf

0

1

C : kw,0

I : mw

pw

vw

xw,0

0

1

Se

F0(t)

v0(t)

I : mb

pb

vb

1

C : kb,c

0 Sf

R : db,w

xc(t)xb,w

vb

Fc

vc

xb,w

qb,cg

+

+−

−

+

BA

B

−

h

−

ADCxU

FU

xU ADC

DAC

DAC

∫

Controller SoC

(b) Representation using bond graphs for the energy conservationpart and block diagrams for the signal processing part

Figure 2: Car wheel model of an electronically controlled suspension system incorporating a semi-active damperand a fast load-leveller [adapted from 9]

electrical elements, also the link to the original physicaldomain gets lost due to the mapping to their electricalcounterparts. The models should rather order the equa-tions to allow for a fast procedural execution (similarto the static scheduling of the SDF MoC), which calcu-lates at each step the unknowns for the next orderedequations and thus solves the equation system explicitly.Bond graphs meet these requirements if the assignedcausalities are exploited. They keep the link to the mod-elled physical domain through the units attached to thepower/energy variables and the element parameters.

The implementation of the bond graph MoC willprofit from the fact that SystemC-AMS is a library ontop of the fully-featured C++ language. The causalityof a bond graph can be completed in a systematic wayat elaboration time using, e.g., the Sequential CausalityAssignment Procedure (SCAP) [9] taking into accountthe known fixed, preferred, or free causality of the ele-ment ports. The user can modify the initial causalityassignments to guide the algorithm and thus to opti-mise its result. The derived computational structureallows to sort in advance the element equations in theright order for model execution. It allows also for someformal checks to audit the model at elaboration time:the number of states and non-states in the system, thepresence of algebraic loops during model execution, orif it is an ill-posed model. This can guide the designerduring model development by providing him more in-sight into the physical system and drawing thus hisattention to potential problems in his design. Algebraicloops can be broken by inserting a one time step delaysimilarly to what is done in the SDF MoC.

For a clear specification of the physical domain, eachvariable needs to be annotated with its measurementunit allowing for dimensional analysis and thus discover-ing of illegal connexion of incompatible ports and calcu-lations involving incompatible quantities. A promisingcandidate to fulfil this requirement is the quantitative

units library, which allows for unit analysis already atcompile time and thus limits its impact on the execu-tion performance and which was recently accepted tobe part of the boost project (http://www.boost.org).

The integration and synchronisation with the otherMoCs, especially the SDF MoC and the Discrete-Event(DE) MoC, which are used to model the signal process-ing/controlling parts of the system, is needed (Figure 3).This might require small adjustments in the synchroni-sation layer of the SystemC-AMS prototype. It shallnot require modifications of the SystemC kernel. Thestrong interaction between the analogue and digitalparts of a SoC requires to take into account discreteswitching of the energy flows inside the bond graphsdue to external signals and thus causality changes dur-ing the model execution. The research in this field ofhybrid bond graphs, which incorporate this local switch-ing capability, is still on-going and needs more effortsto find ways to efficiently reassign causality and toregenerate the computational model at runtime whenjunction switching occurs [2]. Another important aspectis to allow for hierarchical modelling thus implement-ing aspects of word bond graphs [4]. It has to be alsoevaluated in which way multi-port field elements andjunction structures can be supported efficiently by thebond graph extension without sacrificing too much sim-ulation performance.

5 Conclusions and Outlook

This paper proposes the development of a bond graphextension to SystemC-AMS, to improve its modellingcapabilities in the field of energy conserving continuoustime systems. To this end, the current state of the com-peting AMS extensions for SystemC was presented witha focus on the SystemC-AMS prototype. It showed thatthe current solutions for conservative systems modelling

sca_bond_graph_paper.tex 293 2007-06-04 13:31:07Z maehne

. . .

. . .

Under development in SystemC-AMSprototype of OSCI AMS working group

Scope of thiswork

Provided bySystemC

Synchronuousmulti-ratedata flow(SDF)

modellingformalism

Conservativelinear

network (LN)modellingformalism

Bondgraph (BG)modellingformalism

Clustering andstatic

schedulingLinear solver

Causalityassignment

and BG solver

Synchronisation layer

Discreteevent (DE)Models of

Computation(CSP, FSM,

etc.)

SystemC 2.x simulation kernel

. . .

. . .

Figure 3: Architecture of SystemC-AMS showing theMoCs provided by the SystemC kernel and the AMS ex-tension as well as the synchronisation layer

are quite low-level with equation setups and analysismethods similar to classic analogue circuits simulatorslike SPICE, which causes a simulation performancepenalty. The bond graph methodology offers advan-tages for the design and verification of AMS SoCs sinceit unifies the description of electrical and non-electricalsystem components on various levels of abstraction. Thecausality analysis of a bond graph allows to optimisethe model execution and gives the designer valuableinsight into the computational structure of his model.The requirements for the planned bond graph extensionwere sketched out and need to be further detailed toderive a specification to be used in the following designand implementation phases.

Acknowledgements

This work has been funded by the Hasler Stiftung underproject number 2161.

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