Fishwick P.A. (editor) Handbook of Dynamic System Modeling

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25-14 Handbook of Dynamic System Modeling

We can now write down ﬂow balance equations by inspection of the state transition rate diagram. First,

for any state (n

, n

) with n

> 0 and n

> 0wehave

λπ(n

−1, n

) +µ

π(n

+1, n

−1) +µ

π(n

, n

+1) −(λ +µ

+µ

)π(n

, n

) = 0 (25.41)

Similarly, for all states (n

, 0) with n

> 0wehave

λπ(n

−1, 0) +µ

π(n

,1)− (λ + µ

)π(n

,0)= 0 (25.42)

and for states (0, n

) with n

> 0:

π(1, n

−1) +µ

π(0, n

+1) −(λ +µ

)π(0, n

) = 0 (25.43)

and ﬁnally for state (0, 0):

π(0, 1) −λπ(0, 0) = 0 (25.44)

In addition, the state probabilities must satisfy the normalization condition:

∞



i=0

∞



j=0

π(i, j) = 1 (25.45)

The set of Eqs. (25.41)–(25.45) can readily be solved to give

π(n

, n

) = (1 −ρ

)ρ

·(1 −ρ

)ρ

(25.46)

where

, ρ

with the usual stability conditions 0 ≤ρ

< 1 and 0 ≤ρ

< 1. Clearly, ρ

is the trafﬁc intensity at the ﬁrst

node. Moreover, ρ

is the trafﬁc intensity of node 2, since the throughput (departure rate) of node 1,

which is λ, is also the arrival rate at node 2. Observe that if we view each of the two nodes as separate

M/M/1 systems with stationary state probabilities π

) and π

), respectively, we get

) = (1 −ρ

)ρ

, π

) = (1 −ρ

)ρ

We see, therefore, that we have a simple product form solution for this two-node network:

π(n

, n

) = π

) ·π

) (25.47)

In fact, the product form in Eq. (25.47) is a consequence of Burke’s theorem, which also asserts that the

queue length at time t in an M/M/1 system is independent of its departure process prior to t.Itallowsus

to decouple the two nodes, analyze them separately as individual M/M/1 systems, and then combine the

results as in Eq. (25.47). It is straightforward to extend this solution to any open Markovian network with

no customer feedback paths. Let us brieﬂy indicate why customer feedback may create a problem. Suppose

that in a simple M/M/1 system customers completing service are returned to the queue with probability p,

or they depart with probability (1 −p). The difﬁculty here is that the process formed by customers entering

the queue, consisting of the superposition of the external Poisson process and the feedback process, is not

Poisson (in fact, it can be shown to be characterized by a hyperexponential distribution). Remarkably, the

departure process of this system is still Poisson (see also Walrand, 1988).

Returning to the general open network model described above, it was established by Jackson (1963)

that a product form solution still exists even if customer feedback is allowed. This type of model is also

referredtoasaJackson network. What is interesting in this model is that an individual node need not have

Queueing System Models 25-15

Poisson arrivals (due to the feedback effect), yet it behaves as if it had Poisson arrivals and can therefore

still be treated as an M/M/1 system. Thus, we have the general product form solution:

π(n

, n

, ... , n

) = π

) ·π

) ···π

) (25.48)

where π

) is the solution of an M/M/1 queueing system with service rate µ

and arrival rate given

by λ

in the solution of Eq. (25.40). It is worth mentioning that the same result also holds for M/M/m

queueing systems, i.e., systems with m

≥1 servers at node i. To guarantee the existence of the stationary

probability distribution in Eq. (25.48), we impose the usual stability condition for each node:

= r



j=1

< m

(25.49)

The open network model we have considered can be extended to include state-dependent arrival and

service processes. In particular, Jackson considered the case where an external arrival process depends on

the total number of customers (n

+···+n

), and the ith service rate depends on the queue length at

node i (Jackson, 1963). Although the analysis becomes considerably more cumbersome, the stationary

state probability can still be shown to be of the product form variety.

25.7.3 Closed Queueing Networks

A closed queueing network is one with a ﬁnite population of N customers. From a modeling standpoint,

a closed network may be obtained from the open network model of the previous section by setting

= 0 and



j=1

= 1 for all i = 1, ... , M

In this case, no external arrivals occur and no customers can leave the system. Under these conditions, the

state variables of the system, X

, ... , X

, must always satisfy



i=1

=N. Thus, the state space is ﬁnite

and corresponds to the number of placements of N customers among M nodes, given by the binomial

coefﬁcient



M +N − 1

M −1



(M +N − 1)!

(M −1)!N!

In addition, if we balance customer ﬂows as in Eq. (25.40) we get



j=1

, i = 1, ... , M (25.50)

There is an important difference between the set of equations (25.40) and that in Eq. (25.50). In Eq. (25.40),

we have M linearly independent equations, from which, in general, a unique solution may be obtained. In

contrast, the absence of external arrival rate terms r

in Eq. (25.50) results in (M −1) linearly independent

equations only. Thus, the solution of Eq. (25.50) for λ

, ... , λ

involves a free constant. For instance,

suppose we choose λ

to be this constant. We may then interpret λ

, i =1, as the relative throughput of

node i with respect to the throughput of node 1.

It turns out that this class of networks also has a product form solution for its stationary state prob-

abilities π(n

, ... , n

), with the values of n

, ... , n

constrained to satisfy



j=1

=N (Gordon and

Newell, 1967). For simplicity, we limit our discussion here to single-server nodes, although the result also

applies to the more general case where node i consists of m

servers. Solving the ﬂow balance equations in

this case gives:

π(n

, ... , n

) =

C(N)

···ρ

(25.51)

25-16 Handbook of Dynamic System Modeling

where ρ

=λ

/µ

with λ

obtained from the solution of Eq. (25.50) with a free constant (arbitrarily chosen)

and C(N) is a constant dependent on the population size N, which is obtained from the normalization

condition

C(N)



,...,n

···ρ

= 1

Thus, to obtain the stationary state probability distribution in Eq. (25.51) we need to go through several

steps of some computational complexity. First, the solution of the linear equations Eq. (25.50) to determine

the parameters λ

, ... , λ

must be obtained. This solution includes a free constant, which must next

be selected arbitrarily. This selection will not affect the probabilities in Eq. (25.51), but only the values

of the parameters ρ

in the product form. Lastly, we must compute the constant C(N), which is usually

not a trivial computational task. There exist several computationally efﬁcient algorithms to carry out this

computation, e.g., see Buzen (1973) where the following recursive relationship is exploited:

(k) = C

i−1

(k) + ρ

(k − 1), i = 2, ... , M, k = 2, ... , N (25.52)

with initial conditions C

(k) =ρ

, k =1, ... , N and C

(1) =1, i =1, ... , M, from which C(N)is

obtained as C(N) =C

(N).

It can also be shown that the utilization of node i when the population size is N is given by

[1 −π

(0)] = ρ

C(N −1)

C(N)

Expressions for other performance measures can similarly be derived in terms of the parameters ρ

and

C(k), k =1, ..., N.

25.7.3.1 Mean Value Analysis

If we are only interested in obtaining performance measures such as the network throughput and the

mean values of the queue length and system time distributions at nodes, then Mean Value Analysis (MVA),

developed in Reiser and Lavenberg (1980), bypasses the need for computing the normalization constant

C(N). MVA exploits a simple relationship between the average customer system time at a network node

and the average queue length at that node. Speciﬁcally, consider a customer arriving at node i, and let

the average system time the customer experiences at i. Moreover, let

be the average queue length seen

by that arrival. Observe that

where 1/µ

is the mean service time at node i. In other words, the customer’s system time consists of two

parts: his own service time, and the total time required to serve all customers ahead of him. It can be

shown that in a closed queueing network with N customers,

is the same as the average queue length at

i in a network with (N −1) customers. Intuitively, what a typical customer sees is the network without

that customer. Therefore, if we denote by

(N) and

(N) the average queue length and average system

time at node i, respectively, when there are N customers in the network, we obtain the following recursive

equation:

(N) =

[1 +

(N − 1)], i = 1, ..., M (25.53)

with initial condition

(0) =0, i =1, ..., M. In addition, we can use Little’s Law (25.24) twice. First, for

the whole network, we have

N =



i=1



(N)

(N) (25.54)

Queueing System Models 25-17

where 

(N) is the node i throughput when there are N customers in the network and N the ﬁxed number

of customers in the network. Note that 

(N)/

(N) =λ

/λ

for all i, j where λ

, λ

are obtained from

Eq. (25.50), so that 

(N) is obtained from Eq. (25.54) as



(N) = N





i=1





(N)



−1

(25.55)

The second use of Little’s Law is made for each node i and we get

(N) = 

(N)

(N), i = 1, ..., M (25.56)

The four equations (25.53)–(25.56) deﬁne an algorithm through which

(N),

(N), and 

(N) can be

evaluated for various values of N =1, 2, ....

25.7.4 Product Form Networks

The open and closed queueing network models considered thus far are referred to as product form net-

works. Obviously, this is because the stationary state probabilities π(n

, ..., n

) can be expressed as

products of terms as in Eq. (25.48) or Eq. (25.51). The ith such term is determined by parameters of

the ith node only, which makes a decomposition of the network easy and efﬁcient. Even though we

have limited ourselves to Markovian networks, it turns out that there exist signiﬁcantly more com-

plicated types of networks, which are also of the product form variety. The most notable type of

product form network is one often referred to as a BCMP network after the initials of the researchers

Baskett, Chandy, Muntz, and Palacios who studied it (Baskett et al., 1975). This is a closed network

with K different customer classes. Class k, k =1, ..., K, is characterized by its own routing proba-

bilities and service rates and four different types of nodes are allowed: (i) A single-server node with

exponentially distributed service times and µ

=µ

for all classes. In addition, an FCFS queueing dis-

cipline is used. This is the simplest node type which we have used in our previous analysis as well.

(ii) A single-server node and any service time distribution, possibly different for each customer class,

as long as each such distribution is differentiable. The queueing discipline here must be of the processor-

sharing (PS) type (i.e., each customer in queue receives a ﬁxed“time slice”of service in round-robin fashion

and then returns to the queue to wait for more service if necessary; the PS discipline is obtained when

this time slice is allowed to become vanishingly small). (iii) The same as before, except that the queueing

discipline is of the Last-Come-First-Served (LCFS) type with a preemptive resume (PR) capability. This

means that a new customer can preempt (i.e., interrupt) the one in service, with the preempted customer

resuming service at a later time. (iv) A node with an inﬁnite number of servers and any service time

distribution, possibly different for each customer class, as long as each such distribution is differentiable.

In this type of network, the state at each node is a vector of the form X

=[X

, X

, ..., X

], where X

is the number of class k customers at node i. The actual system state is the vector X =[X

, X

, ..., X

]

and, assuming the population size of class k is N

, we must always satisfy the condition



j=1

for

all k. The actual product form solution can be found in Baskett et al. (1975) or several queueing theory

textbooks such as Trivedi (1982).

25.8 Non-Markovian Queueing Systems

Beyond queueing systems whose arrival and service processes are modeled through exponential distribu-

tions, analytical results are very limited. There are several sophisticated techniques for approximating the

distributions of these processes using combinations of exponential ones or by approximating interesting

performance measures by means of the ﬁrst two moments only. These and related techniques for analyzing

complex queueing models are beyond the scope of this chapter and the reader is referred to specialized

books, including Asmussen (2003), Bremaud (1981), Chen and Yao (2000), Kelly (1979), Kleinrock (1975),

25-18 Handbook of Dynamic System Modeling

Neuts (1981), Walrand (1988). We limit ourselves to one well-known and very useful analytical result that

applies to the M/G/1 queueing system. In particular, it is possible to derive a simple expression for the

average queue length E[X] of such a system. This is known as the Pollaczek–Khinchin formula (or PK

formula):

E[X] =

1 −ρ

−

2(1 −ρ)

(1 −µ

) (25.57)

where 1/µ and σ

are the mean and variance, respectively, of the service time distribution and ρ =λ/µ

is the trafﬁc intensity as deﬁned in Eq. (25.9), λ being the Poisson arrival rate. We can immediately see

that for exponentially distributed service times, where σ

=1/µ

, (Eq. 25.57) reduces to the average queue

length of the M/M/1system,E[X] =ρ/(1 −ρ), as in Eq. (25.33).

Finally, it is worth mentioning that many software tools have been developed over the years to facilitate

the process of modeling queueing systems and of estimating performance measures such as throughput,

mean queue lengths, and mean delays. Some are based on discrete event simulation (Extend and SimEvents

are two of the most recent commercial products of this type), while others are based on approximation

methods (RESQ and QNA are such examples). More specialized tools have also been developed for

particular application areas such as communication networks (e.g., Opnet) and manufacturing systems

(e.g., MPX).

References

Asmussen, S., Applied Probability and Queues, 2nd ed., Wiley, New York, 2003.

Baskett, F., Chandy, K. M., Muntz, R. R., and Palacios, R. R., Open, closed, and mixed networks of queues

with different classes of customers, J. ACM, 22(2), 248–260, 1975.

Bremaud, P., Point Processes and Queues, Springer, New York, 1981.

Burke, P. J., The output of a queueing system, Oper. Res., 4, 699–704, 1956.

Buzen, J. P., Computational algorithms for closed queueing networks with exponential servers, Commun.

of ACM, 16(9), 527–531, 1973.

Cassandras, C.G., and Lafortune, S., Introduction to Discrete Event Systems, Kluwer, Norwell, MA, 1999.

Chen, H., and Yao, D.D., Fundamentals of Queueing Networks, Springer, New York, 2000.

Gordon, W. J., and G. F. Newell, Closed queueing systems with exponential servers, Oper. Res., 15(2),

254–265, 1967.

Jackson, J. R., Jobshop-like queueing systems, Manage. Sci., 10(1), 131–142, 1963.

Kelly, F. P., Reversibility and Stochastic Networks, Wiley, New York, 1979.

Kleinrock, L., Queueing Systems, Volume I: Theory, Wiley, New York, 1975.

Lindley, D. V., The theory of queues with a single server, Proc. Cambridge Philos. Soc., 48, 277–289, 1952.

Little, J. D. C., A proof of L =λW, Oper. Res., 9(3), 383–387, 1961.

Neuts, M. F., Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach,Dover,NewYork,

1981.

Reiser, M. and Lavenberg, S. S., Mean-value analysis of closed multichain queueing networks, J. of ACM,

27(2), 313–322, 1980.

Stidham, S., Jr., A last word on L =λW, Oper. Res., 22, 417–421, 1974.

Trivedi, K. S., Probability and Statistics with Reliability, Queuing and Computer Science Applications,

Prentice-Hall, Englewood Cliffs, NJ, 1982.

Walrand, J., An Introduction to Queueing Networks, Prentice-Hall, Englewood Cliffs, NJ, 1988.

Port-Based Modeling of

Engineering Systems in

TermsofBondGraphs

Peter Breedveld

University of Twente

26.1 Introduction .................................................................26-1

Port-Based Modeling versus Traditional

Modeling

•

Dynamic Models of Engineering Systems

26.2 Structured Systems: Physical Components and

Interaction ....................................................................26-4

26.3 Bond Graphs.................................................................26-5

Appearance

•

Elementary Behaviors

•

Causality

26.4 Multiport Generalizations............................................26-21

Multiport Storage Elements

26.5 Conclusion....................................................................26-28

26.1 Introduction

26.1.1 Port-Based Modeling versus Traditional Modeling

To be concise, port-based modeling of dynamic behavior will be introduced from the point of view that

the reader is well aware of traditional modeling techniques as well as the common pitfalls of modeling, viz.:

•

confusion of physical components with ideal concepts (conceptual elements), and

•

confusion of modeling with model manipulation, where modeling is the mere decision process of

which aspects should be taken into account to arrive at a competent model for a speciﬁc prob-

lem context, while model manipulation is any kind of transformation of the representation of a

model into a form that allows better insight in the particular aspect under study (none of these

transformations should change the physics properties of the model).

The key idea in any modeling approach that starts from the a priori knowledge about the physics properties

of the system to be modeled is that a conceptual separation is made between various fundamental behaviors

that are (considered to be) relevant for a given problem context. Although this step is always present, it is

often preceded by a step in which the system to be modeled is subdivided into subsystems on the basis of

aspects of function or conﬁguration. A pump (drive) system, for example, can be seen as an interconnection

of an electric power source (an ampliﬁer), an electric motor, a transmission, and a mechanism for

displacement of a ﬂuid (a gear box and a load). This is the ﬁrst conceptual level of observable, functional

components. The total behavior consists not only of the observable dynamic interaction between these

subsystems, but also of the conceptual interaction between the fundamental behaviors that constitute the

behavior of these subsystems. Already when modeling at this level, it is useful to consider both forms

of interaction from the point of view of bilateral relations, instead of the unidirectional input–output

relations that are often used, thus implicitly assuming no“back-effect”(Figure 26.1). This bilateral relation

26-1

26-2 Handbook of Dynamic System Modeling

Submodel 1 Submodel 2

FIGURE 26.1 Bilateral relation.

FIGURE 26.2 Double RC network.

and its relation to the concept of energy is the key idea behind a port-based modeling approach, as will be

one of the main objectives of this chapter.

To give a rather simple example: when modeling the behavior of a double resistor-capacitor (RC)

network (Figure 26.2) assuming connection to a ﬁt source, e.g., a sine generator, the ﬁrst step is to

identify the components (two resistors and two capacitors). Unlike many other domains of physics, the

fabrication of components applied in electrical circuits is given so much attention that, in a context of

normal operation (i.e., neglecting radiation and high frequencies), the components map during modeling

in a one-to-one way to the conceptual elements with the same name (resistor R and capacitor C) that

represent two basic physical concepts, viz. Ohm’s law for dissipation and electrical storage (of charge and

electrostatic energy), respectively. It is discussed later that the use of this conceptual resistor implies an

implicit assumption that the variations of the temperature are such that they may be neglected in the given

context, such that “dissipatable” free energy can be used instead of globally conserved “true” energy. If

one would identify each RC network as a subsystem and consider the transfer function between the input

and the output voltages, the total transfer function is not equal to the product of these transfer functions,

unless one includes a so-called separation ampliﬁer. This is due to the bilateral relation between all these

elements and these two subsystems in particular.

Note that the adjective“physical”(physical system, physical model, and physical modeling) is not used in

the sense of “concrete”within the contextof dynamic modeling, and thus this text,but in the sense of “obey-

ing the laws of physics.” As of now the terminology “physical properties” will be used to address what has

been called “physics properties” earlier. Consequently, a physical model in this sense can be rather abstract.

In most classical modeling approaches, most of the relations are thought to be unilateral (input–output),

in other words, the back-effect is implicitly considered negligible. In those cases where the back-effect

cannot be neglected, it is thought of as a separate, unilateral relation, often addressed as the feedback.

Next all conceptual behaviors are commonly linked at the conceptual level into a set of differential and

algebraic equations using common variables and balance equations. In principle, the algebraic equations

can be used to eliminate a number of variables, such that a set of (partial) differential equations remains.

In some simple cases (e.g., linear systems), such a set may have an analytical solution. However, in most

cases such an analytical solution does not exist and numerical integration (simulation) is used to obtain

an approximate solution. This is discussed extensively in Chapter 17.

However, if the model is written in terms of equations, an important change in the nature of the model

representation is implicitly made: from a simultaneous, by deﬁnition graphical, representation of a system

model in terms of interconnected subsystems and interconnected conceptual elements, the representation

becomes sequential, in the sense that the equations can only be read one after the other. As the parts

of the actual system that is being described are present simultaneously and take part in the behavior

simultaneously, simultaneous representations generally provide more insight, in particular during the

conceptual phase of the modeling process. Furthermore, when algebraic equations are eliminated, not

only some physically relevant variables are lost but also the (conceptual) structure of the relations, such

Port-Based Modeling of Engineering Systems in Terms of Bond Graphs 26-3

that local changes in the model structure are not possible anymore, even though such elimination may be

very useful for analysis or computational purposes.

Several examples of domain-dependent, simultaneous model representations exist and are frequently

used, in particular in engineering, although at the same time they represent the wish to take the structure

apart for a local analysis per submodel or element. For example, a free-body diagram is used to compute

the net contribution to the rate of change of momentum (dp/dt) on the basis of the forces acting on a

body. For the analysis of electrical circuits similar techniques exist, like (modiﬁed) nodal analysis, mesh

analysis, and incidence matrices, see any standard textbook on circuit analysis (e.g., Hayt et al., 2002).

The bilateral interaction between conceptual elements requires the concept of a port, introduced in

the context of electrical networks by Harold Wheeler (Wheeler and Dettinger, 1949) and extended by

Henry Paynter to all relevant physical domains (Paynter, 1961). Some authors have taken this concept even

beyond physical systems as any relation can be considered bilateral. As a bilateral interaction can be seen

as a combination of two opposite unilateral interactions or signals, two relevant quantities are required to

characterize such a port. This aspect is discussed later in more detail.

Important at this point is that port-based modeling can be distinguished from classical modeling tech-

niques not only by its use of bilateral relations a priori in a simultaneous representation (i.e., graphical), but

also in particular by the fact that no decisions are made about the input–output nature of the constituting

signals of this relation. The relations between the ports are said to be acausal, a terminology to be explained

later and somewhat different from the commonly used meaning of causality and acausality related to the

“arrow of time,” even though it is related in case of the so-called storage ports (cf., the concept of preferred

causality in Section 26.3.3.1). This property of a port not only enables easy reuse of model parts within

other contexts, but also gives immediate feedback to the modeler about his modeling decisions as the

so-called causal properties of ports to be connected not only determine the ﬁnal computational structure,

but may also indicate that certain physical phenomena are not taken into account. This feedback stimulates

the modeler in taking a deliberate decision about what he should include in his model and what not.

26.1.2 Dynamic Models of Engineering Systems

To be precise, the aspects of reality to be modeled are “limited” herein to behavior in time, i.e., dynamics,

of systems that obey the basic principles of classical physics, in the sense that quantum effects play no

role. This means that the dynamic behavior is modeled of macroscopic phenomena that are all supposed

to obey the basic principles of thermodynamics, viz. (global) conservation of energy and positive entropy

production, independent of some speciﬁc domain. The adjective “macroscopic” refers to the nature of

the scope, viz. a thermodynamic approach, rather than the scale of the phenomena: phenomena at the

nanoscale may for instance still be modeled using a macroscopic viewpoint, which combines the kinetic

energy of the random motions into thermal energy using entropy not only as an energy state but also as a

property to express the increase of random motion in terms of the positive entropy production principle.

In contrast, the microscopic view does not collect any motion into thermal energy and this kind of view

may even be applied to objects at a relatively large scale like queueing behavior.

The particular problem context in which the model is made determines the criteria for the selection

of basic phenomena that may determine the relevant behavior of each of the functional or tangible

components. As discussed earlier, the structured set of physical components that constitutes a model,

has to be further abstracted into a conceptually interconnected set of basic behaviors, like storage and

transformation, which will be discussed in more detail later in the form of ideal elements (Section 26.3.2).

It will turn out that this process may be supported by the use of multiport generalizations of the most

basic conceptual elements, in particular, if the relevant phenomena take place in various physical domains.

Such systems are called “multidomain,”“multidisciplinary,”“multiphysics,” and “mechatronic.”

A clear distinction is required between intended behavior and realized behavior, where intended behav-

ior is analyzed by functional modeling during design, while during design evaluation as well as during

troubleshooting the behavior that will be or is actually realized should be analyzed by physical modeling of

all realized aspects, in other words, including the undesired behaviors from a functional point of view. It

26-4 Handbook of Dynamic System Modeling

can be often observed during troubleshooting that the functional models that served their purpose well in

the design phase, fail to provide the required insight for troubleshooting. This means that when a designer

is involved in either concept validation or troubleshooting, he has to make the difﬁcult step of reducing the

priority of what he envisioned on a functional basis, while trying to capture all physical behaviors (to be)

actually realized, even when there is no direct link with the intended function. Making a clear separation

between functional modeling and physical modeling supports this step considerably.

26.2 Structured Systems: Physical Components and Interaction

In this section, a special focus of the port-based approach to modeling is put on structured systems, i.e.,

systems that can be seen as a set of physical or functional components in a structure. Key element of such a

structure is that it represents the relations between the constituting parts or subsystems. As these relations

are bilateral, the (virtual) “points” where these relations can be “attached” to a subsystem have a bilateral

nature too: the ports. In other words, the relations interconnect subsystems into a system via the ports of

the subsystems. The properties of ports related to behavior in terms of classical physics are discussed as

well as the basic concepts that are required to describe the subsystem behavior: state and change (of state).

In contrast with domain-speciﬁc simultaneous representations like free-body diagrams, the analysis of

the model in the port-based approach is not based on tearing the structure apart and performing local

analyses: the structural elements are considered subsystems (nodes of a graph) as well and called junctions

(cf., Section 26.3.2.5).

Within the context of dynamic systems, where the core of the dynamics itself lies in the process of

“storage,” the two relevant quantities or variables in a bilateral relation discussed later in this section are

strongly related to the concept of equilibrium, the balance between the two parts in which a (locally)

conserved quantity or so-called extensive state can be stored: they can either establish the equilibrium by

changing the stored quantity (“equilibrium establishing variable” or “rate of change of extensive state”)

or they determine the equilibrium by forming a balance independent of the extent of the communicating

parts (“equilibrium determining variable” or “intensive state”). For example, in case of diffusion pro-

cesses, the ﬂows of matter of the participating species are the equilibrium-establishing variables (called

generalized ﬂuxes in nonequilibrium thermodynamics) that become zero when the equilibrium is reached.

The concentrations of the participating species are the equilibrium-determining variables (called gener-

alized forces in nonequilibrium thermodynamics) that become equal (zero difference or gradient) when

equilibrium is reached.

The conjugation between these two types of variables that characterize a port is called dynamic conju-

gation. Diffusion phenomena are commonly studied at constant pressure and temperature, thus allowing

energy exchange with the environment that is not considered relevant for the dynamics of the diffusion

process. However, as soon as such assumptions cannot be made and more than one domain (i.e., type

of conserved quantity) needs to be considered, the energy bookkeeping becomes an important means for

consistent modeling, and the dynamic conjugation has to be further constrained to power conjugation

in the sense that the relation between two ports that dynamically interact, describes the power, in other

words, the energy exchange linked to the relation between two connected power ports (in the sequel “port”

will refer to a “power port” unless otherwise indicated). In other words, the power of a port has to be

a function of the two power conjugated variables. The shape of this function is not dictated by nature,

but a common choice is that the product of the two power conjugate variables forms the power of the

interconnection between two power ports called bond. This is a natural consequence of the mathematical

property that a change of energy E, being a function of one or more stored quantities E(q

,…,q

,…,

), can be written as the inner product of the partial derivatives with respect to the stored quantities and

the changes of those stored quantities: dE(q

,…,q

) =



i=1

∂E

∂q

. Domain-dependent examples

of power conjugate variables are voltage and current, force and velocity, pressure and volume ﬂow, and

temperature and entropy ﬂow. The domain-independent terminology for the equilibrium-determining

variable is effort, and ﬂow is used for the equilibrium-establishing variable. In case of common (power)

Port-Based Modeling of Engineering Systems in Terms of Bond Graphs 26-5

Power source Electric motor Transmission Water tank

Centrifugal

pump

␻

␸

FIGURE 26.3 Word bond graph of a pump driven by a DC motor.

FIGURE 26.4 Causal stroke.

bonds, the product of effort and ﬂow is assumed to represent the power, in case of pseudobonds effort

and ﬂow may be related to the power in another manner. For instance, in case of the common variables

used to model thermal behavior, viz. temperature and heat ﬂow, the heat ﬂow itself is the (thermal) power

already. However, it only represents a rate of change of the stored energy as long as the dynamics of other

domains play no role; otherwise, the amount of heat cannot play the role of a proper state variable, as

discussed at length in many introductions to thermodynamics (e.g., Callen, 1960).

Figure 26.3 demonstrates how speciﬁc functional or material parts of an actual system can be represented

in the form of submodels by a short text enclosed by some closed line and identifying them in an insightful

way (often an ellipse is used to distinguish them from the common “blocks” of a block diagram of which

the interface consists of inputs and outputs and where the blocks themselves represent mathematical

operations). These ellipses form the nodes of a graphical representation, a so-called word bond graph,of

which the edges represent the so-called bonds between the ports of these subsystems (Paynter, 1961). The

word bond graph can be seen as the highest level description of a dynamic model, close to the functional

and structural level. The direction of the edges in the form of the half-arrow that is typical for a bond

represents the positive orientation of both ﬂow and power related to that bond.

Another aspect that can be added to a bond is its so-called causal stroke, a little stroke attached to one

end of the bond and perpendicular to it. It represents the computational direction (when chosen) of the

effort signal and consequently also that of the ﬂow signal as the signals of the two conjugate variables are

opposite by deﬁnition (Figure 26.4).

26.3 Bond Graphs

Readers who have been exposed previously to analog models and the bond graph notation in particular

are warned that many introductions start from the misconception that analogies are merely based on

similarity of the underlying differential equations. In this treatise this is considered as an exchange of

cause and effect. The actual source of analogies that lead to the domain independence of both differential

equations and bond graph notation is the mere fact that in human reasoning about dynamic behavior only

a rather limited set of elementary concepts is exploited. This set of ideal concepts is chosen as the backbone

of Section 26.3.2 and maps directly onto the bond graph symbolism. In Section 26.3.3, it is shown how the

concept of computational causality that augments the representation of the physical structure by a bond

graph with a representation of the computational structure, leads to the formulation of a computable

mixed set of differential and algebraic equations.

In models that are solved analytically, it is commonly preferable to eliminate the algebraic equations as

much as possible, as only the structure of the differential equations determines the nature of the described

behavior. In mathematics, this is often taken one step further by scaling or normalizing the variables, which

takesthe model a step further away from its physical interpretation. In the days of analog computing, scaling

and reduction to a minimal form was a necessity and in the early days of digital computing “superﬂuous”