=Paper= {{Paper |id=None |storemode=property |title= A Hyperdense Semantic Domain for Discontinuous Behavior in Physical System Models |pdfUrl=https://ceur-ws.org/Vol-1112/06-paper.pdf |volume=Vol-1112 |dblpUrl=https://dblp.org/rec/conf/models/MostermanSZ13 }} == A Hyperdense Semantic Domain for Discontinuous Behavior in Physical System Models == https://ceur-ws.org/Vol-1112/06-paper.pdf

A Hyperdense Semantic Domain for
Discontinuous Behavior in Physical System
Models

Pieter J. Mosterman, Gabor Simko, Justyna Zander

MathWorks, Vanderbilt University, HumanoidWay

Abstract. Multiple time models have been proposed for the formaliza-
tion of hybrid dynamic system behavior. The superdense notion of time
is a well-known time model for describing event-based systems where sev-
eral events can occur simultaneously. Hyperreals provide a domain for
defining the semantics of hybrid models that is elegantly aligned with
first principles in physics. This paper discusses the value of both time
models and shows how approximating di↵erent physical e↵ects is best
expressed over di↵erent domains. Finally, the formalization and interac-
tion of two types of discontinuities observed in hybrid systems, mythical
modes and pinnacles, are explored. This analysis helps specify seman-
tics that combine continuous-time behavior with discontinuities in the
computational system.

1 Introduction
In recent history, the complexity of engineered systems has grown by leaps and
bounds, largely because of embedded computation. While embedded systems
are well understood and supported by Model-Based Design [14], Cyber-Physical
Systems (CPS) build on a general paradigm of ’openness’ [18] that challenges
the current paradigm of system design. This openness manifests, for example, by
an application that may execute on di↵erent platforms or feature functionality
that may be provided by distinctly separate systems. Because it is open, such a
CPS cannot rely on integration testing (e.g., [20]) as it is part of the traditional
paradigm for embedded system design.
Given the delicate interaction between various component and subsystem
behaviors in their implementation, addressing system integration challenges with
models is not straightforward. In particular, it is essential to create ‘good’ models
of the physics, that is, models that embody correctly the pertinent physical
e↵ects while not giving rise to behaviors that have no physical manifestation. The
desiderata for a formalism to model physical systems thus require domain-specific
models that inherently reflect the laws of physics. Moreover, the models of the
physics must be employable in concert with models of various paradigms such
as those for computational and networking functionality in the overall system.
For system-level studies, physics models are generally well described by conti-
nuous-time behavior (e.g., based on the foundations of thermodynamics [3, 6]).

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A Hyperdense Semantic Domain for Discontinuous Behavior in Physical System Models

At this level, however, physical phenomena that are often part of actuators on the
interface with the information technology domain (e.g., electrical switches, hy-
draulic valves, clutches) typically operate at a time scale too fast to be captured
in continuous detail. Instead, such fast behavior is modeled as discontinuous
change.
The formalization of behaviors in physical system models builds on a seman-
tic domain that combines evolution in a continuous domain, extended with an
integer domain for sequences of mode changes [5, 13]. The resulting R⇥N domain,
so-called superdense time [11], however, is not sufficiently rich to allow a precise
mathematical description of the intricate behavior in physical system models
around discontinuities.1 Specifically, an ontology of behavior in hybrid dynamic
system models of physical systems developed in previous work [15] includes a
class of behaviors called mythical modes [16] that maps well onto superdense
time. However, another class of behaviors called pinnacles requires physical time
to advance during discontinuous change, and, therefore, is not amenable to em-
ploying superdense time as a semantic domain.
Related work [1, 2, 9] has turned to hyperreals from nonstandard analysis [10]
to define semantics of hybrid dynamic systems. In this paper, the hyperreals
are considered as a semantic domain that supports pinnacles. In combination
with an integer domain for mythical modes, this leads to a hyperdense time
domain that supports the various classes of behavior found in hybrid dynamic
system models of physical systems. The mathematical formalization is mapped
onto a computational implementation that allows for generation of consistent
and physically meaningful behavior of interactions between various classes of
discontinuities.
Section 2 presents the notion of continuous-time interacting with discontinu-
ities in physical system models. Bond graphs are the formalism to represent these
phenomena. Further, pinnacles and mythical modes are introduced in detail and
related to the notions of superdense and hyperreal time. Section 3 discusses the
interactions among the di↵erent modes. Semantics of discontinuous change is ex-
plained based on Newton’s cradle modeled as bond graphs. Section 4 concludes.

2 Discontinuities in Physical System Models

At a macroscopic level, physical systems are well modeled as continuous-time
systems [7]. Continuous phenomena that occur at a time scale much faster than
the behavior of interest can be approximated by discontinuities. This section first
introduces bond graphs [17] as a formalism to model the continuous-time behavior
of physical systems. Next, an ideal switching element is added to represent the
discontinuity and form hybrid bond graphs [12].

1
Note that superdense time is typical in a computational approximation of the math-
ematical representation. However, floating point numbers then represent the contin-
uous domain and the approximation of the continuous domain is in fact not dense.

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A Hyperdense Semantic Domain for Discontinuous Behavior in Physical System Models

2.1 Bond Graphs

Across physics domains (e.g., electrical, hydraulic, thermal, chemical, etc.) ther-
modynamics identifies two types of variables subject to dynamic behavior repre-
senting either: (i) extensive quantities or (ii) intensive quantities. The dynamics
of extensities and intensities are related by conduction, that is, when there is a
di↵erence in intensities, a change in extensity follows. For example, a di↵erence
in velocities between two bodies results in a force acting between them that
dp
causes a change in momentum (F = m dv dt = dt ). The change of energy, power,
as the product of the intensity di↵erence, e↵ort, and its corresponding change
in extensity flow then provides a general notion of dynamics across physics do-
mains. For example, v · F equates power much like in the electrical domain the
product of the intensity di↵erence (voltage, v) and change of extensity (current i)
equates power. Consequently, any change in dynamic variable values is the result
of an e↵ort, e, and a flow, f , acting. Moreover, there are two basic energy-based
phenomena: (i) storage of either e↵ort (C) or flow (I) and (ii) dissipation (R).
Finally, ideal sources of e↵ort (Se) and of flow (Sf ) define the model context.
Behavior of the connections then relates the e↵orts and the flows of all in-
teracting phenomena such that the P sum of their product equates 0 (so there is
neither dissipation nor storage), i ei · fi = 0. The two orthogonal implementa-
tions of this are that either all e↵orts are the same while the flows sum to zero,
or the converse. In the electrical domain, this corresponds to either Kirchho↵’s
current law or Kirchho↵’s voltage law. In bond graph terminology these connec-
tions
P are represented by junctions, the former by a 0 junction P (8i6=j ei = ej and
i f i = 0) and the latter by a 1 junction (8 i6 = j f i = f j and i ei = 0).
Introducing discontinuities into bond graphs requires an idealized form of dis-
continuous change in dynamic behavior, which is well represented by a reconfigu-
ration of the junction structure because this structure is ideal. This idealized re-
configuration amounts to a junction between phenomena beingP active or not [19].
In other words, a 0 junction can be active (8i6=j ei = ej and i fi = 0) or not
(8i eiP = 0) and a 1 junction exhibits the dual behavior when active (8i6=j fi = fj
and i ei = 0) or not (8i fi = 0). Note that when a junction is not active, indeed
no power flows across it. These junctions that can change their mode from active
(on) to inactive (o↵) are called controlled junctions.

2.2 The Logic of Discontinuities in Physics Models

A controlled junction is equipped with a finite state machine (FSM) that deter-
mines the junction on or o↵ mode, which involves capturing: (i) how the state of
the FSM maps onto the on and o↵ mode of the junction and (ii) how the phys-
ical quantities map onto transition conditions of the FSM. Continuity of power
implies that discontinuities in physical quantities result from a lack of detail in
modeled phenomena, which come in two classes: (i) storage and (ii) dissipation.
The discontinuous behavior that emerges in turn for each of these is discussed
next.

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A Hyperdense Semantic Domain for Discontinuous Behavior in Physical System Models

Pinnacles Multibody collisions are often modeled by discontinuous velocity
changes. In a hybrid bond graph model, a collision between two bodies, m1 and
m2 , can be modeled as depicted in Fig. 1. The two bodies are modeled as inertias,
I, connected to a common velocity, 1, junction. These junctions represent the
respective velocities, v1 and v2 , which are connected via a common force, 0,
junction. This 0 junction is controlled and when o↵ it exerts force 0 on both
bodies. Upon collision, the 0 junction turns on and it now enforces a velocity
balance such that v1 v2 + v = 0, where v is computed by an ideal flow source,
Sf , as v = v1 v2 , with the ‘-’ superscript referring to signals immediately
preceding the collision.

Fig. 1. Ideally plastic collision

The FSM controlling the on/o↵ mode of the 0 junction switches from o↵ to
on when the bodies make contact ( x > 0) and when they are moving toward
one another ( v > 0). Here the v > 0 is essential to model that there is a
collision as opposed to the bodies only being in contact. As soon as the bodies
move away from one another ( v < 0), the 0 junction switches to o↵, irrespective
of whether the bodies are touching.
During behavior generation, when x > 0 && v > 0 holds, a collision
occurs and the flow source enforcing the velocity di↵erence v becomes P active.
Based
P on this velocity di↵erence and conservation of momentum ( i mi vi =
i m i v i ), the velocities upon collision can be computed. The state of the velocity
of the bodies is then reinitialized and this leads to the condition v < 0 being
satisfied. Thus, a consecutive mode change occurs where the FSM moves to the
o↵ mode again. In the o↵ mode the bodies behave as independent masses, and,
therefore, no further changes in the physical state occur. Since the discrete mode
changes have thus converged, the system proceeds to evolve in continuous time.
The end result is that the bodies m1 and m2 evolve according to a mode of
continuous evolution. With a point in time at which two mode changes occur:
(i) first, a collision mode occurs that necessitates a reinitialization (discontinu-
ous change) and (ii) second, the system changes back to a mode of continuous
evolution. The collision mode that is active only as a reinitialization of physical
state is referred to as a pinnacle [13].

Mythical Mode Change Now, consider two bodies m1 and m2 with m2 at rest
on top of m1 . When at a point in time a large enough external force is exerted
on m1 , m1 will start moving with a corresponding velocity. However, if the force

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A Hyperdense Semantic Domain for Discontinuous Behavior in Physical System Models

is sufficiently large that the breakaway friction force Fbreakaway between m1 and
m2 is exceeded, m2 may remain at rest.
A hybrid bond graph model of such a system is depicted in Fig. 2. An ideal
source of e↵ort exerts a force on m1 because connected to the common velocity
1 junction that represents the velocity of m1 . When on, a controlled 0 junction
connects the 1 junction that represents the velocity of m2 , which forces m1 and
m2 to move with the same velocity. The FSM for the controlled 0 junction shows
that the junction changes to its o↵ mode when the force between m1 and m2
exceeds the breakaway force, F > Fbreakaway . In the o↵ mode, the 0 junction
exerts 0 force on both m1 and m2 , and so they move independently. The FSM also
shows that if the velocity di↵erence between m1 and m2 falls below a threshold
velocity ( v < vth ) the two bodies ‘stick’ to each other again.

Fig. 2. Two bodies with a breakaway force

During behavior generation, initially the 0 junction is in its on mode because
the bodies are at rest with one atop the other and the system evolves in con-
tinuous time. Now, at the point in time where Fin changes discontinuously new
velocities for both m1 and m2 are computed. These velocities, however, may
require a force to be exerted on m2 that causes the condition F > Fbreakaway to
be satisfied and the 0 junction changes to its o↵ mode. In the o↵ mode, if the
velocity di↵erence is sufficiently large, no further mode changes occur and the
system proceeds to evolve in continuous time.
At the point in time at which a discontinuous force is exerted the corre-
sponding velocities and forces are computed and based on the newly computed
values the connection between the two bodies changes mode such that they are
dynamically independent. Since there is no e↵ect of the external force on the
velocity of m2 , in order to arrive at the proper values for reinitialization of v1
and v2 the mode where the external force becomes active while m1 and m2 are
still connected is considered to have no e↵ect on the physical state, which is
referred to as a mythical mode [13].

2.3 Introduction to Superdense Time

Time-event sequence is a semantic domain for describing event-based models.
Intuitively, time-event sequences are instanteneous events separated by non-
negative real numbers that describe time durations between the events. Events

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A Hyperdense Semantic Domain for Discontinuous Behavior in Physical System Models

separated by zero duration are simultaneous, but have a well-defined causal or-
dering.
Superdense time was introduced to represent time-event sequences as func-
tions of time [11]. Superdense time is a totally ordered subset of R+ ⇥ N, where
the non-negative real number represents the real time and the natural number
represents the causal ordering. Simultaneous events at time t are mapped to
(t, 0), (t, 1), . . . superdense time instants such that the ordering of the events is
preserved.
The (total) ordering of superdense time is given by the following definitions:
(t, n) = (t0 , n0 ) , t = t0 ^ n = n0 , and (t, n) < (t0 , n0 ) , t < t0 _ (t = t0 ^ n <
n0 ). Therefore, superdense time is a time model that can be used to describe
simultaneous events as functions of time, while retaining the causality of events.
Mythical modes emerge as an artifact of logical inference to determine a new
mode in which physical state can change. As such, mythical modes do not a↵ect
the dynamic state of a physical system. Moreover, di↵erent logic formulations
may traverse di↵erent mythical modes yet still arrive at the same resulting mode
where physical state changes can occur. Consequently, the logical evaluation
has no corresponding manifestation in the dynamic state of a physical system
and occurs at a single point in time along a logical inferencing dimension. This
behavior corresponds to the superdense semantic domain.

2.4 Introduction to Hyperreal Time
Calculus comprises two di↵erent approaches to capturing infinitely small values:
either through the use of limits, or by the extension of the field of reals with
infinitesimals. An infinitesimal ✏ is any number, such that |✏| < n1 , for any n 2 N.
Intuitively, the idea behind hyperreals is to extend the dense field of R with
infinitely many points around each real number such that any real sentence
that holds for one or more real functions also holds for the hyperreal natural
extensions of these functions [10] (transfer principle).
In the ultrapower construction [8], hyperreals are represented as sequences
of real numbers u1 , u2 , . . . un 2 Rn with real numbers embedded as constant se-
quences (i.e., a real number r is the sequence of r, r, . . . r 2 Rn ). These sequences,
together with elementwise addition and multiplication operations, form a com-
mutative ring but not a field (since the multiplication of two non-zero numbers
could result in zero: 0, 1, 0, . . . ⇥ 1, 0, 1, . . . = 0, 0, . . .). This issue is remedied by
considering equivalence classes of Rn defined by a free ultrafilter U of N.
Let J be a non-empty set. An ultrafilter on J is a nonempty collection U of
subsets of J having the following properties: ; 2 / U ; A 2 U and B 2 U implies
A \ B 2 U ; A 2 U and A ✓ B ✓ J implies B 2 U ; for all A ✓ J, either A 2 U
or J \ A 2 U . For any x 2 J there is a principal ultrafilter {A ✓ J | x 2 A}.
Finally, any non-principal ultrafilter is called a free ultrafilter.
Given an ultrafilter U , an equivalence relation =U can be defined over Rn :
u =U v holds for sequences u = u1 , . . . , un and v = v1 , . . . , vn if and only if
{i | ui < vi } 2 U . The hyperreals are then defined as the quotient of Rn by U ,
⇤
R = Rn /U . Now, ⇤ R is an ordered field for which the transfer principle holds.

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A Hyperdense Semantic Domain for Discontinuous Behavior in Physical System Models

As a semantic domain, hyperreals have the advantage that between any real
time instant there are many ordered time instants. Such extension of time greatly
simplifies the semantic specification of discontinuities, in particular, the descrip-
tion of pinnacles that represent fast physical behaviors where the dynamic state
changes discontinuously. As a result, a pinnacle corresponds to a distinct state
of physical behavior. In physics, such a distinct state corresponds to a distinct
point in time. Because the continuous behavior represented by a pinnacle is con-
sidered to occur infinitely fast, time is considered to advance by an infinitesimal
amount for a pinnacle to implement the physical state change. This behavior
corresponds to the hyperreal domain.
It is a straightforward extension to introduce a hyperdense time model as
a “combination” of the super-dense and hyperreal time models. We define the
hyperdense time ⇤ R+ ⇥ N as the product of the non-negative hyperreals and nat-
ural numbers. Such a time model can be used for representing both infinitesimal
time advancements, as well as establishing a causal ordering at any hyperreal
time instant.

3 Semantics of Discontinuity Behavior

The formalized models of time provide the ingredients for a semantic domain
that is sufficiently rich to formalize the pinnacle and mythical mode behavior at
discontinuities as well as combinations.

3.1 Interacting Pinnacles and Mythical Modes

With superdense time as a semantic domain for mythical modes and hyperreals
for pinnacles, models that engender both build on a combined hyperdense se-
mantic domain. The particular value of such a precise semantic description lies in
the ability to develop consistent computational behavior generation algorithms.
Because of the discreteness of computational values, the semantic domain of
values in computational models can represent neither superdense nor hyperreal
domains. Therefore, the behavior generation algorithms must include sophistica-
tion that addresses the di↵erences between superdense and hyperreal semantic
domains. The computational implementation of each of the semantic domains
and their interaction is described based on an illustrative example.
In Fig. 3(a), a variant of Newton’s Cradle is shown. One of the bodies, m3 ,
has another body, m2 , positioned on top of it. Stiction e↵ects between m2 and m3
cause them to behave as one body with combined mass as long as the breakaway
force between them, Fbreakaway , is not exceeded. A body, m1 , may collide with
m3 according to a perfectly elastic collision, v32 = ✏ v32 , where v is the
di↵erence in velocities (v3 v2 ) after the collision and v32 is the di↵erence in
velocities before the collision.
The bond graph model in Fig. 3(b) shows the three masses each connected
to a common velocity junction, 1, with the velocity of the directly connected
mass on all ports. Common force junctions, 0, connect the 1 junctions and are

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A Hyperdense Semantic Domain for Discontinuous Behavior in Physical System Models

(a) Picture model (b) Bond graph model

Fig. 3. Newton’s Cradle for advanced maneuvers

controlled junctions such that a finite state machine determines their on or o↵
state. A modulated flow source MSf models the collision with restitution ✏ = 0.8.
If the controlled junction 01 is in its on state, this flow source enforces a di↵erence
in velocities of m1 and m3 , possibly accounting for the rigidly connected mass
m2 . If the controlled junction 01 is in its o↵ state, a 0 force is exerted on both
m1 and m3 (possibly accounting for m2 ). In its o↵ state, the controlled junction
02 exerts a 0 force as well, which is when the force at the contact point between
m2 and m3 exceeds the breakaway force. Note that for the case when m2 and
m3 move independently in continuous time no viscous friction is modeled for
clarity purposes. If the di↵erence in velocities between m2 and m3 falls below a
threshold level, the stiction e↵ect becomes active, modeled by 02 changing to its
on state. In the on state, a di↵erence in velocities of m2 and m3 of 0 is enforced.
Upon collision of m1 and m3 , if the di↵erence in velocities v2 and v3 , v23 , is
less than the threshold velocity vth , stiction is active and m2 and m3 behave as
one body with mass m2 +m3 . When m2 +m3 > m1 , m1 will have a return velocity
and start moving in the opposite direction compared to the velocity before the
collision. However, the momentum of m1 may be such that an impulsive force [4]
arises between m2 and m3 that triggers the Fbreakaway transition, causing the
two bodies to move independently. In this case, if m1 = m3 , there is no return
velocity of m1 but instead it acts as in the case of Newton’s Cradle where m3
assumes all of the momentum of m1 while m1 comes to rest. In this case the
velocity of m2 is not a↵ected by the collision.
The importance of semantic domain that combines both superdense as well
as hyperreals is clearly illustrated by this example. While the condition for 02
to switch from on to o↵ occurs in 0 time, the condition for 01 to switch from on
to o↵ occurs in infinitesimal, ✏, time. A critical consequence of this phenomenon
is that, although in reasoning about the system, 01 first changes its state to on,
after which the change of state in 02 to o↵ is determined, the change of state in
01 back to o↵ is not e↵ected until after the change of state in 02 to o↵.
In Table 1 and in Table 2, 0F 13 and 0F 23 are the junctions at a select force
impact, while pm1 , pm2 , and pm3 are the momenta for each of the masses. The
sequences of mode changes are depicted in Table 2, which shows clearly the
di↵erence in e↵ects as the model evolves in superdense and in hyperreal time.
The ability to di↵erentiate between t = htcollide , 1i and t = htcollide + ✏, 0i makes
it possible to distinguish the pinnacle e↵ect of 01 from the mythical mode e↵ect of
02 . Otherwise, 01 would have switched back o↵ simultaneously with 02 switching

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A Hyperdense Semantic Domain for Discontinuous Behavior in Physical System Models

time 0F 13 0F 23 pm1 pm2 pm3
t = htcollide ✏, 0i o↵ on 1 0 0
t = htcollide , 0i on on -0.2 0.6 0.6
t = htcollide + ✏, 0i o↵ on -0.2 0.6 0.6

2
Table 1. Mode changes for vth = 0.1 and Fth = 0.95

time 0F 13 0F 23 pm1 pm2 pm3
t = htcollide ✏, 0i o↵ on 1 0 0
t = htcollide , 0i on on -0.2 0.6 0.6
t = htcollide , 1i on o↵ 0.1 0.9 0.0
t = htcollide + ✏, 0i o↵ o↵ 0.1 0.9 0.0

2
Table 2. Mode changes for vth = 0.1 and Fth = 0.5

o↵. This would either: (i) not allow modeling of inferencing (mythical) modes or
(ii) have the collision e↵ect (incorrectly) computed for m2 and m3 comprising a
combined mass of m2 + m3 .

4 Conclusions
Superdense and hyperreal time notions provide a comprehensive basis for formal-
izing a computational semantics. The interplay between them enables the design
of models that include continuous-time behavior interacting with discontinuities
of physical system models. Such formalization is of great value, in particular
in simulation technologies, because of the benefits in a consistent projection of
behavior according to the laws of physics into a computational representation.
Based on the bond graph modeling formalism, a formalization is developed
for combining continuous-time with discontinuities. The combination provides a
theoretical reference for the computational integration of di↵erent e↵ects of dis-
continuities observed across multiple domains. Moreover, the work allows for a
consistent mapping onto corresponding algorithms that lack hyperreals as execu-
tion domain. Most prominently, the research addresses how to combine behavior
because of logical switching with physics-based switching behavior.
Formalization in this domain often foregoes the collision mode by reinitial-
izing velocities as a transition action in a state machine. Though there is no
fundamental di↵erence in behavior generation, such a representation makes it
exceedingly complicated to attribute a sound theory of physics to discontinuous
behavior. Instead, this paper relates pinnacles and mythical modes to di↵erent
notions of time so as to formalize their interaction in a computational sense.

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A Hyperdense Semantic Domain for Discontinuous Behavior in Physical System Models

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