Multiple time models have been proposed for the formalization of hybrid dynamic system behavior. The superdense notion of time is a well-known time model for describing event-based systems where several 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 interaction of two types of discontinuities observed in hybrid systems, mythical modes and pinnacles, are explored. This analysis helps specify semantics that combine continuous-time behavior with discontinuities in the computational system.
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 [
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
continuous-time behavior (e.g., based on the foundations of thermodynamics [
The formalization of behaviors in physical system models builds on a
semantic domain that combines evolution in a continuous domain, extended with an
integer domain for sequences of mode changes [
Related work [
Section 2 presents the notion of continuous-time interacting with discontinuities 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 explained based on Newton’s cradle modeled as bond graphs. Section 4 concludes. 2
At a macroscopic level, physical systems are well modeled as continuous-time
systems [
Across physics domains (e.g., electrical, hydraulic, thermal, chemical, etc.) thermodynamics identifies two types of variables subject to dynamic behavior representing 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 causes a change in momentum (F = m ddvt = ddpt ). 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 domains. 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 interacting phenomena such that the sum of their product equates 0 (so there is neither dissipation nor storage), Pi ei · fi = 0. The two orthogonal implementations 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 connections are represented by junctions, the former by a 0 junction (8 i6=j ei = ej and Pi fi = 0) and the latter by a 1 junction (8 i6=j fi = fj and Pi ei = 0).
Introducing discontinuities into bond graphs requires an idealized form of
discontinuous change in dynamic behavior, which is well represented by a
reconfiguration of the junction structure because this structure is ideal. This idealized
reconfiguration amounts to a junction between phenomena being active or not [
A controlled junction is equipped with a finite state machine (FSM) that determines 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 physical 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. 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.
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 active. Based on this velocity di↵ erence and conservation of momentum (Pi mivi = Pi mivi), 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
(discontinuous 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 [
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 is su ciently 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.
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 continuous 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 su ciently 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
corresponding 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 [
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 nonnegative real numbers that describe time durations between the events. Events separated by zero duration are simultaneous, but have a well-defined causal ordering.
Superdense time was introduced to represent time-event sequences as
functions of time [
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 [
In the ultrapower construction [
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.
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 description 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 considered 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 natural 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 su ciently 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 semantic 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 sophistication 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 (a) Picture model
(b) Bond graph model 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 [
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 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 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 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 formalizing 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 discontinuities observed across multiple domains. Moreover, the work allows for a consistent mapping onto corresponding algorithms that lack hyperreals as execution 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 reinitializing 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.