A Hybrid Modeling and Simulation Methodology for Dynamic Physical

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A Hybrid Modeling and Simulation Methodology for Dynamic Physical Systems Pieter J Mosterman and Gautam Biswas SIMULATION 2002; 78; 5 DOI: 10.1177/0037549702078001197 The online version of this article can be found at: http://sim.sagepub.com/cgi/content/abstract/78/1/5

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A HYBRID MODELING AND SIMULATION METHODOLOGY FOR DYNAMIC PHYSICAL SYSTEMS

METHODOLOGY

A Hybrid Modeling and Simulation Methodology for Dynamic Physical Systems Pieter J. Mosterman Institute of Robotics and Mechatronics DLR Oberpfaffenhofen D-82230 Wessling, Germany Gautam Biswas Department of Electrical Engineering and Computer Science Box 1679, Station B Vanderbilt University Nashville, TN 37235 This article presents the design and implementation of a robust simulation methodology for hybrid models of physical systems that encompasses behavior patterns that undergo discrete transitions between modes of continuous behavior evolution. This methodology is based on a mathematical framework that captures time scale and parameter abstractions in modeling complex physical phenomena. The formal semantics for characterizing hybrid behaviors is defined in terms of three distinct modes of system operation: (1) continuous, (2) pinnacles, and (3) mythical. A key link is established between the switching transitions and the a priori and a posteriori state vector values, which defines the recursive mode switching functions that govern the interactions between the continuous and discrete components of the system models. The effectiveness of the approach is demonstrated by examples of physical system models that are presented along with a simulator to handle these. Keywords: Hybrid models, hybrid simulation, abstraction of complex physical behaviors, formal semantics, recursive mode switching

1. Introduction The dynamic behavior of physical systems, governed by the principles of continuity of power and conservation of energy [1], evolves in real space as a continuous function of time. System behaviors are often complex and occur at different temporal and spatial scales. At a specified level of interest determined by the task at hand, the system model can be simplified by abstraction to exhibit simpler piecewise continuous behavior. In other situations, physical systems with embedded digital control are designed to operate in multiple configurations or modes. Within each mode, system behavior evolves continuously, but discrete mode changes can occur at points in time, resulting in discontinuities in overall system behavior. These mode changes are governed by system variables crossing threshold values and by external events imposed by the supervisory controllers. Submission Date: November 1998 Accepted Date: August 2001

SIMULATION, Vol. 78, Issue 1, January 2002 5-17 © 2002 The Society for Modeling and Simulation International

Previous work has focused on developing formal methodologies for hybrid modeling and analysis of dynamic physical systems that conform to the physical principle of conservation of state [2, 3, 4]. In this framework, a set of unambiguous and consistent principles was developed that define the interaction between the continuous and discrete modeling formalisms. This article uses the mathematical models developed earlier [3–6] to define formal execution semantics for hybrid simulation algorithms. Consider the electrical circuit shown in Figure 1. For modeling purposes, the manually operated switch Sw and the diode D exhibit ideal behavior; that is, they switch on and off instantaneously.1 When the switch is closed (mode α10 in Fig. 2), the circuit is complete and the inductor draws a current to build up flux, p0. The diode is inactive (off ) in this mode of operation. At the timepoint when the switch is opened, creating an open circuit, the current drawn by the inductor should drop to 0, causing its flux, p0, to discharge instantaneously. The constituent relation for 1. In reality, their time constants of operation are assumed to be much faster than the time constants associated with the rest of the system behavior. Therefore, the switching behavior is modeled to be instantaneous.

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P. J. Mosterman and G. Biswas

Figure 1. Physical system with discontinuities

Figure 2. A series of mode switches may occur

the inductor, VL = L dp , implies an infinite voltage drop is dt generated across the diode by the instantaneous change of flux in the inductor. Continuing this analysis, it is deduced that the diode comes on at the instant its threshold voltage, Vdiode is exceeded. The mode where the switch was open and the diode inactive (mode α00 in Fig. 2) can never occur in real time. If it did, the energy stored as flux in the inductor would be released instantaneously, and the true behavior where the inductor discharges through the diode would not be predicted by the idealized model. To derive the correct behavior from the simplified, idealized model, the flux in the inductor has to be invariant across the instantaneous switching changes in the system. Typically, a diode requires a current flow that exceeds a threshold value, Ith > 0 to remain on. If the inductor has built up a positive flux, the diode comes on when the switch opens. However, if the flux in the inductor is too low to maintain a current above the threshold value, Ith, the diode will switch off instantaneously. But the computed voltage drop when the switch and diode are both off exceeds the threshold voltage, Vdiode, which implies the diode must come on again. This model seems to imply that the system goes into a loop of instantaneous changes (see dashed arrow in Fig. 2). If this were the case, system behavior would no longer evolve in real time, which conflicts with physical principles because behavior of any physical system cannot halt in time. This simple example illustrates a number of characteristics specific to hybrid system modeling and analysis: • Changes in the model configuration may cause discontinuities in system variable values. This complicates the computation of the system state vector when mode transitions occur. In some cases, configuration changes may result in dependencies among state variables causing the dimensions of the state vector to change. • In systems with multiple switching elements, a mode change may trigger a sequence of additional instantaneous changes. This further complicates behavior generation because

6

— the sequence of instantaneous mode changes must terminate in a real mode where behavior continues to evolve continuously and — the state vector in the final mode has to be computed across the sequence of mode changes.

The rest of this article develops these notions into formal mathematical specifications from which we derive the execution semantics for a hybrid simulation algorithm based on physical system principles. Simulation of a falling rod system demonstrates the effectiveness of this approach. Other applications of this approach are discussed in [5, 7, 8].

2. Background A generic hybrid system operates on a domain that combines discrete and continuous dimensions. Behavior in this space is specified by the state vector, xα(t), a piecewise continuous function defined on the α ∈ ℵ and t ∈ ℜ dimensions. Figure 3(a) illustrates that the behavior of such systems may consist of intervals and points (illustrated by hollow circles) on the time line, where system behavior is not defined. On the other hand, hybrid dynamic system behavior [9, 10] evolves in time t with no gaps on the time line and has an established direction of flow. This is illustrated in Figure 3(b). Piecewise continuous behavior on intervals is represented by a well-behaved set of ordinary differential equations, f, called fields. An instance of the set of temporal behaviors associated with a field is called a flow, F. Switching from one flow to another occurs at well-defined points in time when system variable values reach or exceed prespecified threshold values. This defines an interval-point paradigm where flows are piecewise continuous and any discontinuous changes that occur have to be simple; that is, limit values exist at points of discrete switching [11]. Therefore, hybrid dynamic systems consist of three distinct subdomains:

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Figure 4. A hybrid system

Figure 3. A hybrid system (a) and a hybrid dynamic system (b)

• A continuous domain, T, with time, t ∈ T, as the continuous variable. • A piecewise continuous domain, Vα, that specifies variable flow, xα(t), uniquely on the time line. • A discrete domain, I, that captures the set of piecewise continuous domains, Vα, that define system behavior.

Notation similar to Guckenheimer and Johnson [10] is adapted and I specified to be a discrete indexing set, where α ∈ I represents the mode of the system. Fα is a continuous 2 n C flow on a possibly open subset Vα of ℜ , called a chart (Fig. 4).The subdomain of Vα where a continuous time flow occurs is called a patch, Uα ⊆ Vα. The flows constitute the piecewise continuous part of the hybrid system. Behavior of the system is specified by xα(t), a location in chart Vα in mode α at time t. An explicitly defined isolated point that does not embody continuous behavior (e.g., an elastic collision) is called a pinnacle, Pα. The discrete switching function γ βα is defined as a threshold function on Vα. When γ βα ≤ 0 in mode α, the system transits to mode β, defined by the mapping g αβ : Vα → Vβ. The piecewise continuous level curves γ βα = 0 define patch boundaries. If a flow Fα includes the level curve, it contains the boundary point, Bα (see Fig. 4). A hybrid dynamic system is then defined by the 5-tuple2 H = < I, Xα, fα, γ βα , g αβ >,

(1)

2. Guckenheimer and Johnson [10] refer to the respective parts as < Vα, Xα, Fα, h αβ , Tαβ >.

Figure 5. Multiple transitions occur at time point t because the transported point is not in the domain of the new patch

where Xα and fα are the state vector and field in mode α. The other components have been defined above. Trajectories in the system are defined in terms of an initial point xα1(t0). As long as γ αα i1 > 0, ∀αi, behavior evolution from xα1(t0) in α1 is specified by Fα1 . This continues till the first occurrence of a time point ts, where γ αα 21 ( x α 1 (t)) = 0 3 for some α2. Computing x α 1 (t s− ) = lim t ↑ t s Fα 1 (t) the transformation g αα 12 takes the trajectory from x α 1 (t s− ) ∈Vα 1 to x α 2 (t s− ) ∈Vα 2 . The point x α 2 (t s ) g αα 12 ( x α 1 (t s− )) is regarded as a new initial point in mode α2. 3. It is possible that more than one αi satisfies this condition at time t. In this case, the system behavior is nondeterministic.

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SIMULATION 7


If there exists α3 ∈ I, such that γ αα 32 ( x α 2 (t s )) ≤ 0, the tra-

jectory is immediately transferred to g αα 23 ( x α 2 (t s )) ∈Vα 3 . In other words, as Figure 5 illustrates, the trajectory is redirected from Vα 2 to Vα 3 because the transported point x α 2 was outside the new patch U α 2 . A characteristic of hybrid systems is the possibility of a number of these immediate changes occurring before a new patch is arrived at, where again a flow defined by a field governs system behavior [10, 12, 13]. In general, this situation occurs if γ αα kk + 1 transports a trajectory to mode αk+1, and initial point, x α k + 1 (t) is transferred by g αα kk + 1 to a value that results in γ αα kk ++ 21 ≤ 0, i.e.,

g αα kk + 1 ( x α k ) ∉U α k + 1 At this point, the system instantaneously transits to another mode αk+2. These immediate transitions will continue until the system behavior reaches a mode αm where the initial point is within the patch U α m . To deal with these sequences of transitions, Alur et al. [12, 14], Guckenheimer and Johnson [10], and Deshpande and Varaiya [9] propose model semantics based on temporal sequences of abutting intervals. γ αα 1 ( x )

γ αα 2 ( x ) 1

0

g αα 1 ( x )

x 0 → x1 t1 ] [t 0 14 4244 3

0

→

Va0

x 1+ → x 2 t2 ] [t 1 14 4244 3

g αα 2 ( x ) 1

→

K

Va1

γ αα m m −1 g αα m m −1

(x )

(2)

x m+ → x m + 1 , t m +1 ] [t m 1442443

(x )

→

Figure 6. A model of hybrid dynamic systems

Vam

where ti ≤ ti+1 ∀i. This specification includes overlapping intervals; that is, the trajectory has several values at time point ts. To clearly specify the sequence of transitions, these values are supplemented with an index that specifies their order of transition. During a series of discrete switches, (ts,i), (ts,i+1) . . . (ts,n), the trajectory moves through these points in the order specified, with repeated applications of g αβ . A change in the ordering may result in the derivation of different initial points of the new flow. This produces a sequence of transitions of the form x = x α1  &  x = f α 1 ( x, t ) 43  142 α1 

γ αα 2 ( x ) 1

x a 2 = g αα 2 1

(x )

→

γ αα m

(x )

m −1 x α m − 1 = g αα m m −1

→

8

(x )

x = x α2  & = x f  α 2 ( x, t ) 43  142 α2 

γ αα 3 ( x ) 2

x α 2 = g αα 3 ( x )

x = x αm  & . x f =  α m ( x, t ) 42 44 3  14 αm 

→

2

K

3. A Hybrid Model for Physical Systems The conceptual model is extended to a model for dynamic physical systems with hybrid behavior. For physical system models, the event generation function γ is based on continuous signals linked to the state variables in the system. A mode change may result in a change in functional relations between state variables and signals. The recomputed signal values can cause further mode changes. Our work on physical system modeling extends the mathematical hybrid system model in eq. (1) to an implementation model defined by the 9-tuple [6, 15] H = < I, Σ, φ, Xα, Uα, fα, g , hα, γ βα >. α

(3)

(4)

Here, Xα and Uα are the state and input vectors, and field fα represents the continuous model in mode α. The discrete model is represented by I, the discrete indexing set that corresponds to the possible modes in the system; Σ is the set of events that cause mode transitions; and φ is the discrete state transition function. The earlier defined γ and g, and the signal generation function, h, represent the interac-



tions between the discrete and continuous models. Figure 6 shows a schematic representation of the execution of such models. The events σx and σs represent events generated externally and internally, respectively. The variables s are signals that are algebraically related to the state and input variables. Before the components of the model are described in greater detail, an example of a falling rod system is developed to illustrate the modeling and simulation techniques in the rest of the article.

3.1 The Falling Rod System Consider the rigid body collision between a thin rod and a floor, depicted in Figure 7. The rod of length l drops vertically, but it hits the floor at an angle, θ. The free-falling rod is assigned mode α00, as shown in the figure. Upon collision, small deformations of the rod and floor force the vertical velocity of the rod-tip, A, to quickly become 0. If this phenomenon occurs on a time scale much smaller than the time scale of the macroscopic behavior, the deformation effects can be abstracted away. As a result, the model will predict a discontinuous change in vA,y at the point of contact. Coulomb friction [16] between the rod and floor may result in one of the following phenomena on contact: • the rod may stick to the floor and rotate around the point of initial contact (the stuck mode, α01, in Fig. 8), • the rod-tip exerts a force in the horizontal direction x at point A, FA,x, that is larger than the product of the normal force (Fn) and a friction coefficient (µ), that is, |FA,x| > µ Fn, causing the rod to start sliding (the slide mode, α11, in Fig. 8), and • one further mode transition occurs (α11 to α21 in Fig. 8) at the start of the sliding process. At this point, a Coloumb frictional force becomes active and influences motion in the x-direction.

Determining which scenario occurs requires that the forces be calculated at the point of collision. The contact forces are modeled as impulses P [17] and take on the form of Dirac functions (δ) in time. A Dirac function is a pulse with infinite magnitude and finite area that occurs at a point in time.

3.2 The Continuous Model

As an example, the continuous model for the falling rod system (Fig. 7) in the stuck mode, α01, that is, its rotational behavior at the point of contact, A, is expressed as θ ag w& = −Kml+cos ml 2 & & f α 01 : v x = lsinθω v& = −l cos θω&  y

(5)

where m is the rod mass and J is its moment of inertia. vx and vy, the linear velocities of the center of mass, are not state variables since they are algebraic functions of ω given by  v x = lωsinθ .  v y = lωcosθ

(6)

This example illustrates the change in the size of the state vector when the rod moves from the free fall mode, α00, to the contact mode, α01. As discussed earlier, this can produce discontinuous changes in the state vector. In contrast, if the rod bounces back up after contact, the system moves from the contact mode to the free mode, and the state vector increases in size. No discontinuous changes can occur in this case.

3.3 The Discrete Model

Continuous physical system behaviors are governed by energy interactions and typically represented as state space models with the dynamic behaviors expressed as a set of ordinary differential equations (ODEs). For hybrid models, the continuous behavior in each real mode α is expressed as x& (t) = f α ( x α (t), u α (t), t). t ∈ ℜ and α ∈ ℵ. Xα ∈ ℜ is the continuous state vector, p and Uα ∈ ℜ is the vector of input signals. For every continuous mode α, there is one and only one field, fα, that defines system behavior. m

Figure 7. A collision between a thin rod and a floor

Discrete events in hybrid dynamic systems are modeling artifacts attributed to parameter and time scale abstractions. The discrete changes are modeled by the transition function, φ, and transitions are invoked by events in a set Σ. In our compositional modeling approach, we systematically derive φ from a set of independent state machines that define local switching effects. A mode is defined as the combination of the individual states of the state machines. n Theoretically, with n switching functions, 2 different modes are possible, but many of these modes do not have a physical representation. They may, for example, be traversed as mythical modes that occur between two real


SIMULATION 9


Figure 8. Two possible scenarios after collision

modes of operation. An important contribution of our work is to establish execution semantics that handle these sequences of mode changes correctly. The discrete model can be implemented by Petri nets or finite state machines. It is represented as • I = {α0, . . ., αk} is a set of states describing the modes of the system. • Σ = {σ0, . . ., σl} is the set of events that can cause state transitions. Events are generated from signal values in the physical process (Σs), or they can be external control signals (Σx), Σ = Σs × Σ x. • φ: I × Σ → I represents a discrete state transition function that defines the new mode after an event occurs.

3.4 Interactions Interactions between the continuous and discrete models are specified by (1) events generated in the continuous model and (2) mode changes defined by the discrete model. More formally they can be expressed as • S ∈ ℜ , the signals used for event generation. • h : X × U × I → S, returns signals from the input and state variable values in a given mode. • g : X × I → X+, computes the a posteriori state vector, X+, in the new mode from the a priori state vector, X. There may be a discontinuous change from X to X+. • γ : S × S+ → Σs, generates discrete events, Σs, from the signal values. These signal values may be computed from the a priori state vector, S, or the a posteriori state vector, S+. n

10

The function γ generates discrete events when signals cross prespecified threshold values. The collision transition for the falling rod is defined by the following constraints:  y + ≤ 0 ∧ v A, y < 0 ⇒ σ contact . γ:  A + ⇒ σ free  Fn ≤ 0

(7)

The output function, h, computes the values of these signals from the continuous state vector. For the signals used in the collision transition for the falling rod, this yields   y A = ∫ v y dt − lsinθ  = ( + ωcosθ . h:  v A, y m v y l if α 00 0   Fn = m (v& − a ) otherwise y g  

(8)

The generated events applied to the model may indicate that the system changes its mode of continuous operation. When mode switching occurs, the continuous state vector of the system may change. The function g transforms the continuous state vector as mode changes occur. In general, these transformations may be hard to derive, but for physical system models, this function has to satisfy the principle of conservation of state. When the falling rod first makes contact with the floor, conservation of state is applied to derive the state vector transformation function [5]:



ω + = ωJ − ml ( cosθv y2− sinθv x ) J + ml  . g α 01 :  v x+ = lω + sinθ  v + = −lω + cosθ  y

(9)

In mode α00, the rod has three degrees of freedom, and the state mapping does not cause discontinuous changes:

g

α 00

ω + = ω  : v x+ = v x . v + = v y  y

the impulse functions at the point of impact is determined by the state vectors immediately prior to the impact, x, and immediately after the impact, x+. To calculate x+, the principle of conservation of state is applied, which requires that the exchange of momentum between the linear and angular inertias upon impact does not generate or annihilate momentum [4, 5]. Upon contact, mode α01, the linear velocities of the center of mass, vx and vy, are completely determined by the angular velocity, ω:

(10)

4. Abstractions in Physical System Models Previous work [2, 3] formulated a systematic approach to hybrid modeling of dynamic physical systems using a hybrid bond graph formulation. Bond graphs are a language for building topological models of physical systems based on energy exchange between components of the system [8]. System components model physical phenomena, such as energy storage (capacitive and inertial elements), dissipation (resistive elements), and energy transformation between domains (transformer and gyrator elements). Hybrid bond graphs introduce switching mechanisms that cause dynamic changes in the topological model structure during behavior evolution. Schemes were developed to express switching specifications between the configurations (modes) as conditions based on state variables. Switching conditions may be expressed in terms of the state variables immediately before switching occurred (a priori values), or in terms of state variables computed by solving the initial value problem for the newly activated mode (a posteriori values). The a priori and a posteriori values may differ when the dimensions of the state vector change from one mode to another.

4.1 Abstractions Previous work [3, 4, 18] identified two types of abstraction phenomena that lead to discontinuities in physical system models: (1) parameter abstraction and (2) time scale abstraction. In this article, it is shown that switching conditions that result from parameter abstractions have to be in terms of a posteriori values, and conditions due to time scale abstraction have to be in terms of a priori values.

4.1.1 Parameter Abstraction Parameter abstractions occur when small dissipation and storage elements are ignored to simplify the system model. This may cause discontinuous changes in system behavior. Consider the example of the rigid body collision between the rod and the floor from Figure 7. The system needs to be analyzed to determine the mode on contact. This requires calculation of the forces at the point of collision. As discussed, the contact forces are impulses, P, that take on the form of Dirac pulses (δ) in time. The area of

v x+ = lω + sinθ .  + + v y = −lω cosθ

(11)

Conservation of momentum yields the new state vector [2, 4] ω+ =

ωJ − ml( cosθv y − sinθv x ) J + ml 2

,

(12)

where J is the rod inertia, m is its mass, and l is its length. These a posteriori values may be such that the impulses corresponding to the forces FA,x and Fn result in |PA,x| > µ Pn and the rod starts to slide (mode α11). In this mode, the horizontal velocity v x+ is not algebraically dependent on ω, but + the vertical velocity is; that is, v y+ = –lω cosθ holds. Thus, the rod tip moves freely in the x-direction. Its vertical momentum immediately before contact (mode α00) is distributed over its a posteriori (after impact) angular velocity and a posteriori vertical momentum to ensure yA does not change (i.e., the constraint to keep it in contact with the floor holds). In Figure 8, the transition conditions between modes α11 (slide) and α01 (stuck) are defined by automata S, where vth represents a small threshold value. If the continuous state vector in the sliding mode, α11, was computed from the previously inferred mode, α01, the + dependence of v x+ on ω in that mode would result in a horizontal velocity associated with its center of mass, which would keep the rod tip from moving in the x-direction as well. That would be physically incorrect. So the consecutive mode switch to α11 has to occur before the state vector is updated to its a posteriori values, x = x+. Mathematically, this can be represented as  x + = g α 1 ( x)  + x = x  x& = f ( x, t) α1  442 1 44 3

γ αα 2 ( x , x + ) 1

→

α1

γ αα 3 ( x , x + ) 2

→

 x + = g α 2 ( x)     44244 1 3

(13)

α2

α3

+

 x = g ( x)  + , x = x  x& = f ( x, t) α3  442 1 44 3 α3


SIMULATION 11


Figure 10. A collision between two bodies

Figure 9. Coulomb friction causes physical events

where ε, the coefficient of restitution, is typically a function of v1 and v2. Combining eqs. (14) and (15) results in where α2 is a so-called mythical mode [5, 13]. It is clear that switching in this example has to be based on x+ rather than x (see Figs. 7 and 8). Figure 7 shows the constraint that achieves the contact mode of operation. As long as the rod exerts a downward force on the floor, it stays in contact. If the normal force, Fn, becomes negative, the rod disconnects and lifts off the floor. In the sliding mode, α11, vA,x ≠ 0; therefore, there is a Coloumb friction force Ff in the opposite direction. This force changes discontinuously at vA,x = 0 and that causes another immediate mode change to α21. This is illustrated in Figure 9, which shows the three modes of the Coulomb friction characteristic. In mode 1, the horizontal velocity is 0 and the force at the contact point is free but limited by a breakaway force that in general is larger than the constant friction force, Ff , in one of the sliding modes. In mode 2, there is a negative horizontal velocity and a constant friction force acting. Mode 3 is similar.

v 1+ = 1 −∈ v 1 + 1 +2 ∈ v 2 2 .  + 1+ ∈ 1 −∈ v 2 = 2 v 1 + 2 v 2

This algebraic relation only holds at a point in time. Therefore, the switching specifications have to ensure that the collide mode is departed immediately after the state vector is updated, x = x+. Mathematically, this is represented as  x + = g α 1 ( x)  + x = x  x& = f ( x, t) α1  442 1 44 3

v 2+ + v 1+ = v2 + v1,

(14)

where v1 and v2 are the velocities of the masses before collision, and v 1+ and v 2+ are their velocities after collision. In detail, the collision process includes elasticity effects that store and return the kinetic energy of the masses over a short period of time. Because the time scale of this phenomenon is very small compared with the behavior of interest, it can be modeled as an instantaneous change at a point in time governed by the restitution equation: v 2+ – v 1+ = –ε (v2 – v1),

12

(15)

γ αα 2 ( x , x + ) 1

→

α1

γ αα 3 ( x , x + ) 2

→

4.1.2 Time Scale Abstraction Time scale abstractions are applied to behaviors that occur on a small time scale. The behavior is not abstracted away but compressed to occur at a point in time, and this may again cause discontinuous changes in the system behavior. Consider the perfectly elastic collision of two bodies. Mass m1 moving with linear velocity v1 collides with mass m2 moving with velocity v2 (see Fig. 10). In general, the collision phenomenon is governed by the principle of conservation of momentum. Using m1 = m2 yields

(16)

 x + = g α 2 ( x)  + x = x   44244 1 3 α2

(17)

,

α3

+

 x = g ( x)  + x = x  x& = f ( x, t) α3 1442 44 3 α3

where α2 is a pinnacle. Figure 10 illustrates that switching specifications have to be in terms of a priori state variable values. Moreover, a switching specification such as is used for the falling rod to make the bodies disconnect when the force between them becomes negative, F12 < 0, cannot be applied here because on collision F12 > 0, and the constraint would not move the system of bodies into a free mode immediately after collision. Thus, x& = fα(x,t) would be executed, but fα does not exist for the collision mode where behavior is defined by algebraic relations, see eq. (16).

4.1.3 Summary The two types of abstraction have a distinctly different effect on how to formulate switching specifications and introduce the fundamentally different behavior between mythical modes and pinnacles. Parameter and time scale abstractions have been applied to a number of examples. The diode inductor example presented in the Introduction, and analyzed in greater detail in other work [6], is another application of parameter abstraction. Parameter abstractions have also been applied to clutch and braking systems



[7]. A realistic application of time scale abstractions has been in modeling of hydraulic actuators of aircraft attitude control systems [19]. Time scale abstraction collapses behavior during small intervals into points, and the switching model uses a priori state values. In contrast, parameter abstraction abstracts away complex nonlinear behaviors, which are modeled by switching conditions based on a posteriori state values computed by g αβ . The mythical modes that result from these conditions are modeling artifacts and have no real representation, and, therefore, do not affect the state vector, x. This is called the principle of invariance of state [2]. Mythical modes can be replaced by direct transitions to the final real mode (either a pinnacle, Pα, or a continuous mode, F α ). However, finding these direct transitions requires considerable effort and because of its global character needs to be performed whenever local changes to the model are made. A more pragmatic approach is to incorporate systematic techniques in the compositional modeling formalism to deal with these artifacts. Furthermore, translating a system model into a model where only a priori state variable values are used complicates the model verification task considerably. If a posteriori values are used, invariance of state can be conveniently applied for model verification purposes [3].

4.2 Physical Model Semantics A mathematical model can now be presented that embodies the physical abstraction semantics. It relies on a switching function γ βα that depends on values xα prior to the jump, and values x α+ just after the jump. The semantics are specified by the recursive relation between γ βα and g αβ , which takes the form  x α+ k = g αα ki ( x α k ) .  αi +1 +  γ α i ( x α k , x α k ≤ 0)

(18)

Note the αk subscript of x α k in g αα ki ( x α k ). In physical systems, continuous behavior is completely specified by the state. Therefore, the state mapping is independent of the departed mode, that is, g αβ is independent of α. This results in the general sequence  x + = g α 1 ( x)  + x = x  x& = f ( x, t) α1  442 1 44 3

γ αα 2 ( x , x + ) 1

→

α1

γ αα 3 ( x , x + ) 2

→ K

γ αα m

 x + = g α 2 ( x)  + x = x  x& = f ( x, t) α2  442 1 44 3 α2

m −1

(x , x + )

→

(19)

.

αm

+

 x = g ( x)  + x = x  x& = f ( x, t) αm 1442 44 3 αm

In this sequence, each mode, α, may be departed when any of the three assignment statements is executed. The differ-

Figure 11. A general hybrid system

+

ence between eq. (3) and eq. (19) is the use of (x,x ) as argument to γ βα . A block diagram of the resultant operation of a hybrid system corresponding to the 9-tuple model discussed in Section 3 is presented in Figure 11. The model includes the semantic specifications developed in the last section. In relation to the model in Figure 6, there is an additional feedback, x+, from g to γ. Overall, three execution cases can be distinguished (Fig. 12): 1. Mythical mode: This occurs when x+ = g α i (x) leads to γ αα ii + 1 ( x, x + ) ≤ 0. The immediate transition caused by x+ bypasses the integrator (∫) element and the state vector, x, remains unchanged through the transition. As shown in Figure 12(a), mythical modes can occur in the loops (i) φ → h → γ. This corresponds to mythical modes with no discontinuous changes in the state vector. (ii) φ → g → h → γ. This corresponds to mythical modes with discontinuous changes in the state vector. 2. Pinnacle: This occurs when x = x + results in γ αα ii + 1 ( x, x + ) ≤ 0. Updating of x stored by the ∫ element causes a mode transition, and, therefore, mode


SIMULATION 13


Figure 12. Classes of modes of operation

αi only exists at a point in time. In Figure 12(b), pinnacles are shown to arise from the φ → g → ∫ → h → γ loop since they require the state vector to be updated in the ∫ element. 3. Continuous mode: This occurs when x& = f ( x, t) results in γ αα ii + 1 ( x, x + ) > 0. The system goes into a period of continuous evolution, until γ αα ii + 1 ( x, x + ) ≤ 0, and the mode is departed. In Figure 12(c), continuous modes arise from the φ → f → ∫ → h → γ loop while g is not part of this loop because x+ = x during continuous behavior. Pinnacles and continuous modes are also referred to as real modes because they change x stored by the ∫ element. Comparison with the general sequence in eq. (3) shows that the g αβ operation can be associated with the transition or the new mode. In this article, g αβ is associated with modes for two reasons: (1) it affects the energy distribution in the system, and therefore it should correspond to a physical mode of operation, and (2) for mythical mode transitions, the state vector x remains unchanged; therefore, it makes it easier to implement a simulator if g is associated with a mode rather than a transition. 5. The Hybrid Dynamic System Simulator The hybrid dynamic simulator implements the model of Figure 11. The integrator is the core of the continuous simulation component. To simplify the implementation, a forward Euler integrator that approximates derivatives by x& = x k + ∆1 t− x k or xk+1 = f∆t + xk is used, though in principle any solver with a root-finding mechanism can be used. For example, the state equations in eq. (5) would be implemented as

14

ω k + 1 = − cosθ 2k a g ∆t + ω k J + ml  f α 01 : v x , k + 1 = lsinθ k + 1 ω k + 1 v  y , k + 1 = −lcosθ k + 1 ω k + 1

(20)

(ag represents the gravitational constant, the acceleration due to gravity). The h function is implemented in a similar manner (21)  y A+ = y M+ , k + 1 − lsinθ +k + 1  + + + +  v A, x = v x , k + 1 − lsinθ k + 1ω k + 1  v A, y = v y , k + lcosθ k ω k  if α 00 . h:  +  0 +  Fn = m v y , k + 1 − v y , k − a g erwise oth ∆t    + if α 00 0  FA, x =  v x+ , k + 1 − v x , k m otherwise   ∆t

(

)

The derivative terms in Fn and FA,x may produce Dirac pulses and are termed hd. The remaining terms are part of hi. If no discontinuous changes occur (e.g., v y+, k + 1 = vy,k+1), the forces Fn and FA,x can be estimated numerically using the above equation. When discontinuous changes occur (e.g., v y+, k + 1 ≠ vy,k+1), the derivative term is a Dirac pulse of infinite magnitude that dominates all the other terms in the equation. However, the numerical approximation may compute a pulse magnitude that does not dominate the other terms in the equation, and this would result in incorrect simulation results. Therefore, discontinuous changes are tracked separately, and when they occur, their time derivative terms are replaced by Dirac pulses. Algorithm 1 implements the derSignal function. If ∆T = 0 for a pinnacle,



Figure 13. A number of trajectories in phase space of the colliding rod, vth = 0.2, θ = 0.85, l = –1.0, y0 = 1.5

or if hd(x , α) ≠ hd(x, α) for a discontinuous change, the area of the corresponding Dirac pulse is returned. Otherwise, no discontinuous changes have occurred, and the Euler approximation is returned as the value of the derivative. Algorithm 2 is the primary simulation module. The recursive relation, eq. (18), is implemented as function selMode (Algorithm 3), executed twice for each time step: (1) when continuous evolution terminates in a new mode,4 and (2) when pinnacles are traversed, and the ∆T argument in Algorithm 3 is set to 0. +

+ k +1

Algorithm 1 derSignal ( x , x k + 1 , x k , α m , ∆T ) if ∆T = 0 then s = hi(xk+1, αm) else s = (hd(xk+1, αm) – hd(xk, αm))/∆T + hi(xk+1, αm) end if if ∆T = 0 ∨ h d ( x k++ 1 , α m ) ≠ h d ( x k + 1 , α m ) then s + = h d ( x k++ 1 , α m ) − h d ( x k + 1 , α m ) else s + = ( h d ( x k++ 1 , α m ) − h d ( x k , α m )) / ∆T + h i ( x k++ 1 , α m ) end if Algorithm 2 Hybrid Simulation x k + 1 = x( 0); x k++ 1 = x k + 1 ; x k = x k + 1 t = 0;δt ← Simulation time step α m + 1 = selMode( x k++ 1 , x k + 1 , x k , α 0 , 0) if α m + 1 ≠ α 0 then repeat x k + 1 = x k++ 1 α m = α m +1 α m + 1 = selMode( x k++ 1 , x k + 1 , x k , α m , 0) until α m + 1 = α m end if {Initialization Completed} repeat t = t + δt x k + 1 = f ( x k , t)δt x k++ 1 = x k + 1 α m + 1 = selMode( x k++ 1 , x k + 1 , x k , α m , 0)

if α m + 1 ≠ α 0 then repeat x k + 1 = x k++ 1 α m = α m +1 α m + 1 = selMode( x k++ 1 , x k + 1 , x k , α 0 , 0) until α m + 1 = α m end if x k = x k +1 until simulation end Algorithm 3 selMode( x k++ 1 , x k + 1 , x k , α m , ∆T ) a m + 1 = φ(α m , derSignal( x k++ 1 , x k + 1 , x k , α m , ∆T )) if α m + 1 ≠ α m then x k++ 1 = g α m + 1 ( x k + 1 ) ret _ val = selMode( x k++ 1 , x k + 1 , x k , α m + 1 , ∆T ) else ret _ val = α m end if The simulated trajectories of the rod in phase space for three different values of the friction coefficient (µ) are shown in Figure 13. The system is initialized with zero angular and linear velocities (vx, vy, ω) = (0,0,0). Because of the orientation of the coordinate system of the rod, the position of the rod tip, point A, is at a negative value of l. Once the rod is released, flow Fα 00 applies, and the magnitude of its vertical velocity increases in time. When the rod tip, A, touches the floor, the rod may start to slide, governed by flow Fα 21 (happens for µ = 0.26 and µ = 0.265), or it may get stuck and behavior is governed by flow Fα 01 (happens for µ = 0.27). The discontinuous jumps between flows are illustrated in Figure 13. Also, for simulations with µ = 0.26 and µ = 0.265, the sliding mode, α21 is activated immediately after α00 because a force balance computation indicates that the stuck mode α01 is departed instantaneously; that is, it has no real existence at the point of collision. 4. Precision is improved by varying δt and employing a bisectional search.


SIMULATION 15


When sliding, the center of mass of the rod accelerates in the horizontal direction, and the magnitude of the negative velocity at the rod tip decreases. When it falls below a threshold value, transition conditions determine that the rod gets stuck, which implies a mode change to α01 and flow Fα 01 . The transition conditions had to be properly specified so that the system does not go into a loop of instantaneous mode changes (sliding and stuck), which would violate the physical principle that system behavior must evolve in time [2, 3]. 6. Conclusions This article presents a formal mathematical framework for analyzing hybrid behaviors and uses the framework to develop a methodology for simulating behaviors of dynamic physical systems. This extends previous work on compositional modeling of hybrid systems combining bond graph models with local discrete finite state automata [3, 7]. Parameter and time scale abstractions are the key to developing systematic switching specifications that are governed by both a priori and a posteriori state vector values. Hybrid dynamic behaviors are piecewise continuous and combine continuous behavior over intervals of time with pinnacles that represent a real behavior at a point in time (time scale abstraction) and mythical modes that have no real existence on the timeline (artifact of parameter abstraction). The formal model specifications are made up of three components, (1) the continuous model, (2) the discrete model, and (3) the interaction model, and define a methodology for developing hybrid dynamic models of physical systems. A hybrid dynamic simulator is developed as a combination of three algorithms from the mathematical execution semantics. State vectors variable values have to satisfy the principle of invariance of state across mythical mode changes, whereas discontinuous changes in state variables that occur at pinnacles are derived using the principle of conservation of state and explicitly defined interactions with the environment (represented as Dirac pulses). When mode changes occur, global specifications are derived dynamically from local switching functions. The use of local specifications often results in the system going through a sequence of discrete changes, but the modeling task is simplified because compositional modeling principles can be applied. This contrasts other approaches that have been employed for hybrid modeling (e.g., [12]), which require predefined global specifications of continuous system behavior in terms of differential equations. Current research efforts are directed toward extending this methodology to design and analysis of embedded (computerbased) control of complex physical systems [6, 8]. The hybrid bond graph modeling and simulation algorithms have been integrated into a software package called HYBRSIM [20]. This package, implemented in Java, can be accessed on the Web from http://www.op.dlr.de/~pjm/ hybrsim. HYBRSIM establishes a formal framework and serves as a precursor to an object-oriented implementation as part of the Modelica modeling language [21].

16

7. Acknowledgments The authors would like to thank the two anonymous reviewers for their valuable suggestions in improving this article. 8. References [1] Paynter, Henry M. 1961. Analysis and design of engineering systems. Cambridge, MA: MIT Press. [2] Mosterman, Pieter J. 1997. Hybrid dynamic systems: A hybrid bond graph modeling paradigm and its application in diagnosis. PhD dissertation, Vanderbilt University, Nashville, TN. [3] Mosterman, Pieter J., and Gautam Biswas. 1998. A theory of discontinuities in dynamic physical systems. Journal of the Franklin Institute 335B (3): 401-39. [4] Mosterman, Pieter J., and Gautam Biswas. 2000. A comprehensive methodology for building hybrid models of physical systems. Artificial Intelligence 121:171-209. [5] Mosterman, Pieter J., and Gautam Biswas. 1997. Formal specifications for hybrid dynamical systems. Paper presented at IJCAI-97, pp. 568-73, August, Nagoya, Japan. [4] Cellier, F. E., H. Elmqvist, and M. Otter. 1991. Modelling from physical principles. In The control handbook, edited by W. S. Levine, pp. 99-107. Boca Raton, FL: CRC Press. [5] de Vries, Theo J. A., Peter C. Breedveld, and Piet Meindertsma. 1993. Polymorphic modeling of engineering systems. Paper presented at Proceedings of the International Conference on Bond Graph Modeling, pp. 17-22, San Diego, CA. [6] Mosterman, Pieter J., Gautam Biswas, and Janos Sztipanovits. 1998. A hybrid modeling and verification paradigm for embedded control systems. Control Engineering Practice 6:511-21. [7] Mosterman, Pieter J., Gautam Biswas, and Martin Otter. Simulation of discontinuities in physical system models based on conservation principles. Paper presented at SCS Summer Simulation Conference, July, Reno, CA. [8] Mosterman, Pieter J., Feng Zhao, and Gautam Biswas. 1999. Sliding mode model semantics and simulation for hybrid systems. In Hybrid systems V, lecture notes in computer science, edited by P. Antsalelis, W. Kohn, M. Lemmon, A. Nerode, and S. Sastry, pp. 218-37. New York: Springer-Verlag. [9] Deshpande, Akash, and Pravi Varaiya. 1995. Viable control of hybrid systems. In Hybrid systems II, edited by Panos Antsaklis, Wolf Kohn, Anil Nerode, and Shankar Sastry, vol. 999, pp. 128-47. Lecture Notes in Computer Science. New York: Springer-Verlag. [10] Guckenheimer, John, and Stewart Johnson. 1995. Planar hybrid systems. In Hybrid systems II, edited by Panos Antsaklis, Wolf Kohn, Anil Nerode, and Shankar Sastry, vol. 999, pp. 202-25. Lecture Notes in Computer Science. New York: Springer-Verlag. [11] Rudin, W. 1976. Principles of mathematical analysis. 3rd ed. New York: McGraw-Hill. [12] Alur, R., C. Courcoubetis, N. Halbwachs, T. A. Henzinger, P.-H. Ho, X. Nicollin, A. Olivero, J. Sifakis, and S. Yovine. 1994. The algorithmic analysis of hybrid systems. In Proceedings of the 11th International Conference on Analysis and Optimization of Discrete Event Systems, edited by J. W. Bakkers, C. Huizing, W. P. de Roeres, and G. Rozenberg, pp. 331-51. Lecture Notes in Control and Information Sciences 199. New York: Springer-Verlag. [13] Nishida, T., and S. Doshita. 1987. Reasoning about discontinuous change. Paper presented at Proceedings AAAI-87, pp. 643-8, Seattle, WA. [14] Alur, Rajeev, Costas Courcoubetis, Thomas A. Henzinger, and Pei-Hsin Ho. 1993. Hybrid automata: An algorithmic approach to the specification and verification of hybrid systems. In Lecture notes in computer science, edited by R. L. Grossma, A. Nerode, A. P. Ravn, and H. Rischel, vol. 736, pp. 209-29. New York: Springer-Verlag. [15] Lennartson, Bengt, Michael Tittus, Bo Egardt, and Stefan Pettersson. 1996. Hybrid systems in process control. IEEE Control Systems, October, pp. 45-56.



[16] Lötstedt, P. 1981. Coulomb friction in two-dimensional rigid body systems. Zeitschrift für angewandte Mathematik and Mechanik 61:605-15. [17] Brach, Raymond M. 1991. Mechanical impact dynamics. New York: John Wiley. [18] Mosterman, Pieter J., and Gautam Biswas. 1997. Principles for modeling, verification, and simulation of hybrid dynamic systems. Paper presented at Fifth International Conference on Hybrid Systems, pp. 21-27, September, Notre Dame, IN. [19] Mosterman, Pieter J., and Gautam Biswas. 1999. Hybrid automata for modeling discrete transitions in complex dynamic systems. Hybrid Systems: Computation and Control, Lecture Notes in Computer Science, vol. 1569, pp. 178-92. The Netherlands: Springer Verlag. [20] Mosterman, Pieter J., and Gautam Biswas. 1999. A Java implementation of an environment for hybrid modeling and simulation of dynamic physical systems physical systems. Paper presented at Proceedings of ICBGM ’99, pp. 157-62, January, San Francisco, CA. [21] Mosterman, Pieter J. 2001. MASIM—A Hybrid Dynamic Systems Simulator. Technical report DLR-IB-515-01-07, Institute of Robotics and Mechatronics, DLR Oberpfaffenhofen.

Pieter J. Mosterman recently assumed a position as senior research scientist at The MathWorks, Inc., in Natick, MA. Before, he held a research position at the German Aerospace Center (DLR) in Oberpfaffenhofen. He has a Ph.D. degree in electrical and computer engineering from Vanderbilt University in Nashville, TN, and a M.Sc. degree in electrical engineering from the University of Twente, Netherlands. His primary research interests are in computer-automated multi-paradigm modeling with principal applications in training systems and fault detection, isolation, and reconfiguration. For this, he designed several simulation environments, a.o., the Electronics Laboratory Simulator (nominated for The Computerworld Smithsonian Award), a first version of TRANSCEND, HyBrSim, and MASIM. Specific areas of interest are modeling of physical systems, meta-modeling, and model and formalism transformation to achieve a form of higher intelligence in computer-aided control system design (CACSD). An important aspect concerns the behavior generation for heterogeneous models, which requires a hybrid dynamic systems approach. Dr. Mosterman has served on the program committee of several conferences and workshops and organized special

sessions at the 2001 IEEE Conference on Control Applications, the 2000 IEEE International Symposium on CACSD, and Eurosim ’98. He is currently a member of the IFAC Technical Committee on CACSD, chair of the IEEE CSS Action Group on Hybrid Dynamic Systems for CACSD, expert reviewer for the European Commission, editor of SIMULATION: Transactions of The Society for Modeling and Simulation International for the area of mechatronics, and associate editor of the International Journal of Applied Intelligence. He is also guest editor of special issues of ACM TRANSACTIONS ON MODELING AND COMPUTER SIMULATION and IEEE TRANSACTIONS ON CONTROL SYSTEM TECHNOLOGY on the topic of computer automated multi-paradigm modeling.

Gautam Biswas is an associate professor of Computer Science and Engineering, and Management of Technology at Vanderbilt University. He has a Ph.D. in computer science from Michigan State University in East Lansing, MI. He conducts research in intelligent systems with primary interests in hybrid modeling and analysis of complex embedded systems and their applications to diagnosis and fault-adaptive control. He is also involved in developing simulation-based environments for learning and instruction. Other research areas include Hidden Markov Model techniques for clustering of temporal data sequences and decision-theoretic planning and scheduling for intelligent manufacturing systems. His research is currently supported by funding from DARPA, NASA, NSF, and ONR. Dr. Biswas has served on the program committee of a number of conferences. He was chair of the 1997 IJCAI Workshop on Engineering Problems for Qualitative Reasoning, co-chair of the 1996 Principles of Diagnosis Workshop, the 1999 AAAI Spring Symposium on Hybrid Systems, and AI, the 2001 Workshop on Qualitative Reasoning, Senior Program Committee for AAAI-97 and AAAI-98, and Technical Committee co-chair for the 2000 IEEE SMC conference. He is currently an associate editor for the IEEE Transactions on Systems, Man, and Cybernetics, IEEE Transactions on Knowledge and Data Engineering, and the International Journal of Applied Intelligence. He is a senior member of the IEEE Computer Society, ACM, AAAI, and the Sigma Xi Research Society.


SIMULATION 17

A Hybrid Modeling and Simulation Methodology for Dynamic Physical

A Hybrid Modeling and Simulation Methodology for Dynamic Physical

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