Differential Petri Net Models for Industrial Automation ... - IEEE Xplore

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 36, NO. 4, JULY 2006

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Differential Petri Net Models for Industrial Automation and Supervisory Control Isabel Demongodin, Member, IEEE, and Nick T. Koussoulas, Member, IEEE

Abstract—Supervisory control systems play a central role in modern industrial automation. However, control theory has recently made significant advances in modeling mixed continuous/discrete event systems (“hybrid control systems”), whose typical instantiations include the industrial supervisory controller. This article shows how differential Petri nets, a model for hybrid control systems, can be used to represent industrial supervisory systems in a unified way. Typical industrial automation tests can be modeled, whereas the effect of communication protocols and software can be straightforwardly included using conventional Petri nets. Therefore, a global model for the operation of an industrial control system can be formed and its behavior analyzed. Index Terms—Automation, hybrid systems, Petri nets.

I. INTRODUCTION MORE recent point of convergence between industrial automation and control theory has been the objective to deal effectively with supervisory control systems. The benefits from this convergence are mutual: industry can certainly benefit from the theoretical support in terms of better and more reliable design and increased power of analysis, whereas control theory can have a vast and important domain of application. From the part of control theory, the effort has been to create a modeling framework that can include not only the technological process, but also its supervisory scheme(s) (Fig. 1). These two parts are totally different in nature. The technological process has continuous dynamics that are typically modeled by ordinary differential equations. The supervisory mechanism responsible for deciding on commands (e.g., set points) and control policies (e.g., PID, self-tuning) typically consists of logical operations executed via programmable logic controllers, and the corresponding models belong to the class of discrete event dynamic systems (DEDS). Moreover, networking can introduce additional complexity. The resulting overall supervisory control system, being a mixture of continuous time and discrete event dynamic processes, has been termed “hybrid” or “discontinuous.” Modeling, analysis, control, and synthesis of such systems pose a number of challenging problems. Different approaches to modeling have been proposed and, at present, there are already

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Manuscript received November 14, 2003; revised December 2, 2004. This work was partially supported by the European Commission of the European Union through the ESPRIT-8924 program SESDIP-Structural Evaluation and Synthesis of Distributed Industrial Processes. This paper was recommended by Associate Editor P. Borne. I. Demongodin is with the Ecole des Mines de Nantes, France. She is now with LISA (Laboratory of Automation and System Engineering), University. Angers—CNRS FRE 2656, 49000 Angers, France (e-mail: isabel. [email protected]). N. T. Koussoulas is with the Electrical and Computer Engineering Department, University of Patras, Patras, Greece (e-mail: [email protected]. gr). Digital Object Identifier 10.1109/TSMCC.2005.848154

a large number of models for hybrid control systems. The basis of the modeling effort is to start from a model for either part of the hybrid system and extend it so it will be able to account for the dynamics of the other part. Most of the proposed models are based on a DEDS model with a suitable extension for the continuous dynamics representation. An informative overview can be found in [1] and [2], whereas current and past developments and applications are well represented in the series [3]–[7]. Among the proposed DEDS models, Petri nets have a special position because they seem to be more accepted in industry and the applications domain, in general. Extensions of the basic Petri model have already been used to model systems where some kind of continuity or full continuity as first described in hybrid Petri nets is present [8]. In the domain of hybrid systems, more than 15 Petri net-based models have been developed so far, and they are summarized in an excellent survey [9]. Among those models, differential Petri nets (DPNs) [10] constitute an extension of hybrid Petri nets [11], [12] and can represent simultaneously continuous dynamic systems, modeled as systems of ordinary differential equations, and DEDS. This article demonstrates the application of DPN in modeling common industrial supervision tasks. Section II presents an extensive introduction to the DPN model, including the construction of the evolution graph. In section III, we provide DPN models for typical industrial automation tests and monitoring functions. Section IV contains a detailed example of a system whose model consists of switching between two alternative representations dependent on current conditions, and is followed by the Conclusion. II. BACKGROUND ON DIFFERENTIAL PETRI NETS A. Model Definition DPNs extend and combine the advantages of continuoustype Petri net models and those of the discrete Petri net. The novel features of DPNs are the negative real values accepted for place markings and the use of an integration mechanism for the approximate representation of the continuous system. Under the assumption that the continuous system can be represented by a finite number of linear first-order difference state equations, DPNs provide enough power to model a hybrid system in a single graph. Full development of the DPN model can be found in [10]. However, for the convenience of the reader, we present a succinct description. A DPN is composed of two kinds of places and two kinds of transitions: 1) discrete place and discrete transition 2) differential place and differential transition as shown in Fig. 2.

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Fig. 1.


Structure of a typical industrial automation system.

Fig. 2.

Nodes of a differential Petri net.

Fig. 3.

Explicit representation of a discrete implicit differential transition.

Marking of places is similar in principle to that of the hybrid Petri nets, except that in DPNs the marking is allowed to also take negative real values. Another interesting aspect is that to every differential transition we associate a firing speed representing either a variable proportional to a state variable (or a marking of a differential place) or an independent variable. Because a differential transition is always enabled, to discretize the continuous system, we introduce to every differential transition a firing frequency representing the integration step that would be used when carrying out a simulation. According to the Petri net theory, this delay is associated with the implicit discrete transition linked by a discrete place to this differential transition, as shown in Fig. 3. The intrinsic characteristics associated with a differential transition do not allow the definition of autonomous DPNs. In fact, inherent in the differential transition is the notion of time, permitting the discretized “view” of continuous systems with

a certain period (namely, the integration step). Thus, a DPN is actually a timed DPN whose definition follows: Definition: A timed differential Petri net is defined by B = R, f, M0 , T verifying the following conditions: 1) R is a Petri net defined by R = P, T, Pre, Post with a) P : a finite set of places with dim P = n < ∞; b) T : a finite set of transitions with dim T = m < ∞; c) P ∩ T = ∅ and P ∪ T = ∅; d) Pre(Pi , Tj ) is a function that defines arcs from a place to a transition with Pre(Pi , Tj ) ∈ R; e) Post(Pi , Tj ) is a function that defines arcs from a transition to a place with Post(Pi , Tj ) ∈ R. 2) f : P ∪ T → {D, DF }, named “differential function,” indicates for every node whether it is a discrete or differential node; 3) M0 is the initial marking; 4) T is a map, to be called “timing map,” that associates a real number to every transition that can evolve in time, and also a delay for each differential transition. The delay in each differential transition corresponds directly to the integration step that would be used when a simulated evolution of the continuous system was carried out. The choice of the numerical value of this delay is constrained by numerical stability considerations and is a tradeoff between calculation speed and accuracy. More on the selection of integration steps can be found even in elementary books on numerical methods covering the integration of ordinary differential equations. It must be noted that it is possible to use a different delay for each differential transition. This may prove quite helpful in the representation of distributed systems, for instance. The marking vector is composed of two components. The first one, named M D , is a discrete vector where the integer value corresponds to the number of tokens inside discrete places. The second one, named M DF , contains real values corresponding to the continuous markings of differential places. From this decomposition, we can define the global marking vector of a

DEMONGODIN AND KOUSSOULAS: DIFFERENTIAL PETRI NET MODELS FOR INDUSTRIAL AUTOMATION AND SUPERVISORY CONTROL

Fig. 4.

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Node of the evolution graph.

DPN as follows: M D (t) ............. M (t) = M DF (t)   D m1 (t)  ..    .    mD  (t) n d  =   DF     m1 (t)     . ..   mDF n c (t)

introduced in the same way as for general Petri nets, under the constraint that the values corresponding to the discrete part must be integer. 

 mD 1 (t) ∈ N  ..    .    mD  (t) ∈ N n d   with   DF   m1 (t) ∈ R     . ..   mDF n c (t) ∈ R

where nd and nc are the number of discrete places and of differential places, respectively. It should be noted that a number of discrete places and discrete transitions are implicit (see Fig. 3). For these nodes we have that dim P DI = dim T DI = dim T DF , i.e., the number of discrete implicit places is equal to the number of discrete implicit transitions, and both are equal to the number of differential transitions because they serve as “support” for the latter. The marking of the discrete implicit places is always equal to 1 mDI i (t) = 1,

∀t, ∀i ∈ P DI

where P DI is the set of discrete implicit places, with P DI ⊂ P . It should also be noted that when a discrete transition is enabled at a date t, the tokens required to fire this transition are reserved during the associated delay. When the delay is over, the transition is fired, and the tokens (or marking) reserved for firing are removed from the preplaces of this discrete transition, whereas the nonreserved tokens (or marks) are added to the postplaces. Thus, at any time, the present marking for discrete places is the sum of the reserved and the unreserved markings. Reserved marks can be defined in a similar way for the differential places. The corresponding markings are denoted by rD and rDF , respectively. At the time of expiration of the delay in the discrete transition, the marking of the differential place is reduced by the value of the reserved marking. The previous markings are explained more extensively in [10]. Finally, invariants can be

B. Dynamics For applications, it is more important to emphasize the more applied aspects of DPNs such as the evolution graph used to analyze the dynamic behavior. The evolution graph of a DPN can be formed by nodes corresponding to interevent states (also called stable states) and transitions with delay between these nodes corresponding to events. A node is divided in two parts, the first one representing the markings of the discrete places, and the second representing the firing speed vector of the differential transitions and the reserved markings, as shown in Fig. 4. These nodes are linked by arcs and transitions that determine the occurrence of events and the “expended” delay between two consecutive interevent states. The transition associated to this arc is noted as follows: occurrence of events | dk ,

where dk is the elapsed time.

We can also define an incidence matrix W for every DPN W = [Wij ]n ×m where Wij = Post(Pi , Tj ) − Pre(Pi , Tj ). Furthermore, a “fundamental relation” can be formulated from which it is possible to deduce from a given marking at time ti the reachable marking at the date tk tk M (tk ) = M (ti ) + W · σ(tk ) + v(u) du ti

where the characteristic vector σ(t) represents the firing sequence for the discrete transitions and the speeds vector v(t) contains the instantaneous firing speeds of differential transitions.

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Fig. 5.

DPN representation of the system.

C. Example We consider a system whose behavior can be represented by an alternation between the following two models: x(t) ˙ = −3 x(t) ˙ = −0.5x(t). The switching between these two continuous models is governed by the following simple rule. For three time units, the system admits model I, and for the following five time units, model II is valid; then the same cycle repeats. At the switching instant, there occurs a jump of constant and known magnitude in the state of the system. Therefore, the rules are for kT ≤ t < kT +

3T , 8

k = 0, 1, . . . use model x(t) ˙ = −3

3T ≤ t ≤ (k + 1)T, k = 0, 1, . . . set for kT + 8 3T 3T x kT + ← x kT + +4 8 8 and use model x(t) ˙ = −0.5x(t). We choose the integration step h = 0.5, which is crude but effective for demonstration purposes. Considering an initial condition of 3 (marking of P5 in Fig. 5), the DPN model is as shown in Fig. 5: We define the following sets: P D = {P1 , P2 }, discrete places T D = {T1 , T2 }, discrete transitions P DI = {P3 , P4 }, discrete implicit places DI

= {T3 , T4 }, discrete implicit transitions

DF

= {P5 }, differential place

T P

T DF = {T5 , T6 }, differential transitions

Then, the Pre-, Post-, and incidence matrices become   0 0 1 0 0 1 1 0 0 0 0 1   1 0 1 0, Pre =  0 0   0 0 0 1 0 1 0 0 0 0 0 0   1 0 0 0 1 0 0 1 0 0 0 1    Post =  0 0 1 0 1 0    0 0 0 1 0 1 4 0 0 0 −0.5 −3 so we obtain     W = 

1 −1 −1 1 0 0 0 0 4 0 TD

0 0 0 1 0 0 0 0 0 0

 0 0 PD 0 0   0 0   P DI . 0 0 −0.5 −3 } P DF

T DI

T DF

Places P1 and P2 , along with T1 and T2 , realize the model selection rules. Place P5 represents the state variable x(t), and its marking indicates the value of that state. The combinations {T6 , P5 } and {T5 , P5 } realize the two continuous dynamic models I and II, respectively. Each of these continuous models is “supported” by the corresponding discrete implicit nodes {T4 , P4 } and {T3 , P3 }, respectively. Place P1 activates model II, whereas place P2 activates model I. The dynamics of the net are determined by the instantaneous firing speeds [12] of the differential transitions. Thus, for model I, we notice that the dynamics are described by the following relations: v6 (t) = 1,

when T6 is enabled

v6 (t) = 0,

otherwise.

The dynamics of the DPN are basically the dynamics of P5 , which are characterized by the previous relations and the


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(b)

(c) Fig. 6. (Continued.) (b) Compressed evolution graph of the example DPN. (c) System state trajectory (marking of place P 5 ) vs. time of the example DPN.

(a) Fig. 6.

(a) Evolution graph (complete) of the example DPN.

following: v5 (t) = m5 (t) dm5 (t) dm5 (t) = W · v(t) ⇒ = −0.5v5 (t) − 3v6 (t). dt dt

To fully understand the behavior of a particular DPN, its evolution graph must be constructed. Alternatively, a summarized version of the behavior can be given by the compressed evolution graph, when possible. The full and compressed evolution graphs for the DPN of the previous example appear in Fig. 6(a) and (b), respectively. Fig. 6(c) shows the trajectory of the state (actually, the marking of place P5 ). III. DIFFERENTIAL PETRI NETS IN SUPERVISORY CONTROL DPNs constitute a “natural” model for supervisory control systems, where discrete logic (usually in software form or programs for programmable logic controllers) and continuous systems intermix. Within supervisory control, there are a number of functionalities necessary for the supervisor to carry out its

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task(s). An important class of functionalities is the monitoring of the system’s operation and the issuance of alarms or reactions to specific events. Such events may be the simple crossing of signal levels, the crossing of signal levels coming from a specific direction (e.g., up or down, in a scalar signal), and entering or leaving a band-like region (defined by two signal levels). Such events may be the result of normal deviations from set points, disturbances, or failures, and usually there are corresponding prescribed reactions, such as changing of plant model for the computation of control policies, changing of the control mode, and similar counteractions. This information can be useful for decision making in the context of modeling. For example, the overshooting of a set point may signify that a first-order (perhaps dominant pole approximation) model may be inadequate to describe the process and a higher order model must be used. Modeling such actions can be a substantial task in the study of the supervisory system’s behavior and performance. It is thus important to validate the power of DPNs in this setting. We provide below a number of typical testing schemes that are widespread in industrial automation environments. A more formal model-based example can be found in section IV. Before proceeding to the test models, it is useful to introduce a concept that is closely related to invariance. This construct, which we call “forced invariant,” proved useful for modeling purposes. In the forced invariant setting, we introduce a new place that will be the “complement” of the place containing the value of the state variable that needs to be tested. Thus, if m(t) is the marking of the original place, the marking of the new place, will be −m(t). Each input (respectively, output) transition will be duplicated as output (respectively, input) transition of the new place, whereas the new arcs will carry equal weights with the corresponding original arcs. More formally, the complement of a differential place is defined as follows (T DF denotes the set of differential transitions): Definition: To each differential place Pj , with f (Pj ) = DF , a complementary differential place Pj can be associated, with respect to pre- and posttransitions, for which the following are true:

Fig. 7. The complement of a differential place with respect to differential transitions.

Fig. 8.

Level crossing.

Fig. 9.

DPN model of an upward-level crossing.

f (Pj ) = DF Pre(Pj , Tk ) = Post(Tk , Pj ) and Post(Pj , Tk ) = Pre(Tk , Pj ),

∀Tk ∈ T DF

mP j (t) = −mP j (t).

Fig. 10. Downward-level crossing.

This construct is shown in Fig. 7. A. Crossing a Signal Level “Upward” This test refers to the event of a positive signal exceeding a higher positive signal level or of a negative signal exceeding a higher negative signal level (Fig. 8). The corresponding test can be modeled via DPN in the following way. “Output” in Fig. 9 and subsequent figures, refers to the answer of the test: if the place is marked after an affirmative answer to the test, then a transition can be suitably enabled. Finally, note the different time constant h for the timed discrete place in the test section of the DPN. This means that the time constants of

the test(s) can be different (equal or slower, as necessary) than those of the continuous system. B. Crossing a Signal Level “Downward” This condition refers to the event of a positive signal crossing from above a lower signal level or of a negative signal exceeding a lower negative signal level (Fig. 10). Because we want to know whether a marking is inferior to a value, we use the notion of complementary differential place previously defined. This test can be modeled via DPN in the way shown in Fig. 11.


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Fig. 14. Variation outside a band-like region.

Fig. 11.

DPN model of a downward-level crossing.

Fig. 12.

Variation within a band-like region.

Fig. 15. DPN model of testing variation outside a band.

Fig. 16. Using the test result in many locations simultaneously. Fig. 13.

DPN model of testing variation within a band.

E. Remarks C. Variation Within a Region This test refers to the event of a signal taking values within a band-like region, defined by two levels A1 and A2 (signs are not important as long as A1 > A2 ) as in Fig. 12. A2 ≤ m(t) ≤ A1 ,

∀t, A1,2 ∈ R

or, equivalently m(t) ≥ A2

and

−m(t) ≥ −A1 ,

∀t, A1,2 ∈ R.

The last relation leads to the test that can be modeled via DPN in the way shown in Fig. 13. D. Variation Outside a Region This test refers to the event of a signal taking values outside a band-like region, defined by two levels A1 and A2 (signs are not important, again, as long as A1 > A2 ), as in Fig. 14. m(t) ≤ A2

or

m(t) ≥ A1 ,

∀t, A1,2 ∈ R.

The corresponding test can be modeled via DPN as shown in Fig. 15:

1) When the result of any of the previous test is to be used in many locations simultaneously, it is possible to do so in a straightforward and elegant way through the construct shown in Fig. 16. 2) The time constant h is related to the integration step used for the continuous model. It is possible, however, and in fact quite probable in the context of hybrid control systems that various continuous models may be used for the same system (see also the example in the next section) or within the same plant. Each of these models can have its own integration constant. In such a case where more than one time constant is present, we can either duplicate the test construct for each constant, or, more elegantly, duplicate only the timed transition and enable it when the corresponding model is active. 3) As noted earlier, each test can be carried out at its own pace through the introduction of appropriate time constants (marked h in the figures). Thus, the flexibility of the model is not hampered in any way. 4) It is worth noting that it is possible to construct large models by combining the individual tests. Typically, a set

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of tests will have as “target” a set of specific continuous quantities. Then, a model can be constructed for the continuous part, and each test can be realized around the same set of differential places. The example in the next section demonstrates this. 5) For the time being, no tool has been developed specifically supporting DPN models. However, OMOLA, an object-oriented modeling language for representing dynamic systems, has been enhanced successfully to describe and simulate DEDS and continuous time dynamic systems operating concurrently. DPNs have been implemented, validated, and compared with other mixed Petri nets within OMOLA’s companion modeling environment OmSim [13]. IV. EXAMPLE: STATE-DEPENDENT MODEL SWITCHING Consider a continuous dynamic system for which more detailed modeling is required, depending on the value and behavior of the output. The two models are the following:

Fig. 17. Simulation of the example system—A, B, C, D are the times of events (crossing of levels; disturbances), whereas on the upper side the corresponding valid model is indicated.

MODEL I: x(t) ˙ = −0.5x(t) x˙ 1 (t) = x2 (t) MODEL II: x˙ 2 (t) = −0.3x1 (t) − 0.3x2 (t). For model I, the output is x(t), whereas for model II the output is x2 (t), and both quantities correspond to the same physical entity. The rules for selecting the appropriate model are the following: RULE 1: When the output is greater than 1.0 in absolute value, use model I. RULE 2: When the output is less than 0.5 in absolute value, use model II. RULE 3: When the output enters the zone between 0.5 and 1.0 (in absolute value) although it was less than 0.5, (in absolute value) the moment before, continue to use model II. RULE 4: When the output enters the zone between 0.5 and 1.0 (in absolute value), although it was larger than one (in absolute value) the moment before, continue using model I. Fig. 17 presents a typical scenario with two disturbances (not represented in the DPN model) at approximately 2000 and 5500 time units (events B and C). Initially, the system is supposed to follow model I with initial condition x(0) = 1.5. To understand more clearly the behavior of the model, note that the oscillatory behavior corresponds to model II. The previous system, consisting of a set of models and a set of rules on the regions of validity of these models, can be represented via a DPN as shown in the following steps. The two alternate models have the DPN representation as shown in Fig. 18. Place P1 represents the continuous state variable x1 of model II, whereas place P2 represents the continuous state variable x2 of model II, which is identical to the continuous state variable x of model I. Table I collects the information regarding necessary tests. Figs. 19–21 show the implementation of the tests described in Section III for the system of the example. In building the overall

Fig. 18. DPN representation of the alternate models.

TABLE I RULES FOR MODEL SELECTION


Fig. 19.

Implementation of tests 1 and 2 of Table I.

Fig. 20.

Implementation for positive values of tests 3 and 4 of Table I.

Fig. 21.

Implementation for negative values of tests 3 and 4 of Table I.

model, we follow the composition method. We first construct the continuous models. Then, we construct each test according to Table I. Finally, we add the logic that ensures the selection of the correct model for each setting. To fuse the four tests, we

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must bring together the results of the tests and “drive” a suitable logic. This is done in Fig. 22 in a summarized fashion because the complexity of the global model does not permit a satisfactory depiction. Note that the differential places and transitions in

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in any industrial context, as soon as a suitable Petri net model is available for the protocols (and networking, in general). The elementary models presented here can be combined to form larger models to correspond to a real system of any level of complexity. In general, the effect of supervisors, especially of complex ones, and their sensitivity to the operating logic and communications intricacies, can be assessed for the integrated system. Thus, a global analysis of integrated industrial automation systems can be put on a new basis and modifications can be tested out. REFERENCES

Fig. 22.

Model selection based on the results of the tests.

Fig. 22 are of course identical with the corresponding entities in the tests of Figs. 19–21 0.5 ≤ x2 (t) ≤ −0.5

and

x2 (t− ) < 0.5 ⇒ Output A

0.5 ≤ x2 (t) ≤ −0.5

and

x2 (t− ) > 1 ⇒ Output C

−1 ≤ x2 (t) ≤ −0.5

and

x2 (t− ) > −0.5 ⇒ Output B

−1 ≤ x2 (t) ≤ −0.5

and

x2 (t− ) < −1 ⇒ Output D.

[1] M. S. Branicky, “Hybrid systems,” Ph.D. dissertation, Mass. Inst. Technol., Cambridge, 1991. [2] M. S. Branicky, V. S. Borkar, and S. K. Mitter, “A unified framework for hybrid control: Model and optimal control theory,” IEEE Trans. Autom. Cont., vol. 43, no. 1, pp. 31–45, Jan. 1998. [3] R. L. Grossman et al., Hybrid Systems, ser. Lecture Notes in Computer Science. New York: Springer, 1993, vol. 736. [4] P. Antsaklis, W. Kohn, A. Nerode, and S. Sastry, Eds., Hybrid Systems II. ser. Lecture Notes in Computer Science. New York: Springer, 1995, vol. 999. [5] R. Alur, T. A. Henzinger, and E. D. Sontag, Eds., Hybrid Systems III. ser. Lecture Notes in Computer Science. New York: Springer, 1996, vol. 1066. [6] P. Antsaklis, W. Kohn, A. Nerode, and S. Sastry, Eds., Hybrid Systems IV. ser. Lecture Notes in Computer Science. New York: Springer, 1997, vol. 1273. [7] P. J. Antsaklis, W. Kohn, M. Lemmon, A. Nerode, and S. Sastry, Eds., Hybrid Systems V. ser. Lecture Notes in Computer Science. New York: Springer, 1999, vol. 1567. [8] J. Le Bail, H. Alla, and R. David, “Hybrid Petri nets,” Proc. 1st European Control Conference, Grenoble, France, July 1991, pp. 187–191. [9] P. Antsaklis, X. D. Koutsoukos, and J. Zaytoon, “On hybrid control of complex systems: A survey,” Eur. J. Automat., vol. 32, nos. 9–10, pp. 1023–1045, Dec. 1998. [10] I. Demongodin and N. T. Koussoulas, “Differential Petri nets: Representing continuous systems in a discrete event world,” IEEE Trans. Autom. Control, vol. 43, no. 4, pp. 573–579, Apr. 1998. [11] H. Alla and R. David, “Continuous and hybrid Petri nets,” J. Circ. Syst. Comput., vol. 8, no. 1, pp. 159–188, 1998. [12] , Petri Nets & Grafcet—Tools for Modeling Discrete-Event Systems. Englewood Cliffs, NJ: Prentice Hall, 1992. [13] N. Pariset, Representation and Simulation of Hybrid Dynamic Systems— Implementation of Hybrid Petri Nets in Omola. Nantes, France: Univ. Nantes, Jun. 1996, Rapport D.E.A..

V. CONCLUSION The objective of this work was to illustrate the use of the DPN model for hybrid systems, in the context of industrial supervisory control. The DPN model introduces several novel features to the class of hybrid Petri nets to which it belongs and provides a unified representation for hybrid control systems. In terms of modeling capability, it can bring together models for decision making enacted by the supervision, which is based on discrete events, and models for continuous systems whose local controls are affected by the decisions of the supervisor. The Petri net basis endows DPN with the capacity for detailed representation of the supervisory logic, whereas its rules of evolution can capture the finest synchronization properties between the discrete event and the continuous part. Overall, this unified model can help evaluate the integrated system from the control point of view, based on the test structures presented in this article. Furthermore, it is also possible to incorporate the communications point of view

Isabel Demongodin (M’94) received the D.E.A. degree in control engineering and the Ph.D. degree in computer and systems engineering from the University of Montpellier, France, in 1990 and 1994, respectively. She held a postdoctorate position at the Laboratory of Automation and Robotics, University of Patras, Greece, in 1994. From 1995 to 2004, she was an Assistant Professor at the Ecole des Mines de Nantes, France, and in 1999 joined the Research Institute of Communications and Cybernetic of Nantes in the Discrete Event Systems group. Currently she is a full Professor at the University of Angers, France, with the Laboratory of Automation and System Engineering. Her research interests are in the areas of discrete event systems, hybrid systems, and Petri nets.


Nick T. Koussoulas (S’76–M’84) holds five engineering degrees from Greece, France, and the United States, including the Ph.D. degree in applied dynamic systems control from University of California, Los Angeles, in 1984. Since 1990, he has been with the Laboratory for Automation and Robotics of the Electrical and Computer Engineering Department at the University of Patras, Greece, where he is now Professor and Head of the Division of Systems & Automatic Control. He was formerly with Bell Communications Research (Bellcore), Red Bank, NJ, where he worked on modeling and designing communication networks with dynamic, feedback-based routing. His main research interests include modeling and control of hybrid systems, path planning and control of mobile robots and transportation vehicles, and simulation.

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