Skip to main content
SpringerLink
Log in

On fluidization of discrete event models: observation and control of continuous Petri nets

  • Published:
Discrete Event Dynamic Systems Aims and scope Submit manuscript

Abstract

As a preliminary overview, this work provides first a broad tutorial on the fluidization of discrete event dynamic models, an efficient technique for dealing with the classical state explosion problem. Even if named as continuous or fluid, the relaxed models obtained are frequently hybrid in a technical sense. Thus, there is plenty of room for using discrete, hybrid and continuous model techniques for logical verification, performance evaluation and control studies. Moreover, the possibilities for transferring concepts and techniques from one modeling paradigm to others are very significant, so there is much space for synergy. As a central modeling paradigm for parallel and synchronized discrete event systems, Petri nets (PNs) are then considered in much more detail. In this sense, this paper is somewhat complementary to David and Alla (2010). Our presentation of fluid views or approximations of PNs has sometimes a flavor of a survey, but also introduces some new ideas or techniques. Among the aspects that distinguish the adopted approach are: the focus on the relationships between discrete and continuous PN models, both for untimed, i.e., fully non-deterministic abstractions, and timed versions; the use of structure theory of (discrete) PNs, algebraic and graph based concepts and results; and the bridge to Automatic Control Theory. After discussing observability and controllability issues, the most technical part in this work, the paper concludes with some remarks and possible directions for future research.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

Notes

  1. Think, for example, in models of atoms based on wave-particle duality or the uncertainty principle (Bernard Pullman, The Atom in the History of Human Thought, Oxford Univ. Press, 1998).

  2. Before electronic computers became a reality, it is clear that the foundations of Theoretical Computer Science go back to the 1930s with the works of Alan Turing. Moreover, in 1928 Claude Shannon applied Boolean Algebra while developing Switching Theory. Even some years before, in the electromechanical domain, Leonardo Torres Quevedo’s chess player was driven by an automata. Furthermore, it is clear that, for example, the pioneering works of Markov and Erlang belong to the first decades of the XX century. Nevertheless, Automata Theory or Queueing Theory (Operations Research in a broader sense), are identifiable bodies of literature defined by the foundational research of the 1950’s and 1960’s.

  3. For each place in the trap, the minimum weight of the input arcs is greater than or equal to the minimum weight of its output arcs, i.e., ∀ p ∈ Θ such that  ∙  p ≠ ∅ it holds that \(\mathop{min}_{t_i\in{\,\!^\bullet{p}}} {{\boldsymbol{Post}}}[p,t_i]\geq \mathop{min}_{t_o\in{{p}^\bullet}} {{\boldsymbol{Pre}}}[p,t_o]\).

  4. For simulation purposes, in the state equation for \(\boldsymbol{m}_{k+1}\), the noise \(\boldsymbol{v}_k\) is added only if \(\boldsymbol{m}_{k+1}\geq 0\). This means that very close to boundaries the system may be kept as deterministic. In fact, if the system is crowded, i.e., \(\boldsymbol{m}_0\) is big, the probability of getting \(\boldsymbol{m}_{k+1} \not \geq 0\) is very low.

References

  • Ajmone-Marsan M, Balbo G, Bobbio A, Chiola G, Conte G, Cumani A (1989) The effect of execution policies on the semantics and analysis of stochastic Petri nets. IEEE Trans Softw Eng 15(7):832–846

    Article  MathSciNet  Google Scholar 

  • Alla H, David R (1998) Continuous and hybrid Petri nets. J Circuits Syst Comput 8(1):159–188

    Article  MathSciNet  Google Scholar 

  • Altman E, Jiménez T, Koole G (2001) On the comparison of queueing systems with their fluid limits. Probab Eng Inf Sci 15:165–178

    Article  MATH  Google Scholar 

  • Alur R, Coucoubetis C, Henzinger TA, Ho P-H, Nicollin X, Olivero A, Sifakis J, Yovine S (1995) The algorithmic analysis of hybrid systems. Theor Comp Sci 138:3–34

    Article  MATH  Google Scholar 

  • Amer-Yahiaj A, Zerhouni N, El-Moudni A, Ferney M (1996) State variable description and controllability of a class of continuous Petri nets. In: Proc. of the IEEE int. symp. on circuits and systems, Atlanta, USA, pp 68–71

  • Amrah A, Zerhouni N, El-Moudni A (1998) On the control of manufacturing lines modelled by controlled continuous Petri nets. Int J Syst Sci 29(2):127–137

    Article  Google Scholar 

  • Apaydin-Ozkan H, Júlvez J, Mahulea C, Silva M (2009) An efficient heuristics for minimum time control of continuous Petri nets. In: Proc. of the 3rd IFAC conf. on analysis and design of hybrid systems, ADHS09, Zaragoza, Spain, pp 44–49

  • Apaydin-Ozkan H, Júlvez J, Mahulea C, Silva M (2010) A control method for timed distributed continuous Petri nets. In: Proc. of the American control conf. 2010, Baltimore, Maryland, USA. IEEE Press, Piscataway, pp 2593–2600

    Google Scholar 

  • Avram F (1997) Optimal control of fluid limits of queuing networks and stochasticity corrections. In: Mathematics of stochastic manufacturing systems, vol 33. American Mathematical Society, Providence, pp 1–37

    Google Scholar 

  • Balbo G, Silva M (eds) (1998) Performance models for discrete event systems with synchronozations: formalisms and analysis techniques. In: Proc. of human capital and mobility MATCH performance advanced school, Jaca, Spain. Available online: http://webdiis.unizar.es/GISED/?q=news/matchbook

  • Balduzzi F, Giua A, Menga G (2000) First-order hybrid Petri nets: a model for optimization and control. IEEE Trans Robot Autom 16(4):382–399

    Article  Google Scholar 

  • Balluchi A, Benvenuti L, Di Benedetto MD, Sangiovanni-Vincentelli AL (2002) Design of observers for hybrid systems. In: Tomlin CJ, Greenstreet MR (eds) Hybrid systems: computation and control, Lecture notes in computer science, vol 2289. Springer, Berlin, pp 76–89

    Google Scholar 

  • Bemporad A, Morari M (1999) Control of systems integrating logic, dynamics, and constraints. Automatica 35(3):407–427

    Article  MathSciNet  MATH  Google Scholar 

  • Bemporad A, Ferrari-Trecate G, Morari M (2002a) Observability and controllability of piecewise affine and hybrid systems. IEEE Trans Automat Contr 45(10):1864–1876

    Article  MathSciNet  Google Scholar 

  • Bemporad A, Morari M, Dua V, Pistikopoulos EN (2002b) The explicit linear quadratic regulator for constrained systems. Automatica 38(1):3–20

    Article  MathSciNet  MATH  Google Scholar 

  • Benevenuti L, Farina L (2002) Positive and compartmental systems. IEEE Trans Automat Contr 47(2):370–373

    Article  Google Scholar 

  • Bergstra JA, Klop JW (1982) Fixed point semantics in process algebra. Technical report IW 206/82 208, Mathematical Centre, Amsterdam

  • Bertsimas D, Gamarnik D, Tsitsiklis JN (1996) Stability conditions for multiclass fluid queueing networks. IEEE Trans Automat Contr 41(11):1618–1631

    Article  MathSciNet  MATH  Google Scholar 

  • Bolch G, Greiner S, de Meer H, Trivedi K (1998) Queueing networks and Markov chains. Wiley, New York

    Book  MATH  Google Scholar 

  • Brammer R (1972) Controllability in linear autonomous systems with positive controllers. SIAM J Control 10(2):329–353

    Article  MathSciNet  Google Scholar 

  • Brams GW (1983) Réseaux de Petri: Théorie et Pratique. Masson, Paris

    Google Scholar 

  • Bramson M (2008) Stability of queueing networks. In: Lecture notes in mathematics, vol 1950. Springer, New York (also published in: Probability Surveys 5:169–345, 2008)

  • Cabasino MP, Giua A, Seatzu C (2010) Fault detection for discrete event systems using Petri nets with unobservable transitions. Automatica 46(9):1531–1539

    Article  MATH  Google Scholar 

  • Campos J, Silva M (1992) Structural techniques and performance bounds of stochastic Petri net models. In: Rozenberg G (ed) Advances in Petri nets 1992, Lecture notes in computer science, vol 609. Springer, New York, pp 352–391

    Google Scholar 

  • Campos J, Colom JM, Silva M (1992) Improving throughput upper bounds for net based models of manufacturing systems. In: Robotics and flexible manufacturing systems, pp 281–294

  • Cardelli L (2008) On process rate semantics. Theor Comp Sci 391(3):190–215

    Article  MathSciNet  MATH  Google Scholar 

  • Cardoso J, Camargo H (eds) (1999) Fuzziness in Petri nets. In: Studies in fuzziness and soft computing, vol 22. Springer, New York.

  • Cassandras CG (2007) Stochastic flow systems: modeling and sensitivity analysis. In: Cassandras CG, Lygeros J (eds) Stochastic hybrid systems, chapter 6, pp 139–167

  • Cassandras CG, Lafortune S (2008) Introduction to discrete event systems. Springer, New York

    Book  MATH  Google Scholar 

  • Cassandras CG, Pepyne DL, Wardi Y (2001) Optimal control of a class of hybrid systems. IEEE Trans Automat Contr 46(3):398–415

    Article  MathSciNet  MATH  Google Scholar 

  • Champagnat R, Esteban P, Pingaud H, Valette R (1998) Modeling and simulation of a hybrid system through Pr/Tr PN-DAE model. In: Proc. of ADPM’98, Reims, France, pp 131–137

  • Champagnat R, Valette R, Hochon JC, Pingaud H (2001) Modeling, simulation and analysis of batch production systems. Discrete Event Dyn Syst: Theory Appl 11:119–136

    Article  MathSciNet  MATH  Google Scholar 

  • Chen C (1984) Linear system theory and design. Oxford University Press, New York

    Google Scholar 

  • Chen H, Mandelbaum A (1991) Discrete flow networks: bottleneck analysis and fluid approximations. Math Oper Res 16:408–446

    Article  MathSciNet  MATH  Google Scholar 

  • Chen H, Yao DD (2001) Fundamentals of queueing networks. Performance, asymptotics and optimization. Stochastic modelling and applied probability, vol 46. Springer, New York

    Google Scholar 

  • Chiola G, Anglano C, Campos J, Colom JM, Silva M (1995) Operational analysis of timed Petri nets and application to the computation of performance bounds. In: Baccelli F, Jean-Marie A, Mitrani I (eds) Quantitative methods in parallel systems. Springer, New York, pp 161–174

    Google Scholar 

  • Ciocchetta F, Hillston J (2009) Bio-pepa: a framework for the modelling and analysis of biological systems. Theor Comp Sci 410(33–34):3065–3084

    Article  MathSciNet  MATH  Google Scholar 

  • Clarke EM, Emerson EA, Sifakis J (2009) Model checking: algorithmic verification and debugging. Commun ACM 52(11):74–84

    Article  Google Scholar 

  • Cohen G, Gaubert S, Quadrat JP (1998) Algebraic system analysis of timed Petri nets. In: Gunawardena J (ed) Idempotency, Cambridge University Press, pp 145–170

  • Colom JM, Silva M (1991) Improving the linearly based characterization of P/T nets. In: Rozenberg G (ed) Advances in Petri nets 1990, Lecture notes in computer science, vol 483. Springer, New York, pp 113–145

    Google Scholar 

  • Commault C, Dion JM, Trinh DH (2005) Observability recovering by additional sensor implementation in linear structured systems. In: Procs. of the 44th IEEE conf. on decission and control, Seville, Spain, pp 7193–7197

  • Connors D, Feigin G, Yao D (1994) Scheduling semiconductor lines using a fluid network model. IEEE Trans Robot Autom 10(2):88–98

    Article  Google Scholar 

  • Dai JG (1995) On positive Harris recurrence of multiclass queuing networks: a unified approach via fluid limit models. Ann Appl Probab 5(1):49–77

    Article  MathSciNet  MATH  Google Scholar 

  • Dallery Y, Liu Z, Towsley D (1997) Properties of Fork/join queueing networks with blocking under various operating mechanisms. IEEE Trans Robot Autom 13(4):503–518

    Article  Google Scholar 

  • David R, Alla H (1987) Continuous Petri nets. In: Proc. of the 8th European workshop on application and theory of Petri nets, Zaragoza, Spain, pp 275–294

  • David R, Alla H (1990) Autonomous and timed continuous Petri nets. In: Proc. 11th int. conf. on application and theory of Petri nets, Paris, France, pp 367–386

  • David R, Alla H (2010) Discrete, continuous and hybrid Petri nets. Springer, Berlin (revised 2nd edn, 2010)

    Book  Google Scholar 

  • Dennis JB (ed) (1970) Record of the project MAC. In: Conf. on concurrent systems and parallel computation. ACM conf. records, 2–5 June 1970. MIT

  • De Schutter B, van den Boom T (2001) Model predictive control for max-plus-linear discrete event systems. Automatica 37(7):1049–1056

    Article  MATH  Google Scholar 

  • Demongodin I (2001) Generalised batches Petri net: hybrid model for high speed systems with variable delays. Discrete Event Dyn Syst 11(1–2):137–162

    Article  MathSciNet  MATH  Google Scholar 

  • Demongodin I, Giua A (2010) Linear programming techniques for analysis and control of batches Petri nets. In: Procs. of the int. workshop on discrete event systems, WODES’10, pp 4–9

  • Demongodin I, Koussoulas NT (1998) Differential Petri nets: representing continuous systems in a discrete-event world. IEEE Trans Automat Contr 43(4):573–579

    Article  MathSciNet  MATH  Google Scholar 

  • Di Cairano S, Bemporad A, Júlvez J (2009) Event-driven optimization-based control of hybrid systems with integral continuous-time dynamics. Automatica 45(5):1243–1251

    Article  MathSciNet  MATH  Google Scholar 

  • Di Febbraro A, Giua A, Menga G (eds) (2001) Special issue on hybrid Petri nets. Discrete Event Dyn Syst 11(1–2)

  • Ding J (2010) Structural and fluid analysis for large scale PEPA models. With applications to content adaptation systems. PhD thesis, University of Edinburgh

  • Ding J, Hillston J (2010) Fundamental results on fluid approximations of stochastic process algebra models. Perform Eval (submitted)

  • Dion JM, Commault C, van der Woude J (2003) Generic properties and control of linear structured systems: a survey. Automatica 39(7):1125–1144

    Article  MathSciNet  MATH  Google Scholar 

  • Donatelli S, Ribaudo M, Hillston J (1995) A comparison of performance evaluation process algebra and generalized stochastic Petri nets. In: IEEE Petri nets and performance models (PNPM’95), pp 158–168

  • Down D, Meyn SP (1997) Piecewise linear test functions for stability and instability of queueing networks. Queueing Syst 27:205–226

    Article  MathSciNet  MATH  Google Scholar 

  • Dub M, Pipan G, Hanzálek Z (2002) Stock optimization of a kanban-based assembly line. In: Proc. of the 12th int. conf. on flexible automation and intelligent manufacturing, Dresden, Germany, pp 1–10

  • Ezpeleta J, Couvreur JM, Silva M (1993) A new technique for finding a generating family of siphons, traps and st-components. application to coloured Petri nets. In: Rozenberg G (ed) Advances in Petri nets 1993, Lecture notes in computer science, vol 674. Springer, New York, pp 126–147

    Google Scholar 

  • Farina L, Rinaldi S (2000) Positive linear systems. Theory and applications. In: Pure and applied mathematics. Wiley, New York

    Google Scholar 

  • Fleming WH (1988) Future directions in control theory: a mathematical perspective. Report of the Panel on Future Directions in Control Theory, SIAM

  • Forrester JW (1961) Industrial dynamics. MIT Press, Cambridge

    Google Scholar 

  • Forrester JW (1968) Principles of systems. Productivity Press, Cambridge

    Google Scholar 

  • Foss S, Konstantopoulos T (2004) An overview of some stochastic stability methods. J Oper Res Soc Jpn 47(4):275–303

    MathSciNet  MATH  Google Scholar 

  • Galpin V (2010) Continuous approximation of PEPA models and Petri nets. Int J Comput Aided Eng Technol 2(4):324–339

    Article  Google Scholar 

  • Gaujal B, Giua A (2004) Optimal stationary behavior for a class of timed continuous Petri nets. Automatica 40(9):1505–1516

    Article  MathSciNet  MATH  Google Scholar 

  • Genc S, Lafortune S (2007) Distributed diagnosis of place-bordered Petri nets. IEEE Trans Autom Sci Eng 4(2):206–219

    Article  Google Scholar 

  • Gianni D, D’Ambrogio A (2007) A language to enable distributed simulation of extended queueing networks. J Comput 2(4):76–86

    Google Scholar 

  • Giua A, DiCesare F, Silva M (1993) Petri Net supervisors for generalized mutual exclusion constraints. In: 12th IFAC world congress, vol 1, Sidney, Australia, pp 267–270

  • Giua A, Seatzu C, Wardi Y (2010) IPA for continuous Petri nets. In: Procs. of the 10th int. work. on discrete event systems, Berlin, Germany, pp 57–62

    Google Scholar 

  • Habets L, van Schuppen JH (2004) A control problem for affine dynamical systems on a full-dimensional polytope. Automatica 40(1):21–35

    Article  MathSciNet  MATH  Google Scholar 

  • Haddad WM, Chellaboina V (2004) Stability theory for nonnegative and compartmental dynamical systems with time delay. Syst Control Lett 51:355–361

    Article  MathSciNet  MATH  Google Scholar 

  • Harrison JM (2002) Stochastic processing networks and activity analysis. In: Suhov Y (ed) Analytic methods in applied probability, in memory of Fridrik Karpelevich. American Mathematical Society, Providence

    Google Scholar 

  • Hayden RA (2010) Convergence of ODE approximations and bounds on performance models in steady-state. In: 9th workshop on process algebra and stochastically timed activities

  • Hayden RA, Bradley JT (2010) A fluid analysis framework for a Markovian process algebra. Theor Comp Sci 411(22–24):2260–2297

    Article  MathSciNet  MATH  Google Scholar 

  • Heiner M, Gilbert D, Donaldson R (2008) Petri nets for systems and synthetic biology. In: Formal methods for computational systems biology, pp 215–264

  • Hennequin S, Lefebvre D, El-Moudni A (1999) Fuzzy control of variable speed continuous Petri nets. In: Proc. of the 38th conf. on decisions and control (CDC99), Phoenix, USA, pp 1352–1356

  • Hermanns H, Herzog U, Katoen J-P (2002) Process algebra for performance evaluation. Theor Comp Sci 274(1–2):43–87

    Article  MathSciNet  MATH  Google Scholar 

  • Hillston J (2005) Fluid flow approximation of PEPA models. In: Procs. of the second int. conf. on the quantitative evaluation of systems (QEST). IEEE Computer Society, Piscataway, pp 33–43

    Chapter  Google Scholar 

  • Hillston J, Recalde L, Ribaudo M, Silva M (2001) A comparison of the expressiveness of SPA and bounded SPN models. In: Procs. of the 9th int. workshop on Petri nets and performance models. IEEE Computer Society Press, Aachen, pp 197–206

    Chapter  Google Scholar 

  • Hiraishi K (2008) Performance evaluation of workflows using continuous Petri nets with interval firing speeds. IEICE Trans Fundam Electron Commun Comput Sci 91(11):3219

    Article  Google Scholar 

  • Hoare CAR (1985) Communicating sequential processes. Prentice Hall, Englewood Cliffs

    MATH  Google Scholar 

  • Holloway LE, Krogh BH, Giua A (1997) A survey of Petri nets methods for controlled discrete event systems. Discrete Event Dyn Syst 7(2):151–190

    Article  MATH  Google Scholar 

  • Horton G, Kulkarni VG, Nicol DM, Trivedi KS (1998) Fluid stochastic Petri nets: theory, applications, and solution techniques. Eur J Oper Res 105:184–201

    Article  MATH  Google Scholar 

  • Iordache MV, Antsaklis PJ (2006) Supervisory control of concurrent systems. Birkhäuser, Boston

    MATH  Google Scholar 

  • Jacquez JA, Simon CP (1993) Qualitative theory of compartmental systems. SIAM Rev 35(1):43–79

    Article  MathSciNet  MATH  Google Scholar 

  • Jensen K, Kristensen LM (2009) Coloured Petri nets. Modelling and validation of concurrent systems. Springer, Berlin

    Book  Google Scholar 

  • Jiménez E, Recalde L, Silva M (2001) Forrester diagrams and continuous Petri nets: a comparative view. In: Proc. of the 8th IEEE int. conf. on emerging technologies and factory automation (ETFA 2001), pp 85–94

  • Jiménez E, Júlvez J, Recalde L, Silva M (2004) Relaxed continuous views of discrete event systems: Petri nets, forrester diagrams and ODES. In: IEEE int. conf. on systems, man, and cybernetics (SMC). The Hague, The Netherlands

  • Jiménez E, Júlvez J, Recalde L, Silva M (2005) On controllability of timed continuous Petri net systems: the join free case. In: Proc. of the 44th IEEE conf. on decision and control (Joint CDC-ECC), Seville, Spain, pp 7645–7650

  • Jing X, Recalde L, Silva M (2008a) Tracking control of join-free timed continuous Petri net systems under infinite servers semantics. Discrete Event Dyn Syst 18(2):263–283

    Article  MathSciNet  MATH  Google Scholar 

  • Jing X, Recalde L, Silva M (2008b) Tracking control of timed continuous Petri net systems under infinite servers semantics. In: IFAC world congress, Seoul, Korea, pp 3192–3197

  • Juloski ALj, Heemels WPMH, Weiland S (2007) Observer design for a class of piecewise linear systems. Int J Robust Nonlinear Control 17(15):1387–1404

    Article  Google Scholar 

  • Júlvez J, Boel RK (2010) A continuous Petri net approach for model predictive control of traffic systems. IEEE Trans Syst Man Cybern Part A: Syst Humans 40(4):686–697

    Article  Google Scholar 

  • Júlvez J, Recalde L, Silva M (2003) On reachability in autonomous continuous Petri net systems. In: van der Aalst W, Best E (eds) 24th int. conf. on application and theory of Petri nets (ICATPN 2003), Lecture notes in computer science, vol 2679. Springer, Dordrecht, pp 221–240

    Google Scholar 

  • Júlvez J, Bemporad A, Recalde L, Silva M (2004) Event-driven optimal control of continuous Petri nets. In: 43rd IEEE conf. on decision and control (CDC), Paradise Island, Bahamas, pp 69–74

  • Júlvez J, Recalde L, Silva M (2005) Steady state performance evaluation of continuous mono-T-semiflow Petri nets. Automatica 41(4):605–616

    Article  MathSciNet  MATH  Google Scholar 

  • Júlvez J, Recalde L, Silva M (2006) Deadlock-freeness analysis of continuous mono-t-semiflow Petri nets. IEEE Trans Automat Contr 51(9):1472–1481

    Article  Google Scholar 

  • Júlvez J, Jimenez E, Recalde L, Silva M (2008) On observability and design of observers intimed continuous Petri net systems. IEEE Trans Autom Sci Eng 5(3):532–537

    Article  Google Scholar 

  • Kara R, Djennoune S, Loiseau JJ (2006) State feedback control for the manufacturing systems modeled by continuous Petri nets. In: Proc. of information control problems in manufacturing (INCOM), Saint Etienne, France, vol 12, part 1

  • Kara R, Ahmane M, Loiseau JJ, Djennoune S (2009) Constrained regulation of continuous Petri nets. Nonlinear Anal: Hybrid Syst 3(4):738–748

    Article  MathSciNet  MATH  Google Scholar 

  • Kelly FP, Zachary S, Ziedins I (eds) (1996) Stochastic networks: theory and applications. Royal Statistical Society LNS, 4. Oxford University Press, Oxford (repr. 2002)

    Google Scholar 

  • Kleinrock L (1975) Queueing systems. vol I. Wiley, New York

    MATH  Google Scholar 

  • Kleinrock L (1976) Queuing systems, vol II: computer applications. Wiley, New York

    Google Scholar 

  • Kloetzer M, Belta C (2008) A fully automated framework for control of linear systems from temporal logic specifications. IEEE Trans Automat Contr 53(1):287–297

    Article  MathSciNet  Google Scholar 

  • Kloetzer M, Mahulea C, Belta C, Silva M (2010) An automated framework for formal verification of timed continuous Petri nets. IEEE Trans Ind Inf 6(3):460–471

    Article  Google Scholar 

  • Kumar S, Kumar PR (2001) Queueing network models in the design and analysis of semiconductor wafer fabs. IEEE Trans Robot Autom 17(5):548–561

    Article  Google Scholar 

  • Kurtz TG (1970) Solutions of ordinary differential equations as limits of pure jump Markov processes. J Appl Probab 7(1):49–58

    Article  MathSciNet  MATH  Google Scholar 

  • Lefebvre D (1999) Feedback control designs for manufacturing systems modelled by continuous Petri nets. Int J Syst Sci 30(6):591–600

    Article  MATH  Google Scholar 

  • Lefebvre D (2001) Estimation of the firing frequencies in discrete and continuous Petri nets models. Int J Syst Sci 32(11):1321–1332

    Article  MathSciNet  MATH  Google Scholar 

  • Lefebvre D, Catherine D, Leclerq E, Druaux F (2007) Some contributions with Petri nets for the modelling, analysis and control of HDS. In: Nonlinear Analysis: Hybrid Systems, vol 1(4), pp 451–465

  • Lefebvre D, Leclerq E, Khalij L, De Cursi E, El Akchioui N (2009) Approximation of MTS stochastic Petri nets steady state by means of continuous Petri nets: a numerical approach. In: Proc. of the 3rd IFAC conf. on analysis and design of hybrid systems, Zaragoza, Spain, pp 62–67

  • Lefebvre D, Leclerq E, El Akchioui N, Khalij L, De Cursi E (2010) A geometric approach for the homothetic approximation of stochastic Petri nets. In: Proc. of the 10th IFAC workshop on discrete event systems, Berlin, Germany, pp 245–250

  • Liu Y, Gong W (2002) Perturbation analysis for stochastic fluid queueing systems. Discrete Event Dyn Syst 12(4):391–416

    Article  MathSciNet  MATH  Google Scholar 

  • Liu B, Guo Y, Kurose J, Towsley D, Gong W (1999) Fluid simulation of large scale networks: issues and tradeoffs. In: Procs. of the int. conf. on parallel and distributed processing techniques and applications, pp 2136–2142

  • Luenberger DG (1971) An introduction to observers. IEEE Trans Autom Contr 16(6):596–602

    Article  MathSciNet  Google Scholar 

  • Luenberger DG (1979) Introduction to dynamic systems. Theory, models, and applications. Wiley, New York

    MATH  Google Scholar 

  • Mahulea C, Cabasino MP, Giua A, Seatzu C (2008a) A state estimation problem for timed continuous Petri nets. In: 2007 46th IEEE conf. on decision and control. IEEE, Piscataway, pp 1770–1775

  • Mahulea C, Giua A, Recalde L, Seatzu C, Silva M (2008b) Optimal model predictive control of timed continuous Petri nets. IEEE Trans Automat Contr 53(7):1731–1735

    Article  MathSciNet  Google Scholar 

  • Mahulea C, Ramirez A, Recalde L, Silva M (2008c) Steady state control reference and token conservation laws in continuous Petri net systems. IEEE Trans Autom Sci Eng 5(2):307–320

    Article  Google Scholar 

  • Mahulea C, Recalde L, Silva M (2009a) Basic server semantics and performance monotonicity of continuous Petri nets. Discrete Event Dyn Syst 19(2):189–212

    Article  MathSciNet  MATH  Google Scholar 

  • Mahulea C, Seatzu C, Cabasino MP, Recalde L, Silva M (2009b) Observer design for untimed continuous Petri nets. In: American control conf., St. Louis, Missouri, USA, pp 4765–4770

  • Mahulea C, Recalde L, Silva M (2010) Observability of continuous Petri nets with infinite server semantics. Nonlinear Anal: Hybrid Syst 4(2):219–232

    Article  MathSciNet  MATH  Google Scholar 

  • Merlin P, Faber DJ (1976) Recoverability of communication protocols. IEEE Trans commun 24(9):1036–1043

    Article  MATH  Google Scholar 

  • Milner R (1980) A calculus of communicating systems. In: Lecture notes in computer science, vol 92. Springer, Berlin

    Google Scholar 

  • Misra V, Gong W, Towsley DF (2000) Fluid-based analysis of a network of AQM routers supporting TCP flows with an application to RED. In: Proc. SIGCOMM, pp 151–160

  • Mitra D (1988) Stochastic theory of a fluid model of producers and consumers coupled by a buffer. Adv Appl Probab 20:646–676

    Article  MATH  Google Scholar 

  • Molloy MK (1982) Performance analysis using stochastic Petri nets. IEEE Trans Comput 31(9):913–917

    Article  Google Scholar 

  • Moss FH, Segall A (1982) An optimal control approach to dynamic routing in networks. IEEE Trans Automat Contr 27(2):329–339

    Article  MathSciNet  MATH  Google Scholar 

  • Murata T (1989) Petri nets: properties, analysis and applications. Proc IEEE 77(4):541–580

    Article  Google Scholar 

  • Newell GF (1971) Applications of queuing theory. Chapman and Hall, London (2nd edn, 1982)

    Google Scholar 

  • Nicolis G, Prigogine I (1971) Fluctuations in nonequilibrium systems. Proc Natl Acad Sci USA 68:2102–2107

    Article  MathSciNet  MATH  Google Scholar 

  • Ogata K (2001) Modern control engineering, 4th edn. Prentice Hall, Englewood Cliffs

    Google Scholar 

  • Peterson JL (1981) Petri net theory and the modeling of systems. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  • Petri CA (1962) Communication with automata. Supplement 1 to Technical Report RADC-TR-65-377, vol 1, Griffiss Air Force Base, New York 1966. [Originally published in German: Kommunikation mit Automaten, University of Bonn]

  • Recalde L, Silva M (2001) Petri nets fluidification revisited: semantics and steady state. APII-JESA 35(4):435–449

    Google Scholar 

  • Recalde L, Teruel E, Silva M (1998) On linear algebraic techniques for liveness analysis of P/T systems. J Circuits Syst Comput 8(1):223–265

    Article  MathSciNet  Google Scholar 

  • Recalde L, Teruel E, Silva M (1999) Autonomous continuous P/T systems. In: Kleijn J, Donatelli S (eds) Application and theory of Petri nets 1999, Lecture notes in computer science, vol 1639. Springer, Berlin, pp 107–126

    Chapter  Google Scholar 

  • Recalde L, Haddad S, Silva M (2010) Continuous Petri nets: expressive power and decidability issues. Int J Found Comput Sci 21(2):235–256

    Article  MathSciNet  MATH  Google Scholar 

  • Sauer C, MacNair E, Kurose J (1982) The research queueing package: past, present and future. In: Proc. of the national computer conf., Houston, USA, pp 273–280

  • Seatzu C, Mahulea C, Cabasino MP, Silva M (2009a) Fault diagnoser design for untimed continuous Petri nets. In: Control applications (CCA) & intelligent control (ISIC), 2009. IEEE, Piscataway, pp 1598–1604

    Google Scholar 

  • Seatzu C, Cabasino MP, Mahulea C, Silva M (2010b) New results for fault detection of untimed continuous Petri nets. In: Procs. of the 48th IEEE conf. on decision and control, 2009 held jointly with the 2009 28th Chinese control conf. CDC/CCC 2009, pp 6952–6957

  • Silva M (1985) Las redes de Petri: en la automática y la informática. AC, Madrid (2nd edn AC/Thomson 2002)

  • Silva M (1987) Towards a synchrony theory for P/T nets. In: Voss K et al (eds) Concurrency and nets. Springer, Berlin, pp 435–460

    Google Scholar 

  • Silva M (1993) Introducing Petri nets. In: Practice of Petri nets in manufacturing. Chapman & Hall, London, pp 1–62

    Google Scholar 

  • Silva M, Colom JM (1987) On the structural computation of synchronic invariants in P/T nets. In: Proc. of the 8th European workshop on application and theory of Petri nets, Zaragoza, Spain, pp 237–258

  • Silva M, Colom J (1988) On the computation of structural synchronic invariants in P/T nets. In: Lecture Notes in Computer Science. vol 340, Advances in Petri nets 1988, Springer, Berlin, pp 386–417

    Google Scholar 

  • Silva M, Recalde L (2002) Petri nets and integrality relaxations: a view of continuous Petri net models. IEEE Trans Syst Man Cybern 32(4):314–327

    Article  Google Scholar 

  • Silva M, Recalde L (2003) Unforced continuous Petri nets and positive systems. In: Benvenuti L, De Santis A, Farina L (eds) Positive systems. Procs. of the first multidisciplinary int. symp. on positive systems: theory and applications (POSTA 2003), Rome, Italy, LNCIS, vol 294. Springer, Berlin, pp 55–62

    Google Scholar 

  • Silva M, Recalde L (2004) On fluidification of Petri net models: from discrete to hybrid and continuous models. Annu Rev Control 28:253–266

    Article  Google Scholar 

  • Silva M, Teruel E (1996) A systems theory perspective of discrete event dynamic systems: the Petri net paradigm. In: Borne P, Gentina JC, Craye E, El Khattabi S (eds) Symp. on discrete events and manufacturing systems. CESA ’96 IMACS multiconference, Lille, France, pp 1–12

  • Silva M, Teruel E (1998) DEDS along their life cycle. Interpreted extensions of Petri nets. In: IEEE int. conf. on systems, man and cybernetics, La Jolla, San Diego, CA, USA (http://webdiis.unizar.es/GISED/sites/default/files/2.Syst-Theor_San_Diego98.pdf)

  • Silva M, Teruel E, Colom JM (1998) Linear algebraic and linear programming techniques for the analysis of net systems. In: Rozenberg G, Reisig E (eds) Lectures in Petri nets. I: Basic models, Lecture notes in computer science, vol 1491. Springer, Berlin, pp 309–373

    Google Scholar 

  • Sun G, Cassandras CG, Panayiotou CG (2004) Perturbation analysis of multiclass stochastic fluid models. Discrete Event Dyn Syst 14(3):267–307

    Article  MathSciNet  MATH  Google Scholar 

  • Sussmann HJ, Willems JC (1997) 300 years of optimal control: from the brachystochrone to the maximum principle. IEEE Control Syst Mag 17(3):32–44

    Article  Google Scholar 

  • Teruel E, Silva M (1996) Structure theory of equal conflict systems. Theor Comp Sci 153(1–2):271–300

    Article  MathSciNet  MATH  Google Scholar 

  • Trivedi K, Kulkarni VG (1993) FSPNs: fluid stochastic Petri nets. In: Ajmone Marsan M (ed) Application and theory of Petri nets 1993, Lecture notes in computer science, vol 691. Springer, Berlin, pp 24–31

    Google Scholar 

  • Valk R, Girault C (eds) (2003) Petri nets for systems engineering. A guide to modeling, verification, and applications. Springer, Berlin

    Google Scholar 

  • Vázquez CR, Silva M (2009a) Performance control of Markovian Petri nets via fluid models: a stock-level control example. In: 5th IEEE conf. on automation science and engineering (IEEE CASE), Bangalore, India, pp 30–36

  • Vázquez CR, Silva M (2009b) Piecewise-linear constrained control for timed continuous Petri nets. In: Proc. of the 48th IEEE conf. on decision and control (CDC), Shanghai, China, pp 5714–5720

  • Vázquez CR, Silva M (2010) Timing-dependent boundedness and liveness in continuous Petri nets. In: 10th int. workshop on discrete event systems (WODES), Berlin, Germany, pp 10–17

  • Vázquez CR, Ramírez A, Recalde L, Silva M (2008a) On controllability of timed continuous Petri nets. In: Egerstedt M, Mishra B (eds) Proc. of the 11th int. workshop hybrid systems: computational and control (HSCC08), LNCS, vol 4981, pp 528–541

  • Vázquez CR, Recalde L, Silva M (2008b) Stochastic–continuous state approximation of Markovian Petri net systems. In: Proc. of the 47th ieee conf. on decision and control (CDC), Cancún, México, pp 901–906

  • Vázquez CR, Sutarto HY, Boel R, Silva M (2010a) Hybrid Petri net model of a traffic intersection in an urban network. In: IEEE multiconf. on systems and control, Yokohama, Japan, pp 658–664

  • Vázquez CR, van Schuppen JH, Silva M (2010b) A modular-coordinated control for continuous Petri nets. Technical report, Universidad de Zaragoza

  • Vernon M, Zahorjan J, Lazowska ED (1987) A comparison of performance Petri nets and queueing network models. In: 3rd int. workshop on modelling techniques and performance evaluation. Paris, France

  • Walter G, Contreras M (1999) Compartmental modeling with networks. Birkhauser, Boston

    Book  MATH  Google Scholar 

  • Wang L, Mahulea C, Júlvez J, Silva M (2010) Minimum-time control for structurally persistent continuous Petri nets. In: Conf. on decision and control CDC, Atlanta, USA, pp 2771–2776

  • Whitt W (2002) Stochastic process limits. An introduction to stochastic-process limits and their application to queues. In: Series in operations research and financial engineering. Springer, New York

    Google Scholar 

  • Xie X (2002) Fluid-stochastic-event graphs for evaluation and optimization of discrete-event systems with failures. IEEE Trans Robot Autom 18(3):360–367

    Article  Google Scholar 

  • Xu X, Antsaklis PJ (2003) Results and perspectives on computational methods for optimal control of switched systems. In: Hybrid systems: computation and control. Springer, New York, pp 540–555

    Chapter  Google Scholar 

  • Yao C, Cassandras CG (2009) Perturbation analysis and optimization of multiclass multiobjective stochastic flow models. In IEEE int. conf. on decision and control, CDC, Shanghai, China, pp 914–919

Download references

Acknowledgements

This work benefits from the comments of the following colleagues and friends: R. David (INP de Grenoble), A. Giua (Univ. di Cagliari), E. Jiménez (Univ. de La Rioja), A. Ramírez (Cinvestav, Guadalajara, Mx) and J. Zaytoon (Univ. de Reims-Champagne-Ardennes). All of them and the anonymous reviewers are gratefully acknowledged. This work has been partially supported by the European Community’s Seventh Framework Program under project DISC (Grant Agreement n. INFSO-ICT-224498), by CICYT - FEDER grant DPI2010-20413 and by Fundación Aragón I+D.

Author information

Authors and Affiliations

  1. Aragón Institute of Engineering Research (I3A), University of Zaragoza, Maria de Luna 1, 50018, Zaragoza, Spain

    Manuel Silva, Jorge Júlvez, Cristian Mahulea & C. Renato Vázquez

Authors
  1. Manuel Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jorge Júlvez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Cristian Mahulea
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. C. Renato Vázquez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manuel Silva.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Silva, M., Júlvez, J., Mahulea, C. et al. On fluidization of discrete event models: observation and control of continuous Petri nets. Discrete Event Dyn Syst 21, 427–497 (2011). https://doi.org/10.1007/s10626-011-0116-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10626-011-0116-9

Keywords

Search

Navigation