Discrete Event Dynamic Systems

, Volume 18, Issue 2, pp 263–283 | Cite as

Tracking Control of Join-Free Timed Continuous Petri Net Systems under Infinite Servers Semantics

  • Jing Xu
  • Laura Recalde
  • Manuel Silva


A new low-and-high gain algorithm is presented for tracking control of a subclass of timed continuous Petri Net (contPN) systems working under infinite servers semantics. The inherent properties of timed contPN determine that the control signals must be non-negative and upper bounded by functions of system states. In the proposed control approach, LQ theory is first used to design a low-gain controller such that the control signals satisfy the input constraints. Based on the low-gain controller, a high-gain term is further added to fully employ available control energy, and control performance can be improved consequently. In order to guarantee global tracking convergence and smoothness on the tracking target, a mixed trajectory (state step and ramp) is used instead of a pure step reference signal. The new tracking target is designed to ensure the existence of the low-gain controller and possible fast system response concurrently. Rigorous proof based on Lyapunov function is provided to guarantee that for a conservative and strongly connected Join-Free (JF) timed contPN system, the proposed algorithm can ensure the global asymptotical convergence of both system states and control signals.


Join-free timed continuous Petri net Tracking control Input constraints Convergence analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bemporad A, Morari M, Dua V, Pistikopoulos E (2002) The explicit linear quadratic regulator for constrained systems. Automatica 38(1):3–20MATHCrossRefMathSciNetGoogle Scholar
  2. Bemporad A, Torrisi F, Morari M (2000) Performance analysis of piecewise linear systems and model predictive control systems. In: Proceedings of the 39th IEEE conference on decision and control. Sydney, Australia, pp 4957–4962Google Scholar
  3. Chen BM, Lee TH, Peng K, Venkataramanan V (2003) Composite nonlinear feedback control for linear systems with input saturation: theory and an application. IEEE Trans Automat Contr 48(3):427–439CrossRefMathSciNetGoogle Scholar
  4. David R, Alla H (2005) Discrete, continuous, and hybrid Petri nets. SpringerGoogle Scholar
  5. Dicesare F, Harhalakis G, Proth JM, Silva M, Vernadat FB (1993) Practice of Petri nets in manufacturing. Chapman & Hall, London, UKGoogle Scholar
  6. Farina L, Rinaldi S (2000) Positive linear systems. Theory and applications, pure and applied mathematics. John Wiley and Sons, New YorkGoogle Scholar
  7. Giua A, Mahulea C, Recalde L, Seatzu C, Silva M (2006a) Optimal control of continuous Petri nets via model predictive control. In: 8th international workshop on discrete event systems, WODES 06. IEEE Computer Society Press, Ann Arbor, USA, pp 235–241CrossRefGoogle Scholar
  8. Giua A, Mahulea C, Recalde L, Seatzu C, Silva M (2006b) Optimal control of timed continuous Petri nets via explicit MPC. In: Commault C, Marchand N (eds) Proceedings of the 2nd international symposium on positive systems: theory and applications (POSTA 2006), vol 341 of LNCIS. Springer, Grenoble, France, pp 383–390Google Scholar
  9. Habets LCGJM, Collins PJ, van Schuppen JH (2006) Reachability and control synthesis for piecewise-affine hybrid systems on simplices. IEEE Trans Automat Contr 51(6):938–948CrossRefGoogle Scholar
  10. Johansson M (2003) Piecewise linear control systems. LNCIS 284. SpringerGoogle Scholar
  11. Lin Z, Pachter M, Banda S (1998) Toward improvement of tracking performance – nonlinear feedback for linear systems. Int J Control 70(1):1–11MATHCrossRefMathSciNetGoogle Scholar
  12. Mahulea C, Ramírez A, Recalde L, Silva M (2007) Steady-state control reference and token conservation laws in continuous Petri net systems. Accepted for publication in IEEE Trans. on Automation Science and EngineeringGoogle Scholar
  13. Mahulea C, Recalde L, Silva M (2005) Optimal observability for continuous Petri nets. In: Proceedings of the 16th IFAC world congress, CDROM. Prague, Czech RepublicGoogle Scholar
  14. Mahulea C, Recalde L, Silva M (2006a) On performance monotonicity and basic servers semantics of continuous Petri nets. In: WODES 06: 8th international workshop on discrete event systems. Ann Arbor, USAGoogle Scholar
  15. Mahulea C, Giua A, Recalde L, Seatzu C, Silva M (2006b) On sampling continuous timed Petri nets: reachability “equivalence” under infinite servers semantics. In: ADHS 06: 2nd IFAC Conference on Analysis and Design of Hybrid Systems. Alghero, Italy, pp 37–43Google Scholar
  16. Recalde L, Teruel E, Silva M (1999) Autonomous continuous P/T systems. In: Donatelli S, Kleijn J (eds) Application and theory of Petri nets 1999, vol 1639 of LNCS. Springer, pp 107–126Google Scholar
  17. Saberi A, Lin Z, Teel AR (1996) Control of linear system with saturating actuators. IEEE Trans Automat Control 41(3):368–378MATHCrossRefMathSciNetGoogle Scholar
  18. Silva M, Recalde L (2002) Petri nets and integrality relaxations a view of continuous Petri nets. IEEE Trans Syst Man Cybern 32(4):314–327CrossRefGoogle Scholar
  19. Silva M, Recalde L (2004) On fluidification of Petri net models: from discrete to hybrid and continuous models. Annu Rev Control 28(2):253–266CrossRefGoogle Scholar
  20. Silva M, Recalde L (2005) Continuization of timed Petri nets: from performance evaluation to observation and control. In: Proceedings of the 26th international conference on application and theory of petri nets and other models of concurrency. Springer-VerlagGoogle Scholar
  21. Silva M, Teruel E, Colom JM (1998) Linear algebraic and linear programming techniques for the analysis of P/T net systems. LCNS 1(1941):309–373Google Scholar
  22. Sun Z, Ge SS (2005) Switched linear systems. Control and design, communications and control engineering. SpringerGoogle Scholar
  23. Teruel E, Colom JM, Silva M (1997) Choice-free Petri nets: a model for deterministic concurrent systems with bulk services and arrivals. IEEE Trans Syst Man Cybern 27(1):73–83CrossRefGoogle Scholar
  24. Turner MC, Postlethwaite I, Walker DJ (2000) Non-linear tracking control for multivariable constrained input linear systems. Int J Control 73(12):1160–1172MATHCrossRefMathSciNetGoogle Scholar
  25. Wredenhagen GF, Bélanger PR (1994) Piecewise-linear LQ control for systems with input constraints. Automatica 30(3):403–416MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Instituto de Investigación en Ingeniería de Aragón (I3A), Departamento de Informática e Ingeniería de Sistemas, Centro Politécnico SuperiorUniversidad de ZaragozaSaragossaSpain

Personalised recommendations