Abstract
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.
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References
Bemporad A, Morari M, Dua V, Pistikopoulos E (2002) The explicit linear quadratic regulator for constrained systems. Automatica 38(1):3–20
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 4957–4962
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):427–439
David R, Alla H (2005) Discrete, continuous, and hybrid Petri nets. Springer
Dicesare F, Harhalakis G, Proth JM, Silva M, Vernadat FB (1993) Practice of Petri nets in manufacturing. Chapman & Hall, London, UK
Farina L, Rinaldi S (2000) Positive linear systems. Theory and applications, pure and applied mathematics. John Wiley and Sons, New York
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 235–241
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 383–390
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):938–948
Johansson M (2003) Piecewise linear control systems. LNCIS 284. Springer
Lin Z, Pachter M, Banda S (1998) Toward improvement of tracking performance – nonlinear feedback for linear systems. Int J Control 70(1):1–11
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 Engineering
Mahulea C, Recalde L, Silva M (2005) Optimal observability for continuous Petri nets. In: Proceedings of the 16th IFAC world congress, CDROM. Prague, Czech Republic
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, USA
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 37–43
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 107–126
Saberi A, Lin Z, Teel AR (1996) Control of linear system with saturating actuators. IEEE Trans Automat Control 41(3):368–378
Silva M, Recalde L (2002) Petri nets and integrality relaxations a view of continuous Petri nets. IEEE Trans Syst Man Cybern 32(4):314–327
Silva M, Recalde L (2004) On fluidification of Petri net models: from discrete to hybrid and continuous models. Annu Rev Control 28(2):253–266
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-Verlag
Silva M, Teruel E, Colom JM (1998) Linear algebraic and linear programming techniques for the analysis of P/T net systems. LCNS 1(1941):309–373
Sun Z, Ge SS (2005) Switched linear systems. Control and design, communications and control engineering. Springer
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):73–83
Turner MC, Postlethwaite I, Walker DJ (2000) Non-linear tracking control for multivariable constrained input linear systems. Int J Control 73(12):1160–1172
Wredenhagen GF, Bélanger PR (1994) Piecewise-linear LQ control for systems with input constraints. Automatica 30(3):403–416
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This work was partially supported by CICYT-FEDER projects DPI2003-06376 and DPI2006-15390.
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Xu, J., Recalde, L. & Silva, M. Tracking Control of Join-Free Timed Continuous Petri Net Systems under Infinite Servers Semantics. Discrete Event Dyn Syst 18, 263–283 (2008). https://doi.org/10.1007/s10626-007-0034-z
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DOI: https://doi.org/10.1007/s10626-007-0034-z