Optimal Observability for Continuous Petri Nets

Applications and Theory of Petri Nets 2005

2005

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Abstract. State explosion is a fundamental problem in the analysis and synthesis of discrete event systems. Continuous Petri nets can be seen as a relaxation of discrete models allowing more efficient (in some cases polynomial time) analysis and synthesis algorithms. Nevertheless computational costs can be reduced at the expense of the analyzability of some properties. Even more, some net systems do not allow any kind of continuization. The present work first considers these aspects and some of the alternative formalisms usable for continuous relaxations of discrete systems. Particular emphasis is done later on the presentation of some results concerning performance evaluation, parametric design and marking (i.e., state) observation and control. Even if a significant amount of results are available today for continuous net systems, many essential issues are still not solved. A list of some of these are given in the introduction as an invitation to work on them.

Section: Observationmentioning

confidence: 97%

Section: Observationmentioning

confidence: 99%

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Continuization of Timed Petri Nets: From Performance Evaluation to Observation and Control

Applications and Theory of Petri Nets 2005

2005

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“…In Mahulea et al (2005) it is shown that if the net has attributions, i.e., p ∈ P with | • p| ≥ 2, observability can be lost. In this case, a pole-zero cancelation can appear, which happens for very specific values of λ, i.e., the denominator and the numerator of the transfer function vector between the input flow in places and the outputs:…”

Section: Generic Observabilitymentioning

confidence: 99%

“…In Mahulea et al (2005) an interpretation of the loss of observability in the case of join free (JF) contPN systems is given, when the system contains attributions. It is pointed out that observability cannot be checked locally and can be lost for some specific values of the firing rates of the transitions.…”

Section: Introductionmentioning

confidence: 99%

Observability of Timed Continuous Petri Nets: A Class of Hybrid Systems

Mahulea

IFAC Proceedings Volumes

2008

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Timed continuous Petri net systems with infinite server semantics are piecewise linear systems. This paper addresses several problems regarding the state observability of these systems. We assume that the initial marking/state is not known and measuring some places we want to estimate all the others. First, a study of the different linear systems corresponding to a continuous Petri net system is performed. It is shown that in some cases, some of them are redundant, and so can be disregarded. The notion of distinguishable configurations is introduced. It helps to give a necessary and sufficient criterion for the observability in infinitesimal time. Using results from linear structured systems (Commault et al. (2005)), the concept of generic observability is introduced and it is studied in the case of join free nets.

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

Discrete Event Dyn Syst

2008

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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 lowgain 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. NomenclatureR + denotes the set of non-negative real numbers; R n denotes the Euclidean space of dimension n; R +n denotes a subset of R n , in which the vectors are composed of nonnegative real numbers; M = {1, · · · , m} and N = {1, · · · , n}; For a given vector a ∈ R n (or a b ∈ R n ), a i (or a b,i ) denotes the i-th element of a (or a b ); For a given matrix A ∈ R n×m (or A b ∈ R n×m ), a i (or a b,i ) denotes the i-th row of A (or A b ) except special indication; Given two vectors a 1 ∈ R n and a 2 ∈ R n , the i-the element of min{a 1 , a 2 } is defined as min{a 1,i , a 2,i }; a 1 ≤ a 2 means, ∀i ∈ N, a 1,i ≤ a 2,i ; Given a finite set S, |S| denotes the size of the set S.