Abstract
State estimation of dynamic systems is a resort often used when only a subset of the state variables can be directly measured; observers are the entities computing the system state from the knowledge of its internal structure and its (partially) measured behaviour. The problem of discrete event systems (DES) estimation has been addressed in (Ramirez, 2003) and (Giua 2003); in these works the marking of a Petri net (PN) model of a partially observed event driven system is computed from the evolution of its inputs and outputs. The state of a system can be also inferred using the knowledge on the duration of activities. However this task becomes complex when, besides the absence of sensors, the durations of the operations are uncertain; in this situation the observer obtains and revise a belief that approximates the current system state. Consequently this approach is useful for non critical applications of state monitoring and feedback in which an approximate computation is allows. The uncertainty of activities duration in DES can be handled using fuzzy PN (FPN) (Murata, 1996), (Cardoso, 1999), (Hennequin, 2001), (Pedrycz, 2003), (Ding, 2005); this PN extension has been applied to knowledge modelling (Chen, 1990), (Koriem, 2000), (Shen, 2003), planning (Cao, 1996), reasoning (Gao, 2003) and controller design (Andreu, 1997), (Leslaw, 2004). In these works the proposed techniques include the computation of imprecise markings; however the class of models dealt does not include strongly connected PN for the modelling of cyclic behaviour. In this article we address the problem of state estimation of DES for calculating the fuzzy marking of a Fuzzy Timed Petri Net (FTPN); for this purpose a set of matrix expressions for the recursive computing the current fuzzy marking is developed. The article focuses on FTPN whose structure is a Marked Graph (called Fuzzy Timed Marked Graph -FTMG) because it allows showing intuitively the problems of the marking estimation in exhibiting cyclic behaviour.
TopIntroduction
State estimation of dynamic systems is a resort often used when only a subset of the state variables can be directly measured; observers are the entities computing the system state from the knowledge of its internal structure and its (partially) measured behaviour. The problem of discrete event systems (DES) estimation has been addressed in (Ramirez, 2003) and (Giua 2003); in these works the marking of a Petri net (PN) model of a partially observed event driven system is computed from the evolution of its inputs and outputs.
The state of a system can be also inferred using the knowledge on the duration of activities. However this task becomes complex when, besides the absence of sensors, the durations of the operations are uncertain; in this situation the observer obtains and revise a belief that approximates the current system state. Consequently this approach is useful for non critical applications of state monitoring and feedback in which an approximate computation is allows.
The uncertainty of activities duration in DES can be handled using fuzzy PN (FPN) (Murata, 1996), (Cardoso, 1999), (Hennequin, 2001), (Pedrycz, 2003), (Ding, 2005); this PN extension has been applied to knowledge modelling (Chen, 1990), (Koriem, 2000), (Shen, 2003), planning (Cao, 1996), reasoning (Gao, 2003) and controller design (Andreu, 1997), (Leslaw, 2004).
In these works the proposed techniques include the computation of imprecise markings; however the class of models dealt does not include strongly connected PN for the modelling of cyclic behaviour. In this article we address the problem of state estimation of DES for calculating the fuzzy marking of a Fuzzy Timed Petri Net (FTPN); for this purpose a set of matrix expressions for the recursive computing the current fuzzy marking is developed. The article focuses on FTPN whose structure is a Marked Graph (called Fuzzy Timed Marked Graph -FTMG) because it allows showing intuitively the problems of the marking estimation in exhibiting cyclic behaviour.
Key Terms in this Chapter
Marked Graph: It is a Petri Net subclass in which every place has only one input transition and one output transition.
State Estimation: It is the inference process that determines the current state of a system from the knowledge of sequences of inputs and outputs.
Fuzzy Logic: It is a Knowledge representation technique and computing framework whose approach is based on degrees of truth rather than the usual “true” or “false” of classical logic.
System Monitoring: It is a surveillance process on measurable events and/or outputs of a system; it is often used a reference model that specifies a reasonable good behavior. Deviations from the reference are analyzed and determined if there exist a fault. This process is included as a part of a fault diagnosis process.
Fuzzy Petri Nets: It is a family of formalisms extending Petri nets by the inclusion of fuzzy sets representing usually uncertainty of time elapses.
Imprecise Marking: The imprecise localization of tokens within places of a FTPN; it is computed as a possibility distribution.
Petri Nets: It is a family of formalisms for modeling and analysis of concurrent DES allowing intuitive graphical descriptions and providing a simple but sound mathematical support. A timed Petri net includes information about the duration of the modeled activities.
State Machine: It is a Petri Net subclass in which every transition has only one input place and one output place.
Discrete Events Systems: It is the class of systems whose behavior is characterized by successions of states delimited by asynchronous events. Most of these systems have been man made.