Fuzzy Approximation of DES State

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.