In this paper, we propose an algebraic approach to investigate the diagnosis of partially observable labeled Petri nets based on state estimation on a sliding window of a predefined length h. Given an observation, the resulting diagnosis state can be computed while solving integer linear programming problems with a reduced subset of basis markings. \blue{The proposed approach consists in exploiting} a subset of h observations at each estimation step, which provides a partial diagnosis relevant to the current observation window. This technique allows a status update with a "forgetfulness" of past observations and enables distinguishing repetitive and punctual faults. The complete diagnosis state can be defined as a function of the partial diagnosis states interpreted on the sliding window. As \blue{the} analysis shows that some basis markings can present an inconsistency with a future evolution, which possibly implies unnecessary computations of basis markings, a withdrawal procedure of these \blue{irrelevant} basis markings based on linear programming is proposed.
The paper deals with a filter design for nonlinear continuous stochastic systems with discrete-time measurements. The general recursive solution is given by the Fokker-Planck equation (FPE) and by the Bayesian rule. The stress is laid on the computation of the predictive conditional probability density function from the FPE. The solution of the FPE and its integration into the estimation algorithm is the cornerstone for the whole recursive computation. A new usable numerical scheme for the FPE is designed. In the scheme, the separation technique based on the upwind volume method and the finite difference method for hyperbolic and parabolic part of the FPE is used. It is supposed that separation of the FPE and choice of a suitable numerical method for each part can achieve better estimation quality comparing to application of a single numerical method to the unseparated FPE. The approach is illustrated in some numerical examples.