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.
The paper makes a sketch of an SDOF system response analysis subjected to a random excitation having a form of the additive Poisson driven independent random impulses. A special generalised Fokker-Planck equation having a form of an integro-differential equation is presented together with boundary and initial conditions. Later the Galerkin-Petrov process as a method of a numerical solution of the respective evolutionary integro-differential equation for the probability density function (PDF) is presented in general. Various analytic and semi-analytic solution methods have been developed for various systems to obtain results requested. However numerical approaches offer a powerful altemative. In particular the Finite Element Method (FEM) seems to be very effective. Shape and weighting functions for purposes of a numerical solution procedure are carred out and corresponding ordinary differential system for PDF values in nodes is deduced. As a demonstration particular SDOF systems are investigated. Resulting PDFs are analysed and mutually compared. and Obsahuje seznam literatury
The Fokker-Planck (FP) equation is frequently used when the response of the dynamic system subjected to additive and/or multiplicative ramdom noises is investiagted. It provides the probability density function (PDF) representing the key information for further study of the dynamic system. Various analytic and semi-analytic solution methods have been developed for various systems to obtain results requested. However numerical approaches offer a powerful alternative. In particular the Finite Element Method (FEM) seems to be very effective. A couple of single dynamic linear/non-linear systems under additive and multiplicative random excitations are discussed using FEM as a solution tool of the FP equation. and Obsahuje seznam literatury