# Probability and Statistics Seminar

### Accurate implementations of nonlinear Kalman-like filtering methods with application to chemical engineering

A goal in many practical applications is to combine a priori knowledge about a physical system with experimental data to provide on-line estimation of states and/or parameters of that system. The time evolution of the (hidden) state is modeled by using dynamic system which is perturbed by a certain process noise. This noise is used for modeling the uncertainties in the system dynamics. The term optimal filtering traditionally refers to a class of methods that can be used for estimating the state of a time-varying system which is indirectly observed through noisy measurements. In this talk, we discuss the development of advanced Kalman-like filtering methods for estimating continuous-time nonlinear stochastic systems with discrete measurements. We starts with a brief overview of existing nonlinear Bayesian methods [1]. Next, we focus on the numerical implementation of the Kalman-like filters (the Extended Kalman filter, the Unscented Kalman filter and Cubature Kalman filter) for estimating the state of continuous-discrete models [2]. The standard approach implies that the Euler-Maruyama method is used for discretization of the underlying (process) stochastic differential equation (SDE). To reduce the discretization error, some subdivisions might be additionally introduced in each sampling interval. Some modern continuous-time filtering methods are developed by using a higher order methods, e.g. see the cubature Kalman filter based on the Ito-Taylor expansion for discretizing the underlying SDE in [3]. However, all resulted implementations are the fixed step size methods and they do not allow for a proper processing of long and irregular sampling intervals (e.g. when missing measurements are appeared). An alternative methodology is to derived the moment differential equations, first. Next, the resulted ordinary differential equations (ODEs) are solved by modern ODE solvers. This approach allows for using variable step size solvers and copes with long/irregular sampling intervals accurately. Besides, we use the ODE solvers with global error control that improves the estimation quality further [4]. As a numerical example we consider the batch reactor model studied in chemical engineering literature [5].