The Extended Kalman Filter (EKF) provides an efficient method for generating approximate maximum-likelihood estimates of the state of a discrete-time nonlinear dynamical system (see Chapter 1). The filter involves a recursive procedure to optimally combine noisy observations with predictions from the known dynamic model. A second use of the EKF involves estimating the parameters of a model (., neural network) given clean training data of input and output data (see Chapter 2). | Kalman Filtering and Neural Networks Edited by Simon Haykin Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-36998-5 Hardback 0-471-22154-6 Electronic 5 DUAL EXTENDED KALMAN FILTER METHODS Eric A. Wan and Alex T. Nelson Department of Electrical and Computer Engineering Oregon Graduate Institute of Science and Technology Beaverton Oregon . INTRODUCTION The Extended Kalman Filter EKF provides an efficient method for generating approximate maximum-likelihood estimates of the state of a discrete-time nonlinear dynamical system see Chapter 1 . The filter involves a recursive procedure to optimally combine noisy observations with predictions from the known dynamic model. A second use of the EKF involves estimating the parameters of a model . neural network given clean training data of input and output data see Chapter 2 . In this case the EKF represents a modified-Newton type of algorithm for on-line system identification. In this chapter we consider the dual estimation problem in which both the states of the dynamical system and its parameters are estimated simultaneously given only noisy observations. 123 124 5 DUAL EXTENDED KALMAN FILTER METHODS To be more specific we consider the problem of learning both the hidden states xk and parameters w of a discrete-time nonlinear dynamical system Xk 1 - F Xk Uk w vk 5 yk- H xk w nk where both the system states xk and the set of model parameters w for the dynamical system must be simultaneously estimated from only the observed noisy signal yk. The process noise vk drives the dynamical system observation noise is given by nk and Uk corresponds to observed exogenous inputs. The model structure F - and H - may represent multilayer neural networks in which case w are the weights. The problem of dual estimation can be motivated either from the need for a model to estimate the signal or in other applications from the need for good signal estimates to estimate the model. In general applications can be divided into the tasks
