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Kalman Filtering and Neural Networks - Chapter 1: KALMAN FILTERS

The celebrated Kalman filter, rooted in the state-space formulation of linear dynamical systems, provides a recursive solution to the linear optimal filtering problem. It applies to stationary as well as nonstationary environments. The solution is recursive in that each updated estimate of the state is computed from the previous estimate and the new input data, so only the previous estimate requires storage. | 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 1 KALMAN FILTERS Simon Haykin Communications Research Laboratory McMaster University Hamilton Ontario Canada haykin@mcmaster.ca 1.1 INTRODUCTION The celebrated Kalman filter rooted in the state-space formulation of linear dynamical systems provides a recursive solution to the linear optimal filtering problem. It applies to stationary as well as nonstationary environments. The solution is recursive in that each updated estimate of the state is computed from the previous estimate and the new input data so only the previous estimate requires storage. In addition to eliminating the need for storing the entire past observed data the Kalman filter is computationally more efficient than computing the estimate directly from the entire past observed data at each step of the filtering process. In this chapter we present an introductory treatment of Kalman filters to pave the way for their application in subsequent chapters of the book. We have chosen to follow the original paper by Kalman 1 for the 1 2 1 KALMAN FILTERS derivation see also the books by Lewis 2 and Grewal and Andrews 3 . The derivation is not only elegant but also highly insightful. Consider a linear discrete-time dynamical system described by the block diagram shown in Figure 1.1. The concept of state is fundamental to this description. The state vector or simply state denoted by xk is defined as the minimal set of data that is sufficient to uniquely describe the unforced dynamical behavior of the system the subscript k denotes discrete time. In other words the state is the least amount of data on the past behavior of the system that is needed to predict its future behavior. Typically the state xk is unknown. To estimate it we use a set of observed data denoted by the vector yk. In mathematical terms the block diagram of Figure 1.1 embodies the following pair of