Introduction to Kalman Filtering

This document is an introduction to Kalman optimal Filtering applied to linear systems. It is assumed that the reader is already aware of linear servo-loop theory, frequency-domain Filtering (continuous and discrete-time) and state-space approach to represent linear systems. Generally, Filtering consists in estimating a useful information (signal) from a measurement (of this information) perturbed by a noise. Frequency-domain Filtering assumes that a frequency-domain separation exists between the frequency response of the useful signal and the frequency response of the noise. Then, frequency-domain Filtering consists in seeking a transfer function fitting a template on its magnitude response (and too much rarely, on its phase response). Kalman optimal filtering aims to estimate the state vector of a linear system (thus, this state is the useful information) and this estimate is optimal w.r.t. an index performance: the sum of estimation error variances for all state vector components. First of all, some backgrounds on random variables and signals are required then, the assumptions, the structure and the computation Kalman Filter could be introduced. In the first chapter, we remind the reader how a random signal can be characterized from a mathematical point of view. The response of a linear system to a random signal will be investigated in an additional way to the more well-known response of a linear system to a deterministic signal (impulse, step, ramp, ... responses). In the second chapter, the assumptions, the structure, the main parameters and properties of Kalman Filter will be defined. The reader who wish to learn tuning methodology of the Kalman filtering can directly start the reading at chapter 2. But the reading of chapter 1, which is more cumbersome from a theoritical point of view, is required if one wishes to learn basic principles in random signal processing, on which is based Kalman Filtering. There are many applications of Kalman Filtering in aeronautics and aerospace engineering. As Kalman filter provides an estimate of plant states from an a priori information of the plant behaviour (model) and from real measurement, Kalman Filter will be used to estimate initial conditions (ballistics), to predict vehicle position and trajectory (navigation) and also to implement control laws based on a state feedback and a state estimator (LQG: Linear Quadratic Gaussian control). The signal processing principles on which is based Kalman Filter will be also very useful to study and perform test protocols, experimental data processing and also parametric identification, that is the experimental determination of some plant dynamic parameters