Page 93 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

82 STATE ESTIMATION

measurement model (in fact, each possible word has its own state space
model).
The outline of the chapter is as follows. Section 4.1 gives a framework
for estimation in dynamic systems. It introduces the various concepts,
notations and mathematical models. Next, it presents a general scheme
to obtain the optimal solution. In practice, however, such a general
scheme is of less value because of the computational complexity involved
when trying to implement the solution directly. Therefore, the general
approach needs to be worked out for different cases. Section 4.2 is
devoted to the case of continuous state variables. Practical solutions
are feasible if the models are linear-Gaussian (Section 4.2.1). If the
model is not linear, one can resort to suboptimal methods (Section 4.2.2).
Section 4.3 deals with the discrete state case. The chapter finalizes
with Section 4.4 which contains an introduction to particle filtering.
This technique can handle nonlinear and non-Gaussian models
covering the continuous and the discrete case, and even mixed cases
(i.e. combinations of continuous and discrete states).
The chapter confines itself to the theoretical aspects of state estima-
tion. Practical issues, like implementations, deployment, consistency
checks are dealt with in Chapter 8. The use of MATLAB is also deferred
to that chapter.

4.1 A GENERAL FRAMEWORK FOR ONLINE
ESTIMATION

Usually, the estimation problem is divided into three paradigms:

. online estimation (optimal filtering)
. prediction
. retrodiction (smoothing, offline estimation).

Online estimation is the estimation of the present state using all the
measurements that are available, i.e. all measurements up to the present
time. Prediction is the estimation of future states. Retrodiction is the
estimation of past states.
This section sets up a framework for the online estimation of the states
of time-discrete processes. Of course, most physical processes evolve in
the continuous time. Nevertheless, we will assume that these systems
can be described adequately by a model where the continuous time is
reduced to a sequence of specific times. Methods for the conversion from