6 Fundamentals of Filtering
183
concepts needed to have a comprehensive understanding of filtering theory. For a
more formal mathematical treatment, the reader should refer to the classical work
from Jazwinski [29].
The remainder of the chapter is structured as follows. Section 6.1 will present the
framework of state estimation problems for time-continuous systems and introduce
the needed working mathematical concepts. As the probabilistic approach is
pursued, Sect. 6.2 will discuss the exact and approximated methods for propagating
probability distributions through generic nonlinear transformations, which in this
setting are the dynamical equations and the observation model. Finally, Sect. 6.3
introduces several practical algorithms to compute the (often approximated) solution
of the filtering problem, each more suitable under different working conditions and
assumptions. To summarise the chapter, Sect. 6.4 provides a final overview of the
presented topics.
6.1.1 Building Blocks
Generally in aerospace applications, ‘filtering’ and ‘state estimation’ are interchangeable to indicate the process of propagating the state distribution knowledge
through a dynamical model, and updating this estimate when new observations
are available, potentially decreasing (filtering) the noise contributions. On the
other hand, in mathematical applications, the filtering problem defines only the
update step, when prior information and noisy measurements are combined. For the
notation employed in this book, a filter is a model handling both the propagation and
the update; therefore, the two aforementioned names shall be considered synonyms.
Modern filtering theory employs a general framework to handle a great variety
of mathematical problems. A requirement of the system to be filtered is to be a
stochastic process with possibly hidden (unobserved) states in the most general
cases. In addition, some a priori estimate of the initial state should be available.
Therefore, three main elements are necessary components to define a filter:
• Dynamical model: a set of equations mapping the state at time t i to the state at
time t j . Generally in the continuous-time case, the dynamical model is described
by a set of differential equations. In line with the applications outlined, the
conventional set of dynamical equations will be described as first-order ordinary
differential equations in the state space form:
˙
x = f(t, x) + G(t, x)w ,
(6.1)
where f is the deterministic system dynamics, w is a white Gaussian noise and
G is its coefficient matrix. Usually, the stochastic term is introduced to model
possible approximation errors. It is worth clarifying that with this formulation we
intend to encompass the full range of dynamical models and we do not restrict to
natural dynamics only. For example, while control forces u are usually included
in this notation writing ˙
x = f(t, x, u), in this chapter the explicit dependency
183
concepts needed to have a comprehensive understanding of filtering theory. For a
more formal mathematical treatment, the reader should refer to the classical work
from Jazwinski [29].
The remainder of the chapter is structured as follows. Section 6.1 will present the
framework of state estimation problems for time-continuous systems and introduce
the needed working mathematical concepts. As the probabilistic approach is
pursued, Sect. 6.2 will discuss the exact and approximated methods for propagating
probability distributions through generic nonlinear transformations, which in this
setting are the dynamical equations and the observation model. Finally, Sect. 6.3
introduces several practical algorithms to compute the (often approximated) solution
of the filtering problem, each more suitable under different working conditions and
assumptions. To summarise the chapter, Sect. 6.4 provides a final overview of the
presented topics.
6.1.1 Building Blocks
Generally in aerospace applications, ‘filtering’ and ‘state estimation’ are interchangeable to indicate the process of propagating the state distribution knowledge
through a dynamical model, and updating this estimate when new observations
are available, potentially decreasing (filtering) the noise contributions. On the
other hand, in mathematical applications, the filtering problem defines only the
update step, when prior information and noisy measurements are combined. For the
notation employed in this book, a filter is a model handling both the propagation and
the update; therefore, the two aforementioned names shall be considered synonyms.
Modern filtering theory employs a general framework to handle a great variety
of mathematical problems. A requirement of the system to be filtered is to be a
stochastic process with possibly hidden (unobserved) states in the most general
cases. In addition, some a priori estimate of the initial state should be available.
Therefore, three main elements are necessary components to define a filter:
• Dynamical model: a set of equations mapping the state at time t i to the state at
time t j . Generally in the continuous-time case, the dynamical model is described
by a set of differential equations. In line with the applications outlined, the
conventional set of dynamical equations will be described as first-order ordinary
differential equations in the state space form:
˙
x = f(t, x) + G(t, x)w ,
(6.1)
where f is the deterministic system dynamics, w is a white Gaussian noise and
G is its coefficient matrix. Usually, the stochastic term is introduced to model
possible approximation errors. It is worth clarifying that with this formulation we
intend to encompass the full range of dynamical models and we do not restrict to
natural dynamics only. For example, while control forces u are usually included
in this notation writing ˙
x = f(t, x, u), in this chapter the explicit dependency