Chapter 6
Fundamentals of Filtering
Cristian Greco and Massimiliano Vasile
Abstract Accurately estimating the state of a dynamical system is of fundamental
importance in a variety of applications, from engineering challenges to everyday
life. This task is complex because uncertainties typically affect the dynamical
behaviour as well as the available observations of the (hidden) state. The goal
of this chapter is to provide a comprehensive overview, from the probabilistic
problem statement to methods for its solution. In particular the focus will be on
filtering problems for time-continuous state evolution equations and time-discrete
observations. It will be shown that, except for very few cases, the filtering problem
has no closed-form solution, which is generally infinite-dimensional. Hence, several
practical algorithms to find an approximate solution are presented.
Keywords State estimation · Uncertainty propagation · Inference · Navigation ·
Filtering algorithms
6.1 The State Estimation Problem
Filtering theory addresses the problem of estimating the state of a system given
an uncertain knowledge of its dynamical equations and noisy indirect observations.
Being a mathematical branch of the general stochastic processes theory, it finds
wide application in several fields spanning from engineering to physics, from
biology to medical sciences. Among the most notable examples of application
there are the Global Positioning System (GPS) technology, brain imaging methods,
signal processing techniques, the prediction of the evolution of (potential) infectious
diseases or environmental catastrophes and any other system modelled by stochastic
differential equations [51].
Modern filtering techniques combine the dynamical and measurement information in order to optimally estimate a system state or some model parameters, where
C. Greco () · M. Vasile
University of Strathclyde, Glasgow, UK
e-mail: c.greco@strath.ac.uk; massimiliano.vasile@strath.ac.uk
© Springer Nature Switzerland AG 2021
M. Vasile (ed.), Optimization Under Uncertainty with Applications to Aerospace
Engineering, https://doi.org/10.1007/978-3-030-60166-9_6
181
Fundamentals of Filtering
Cristian Greco and Massimiliano Vasile
Abstract Accurately estimating the state of a dynamical system is of fundamental
importance in a variety of applications, from engineering challenges to everyday
life. This task is complex because uncertainties typically affect the dynamical
behaviour as well as the available observations of the (hidden) state. The goal
of this chapter is to provide a comprehensive overview, from the probabilistic
problem statement to methods for its solution. In particular the focus will be on
filtering problems for time-continuous state evolution equations and time-discrete
observations. It will be shown that, except for very few cases, the filtering problem
has no closed-form solution, which is generally infinite-dimensional. Hence, several
practical algorithms to find an approximate solution are presented.
Keywords State estimation · Uncertainty propagation · Inference · Navigation ·
Filtering algorithms
6.1 The State Estimation Problem
Filtering theory addresses the problem of estimating the state of a system given
an uncertain knowledge of its dynamical equations and noisy indirect observations.
Being a mathematical branch of the general stochastic processes theory, it finds
wide application in several fields spanning from engineering to physics, from
biology to medical sciences. Among the most notable examples of application
there are the Global Positioning System (GPS) technology, brain imaging methods,
signal processing techniques, the prediction of the evolution of (potential) infectious
diseases or environmental catastrophes and any other system modelled by stochastic
differential equations [51].
Modern filtering techniques combine the dynamical and measurement information in order to optimally estimate a system state or some model parameters, where
C. Greco () · M. Vasile
University of Strathclyde, Glasgow, UK
e-mail: c.greco@strath.ac.uk; massimiliano.vasile@strath.ac.uk
© Springer Nature Switzerland AG 2021
M. Vasile (ed.), Optimization Under Uncertainty with Applications to Aerospace
Engineering, https://doi.org/10.1007/978-3-030-60166-9_6
181