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C. Greco and M. Vasile
observations and after an observation update will retain a normal distribution
(see Sect. 6.2.1). This precious characteristic and the simple combination rules
of Gaussian distributions result in a closed-form exact solution of the filtering
equations called Kalman filter (KF), introduced in Sect. 6.3.1. However, in the
majority of real-world applications, the involved transformations are nonlinear, and
a closed-form solution does not exist in the general case. Nonetheless, the Gaussian
approximation of the conditional density proves sufficient for a wide range of
practical applications. To approximate the Gaussian evolution through a nonlinear
transformation, the techniques introduced in the previous section shall be used.
Specifically, the extended Kalman filter (EKF), presented in Sect. 6.3.2, expands
the nonlinear function in Taylor series and retains only the first terms, whereas the
unscented Kalman filter (UKF), described in Sect. 6.3.3, approximates the posterior
distribution using the unscented transformation deterministic approach.
The basic idea of approximating the probability density function as being
normally distributed has been embedded in the general framework of Gaussian
filtering. This denomination encloses a family of algorithms employing moment
matching approximations, and usually explicit cubature rules for computing the
integrals required by the expectation operator. As it turns out, the general Gaussian
filtering framework can be seen as a generalisation of the Kalman filter and some of
its extensions presented in this chapter. This framework will be shortly outlined in
Sect. 6.3.4.
When the assumption of normal densities is too restrictive or not representative,
other techniques should be employed to approximate the underlying real distribution
in a finite-dimensional basis. Among the several existing methods, the particle
filter employs sampling methods, hence resulting in a discrete distribution. As
a sampling-based method, the particle filter is highly flexible and capable of
approximating posterior distribution of any nature, when the number of samples
is selected appropriately. This filter will be introduced in Sect. 6.3.5.
6.3.1 Kalman Filter
In the linear case, the filtering model is described by the linear equations of
motion and observation model. In general, linear systems are rather easy to
characterise and often allow closed-form solutions. On the other hand, they can
model only simplified problems, as real-world systems generally involve complex
nonlinearities. Nonetheless, the closed-form solution of an associated linear system
can be used to construct an approximation of the original one, or in general can
provide useful insight in some of its properties.
The Kalman filter is the closed-form algorithm for the evolution of the conditional probability density in a sequential linear filtering problem. The model for the
linear filtering problem definition (see Sect. 6.1.1) is formulated as:
C. Greco and M. Vasile
observations and after an observation update will retain a normal distribution
(see Sect. 6.2.1). This precious characteristic and the simple combination rules
of Gaussian distributions result in a closed-form exact solution of the filtering
equations called Kalman filter (KF), introduced in Sect. 6.3.1. However, in the
majority of real-world applications, the involved transformations are nonlinear, and
a closed-form solution does not exist in the general case. Nonetheless, the Gaussian
approximation of the conditional density proves sufficient for a wide range of
practical applications. To approximate the Gaussian evolution through a nonlinear
transformation, the techniques introduced in the previous section shall be used.
Specifically, the extended Kalman filter (EKF), presented in Sect. 6.3.2, expands
the nonlinear function in Taylor series and retains only the first terms, whereas the
unscented Kalman filter (UKF), described in Sect. 6.3.3, approximates the posterior
distribution using the unscented transformation deterministic approach.
The basic idea of approximating the probability density function as being
normally distributed has been embedded in the general framework of Gaussian
filtering. This denomination encloses a family of algorithms employing moment
matching approximations, and usually explicit cubature rules for computing the
integrals required by the expectation operator. As it turns out, the general Gaussian
filtering framework can be seen as a generalisation of the Kalman filter and some of
its extensions presented in this chapter. This framework will be shortly outlined in
Sect. 6.3.4.
When the assumption of normal densities is too restrictive or not representative,
other techniques should be employed to approximate the underlying real distribution
in a finite-dimensional basis. Among the several existing methods, the particle
filter employs sampling methods, hence resulting in a discrete distribution. As
a sampling-based method, the particle filter is highly flexible and capable of
approximating posterior distribution of any nature, when the number of samples
is selected appropriately. This filter will be introduced in Sect. 6.3.5.
6.3.1 Kalman Filter
In the linear case, the filtering model is described by the linear equations of
motion and observation model. In general, linear systems are rather easy to
characterise and often allow closed-form solutions. On the other hand, they can
model only simplified problems, as real-world systems generally involve complex
nonlinearities. Nonetheless, the closed-form solution of an associated linear system
can be used to construct an approximation of the original one, or in general can
provide useful insight in some of its properties.
The Kalman filter is the closed-form algorithm for the evolution of the conditional probability density in a sequential linear filtering problem. The model for the
linear filtering problem definition (see Sect. 6.1.1) is formulated as: