182
C. Greco and M. Vasile
the concept of optimality will be formalized in Sect. 6.1.3. This process is nothing
more than an inversion problem of the measurements equations y = h(t, x) + —
where y and x are, respectively, the observation and state vector, h is the observation
model function and is the measurement error—taking into account an initial
estimate of the state and the equations governing the state evolution. Historically,
this step has been tackled from two conceptually different perspectives:
• Probabilistic approach: the errors in the dynamics and the observations are
characterized as random variables with known probability distributions. Hence,
the goal is to compute the conditional probability p(x k |y 1:k ), whose knowledge
represents the complete solution to the filtering problem. Indeed, in the general
nonlinear case, the filter state is infinite-dimensional and represented by the conditional density function [29]. Assumptions need to be made on the distributions
of random variables x and y to obtain a finite-dimensional solution, as will be
shown in Sect. 6.3. Out of the conditional probability density function, common
choices for the optimal estimate of the state are the distribution mean, median,
mode and so on.
• Statistical approach: the errors in the dynamics and the observations are considered as unknown but deterministic; the goal is then to minimise a chosen
performance index which is a function of the deviation between the obtained
measurements and the computed observations (through the deterministic part
of the observation model). The dynamical equations are then formulated as
constraints of the minimisation procedure. A well-known statistical approach for
which this optimisation step has a closed-form solution is the (possibly recursive)
weighted least squares method [58] which minimizes the sum of the weighted
square deviations. In general, different performance indices result in different
estimates of the state.
In general, this chapter pursues the probabilistic approach to state estimation in its
theoretical developmentHowever, when looking at the practical implementation of
filtering methods, often this boundary starts to fade. This results from the different
possible derivations and interpretations of the same method. As an example, the
time-discrete linear Kalman filter (see Sect. 6.3.1) can be derived starting either from
a probabilistic reasoning [29] or from a statistical approach [58], leading to the same
final algorithm.
Often, engineering books focus on the latter approach as it gives a simpler
high-level interpretation to the filtering step, i.e. as a constrained optimisation.
Moreover, it requires little or no knowledge of probability and allows for a direct
focus on the filter implementation. On the other hand, the probabilistic approach
provides solid mathematical justification to each assumption and step at the cost
of increased theoretical complexity. Indeed, the knowledge of probability theory
is essential and an understanding of stochastic differential equations often useful.
In this chapter the latter approach will be pursued to provide a comprehensive
awareness of filtering theory and the advantages and approximations resulting from
a selected filter. Nonetheless, the following development will attempt to simplify
the mathematically formal description retaining the minimum set of fundamental
C. Greco and M. Vasile
the concept of optimality will be formalized in Sect. 6.1.3. This process is nothing
more than an inversion problem of the measurements equations y = h(t, x) + —
where y and x are, respectively, the observation and state vector, h is the observation
model function and is the measurement error—taking into account an initial
estimate of the state and the equations governing the state evolution. Historically,
this step has been tackled from two conceptually different perspectives:
• Probabilistic approach: the errors in the dynamics and the observations are
characterized as random variables with known probability distributions. Hence,
the goal is to compute the conditional probability p(x k |y 1:k ), whose knowledge
represents the complete solution to the filtering problem. Indeed, in the general
nonlinear case, the filter state is infinite-dimensional and represented by the conditional density function [29]. Assumptions need to be made on the distributions
of random variables x and y to obtain a finite-dimensional solution, as will be
shown in Sect. 6.3. Out of the conditional probability density function, common
choices for the optimal estimate of the state are the distribution mean, median,
mode and so on.
• Statistical approach: the errors in the dynamics and the observations are considered as unknown but deterministic; the goal is then to minimise a chosen
performance index which is a function of the deviation between the obtained
measurements and the computed observations (through the deterministic part
of the observation model). The dynamical equations are then formulated as
constraints of the minimisation procedure. A well-known statistical approach for
which this optimisation step has a closed-form solution is the (possibly recursive)
weighted least squares method [58] which minimizes the sum of the weighted
square deviations. In general, different performance indices result in different
estimates of the state.
In general, this chapter pursues the probabilistic approach to state estimation in its
theoretical developmentHowever, when looking at the practical implementation of
filtering methods, often this boundary starts to fade. This results from the different
possible derivations and interpretations of the same method. As an example, the
time-discrete linear Kalman filter (see Sect. 6.3.1) can be derived starting either from
a probabilistic reasoning [29] or from a statistical approach [58], leading to the same
final algorithm.
Often, engineering books focus on the latter approach as it gives a simpler
high-level interpretation to the filtering step, i.e. as a constrained optimisation.
Moreover, it requires little or no knowledge of probability and allows for a direct
focus on the filter implementation. On the other hand, the probabilistic approach
provides solid mathematical justification to each assumption and step at the cost
of increased theoretical complexity. Indeed, the knowledge of probability theory
is essential and an understanding of stochastic differential equations often useful.
In this chapter the latter approach will be pursued to provide a comprehensive
awareness of filtering theory and the advantages and approximations resulting from
a selected filter. Nonetheless, the following development will attempt to simplify
the mathematically formal description retaining the minimum set of fundamental