IEEE Aerospace and Electronic Systems Magazine - December 2020 - 17

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measurement nonlinear vector transformations, and the
vectors wk , vk represent unknown state and measurement
noises with known descriptions in the form of the probability density functions (PDFs) pðwk Þ, pðvk Þ, respectively.
The initial state PDF pðx0 Þ is known and independent
of the noises. The state equation (1) models time behavior
of the state and the measurement equation (2) gives a relation between the sought state and available measurement.
In navigation applications, the system may refer to a
navigated vehicle, the state xk to sought vehicle navigation information (i.e., vehicle position, velocity, attitude,
and heading), the input uk to, e.g., the readings of an inertial measurement unit (IMU), and the measurement zk ,
e.g., to the readings of a global navigation satellite system
(GNSS) receiver. Then, the nonlinear function fk ðÁÞ in (1)
stems from dynamic or kinematic laws, hk ðÁÞ in (2) models a relation between the receiver output and state variables, and the state and measurement noise PDFs pðwk Þ,
pðvk Þ are determined by the noise properties of the IMU
and GNSS receiver [3], [4].

STATE ESTIMATION AND BAYESIAN RECURSIVE
RELATIONS
The goal of state estimation, or more precisely filtering, is
to find an estimate of the state xk on the basis of a
sequence of all measurements zk ¼ ½z0 ; z1 ; . . . ; zk  up to
the time instant k and the model (1), (2). In particular, in
the Bayesian formulation of state estimation, the goal is to

find the PDF of the state xk conditioned on the measurements zk , i.e., the estimate in the form of the conditional
PDF pðxk jzk Þ; 8k, is sought. The conditional PDF provides
a full description of the state estimate.
In the Bayesian framework, the general solution for
state estimation is given by the Bayesian recursive relations
(BRRs) for the computation of the conditional PDFs,3 [5]
pðxk jzkÀ1 Þpðzk jxk Þ
pðzk jzkÀ1 Þ
Z
pðxkþ1 jzk Þ ¼ pðxkþ1 jxk Þpðxk jzk Þdxk

pðxk jzk Þ ¼

(3)
(4)

where pðxk jzk Þ is the filtering PDF computed by Bayes'
rule (3) and pðxk jzkÀ1 Þ is the one-step predictive PDF
computed by the Chapman-Kolmogorov equation (4).
The PDFs pðxkþ1 jxk Þ and pðzk jxk Þ are the state transition
PDF and measurement PDF unequivocally obtained from
the model (1) and (2), respectively. The PDF
Z
pðzk jzkÀ1 Þ ¼ pðxk jzkÀ1 Þpðzk jxk Þdxk
(5)
is the one-step predictive PDF of the measurement. The
Bayesian recursion (3), (4) starts from the system's initial
condition pðx0 jzÀ1 Þ ¼ pðx0 Þ. The Bayesian recursion is
illustrated in Figure 1.

STATE ESTIMATION METHODS: AN OVERVIEW
The BRRs are complex functional relations about which
exact solution is possible for a limited set of models only.
In other cases, an approximate solution to the BRRs must
be employed. These approximate methods can be divided,
3

Figure 1.
Illustration of the Bayesian recursion.

DECEMBER 2020

Considering the model (1), (2), the BRRs (3), (4) should be conditioned also on available sequence of the input uk ; 8k. However, for
the sake of notational simplicity, the input signal is assumed to be
implicitly part of the condition and it is not explicitly stated, i.e.,
pðxkþ1 jxk Þ ¼ pðxkþ1 jxk ; uk Þ, pðxk jzk Þ ¼ pðxk jzk ; ukÀ1 Þ, and
pðxkþ1 jzk Þ ¼ pðxkþ1 jzk ; uk Þ.

IEEE A&E SYSTEMS MAGAZINE

17
IEEE Aerospace and Electronic Systems Magazine - December 2020

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