Position, Navigation, and Timing Technologies in the 21st Century

and

(36.3)

While this is the most general case, it should be noted that most online algorithms would only be concerned with the conditional state estimate at the current epoch. For this situation, Eq. 36.1would be represented as

(36.4)

In the next section, we will present the typical recursive estimation framework which will serve as the foundations for developing the forthcoming nonlinear recursive estimation strategies to follow.

In a typical recursive estimation framework, the system is represented using a process model and one (or more) observation models. The process model represents the internal dynamics of the system and can be expressed as a nonlinear, stochastic difference equation of the form

(36.5)

where x kis the state vector at time k ∈ ℕ, and w k − 1is the process noise random vector at time k – 1. External observations regarding the system state are represented by an observation model. The generalized observation model is a function of both the system state and a random vector representing the observation errors:

(36.6)

In the above equation, z kis the observation at time k , and v kis the random observation error vector at time k . The objective of the recursive estimator is to estimate the posterior pdf of the state vector, conditioned on the observations

(36.7)

where ℤ kis the collection of observations up to, and including, time k . This is accomplished by performing two types of transformations on the state pdf, propagation and updates. The result is a filter cycle given by

(36.8)

Note the introduction of the a priori pdf given by

(36.9)

Further examination of the propagation and update cycle in Eq. 36.8provides insights into how our system knowledge and observations are incorporated into our understanding of the state vector. To begin, we consider the propagation step from epoch k – 1 to k . Time propagation begins with the posterior pdf p ( x k − 1| ℤ k − 1). The process model defined in Eq. 36.5is used to define the transition pdf p ( x k| x k − 1), which can then be used to calculate the a priori pdf at time k via the Chapman–Kolmogorov equation [2]:

(36.10)

Examination of the process model ( Eq. 36.5) shows that the propagated state vector is a first‐order Gauss–Markov random process and is dependent only on the previous state vector and the process noise vector. As a result, we can express the transition probability, which is independent of the prior observation, as

(36.11)

Substituting Eqs. 36.11into 36.10results in the propagation relationship

(36.12)

An observation at time k can be incorporated by considering the posterior pdf p ( x k| ℤ k), which, given the definition of our observation sequence in Eq. 36.3, can be expressed equivalently as

(36.13)

Applying Bayes’ rule to Eq. 36.13yields

(36.14)

Observing the form of the previously defined observation, Eq. 36.6shows that z kis independent of ℤ k − 1, and thus Eq. 36.14can be simplified to

(36.15)

As a final note, we observe that the normalizing term in the denominator, known as the evidence, can be expressed in a more directly obvious form by de‐marginalizing about the state vector as follows:

(36.16)

Thus, we have presented the mathematical form of both the propagation ( Eq. 36.12) and update ( Eq. 36.16) actions on the pdf representing the state random vector.

For a specific class of problems (e.g. linear Gaussian systems), the above equations can be solved in closed form. In this case, the generalized process model ( Eq. 36.5) simplifies to

(36.17)

where is the state transition matrix from time k − 1 to k , and w k − 1is a zero‐mean, white Gaussian sequence with covariance Q k. Similarly, the generalized observation model 36.6simplifies to

(36.18)

where H kis the observation influence matrix at time k , and v kis a zero‐mean, white Gaussian sequence with covariance R k.

Thus, both the a priori and posterior pdfs can be represented as the following Gaussian densities, respectively:

(36.19)