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

(36.20)

where represents a Gaussian density with μ mean and Λ covariance. In addition, the plus and minus superscripts are used to express an a priori or a posteriori quantity, respectively. Substituting the linear process model ( Eq. 36.17) into our propagation relationship ( Eq. 36.12) results in the linear Kalman filter propagation equations

(36.21)

(36.22)

Furthermore, substituting the linear observation model ( Eq. 36.18) into our update relationship ( Eq. 36.16) results in the linear Kalman filter update equations:

(36.23)

(36.24)

where z kis the realized measurement observation, and K kis the Kalman gain at time k :

(36.25)

and S kis the residual covariance matrix, given by

(36.26)

In many cases, systems can be accurately represented by linear Gaussian models. Unfortunately, there are a number of systems where these models are not adequate. This motivates the development of various algorithms that attempt to solve these equations for various classes of problems.

In the next section, we will present the fundamental concepts which will be used to derive various recursive nonlinear estimators.

In the previous sections ( Section 36.2.1), we have developed the generalized theory for recursive estimation problems. The theory is based on the fundamental need to determine the pdf of the state vector at an epoch of interest, conditioned on the observations up to, and including, the current epoch. Complete knowledge of the conditional state pdf represents maximum possible knowledge of the system. This is, in fact, the normal state of affairs for Gaussian systems, as the pdf can be completely described by a mean and covariance.

Consider a Gaussian random vector xwith mean and covariance and P x, respectively. As mentioned previously, for Gaussian densities, these two parameters are sufficient to completely describe the full pdf of the random vector.

Next consider a linear transformation from xto ywhich is governed by the transformation matrix H. The resulting equation for yis given by

(36.27)

In this case, the transformed random vector, y, can be shown to be a Gaussian random vector with mean and covariance

(36.28)

(36.29)

This preservation of Gaussian nature when transformed via linear operations is an important property of Gaussian densities that makes the linear Kalman filter relatively simple to implement.

Now consider a generalized nonlinear transformation

(36.30)

In this case, the density of ycan become difficult to calculate exactly. While we will address this issue in more detail later in the chapter, generally speaking, the resulting density function is clearly non‐Gaussian, thus limiting the performance of the linear Kalman filter algorithm. Nonlinear estimators attempt to maintain a higher‐fidelity estimate of the overall density function as it transforms over time.

In the next section, we present our first class of estimators designed to support systems with non‐Gaussian pdfs.

One approach for modeling systems with non‐Gaussian pdfs is the use of composite random variables expressed as a sum of Gaussian random variables. The generalized Gaussian sum can be expressed as

(36.31)

where w [j]is a scalar weighting factor, y jis a Gaussian random variable with mean , and covariance . These individually weighted Gaussian random variables can represent the overall distribution of the state vector. An example of a density function created using a sum of Gaussian random variables is shown in Figure 36.1.

One implementation of the Gaussian sum filtering approach is known as multiple model adaptive estimation (MMAE). The MMAE filter uses a weighted Gaussian sum to address the situation where unknown or uncertain parameters exist within the system model. Some examples of these types of situations include modeling discrete failure modes, unknown structural parameters, or processes with multiple discrete modes of operation (e.g. “jump” processes).

Figure 36.1 Gaussian sum illustration. The random variable x sumis represented by a weighted sum of three individual Gaussian densities. In this example, x sum= 0.25 x 1+ 0.5 x 2+ 0.25 x 3.

Consider our standard linear Gaussian process and observation models, repeated from Eqs. 36.17and 36.18for clarity: