State Models for Stochastic Processes (GPS) Part 2

Scalar Gauss-Markov Process

The scalar Gauss-Markov process refers to the special case of eqn. (4.79) where the state, input, and output are each scalar variables:

where w is Gaussian white noise with PSD denoted by a2. In eqn. (4.90), the parameter t is the correlation time. This process has already been used in Examples 4.14 and 4.15.

Eqn. (4.90) is written in its most general form. The PSD of y is

which shows that once the correlation time t is determined, the value of determines the product of the parameters H, G, andOften, one or two of these three parameters is arbitrarily set to one with the remaining parameter values selected to achieve the desired value at

Compound Augmented States

The examples of the previous sections each contained a single type of error. In more realistic situations, several forms of error may be present. This section presents an example involving compound error models augmented to simplified INS error equations.

Example 4.18 The ideal dynamics of a one dimensional INS implemented in inertial space are

Two sensors are available. The first sensor measures position. The second sensor measures acceleration.

The position measurement model is

whererepresents Gaussian white noise andis the scalar Gauss-Markov process

The implemented INS is described as

where the accelerometer model is

whereis a Gaussian random-constant scale-factor error,is a constant plus random walk Gaussian bias error, and va is Gaussian white noise. The predicted output at any time is calculated as

The differential equations for the error variables are found by subtracting eqns. (4.94-4.95) from eqns. (4.91-4.92)

with each error term defined asThe model for the residual measurementis defined by subtracting eqn. (4.96) from eqn. (4.93)

The augmented error state equations are then defined to be

These error equations have the form of eqn. (4.79) with Gaussian white noise inputs.

This example has considered a hypothetical one dimensional INS with only two sensors. The state of the original INS was two and three error states were appended. In realistic navigation systems in a three dimensional world with many more sensed quantities, it should be clear that the dimension of the state of the error model can become quite large. A

When designing a system that will be implemented in a real-time application, there is usually a tradeoff required between reasonable cost and computation time and the desire for accurate modeling. Even in the above single axis example, the dimension of the augmented error state vector Sx is large (i.e., 5) relative to the dimension of the original state x = [p, of the INS (i.e., 2), and several more error states could be included in the quest for modeling accuracy. In realistic applications, the dimension of the augmented state vector is potentially quite large, often too large for a realtime system. Therefore, the design may result in two models. The most complex model that accounts for all error states considered to be significant is referred to as the truth model. A simplified Design model may be constructed from the truth model by eliminating or combining certain state variables. The art is to develop a design model small enough to allow its use in practical realtime implementations without paying a significant performance penalty in terms of state estimation accuracy.

Time-propagation of the Mean

From eqn. (4.65), if the mean of the state vector is known at some time k0 and

Intuitively, this formula states that since nothing is known a priori about the specific realization of the process noise for a given experiment, the mean of the state is propagated according to the state model:

The equation foris found by analysis similar to that for

Time-propagation of the Variance

The discrete-time error state covariance matrix is defined as

where

The state covariance at time k + 1

can be simplified using eqns. (4.65) and (4.98) to determine eqn. (4.99) for propagating the state error covariance through time:

Note that eqns. (4.97) and (4.99) are valid for any zero mean noise process satisfying the assumption of eqn. (4.62). No other assumptions were used in the derivations. If wk and vk happen to be zero mean Gaussian processes, then xk is also a Gaussian stochastic process with mean and variance given by eqns. (4.97) and (4.99).

By limiting arguments,the covariance propagation for the continuous-time system

is described by

Example 4.19 Consider the scalar Gauss-Markov process with

withwhere the PSD of the white noise process w isLetting

Eqn. (4.101) has the solution

which can be verified by direct substitution. In steady state, which is useful in applications involving scalar Gauss-Markov processes when the steady-state covarianceand the correlation timeare known andis to be determined.