As motivation for the study of stochastic systems, consider the error dynamics for a single channel of a tangent plane INS, as depicted in Figure 4.4 and the following equation
where
represents error in the navigation state. The error model is driven by two sources of error: accelerometer error ea and gyro error eg. These errors originate in the sensors, but appear in the error model as process noise. The error at each source may take several forms: white noise, correlated (colored) noise, bias error, scale factor error, etc. After any deterministic error sources have been compensated or modeled, the remaining instrument errors will be modeled as random inputs. The use of random variables to model the error sources is reasonable, since although the analyst may not be able to specify the value of the various error sources at some future time, it is usually possible to specify the dynamic nature and statistics of the error sources as a function of time. It is also reasonable to restrict our attention to linear stochastic systems, since the primary focus of the stochastic analysis in navigation systems will be on linearized error dynamics.
Figure 4.4: Block diagram for single channel INS error model.
The primary objectives for this section are to present: (1) how to account for stochastic sources of error in state space dynamic system modeling; and, (2) how the mean and covariance of system errors propagate in linear stochastic systems. This section presents various results for linear, state space systems with random inputs that will be used throughout the remainder of the topic.
Standard Model
The model for a finite-dimensional linear continuous-time system with stochastic inputs can be represented as
where w(t) and v(t) are random variables. The random variable w is called the process noise. The random variable v is called the measurement noise. At each time, x(t) and y(t) are also random variables. The designation random variable implies that although the value of the random variable at any time is not completely predictable, the statistics of the random variable may be known. Assuming that the model is constructed so that v(t) and w(t) are white, the mean and covariance of the random variables w(t) and v(t) will be denoted
In the analysis that follows, it will often be accurate (and convenient) to assume that the process and measurement noise are independent of the current and previous state
and independent of each other
In most applications, the random processes w(t) and v(t) will be assumed to be Gaussian.
The linear discrete-time model is
The mean and covariance of the random variables
and
will be denoted
Assumptions corresponding to eqns. (4.62-4.64) also apply to the discrete-time model.
In the analysis to follow, the time argument of the signals will typically be dropped, to simplify the notation.
Stochastic Systems and State Augmentation
Navigation system error analysis will often result in equations of the form
where
represent instrumentation error signals and
represents the error in the nominal navigation state. This and subsequent sections of this topic will present a set of techniques which will allow the error signals
and
to be modeled as the outputs of linear dynamic systems
and
with the noise processes
being accurately modeled as white noise processes. By the process of state augmentation, equations (4.70-4.75) can be combined into the state space error model
which is in the form of eqn. (4.57) and driven only by white noise processes.
In these equations, the augmented state is defined as
The measurements of the augmented system are modeled as
which are corrupted only by additive white noise.
For the state augmented model to be an accurate characterization of the actual system, the state space parameters
corresponding to the appended error models and the statistics of the driving noise processes must be accurately specified. Detailed examples of augmented state models are presented in Sections 4.6.3 and 4.9. Section 4.6.3 also discusses several basic building blocks of the state augmentation process.
Gauss-Markov Processes
For the finite dimensional state space system
where w(t) and v(t) are stochastic processes, if both w and v are Gaussian random processes, then the system is an example of a Gauss-Markov process . Since any linear operation performed on a Gaussian random variable results in a Gaussian random variable, the state x(t) and system output y(t) will be Gaussian random variables. This is a very beneficial property as the Normal distribution is completely described by two parameters (i.e., the mean and covariance) which are straightforward to propagate through time, as is described in Section 4.6.4 and 4.6.5. The purpose of this section is to present several specific types of Gauss-Markov processes that are useful for error modeling.
Random Constants
Some portions of instrumentation error (e.g., scale factor) can be accurately represented as constant (but unknown) random variables. If some portion of the constant error is known, then it can be compensated for and the remaining error can be modeled as an unknown constant. An unknown constant is modeled as
This model states that the variable x is not changing and has a known initial variance
Example 4.16 An acceleration measurement
is assumed to be corrupted by an unknown constant bias
The bias is specified to be zero mean with an initial error variance of
1. What
is an appropriate state space model for the accelerometer output
2. What is the appropriate state space model when this accelerometer error model is augmented to the tangent plane single channel error model of eqn. (3.91)? Assume that the constant bias is the only form of accelerometer error and that
Since the bias is assumed to be constant, an appropriate model is
with initial condition
Note that there is no process noise in the bias dynamic equation. Since no other forms of error are being modeled,
Therefore, the augmented single channel error model becomes
Note that for eqn. (4.82), if either a position or velocity measurement were available, an observability analysis will show that the system is not observable. The tilt and accelerometer bias states cannot be distinguished. See Exercise 4.15. A
Brownian Motion (Random Walk) Processes
The Gauss-Markov process x(t) defined by
with
is called a Brownian motion or random walk process. The mean of x can be calculated as
By the definition of variance in eqn. (4.22), the covariance function for x can be calculated as
which yields
In navigation modeling, it is common to integrate the output of a sensor to determine particular navigation quantities. Examples are integrating an accelerometer output to determine velocity or integrating an angular rate to determine an angle, as in Example 4.2. If it is accurate to consider the sensor error as white random noise, then the resulting error equations will result in a random walk model.
As with white noise, the units of the random walk process often cause confusion. Note that if x is measured in degrees, then varx (t) has units of deg2. Therefore, the units of
which makes sense given that
is the PSD of a white angular rate noise process.
One often quoted measure of sensor accuracy is the random walk parameter. For example, a certain gyro might list its random walk parameter to be 4.0deg
By eqn. (4.84), the random walk parameter for a
sensor quantifies the rate of growth of the integrated (properly compensated) sensor output as a function of time. The
in the denominator of the specification reflects that the standard deviation and variance of the random walk variable grow with the square root of time and linearly with time, respectively. When the specification states that the angle random walk parameter is .
then
For a random walk process, the state space model is
where
The transfer function corresponding to eqns. (4.83) and (4.86) is
so by eqn. (4.49), the PSD of x is
Example 4.17 An accelerometer output is known to be in error by an unknown slowly time-varying bias
Assume that the ‘turn-on’ bias is accurately known and accounted for in the accelerometer calibration; and, that it is accurate to model the residual time-varying bias error as a random walk:
where
represents Gaussian white noise and
is the PSD of
The parameter
is specified by the manufacturer.
The random walk bias model to a single channel of the inertial frame INS error model:
yields the following augmented state model
This example has made the unrealistic assumption that the turn-on bias is exactly known so that a proper random walk model could be used. In a realistic situation where the initial bias is not perfectly known, a constant plus random walk error model would be identical to the equations shown above with Pb (0) specified to account for the variance in the initial bias estimate.
