Statistical Fault Detection with Applications to IMU Disturbances

This thesis deals with the problem of detecting faults in an environment where the measurements are affected by additive noise. To do this, a residual sensitive to faults is derived and statistical methods are used to distinguish faults from noise. Standard methods for fault detection compare a batch of data with a model of the system using the generalized likelihood ratio. Careful treatment of the initial state of the model is quite important, in particular for short batch sizes. One method to handle this is the parity-space method which solves the problem by removing the influence of the initial state using a projection. In this thesis, the case where prior knowledge about the initial state is available is treated. This can be obtained for example from a Kalman filter. Combining the prior estimate with a minimum variance estimate from the data batch results in a smoothed estimate. The influence of the estimated initial state is then removed. It is also shown that removing the influence of the initial state by an estimate from the data batch will result in the parity-space method. To model slowly changing faults, an efficient parameterization using Chebyshev polynomials is given. The methods described above have been applied to an Inertial Measurement Unit, IMU. The IMU usually consists of accelerometers and gyroscopes, but has in this work been extended with a magnetometer. Traditionally, the IMU has been used to estimate position and orientation of airplanes, missiles etc. Recently, the size and cost has decreased making it possible to use IMU:s for applications such as augmented reality and body motion analysis. Since a magnetometer is very sensitive to disturbances from metal, such disturbances have to be detected. Detection of the disturbances makes compensation possible. Another topic covered is the fundamental question of observability for fault inputs. Given a fixed or linearly growing fault, conditions for observability are given. The measurements from the IMU show that the noise distribution of the sensors can be well approximated with white Gaussian noise. This gives good correspondence between practical and theoretical results when the sensor is kept at rest. The disturbances for the IMU can be approximated using smooth functions with respect to time. Low rank parameterizations can therefore be used to describe the disturbances. The results show that the use of smoothing to obtain the initial state estimate and parameterization of the disturbances improves the detection performance drastically.