Kalman Filter-Based Fault Detector
In control engineering, fault detection is a process to detect the occurrence of faults, which are unexpected or unwanted or intolerable behaviours of the system by using run-time measurements. This process in general generates alarm signals that indicate the early occurrence of faults or a trace that faults may have existed inside the system. Several research papers have been written on this topic, see e.g. [KFB1] and reference therein.
In particular, one of the pioneering works for Failure Detection and Identification is the one in [KBF2], where the problem of detecting and identifying control system component failures in linear time-invariant systems has been addressed. are given in the paper to detect and uniquely identify a component failure both in the case when system components can fail simultaneously, and in the case when they fail only one at a time. However, this approach, and the ones derived within this concept, suffer of a main issue: they assume that a mathematical model of the system is available.
Nowadays, it is well known that deriving a mathematical model for complex systems can be cost- and time-prohibitive. Thus, the research community started to investigate the so-called data-driven methods, where the information, and even a model, of the system are extracted from previously collected historical data. In this respect, there are various methods which can be deployed to track faults. One among them is the approach based on the Principal Component Analysis (PCA) [KBF3], also enforced with the Kalman filter, presented in [KBF4] and [KBF5].
In [KFB4] the authors presented an output-only damage detection technique based on time-series models. They applied the PCA algorithm to the historical dataset in order to reduce the data size. Then, they fitted the data with an Auto Regressive (AR) model and calculated the Probability Density Function (PDF) of the damage feature, which was obtained by taking the ratio between the variances of prediction errors considering references and healthy dataset. A system was finally declared as faulty if the distance between the peak of the corresponding PDF and the healthy PDF was larger than the tolerance bound.
Such an approach has been then modified in [KFB5] by introducing a novel form of damage detection algorithm based on Recursive Principal Component Analysis (RPCA) together with the Time Varying Auto Regressive Model (TVAR), also leveraging the Kalman filter. In particular, they first used PCA on the dataset to project the dataset to its own orthogonal space, and then generated time series models based on the responses and fitted a TVAR model into the projected dataset.
- Fault detection performanceSimplicity in the implementation
- Good model accuracy and good performance in terms of faults detected when the system exhibit a linear behavior
- PCA-based approach works well only with linear systems
- Lack of a bridge between the model-based approaches and Machine Learning to define data-driven fault detectors
- [KFB1] Ding, S. X. (2008). Model-based fault diagnosis techniques: design schemes, algorithms, and tools. Springer Science & Business Media.
- [KBF2] Massoumnia, M. A., Verghese, G. C., & Willsky A. S. (1989). Failure Detection and Identification. IEEE Transactions on Automatic Control, 34(3), 316 –321.
- [KFB3] Jolliffe, I. T. (1986). Principal component analysis. Springer, New York, NY.
- [KFB4] Lakshmi, K., & Rama Mohan Rao, A. (2014). A robust damage-detection technique with environmental variability combining time-series models with principal components. Nondestructive Testing and Evaluation, 29(4), 357-376.
- [KFB5] Bhowmik, B., Hazra, B., & Pakrashi, V. (2018). Real time damage detection using recursive principal components and time varying auto-regressive modeling. Mechanical Systems and Signal Processing, 101, 549-574.