Input design and data-driven approaches based on convex optimization for fault diagnosis in linear systems

The complexity of automated systems has grown considerably during the past decades. This convolutes the observation of possible faults in these systems. If not being revealed timely, such faults can lead to catastrophic failures. As a result, there is a continuous interest in sophisticated fault diagnosis techniques. Since it is generally desired to diagnose faults in the earliest possible stages, computational challenges are imposed on the algorithms. Whereas the field of fault diagnosis comprises of a large variety of techniques in various categories, these computational challenges appear to emerge wide-ranging.

At the same time, convex optimization has developed as a valuable tool to solve a large variety of mathematical problems with computational efficiency. This computational efficiency is achieved by exploiting favorable structures of the problem. Depending on the specific problem, these structures vary in difficulty to be recognized or arranged. Moreover, some problems lead to a convex optimization problem naturally, while other problems first need some kind of relaxation or sequential process in order to employ convex optimization.

This thesis explores how convex optimization can be utilized in order to solve fault diagnosis problems with computational efficiency. The state-of-the-art is studied for multiple computationally challenging categories of fault diagnosis: online input design approaches, diagnosis of many concurrent faults, and data-driven approaches. First, online input design approaches facilitate fault diagnosis by computing discriminating input sequences during system operation. Since the input is calculated in real-time those approaches allow only limited computational effort, whereas adequate input determination typically appears to be nontrivial. In this contribution it is shown that an established upper bound on the error probability for linear candidate models with Gaussian noise is concave in the most challenging discrimination conditions. This finding allows to use sequential convex programs for online determination of a discriminating input with low computational effort.

The second contribution in this thesis regards the cantilever dynamics in high-speed atomic force microscopy. Due to the oscillatory behavior above the scrutinized sample, the cantilever typically has intermittent physical contact with the sample. This leads to a large number of (dynamically dependent) impulsive faults. Instead of performing an intractable explicit examination of all (combinations of) hypotheses, this contribution applies sparse estimation as a convex optimization method in order to diagnose these concurrent faults. In a simulation study, the resulting effect on the sample height reconstruction is discernible both qualitatively and quantitatively with respect to the conventional approach to sample height reconstruction in atomic force microscopy.

The third contribution introduces a novel problem formulation for model-free data-driven fault diagnosis. Instead of separate time periods for system identification and fault diagnosis in typical data-driven approaches, model-free data-driven fault diagnosis aims for the simultaneous system identification and fault diagnosis from one single data set. Whereas this is originally a non-convex bilinear problem, a proposed solution reformulates it as a convex optimization problem using a so-called lifting technique. Furthermore, online evaluation of this optimization problem is facilitated by a developed recursive implementation. The proposed methodology is tested both on simulation data and real-life flight test data.

By demonstrating the potential of convex optimization to a deliberate selection of fault diagnosis problems, this thesis serves as a source of inspiration for solving a wider variety of fault diagnosis problems efficiently. Furthermore, various elements related to convex optimization and its recursive implementation presented in this thesis have additional relevance to the general field of control science beyond fault diagnosis. Future applications of the presented methodology can arise for instance in the data-driven control in the presence of disturbances, or recursive blind deconvolution of real-time image sequences.