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In this draft, fault diagnosis in nonlinear dynamic systems is addressed. The
objective of this work is to establish a framework, in which not only
model-based but also data-driven and machine learning based fault diagnosis
strategies can be uniformly handled. Instead of the well-established
input-output and the associated state space models, stable image and kernel
representations are adopted in our work as the basic process model forms. Based
on it, the nominal system dynamics can then be modelled as a lower-dimensional
manifold embedded in the process data space. To achieve a reliable fault
detection as a classification problem, projection technique is a capable tool.
For nonlinear dynamic systems, we propose to construct projection systems in
the well-established framework of Hamiltonian systems and by means of the
normalised image and kernel representations. For nonlinear dynamic systems,
process data form a non-Euclidean space. Consequently, the norm-based distance
defined in Hilbert space is not suitable to measure the distance from a data
vector to the manifold of the nominal dynamics. To deal with this issue, we
propose to use a Bregman divergence, a measure of difference between two points
in a space, as a solution. Moreover, for our purpose of achieving a
performance-oriented fault detection, the Bregman divergences adopted in our
work are defined by Hamiltonian functions. This scheme not only enables to
realise the performance-oriented fault detection, but also uncovers the
information geometric aspect of our work. The last part of our work is devoted
to the kernel representation based fault detection and uncertainty estimation
that can be equivalently used for fault estimation. It is demonstrated that the
projection onto the manifold of uncertainty data, together with the
correspondingly defined Bregman divergence, is also capable for fault
detection.