Quadratic boundedness is a notion of stability that
is adopted to investigate the design of observers for dynamic
systems subject to bounded disturbances. We will show how
to exploit such observers for the purpose of fault detection.
Toward this end, first of all we present the naive application of
quadratic boundedness to construct state observers for linear
time-invariant systems with state augmentation, i.e., where
additional variables may be introduced to account for the
occurrence of a fault. Then a Luenberger observer is designed
to estimate the augmented state variable of the system in such
a way to detect the fault by using a convenient threshold
selection. Finally, such an approach is extended to piecewise
affine systems by presenting a hybrid Luenberger observer and
its related design based on quadratic boundedness. The design
of all the observers for both linear time-invariant and piecewise
affine systems can be done by using linear matrix inequalities.
Simulation results are provided to show the effectiveness of the
proposed approach.
