  This paper proposes a nonlinear estimator for the robust reconstruction of
process and sensor faults for a class of uncertain nonlinear systems. The
proposed fault estimation method augments the system dynamics with an
ultra-local (in time) internal state-space representation (a finite chain of
integrators) of the fault vector. Next, a nonlinear state observer is designed
based on the known parts of the augmented dynamics. This nonlinear filter
(observer) reconstructs the fault signal as well as the states of the augmented
system. We provide sufficient conditions that guarantee stability of the
estimation error dynamics: firstly, asymptotic stability (i.e., exact fault
estimation) in the absence of perturbations induced by the fault model mismatch
(mismatch between internal ultra-local model for the fault and the actual fault
dynamics), uncertainty, external disturbances, and measurement noise and,
secondly, Input-to-State Stability (ISS) of the estimation error dynamics is
guaranteed in the presence of these perturbations. In addition, to support
performance-based estimator design, we provide Linear Matrix Inequality (LMI)
conditions for $\mathcal{L}_2$-gain and $\mathcal{L}_2-\mathcal{L}_\infty$
induced norm and cast the synthesis of the estimator gains as a semi-definite
program where the effect of model mismatch and external disturbances on the
fault estimation error is minimized in the sense of $\mathcal{L}_2$-gain, for
an acceptable $\mathcal{L}_2-\mathcal{L}_\infty$ induced norm with respect to
measurement noise. The latter result facilitates a design that explicitly
addresses the performance trade-off between noise sensitivity and robustness
against model mismatch and external disturbances. Finally, numerical results
for a benchmark system illustrate the performance of the proposed
methodologies.
