  We study the problem of designing interval-valued observers that
simultaneously estimate the system state and learn an unknown dynamic model for
partially unknown nonlinear systems with dynamic unknown inputs and bounded
noise signals. Leveraging affine abstraction methods and the existence of
nonlinear decomposition functions, as well as applying our previously developed
data-driven function over-approximation/abstraction approach to over-estimate
the unknown dynamic model, our proposed observer recursively computes the
maximal and minimal elements of the estimate intervals that are proven to
contain the true augmented states. Then, using observed output/measurement
signals, the observer iteratively shrinks the intervals by eliminating
estimates that are not compatible with the measurements. Finally, given new
interval estimates, the observer updates the over-approximation of the unknown
model dynamics. Moreover, we provide sufficient conditions for uniform
boundedness of the sequence of estimate interval widths, i.e., stability of the
designed observer, in the form of tractable (mixed-)integer programs with
finitely countable feasible sets.
