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.
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