A simultaneous input and state interval observer is presented for Lipschitz continuous nonlinear systems with unknown inputs and bounded noise signals for the case when the direct feedthrough matrix has full column rank. The observer leverages the existence of bounding decomposition functions for mixed monotone mappings to recursively compute the maximal and minimal elements of the estimate intervals that are compatible with output/measurement signals, and are proven to contain the true state and unknown input. Furthermore, we derive a Lipschitz-like property for decomposition functions, which provides several sufficient conditions for stability of the designed observer and boundedness of the sequence of estimate interval widths. Finally, the effectiveness of our approach is demonstrated using an illustrative example.
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