  We address the problem of designing simultaneous input and state interval
observers for Lipschitz continuous nonlinear systems with rank-deficient
feedthrough, unknown inputs and bounded noise signals. Benefiting from the
existence of nonlinear decomposition functions and affine abstractions, our
proposed observer recursively computes the maximal and minimal elements of the
estimate intervals that are proven to contain the true states and unknown
inputs. Moreover, we provide necessary and sufficient conditions for the
existence and sufficient conditions for the stability (i.e., uniform
boundedness of the sequence of estimate interval widths) of the designed
observer, and show that the input interval estimates are tight, given the state
intervals and decomposition functions.
