The use of connected devices or humans-in-the-loop for measuring outputs of dynamical systems inevitably produces time-varying measurement delays. These delays can lead to instability or severe degradation of system performance. In this paper, linear matrix inequality-based sufficient conditions are proposed for the design of state and unknown input observers based on delayed measurements for a class of nonlinear systems, where the nonlinearities are characterized by incremental multiplier matrices. The proposed observer is guaranteed to perform at specified operational levels in the presence of unknown exogenous inputs acting on the states and measurement outputs. Sufficient conditions are also provided for the estimation of these unknown inputs to a specified degree of accuracy. The potential of the proposed approach is illustrated via estimation of enzyme kinetics. (C) 2018 Elsevier Ltd. All rights reserved.