Continuously-implemented sliding-mode adaptive unknown-input observers under noisy measurements

We propose an estimator for nonlinear systems with unmatched unknown inputs and under measurement noise. The estimator design is based on the combination of observer design for descriptor systems, sliding-modes theory and adaptive control. The estimation of the measurement noise is achieved thanks to the transformation of the original system into a singular form where the measurement noise makes part of the augmented state. Two adaptive parameters are updated online, one to compensate for the unknown bounds on the states, the unkown inputs and the measurement noise and a second one to compensate for the effect of the nonlinearities. To join robust state estimation and unknown-inputs reconstruction, our approach borrows inpiration from sliding-mode theory however, all signals are comtinuously implemented. We demonstrate that both state and unknown-inputs estimation are achieved up to arbitrarily small tolerance. The utility of our theoretical results is illustrated through simulation case-studies.