Adaptive Observer for Systems with Distributed Output Delay

Abstract We are considering the problem of sampled-data observer design for nonlinear time-varying delay systems that are state- and parameter-affine. The novelty is that the system is subject to both distributed delay and parameter uncertainty. A Kalman-like observer is developed to deal with both state and parameter estimation. Its main components are: (i) a time-varying-gain state-estimator involving both output and parameter rate injections; (ii) a distributed-nature adaptive output-predictor that compensates for delay and output sampling delay; (iii) an optimized parameter-estimator that copes with parameter uncertainty, optimization is intended in the sense that we make use of all available information, unlike similar existing estimators. The resulting observer is shown to be exponentially convergent, for small delays and sampling intervals, provided the input signal is sufficiently exciting. The analysis is performed using a Lyapunov-Krasovskii functional, Halanay’s lemma, Wirtinger’s inequality and other tools.