Sampled-data adaptive observer for state and parameter affine systems with distributed and discrete output delays

We are considering the problem of sampled-data observer design for nonlinear time-varying systems that are state-affine. The novelty lies in that both distributed and discrete delays are considered in the output equation. The latter is also subject to a parameter uncertainty of nonaffine nature due to output sampling. Interestingly, all system delays are modeled using a single distributed representation, involving a distribution function, allowing thus for a unified treatment of delays. A Kalman-like observer is developed to cope with both state and parameter uncertainty. 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 compensate for all delay effects, including that of output sampling; (iii) an parameter-estimator that is optimized in the sense that it makes use of all available information. 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.