Adaptive observer with finite-time convergence for a nonlinear nonstationary system with parametric uncertainty and output delay

The problem of estimating an unmeasurable state vector in finite time is solved for a class of nonlinear non‑stationary dynamic systems with parametric uncertainties and output delay. The class of systems under consideration is characterized by a unit relative degree and the presence of additive nonlinearities that exhibit a nonlinear dependence on the measured output signal and a linear dependence on the vector of unknown parameters and the state vector. The non‑stationarity of the system is determined by the explicit time dependence of the system’s dynamics matrix. The magnitude of the delay is assumed to be known. An approach to solving the problem is proposed based on a three‑stage procedure. In the first stage, an input‑invariant observer is constructed, which provides state estimation with the delay and finite convergence time, generating auxiliary estimates. The estimation error model is reduced to a linear regression; to determine its parameters, a parametric identification method with finite convergence time is used, based on dynamic expansion and mixing of the regressor. In the second stage, an algorithm for estimating the vector of unknown parameters in finite time is synthesized based on the obtained estimates. To achieve this, the original system is transformed into a linear regression model, to which an identification method with finite convergence time is applied. At the final stage, a filtering procedure is introduced, reducing the problem of estimating the state vector of the original nonlinear system to the identification of linear regression parameters, which is also carried out using the method of dynamic expansion and mixing of the regressor. The obtained theoretical results ensure that the estimates of the state vector and unknown parameters converge to their true values in finite time. The effectiveness of the proposed method is confirmed by presented results of computer simulation.