Design of Predictive Adaptive Observer for Systems with Simultaneous Input and State Delays with Uncertainty

In this paper, a delay compensation for a class of unknown systems with simultaneous state and input delays through the predictive adaptive observer is investigated. Despite the unknown term, an adaptive state observer is designed to have an exact prediction. The upper bounds of uncertainty with time varying delay are considered unknown, which do not require to be known for the design. Then, the second observer is introduced for simultaneous delay compensation in state and input that input delay can be optionally longer than the delay in the state. Besides, in this approach, the numerical solution is not implied for delay compensation of input in uncertainty system. Based on Lyapunov theory, it has been proved, the input delay in systems with the unknown term can compensate by means of designed observers, while the estimation error is small and bounded. For more details, we also show a numerical example that indicates the proposed approaches’ impact.