Immersion and invariance observers with time-delayed output measurements

We address the problem of constructing (globally) convergent, (reduced-order) observers for general nonlinear systems when the output measurements are subject to constant time-delays. Immersion and invariance (I&I) techniques are used to derive a general tool for constructing I&I observers in the presence of time-delays. We show that an asymptotic estimate of the unknown states can be obtained by rendering attractive an appropriately selected invariant manifold in the extended state space. In this manuscript, the observer may play two different roles. On the one hand, it may be used to reconstruct a delayed version of the unmeasured state from measurements of the available delayed outputs. We show that if the time-delay is known, standard I&I techniques can be directly applied to estimate the delayed unmeasured states. In this case, we refer to the observer as a retarded immersion and invariance observer. On the other hand, the observer may be used to reconstruct both the delay-free unmeasured states and the delay-free output from measurements of the delayed output. In this case, we refer to it as an immersion and invariance predictor. Two examples with chaotic oscillators are presented to show the performance of the observers.