Predictor-Based Output Feedback for Nonlinear Delay Systems

We provide two solutions to the heretofore open problem of stabilization of systems with arbitrarily long delays at the input and output of a nonlinear system using output feedback only. Both of our solutions are global, employ the predictor approach over the period that combines the input and output delays, address nonlinear systems with sampled measurements and with control applied using a zero-order hold, and require that the sampling/holding periods be sufficiently short, though not necessarily constant. Our first approach considers general nonlinear systems for which the solution map is available explicitly and whose one-sample-period predictor-based discrete-time model allows state reconstruction, in a finite number of steps, from the past values of inputs and output measurements. Our second approach considers a class of globally Lipschitz strict-feedback systems with disturbances and employs an appropriately constructed successive approximation of the predictor map, a high-gain sampled-data observer, and a linear stabilizing feedback for the delay-free system. We specialize the second approach to linear systems, where the predictor is available explicitly. We provide two illustrative examples-one analytical for the first approach and one numerical for the second approach.

💡 Research Summary

The paper tackles the long‑standing open problem of stabilizing nonlinear systems that suffer from arbitrarily long input and output delays when only sampled output measurements are available. The authors propose two global predictor‑based output‑feedback designs that work under a zero‑order‑hold implementation of the control input and require only that the sampling/holding period be sufficiently small (the period need not be constant).

First Approach – Explicit Solution Map
For systems whose solution map φ(t,u) can be expressed analytically, the authors construct a one‑sample‑period discrete‑time predictor model F. Under a complete observability assumption on this discrete model (Hypothesis H3), the past input and output data allow reconstruction of the delayed state x(t‑r) in a finite number of steps. The reconstructed state is then fed into a continuous‑time predictor operator P that estimates the future state x(t+τ). Finally, any globally stabilizing feedback law k(·) designed for the delay‑free plant is applied to the predicted state. This three‑stage scheme (observer → predictor → feedback) yields global asymptotic stability; if the delay‑free plant admits dead‑beat stabilization, the same dead‑beat performance is achieved with a single sampling interval. An analytical example with a three‑dimensional feed‑forward system demonstrates the method.

Second Approach – Globally Lipschitz Strict‑Feedback Systems
For a broader class of systems written in strict‑feedback form \