# Sampled-Data Observers for Delay Systems and Hyperbolic PDE-ODE Loops

## TL;DR

This paper develops a method for designing sampled-data observers and stabilizers for systems with delays and PDE-ODE loops, ensuring stability with appropriate sampling periods using Lyapunov and small-gain techniques.

## Contribution

It introduces a novel observer modification with an inter-sample predictor that maintains exponential stability despite sampling delays.

## Key findings

- Robust exponential stability is achieved under certain sampling period constraints.
- The approach applies to delay systems, transport PDEs, and PDE-ODE interconnections.
- Validated through examples including chemical reactor stabilization.

## Abstract

This paper studies the problem of designing sampled-data observers and observer-based, sampled-data, output feedback stabilizers for systems with both discrete and distributed, state and output time-delays. The obtained results can be applied to time delay systems of strict-feedback structure, transport Partial Differential Equations (PDEs) with nonlocal terms, and feedback interconnections of Ordinary Differential Equations with a transport PDE. The proposed design approach consists in exploiting an existing observer, which features robust exponential convergence of the error when continuous-time output measurements are available. The observer is then modified, mainly by adding an inter-sample output predictor, to compensate for the effect of data-sampling. Using Lyapunov stability tools and small-gain analysis, we show that robust exponential stability of the error is preserved, provided the sampling period is not too large. The general result is illustrated with different examples including state observation and output-feedback stabilization of a chemical reactor.

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Source: https://tomesphere.com/paper/1907.06691