# Robust Stability of Discrete-time Disturbance Observers: Understanding Interplay of Sampling, Model Uncertainty and Discrete-time Designs

## TL;DR

This paper analyzes the robust stability of discrete-time disturbance observers in sampled-data systems, emphasizing the effects of sampling zeros, model uncertainty, and discretization methods, and provides design guidelines for stability.

## Contribution

It introduces a root-location based framework for stability analysis that accounts for various discretization methods and large uncertainties, improving robustness understanding.

## Key findings

- Fast sampling ensures necessary and sufficient stability conditions.
- Sampling zeros can destabilize systems if not properly managed.
- Design guidelines for Q-filter and model selection enhance robustness.

## Abstract

In this paper, we address the problem of robust stability for uncertain sampled-data systems controlled by a discrete-time disturbance observer (DT-DOB). Unlike most of previous works that rely on the small-gain theorem, our approach is to investigate the location of the roots of the characteristic polynomial when the sampling is performed sufficiently fast. This approach provides a generalized framework for the stability analysis in the sense that (i) many popular discretization methods are taken into account; (ii) under fast sampling, the obtained robust stability condition is necessary and sufficient except in a degenerative case; and (iii) systems of arbitrary order and of large uncertainty can be dealt with. The relation between sampling zeros---discrete-time zeros that newly appear due to the sampling---and robust stability is highlighted, and it is explicitly revealed that the sampling zeros can hamper stability of the overall system when the Q-filter and/or the nominal model are carelessly selected in discrete time. Finally, a design guideline for the Q-filter and the nominal model in the discrete-time domain is proposed for robust stabilization under the sampling against the arbitrarily large (but bounded) parametric uncertainty of the plant.