Dirty derivatives for output feedback stabilization

TL;DR

This paper proves the stability of linear control systems using dirty derivatives instead of observers, highlighting their noise-attenuation benefits and extending their application to adaptive control with unknown gains.

Contribution

It provides the first Lyapunov-based stability proof for dirty derivatives replacing derivatives of different orders in LTI systems and demonstrates their use in adaptive output feedback control.

Findings

Stability is guaranteed when using dirty derivatives in LTI systems.

Dirty derivatives effectively replace observers for output feedback stabilization.

Application to adaptive control with unknown gains is feasible.

Abstract

Dirty derivatives are routinely used in industrial settings, particularly in the implementation of the derivative term in PID control, and are especially appealing due to their noise-attenuation and model-free characteristics. In this paper, we provide a Lyapunov-based proof for the stability of linear time-invariant control systems in controller canonical form when utilizing dirty derivatives in place of observers for the purpose of output feedback. This is, to the best of the authors' knowledge, the first time that stability proofs are provided for the use of dirty derivatives in lieu of derivatives of different orders. In the spirit of adaptive control, we also show how dirty derivatives can be used for output feedback control when the control gain is unknown.