An SOS method for the design of continuous and discontinuous differentiators

ABSTRACT

Given a (differentiable) signal, it is an important task for many applications to estimate on line its derivatives. Some well-known algorithms to solve this problem include the (continuous) high-gain observers and (discontinuous) Levant's exact differentiators. With exception of the linear high-gain observers, a systematic design of the gains of nonlinear differentiators is, in general, a difficult task and an open research field. In this work, we propose a novel method for the gain-tuning of a family of homogeneous differentiators which estimate the time-derivatives of a signal in finite-time. We show that the stability analysis and the gain-tuning of such differentiators can be done under a unified Lyapunov function framework, and it is converted to a sum of squares problem, that can be solved using LMIs, much in the same spirit of the linear systems.

KEYWORDS:

• Finite-time differentiators
• Lyapunov functions
• homogeneous systems
• sum of squares problem
• LMIs

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. For definitions of strong and weak stability, see, for example, Filippov (1988).

2. For a more detailed study of SOS forms, see, for example, Marshall (2008)

Additional information

Funding

The financial support was from PAPIIT-UNAM (Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica) [project IN113617]; Fondo de Colaboración II-FI UNAM [project IISGBAS-100-2015]; CONACYT (Consejo Nacional de Ciencia y Tecnología) [project 241171], [CVU 267513], [CVU 371652].