Non-asymptotic numerical differentiation: a kernel-based approach

ABSTRACT

The derivative estimation problem is addressed in this paper by using Volterra integral operators which allow to obtain the estimates of the time derivatives with fast convergence rate. A deadbeat state observer is used to provide the estimates of the derivatives with a given fixed-time convergence. The estimation bias caused by modelling error is characterised herein as well as the ISS property of the estimation error with respect to the measurement perturbation. A number of numerical examples are carried out to show the effectiveness of the proposed differentiator also including comparisons with some existing methods.

KEYWORDS:

• Linear integral operators
• numerical differentiation
• non-asymptotic identification
• state estimation
• Fredholm–Volterra integral equations

Acknowledgment

This work has been partially supported by the European Union's Horizon 2020 Research and Innovation Programme under grant agreement No 739551 (KIOS CoE).

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Peng Li http://orcid.org/0000-0002-8217-1326

Gilberto Pin http://orcid.org/0000-0001-7263-3681

Giuseppe Fedele http://orcid.org/0000-0003-0273-780X

Thomas Parisini http://orcid.org/0000-0001-5396-9665

Notes

1 For the sake of clarity, we eliminate the time-dependence of the time-varying matrix $\mathrm{\Gamma }\left(t\right)$, the kernels $K(t,t)$ and their derivatives $K_h^{\left(i\right)}(t,t),\mathrm{\forall }t\in \{1,2\},\mathrm{\forall }h\in \{1,2\}$.

2 For notation simplicity, we eliminate the time-dependence of $e_y\left(t\right)$ and $E\left(t\right)$.