An arbitrary-order differentiator design paradigm with adaptive gains

ABSTRACT

Higher order sliding-mode differentiators have received a great deal of attention in the literature. For the case of reconstructing the first derivative, theoretical convergence conditions for the differentiator are available from which differentiator parameters may be selected. For the case of higher order derivatives, some parameter settings have been suggested for differentiators of certain order but there is no tuning algorithm available to determine convergent parameters for differentiators of arbitrary order. Whilst recognising the strong theoretical properties of sliding-mode differentiators, practitioners report difficulties in achieving wide envelope performance from a single set of differentiator parameters. This paper proposes a constructive design paradigm to generate differentiator parameters which is seen to provide a natural framework to facilitate simple online adaptation of the chosen gains. Simulation experiments as well as experimental results are presented to demonstrate the proposed approach.

KEYWORDS:

• Arbitrary-order differentiator
• sliding mode
• adaptive gains

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Note that in this article complex roots are not considered.

2. A constant step size of 10⁻⁴ s was used during the simulation studies presented in this article. The initial values were selected as z ₀(0) = 5 and z ₁(0) = 0.72.

3. The dependency of the inductance L on the ball position x ₁ is neglected in this model.

4. This Lyapunov function does not satisfy the properties typically required of Lyapunov functions. A detailed discussion of this is presented in Moreno and Osorio (2012) and references therein.