Second-order sliding-mode differentiators: an experimental comparative analysis using Van der Pol oscillator

ABSTRACT

This article provides a comparative study of four different second-order sliding-mode (SOSM) differentiators proposed in the literature, namely, standard higher-order sliding-mode (HOSM) differentiator, non-homogeneous HOSM differentiator, uniform robust exact differentiator and hybrid fixed-time differentiator. Based on sliding-mode principles, these differentiators can provide robust exact differentiation with finite/fixed-time convergence. First, a comprehensive summary of the different methods is provided. Then, the differentiators are applied experimentally to estimate the states of a Van der Pol oscillator. Through experiments, it is shown that the different differentiators outperformed a Kalman-like observer, high-gain differentiator and extended Kalman filter. Finally, some suggestions are provided on the selection of SOSM differentiators for various applications.

KEYWORDS:

• Sliding-mode differentiation
• Van der Pol oscillator
• real-time estimation

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. For details about the ADC performance of dSPACE 1104 board, please consult (DS1104, 2006, p. 159).

2. By choosing sufficiently small ρ, the maximum error magnitude may be reduced. However, this will increase the convergence time significantly.

3. When, the first time, the estimation error entered the region $|x_2-{\widehat{x}}_2|\le 0.1$ and stayed there for a while, we considered that time as the convergence time. For the purpose of convergence time estimation, Figure 6 was used. For some differentiators/observers, the estimation error left this region later on. This is quite natural because the oscillator took some time to reach the stable limit cycle.

Additional information

Funding

Consejo Nacional de Ciencia y Tecnología [grant number 270504].