Minimizing the Homogeneous $\mathcal{L}_2$-Gain of Homogeneous Differentiators
arxiv(2023)
摘要
The differentiation of noisy signals using the family of homogeneous
differentiators is considered. It includes the high-gain (linear) as well as
robust exact (discontinuous) differentiator. To characterize the effect of
noise and disturbance on the differentiation estimation error, the generalized,
homogeneous $\mathcal{L}_2$-gain is utilized. Analog to the classical
$\mathcal{L}_p$-gain, it is not defined for the discontinuous case w.r.t.
disturbances acting on the last channel. Thus, only continuous differentiators
are addressed. The gain is estimated using a differential dissipation
inequality, where a scaled Lyapunov function acts as storage function for the
homogeneous $\mathcal{L}_2$ supply rate. The fixed differentiator gains are
scaled with a gain-scaling parameter similar to the high-gain differentiator.
This paper shows the existence of an optimal scaling which (locally) minimizes
the homogeneous $\mathcal{L}_2$-gain estimate and provides a procedure to
obtain it. Differentiators of dimension two are considered and the results are
illustrated via numerical evaluation and a simulation example.
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