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The problem of differentiating a function with bounded second derivative in the presence of bounded measurement noise is considered in both continuous-time and sampled-data settings. Fundamental performance limitations of causal differentiators, in terms of the smallest achievable worst-case differentiation error, are shown. A robust exact differentiator is then constructed via the adaptation of a single parameter of a linear differentiator. It is demonstrated that the resulting differentiator exhibits a combination of properties that outperforms existing continuous-time differentiators: it is robust with respect to noise, it instantaneously converges to the exact derivative in the absence of noise, and it attains the smallest possible—hence optimal—upper bound on its differentiation error under noisy measurements. For sample-based differentiators, the concept of quasi-exactness is introduced to classify differentiators that achieve the lowest possible worst-case error based on sampled measurements in the absence of noise. A straightforward sample-based implementation of the proposed linear adaptive continuous-time differentiator is shown to achieve quasi-exactness after a single sampling step as well as a theoretically optimal differentiation error bound that, in addition, converges to the continuous-time optimal one as the sampling period becomes arbitrarily small. A numerical simulation illustrates the presented formal results. [ABSTRACT FROM AUTHOR]

In this paper, a fast decomposition algorithm of positive and negative sequence components is derived first. And based on analyzing the interaction between sampling noise and time (angle) delay, a sampling-noise-optimized decomposition algorithm is proposed. This optimized method can identify positive and negative sequence components quickly and limit the error caused by sampling noise. Compared with existing methods, the new algorithm has a rapid response speed and greatly reduces the sampling noise level under the same response time. Therefore it is more suitable for those time-critical applications. Simulation results are in correspond with analytical conclusions and the validity of the method is examined. • Noise-optimized fast decomposition method of positive & negative sequence components. • Faster response speed and relatively lower sampling noise level. • A generic algorithm for different time-critical applications. [ABSTRACT FROM AUTHOR]