Optimal robust exact differentiation via linear adaptive techniques

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