An Exact Robust High-Order Differentiator With Hyperexponential Convergence

A linear time-varying state observer is presented for a chain of integrators having bounded disturbances in the last equation. It is demonstrated that in the noise-free setting, for the continuous-time realization, the estimation error converges to zero with a hyperexponential rate (faster than any exponential) uniformly in the disturbance. An implicit discretization scheme of the observer is proposed, which in the discrete time preserves all the main properties of the continuous-time counterpart. In addition, the discrete-time estimation error is robustly stable with respect to the measurement noise. The efficiency of the suggested observer is illustrated through comparison with a linear high-gain observer and a sliding-mode high-order differentiator.