On continuous time differentiation observers

The so-called high gain observers are alternatives to the longstanding problem of estimating the state (or parameters) of general nonlinear systems. They consist of a linear infinite impulse response filter, and are made to work mainly by cranking their gain so as to overcome the adverse nonlinearities of the plant being observed. Their main point is the simplicity of the design. But they suffer from a structural drawback: by cranking the gain in order to reduce the residual estimation error, at the same time, the user is increasing the overshoots of the estimates in presence of singularities. Singularity refers here to initial conditions and measurements uncertainties. Assuming that measurement uncertainties may be adequately filtered out it remains that the inherent peaking behavior cannot be remedied without a redesign of the observer gain. In this communication we emphasize the fact that reduced peaking feature may be obtained merely by using a full dynamic gain. The gain is obtained by solving a differential equation either offline or online. The observer thus obtained is a continuous time counterpart of numerical differentiation observers which motivates the title of this communication. The observer is used to simulate the online estimation of the state variable of an academic single link flexible joint robot which involves differentiation of the measurements up to 3 times.