Super-twisting observer for second-order systems with time-varying coefficient

Discontinuous Observers, in particular those based in the Super-Twisting Algorithm, are able to estimate the unknown input of a system in finite time when the relative degree is one for all the time and the input is a Lipschitz function of time. The authors extend in this study this result for the case when the relative degree is not well defined for all the time because of the fact that the unknown input has a time-varying coefficient that can be zero for some time intervals and that can also change its sign. The authors propose a discontinuous observer able to estimate the input in finite time and despite of its bounded but unknown velocity of change during the times the relative degree is well defined and one. They also provide a Lyapunov-like analysis to show the convergence of the observer using multiple instead of a single Lyapunov function. It is shown that if the signal to be estimated is Persistently Exciting and its number of changes of signs is bounded in any bounded interval of time, then the observer converges globally, uniformly and in finite time to the true value. The authors use the observer as estimator of a time-varying parameter and illustrate in an example its performance.