Bl-homogeneous sliding-mode observers for SISO linear time-varying systems with an unknown input

In this article, a family of state observers is proposed for a class of SISO linear systems with time -varying coefficients and one unknown input satisfying the structural property of uniform strong observability. It is proved that, if the unknown input is bounded, the observer converges globally and at least asymptotically to the true states of the system. Furthermore, depending on an adequate selection of parameters and gains in the observer, the convergence is global and theoretically exact, in finite -time or even more, in fixed -time, with an attainable arbitrary convergence time. The effectiveness of the observation scheme is illustrated in a practical way on a laboratory crane with movement in two dimensions. The formal proofs of the results are performed with a Lyapunov approach and applying arguments of homogeneity in the bi-limit.