Semi-global Finite-time Observers for a Class of Non-lipschitz Systems

Recently, the research on finite-time observers for nonlinear systems has made great progress [1, 2, 3]. However, some of these observers are not continuous such as sliding mode observers. The continuity property and its importance in finite-time stability are realized in [4, 5]. It is also interesting to point out that continuous observers are realized to be different and unique in the nonlinear context [6, 7]. For instance, the linearized observability is no more necessary for the existence of a continuous observer [7]. There are some results on finite-time continuous observers [3, 8, 9, 10, 11]. In [12], a finite-time observer was constructed for a class of observer error linearizable systems. More recently, we presented a general result: a uniformly observable and globally Lipschitz single output nonlinear system admits semi-global finite-time observers [13], and global ones [14, 15]. However, the results in [12, 13, 14, 15] are of existence type: one of the design parameters has to be chosen to be smaller than but sufficiently close to 1. In this paper, the restriction on this design parameter is relaxed. Then, semi-global finite-time observers are designed for a broader class of systems. The class of systems are also characterized by the non-Lipschitz type of conditions with mixed case and varying rational powers of the increments. They include thus as a proper subset of the class of systems that were found to admit global asymptotic observers in [17, 18]. Compared with [17], the observer that we design is finite-time convergent, but in a semi-global way. As in [17], we restrict our attention to estimating the solutions of systems that exist globally in positive time. 2 Semi-global finite-time observers