Semi-global finite-time observers for nonlinear systems

Abstract

a b s t r a c tIt is well known that high gain observers exist for single output nonlinear systems that are uniformly observable and globally Lipschitzian. Under the same conditions, we show that these systems admit semi-global and finite-time converging observers. This is achieved with a derivation of a new sufficient condition for local finite-time stability, in conjunction with applications of geometric homogeneity and Lyapunov theories.

Global finite-time observers for non linear systems

“…According to the proof of the main theorem of [20], it follows that for every compact set U containing the origin, there exists θ 3 > 0 and ε 2 > 0 such that for θ ≥ θ 3 and α ∈ (1 − ε 2 , 1), U ⊂ Ω, where Ω is the domain of attraction of the observer. But according to [3], there exists δ 0 > 0 such that:…”

“…Such one has been designed for linear systems in [18] extended to linear time-varying system in [19]. A global finite time observer for a linearizable system via input output injection is constructed in [9] and extended to uniformly observable systems in [20] in a semi-global way. Let us note that such output injection structure : a linear part plus a homogeneous part can be derived from the results obtained by Andrieu-Praly-Astolfi ( [21,22] using the concept of homogeneity in the bi-limit : finite time convergence being obtained if the homogeneity degree in the 0-limit is negative.…”

“…The proof of finite-time stability is based on homogeneity property (specifically on Theorem 2). Then, a semi-global observer was constructed by Shen and Xia in [20] based on the Perruquetti et al observer. This result is recall here: Theorem 3.…”

Globally uniformly ultimately bounded observer design for a class of nonlinear systems with sampled and delayed measurements

“…A class of high gain observers were also proposed for nonlinear systems by using a gain update law which is dependent on the system output in [4]. In [9,10,14,15], the authors have studied the finite-time observers design for nonlinear systems with lower triangular form. However, the whole above results are studied based on continuous time analysis.…”

Finite-time output stabilization of the double integrator

“…The reader may found additional properties and results on homogeneity in [6], [7], [8], [9], [10]. The homogeneity property was used many times to design FTS state controls [11], [12], [13], [14], [15], [16], FTS observers [17], [18] and FTS output feedback [19], [20].…”