Global complete observability and output-to-state stability imply the existence of a globally convergent observer Alessandro Astolfi Electrical Engineering Dept. Imperial College London Exhibition Road, London SW7 2BT, UK a.astolfi@imperial.ac.uk Laurent Praly Centre Automatique et Syst` emes ´ Ecole des Mines de Paris 35 Rue Saint Honor´ e, 77305 Fontainebleau, France Laurent.Praly@ensmp.fr Abstract— In this paper we consider systems which are globally completly observable and output-to-state stable. The former property guarantees the existence of coordinates such that the dynamics can be expressed in observability form. The latter property guarantees the existence of a state norm observer and therefore nonlinearities bounding function and local Lipschitz bound. Both allow us to build an observer from an approximation of an exponentially attractive invariant manifold in the space of the system state and an output driven dynamic extension. The state of this observer has the same dimension as the state to be observed. Its main interest is to provide convergence to zero of the estimation error within the domain of definition of the solutions. I. I NTRODUCTION We consider a globally completely observable system whose dynamics can be represented globally by:          ˙ x 0 = x 1 , . . . ˙ x n-1 = x n , ˙ x n = f n (x 0 ,...,x n ) , (1) where f n is Lipschitz continuous. For such a system, we wish to establish the existence of a global observer when the only available measurement is: y = x 0 . Such a problem has received a lot of attention from a wide variety of view points. The route we follow here takes its starting point in a contribution 1 of Kazantzis and Kravaris. In [7], they have generalized, to the nonlinear case, Luenberger’s early ideas proposed in [9] for linear systems (see also [2, Section 7.4 method II]). However their analysis is a local one and requires too stringent assumptions aiming at getting an analytic observer. Our intent here is to remove these extra assumptions and to deal with the global case. For the latter, we need to add an assumption besides global complete observability. Assumption 1 (see [8], [11]): The system (1) is output- to-state stable, i.e. there exist C 1 non-negative functions γ 1 , γ 2 and V satisfying: |x|≤ γ 2 (V (x)) , (2) 1 This contribution has been extended in various ways by Kazantzis and Kravaris themselves but also by Xiao and Krener (see [13] and the references therein). But they remain in the same context of looking for a C ∞ observer or at least one admitting a formal power series representation. and: ˙   V (x) ≤-V (x)+ γ 1 (x 0 ) . (3) With (2) in Assumption 1 and the continuity of f n , there exists a C 1 non-decreasing function γ , lower bounded by 1 say, and satisfying: |x 0 | + ... + |x n | + |f n (x 0 ,...,x n )|≤ γ (V (x)) . (4) It follows that, by defining a new time τ as the solution of 2 : ˙ τ = γ (V (x)) , τ (0) = 0 , and by denoting: ˚a = da dτ = ˙ a γ (V (x)) , the system:                      ˚x 0 = x 1 γ (V (x)) , . . . ˚x n-1 = x n γ (V (x)) , ˚x n = f n (x 0 ,...,x n ) γ (V (x)) (5) is complete. Actually its solutions do not grow faster than |τ | both forward and backward in the new time τ . As a consequence, for any strictly Hurwitz p × p matrix A and any p vector B, the function given as follows is well defined and continuous (see [4, Th´ eor` eme 3.149]): R(x)=  0 -∞ exp(-Aτ ) Bx 0 (τ ) dτ , (6) where, with the notation: x =(x 0 ,...,x n ) , x 0 (τ ) is the first component of the solution x(τ ) of (5), issued from x. Our interest in R comes from the fact that: z = R(x) defines a globally attractive invariant manifold of the system (5) coupled with: ˚z = Az + Bx 0 . (7) 2 We get τ (t) ≥ t for all positive t.