A High Gain Observer for a Class of Implicit Systems Abdellatif El Assoudi LCPI, Departement GE, ENSEM Univ. Hassan II Ain Chock B.P 8118, Oasis Casablanca Morocco Email: a.elassoudi@ensem-uh2c.ac.ma El hassane El Yaagoubi LCPI, Departement GE, ENSEM Univ. Hassan II Ain Chock B.P 8118, Oasis Casablanca Morocco Email: e.elyaagoubi@ensem-uh2c.ac.ma Hassan Hammouri LAGEP, UMR CNRS 5007 Univ. Lyon I - ESCPE-LYON, Bat. 308G 43 Bd du 11 novembre 1918 69622 Villeurbanne Cedex, France Email: hammouri@lagep.cpe.fr Abstract— The high gain observer for dynamical systems described by ordinary differential equations is widely dis- cussed in the literature, see for instance [1], [2], [3], [4],[5], [6], [7], [8], [9], [10], [11], [12]. The aim of this paper is to extend this observer design to a class of differential- algebraic systems. In practice, the computation of solutions of differential-algebraic equations requires the combination of an ordinary differential equations (O.D.E.) routine together with an optimization algorithm. Therefore, a natural way permitting to estimate the state of such a system is to design a procedure based on a similar numerical algorithm. Beside some numerical difficulties, the drawback of such a method lies in the fact that it is not easy to establish a rigorous proof of the convergence of the observer. The main result of this paper is stated in section 3. It consists in showing that the state estimation problem for a class of differential-algebraic systems can be achieved by using an observer having an O.D.E. structure on some R N . Keywords: Nonlinear system, implicit system, high gain observer. I. I NTRODUCTION AND PROBLEM STATEMENT In this paper, the following class of implicit systems is considered: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ˙ x = f 0 (x, z)+ m  i=1 u i f i (x, z) ϕ(x, z)=0 y = h(x, z) (1) where y ∈ R p , u =(u 1 ,...,u m ) ∈ R m , (x, z) ∈ R n × R d , the f i ’s, h and ϕ =(ϕ 1 ,...,ϕ d ) T are assumed to be sufficiently smooth and: ∂ϕ ∂z     x,z is of full rank ∀(x, z) ∈M (2) where M is the set of zeros of ϕ: M =  (x, z) ∈ R n × R d , s.t. ϕ(x, z)=0  (3) Remark 1: Condition (2) implies: i) the local uniqueness of solutions z of ϕ(x, z)=0, for every x. ii) M is a smooth submanifold of R n × R d . In the case where the solution z of ϕ(x, z)=0 can be explicitly expressed as z = ψ(x), system (1) becomes a system of O.D.E. Hence, an observer can be formulated as a system of O.D.E. Otherwise, one may ask if there exists an observer that can be described by ordinary differential equations. In the sequel, we will use the following defini- tion. Definition 1: A non initialized (resp. an initialized) exponential observer for system (1), with input u and output y, is a dynamical system of the form:  ˙ ω = Γ(ω, u, y) ω(0) ∈ Ω ⊂ R N (4) for which there exists a map Ξ = (Ξ 1 , Ξ 2 ) from R N into R n × R d such that Ξ 1 (ω(t)) − x(t) together with Ξ 2 (ω(t)) − z(t) exponentially converge to 0, as t →∞, where Ω is such that Ξ(Ω) contains an open set containing M (resp. Ξ(Ω) = M). Noticing that in practice, an initialized observer only works if the measurements are not noisy, and that the initial state of the observer satisfies the constraint ϕ(Ξ(ω)) = 0. This paper is organized as follows: In section 2, we will give an initialized high gain observer. This observer construction is based on a triangular structure containing this proposed in [8], [12]. In section 3, we robustify the above observer in order to obtain a non initialized one. II. I NITIALIZED HIGH GAIN OBSERVER BASED ON A TRIANGULAR STRUCTURE Given a nonlinear system: ⎧ ⎪ ⎨ ⎪ ⎩ ˙ ζ = F (ζ,u)= F 0 (ζ )+ m  i=1 u i F i (ζ ) y = H(ζ ) (5) where the input u ∈ R m , the state ζ ∈N a smooth manifold of dimension n, the output y ∈ R p , F is a smooth vector field with respect to these arguments. Noticing that the class of systems (1) forms a particular Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005 ThA02.5 0-7803-9568-9/05/$20.00 ©2005 IEEE 6359