A High Gain Observer for a Class of Implicit Systems

The high gain observer for dynamical systems described by ordinary differential equations is widely discussed 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.