A FOLIATION FOR UNKNOWN INPUTS RECONSTRUCTION S. Chaib, D. Boutat, A. Benali, F. Kratz LVR ENSI de Bourges 88 Bd Lahitolle 18020 Bourges France (salim.chaib, driss.boutat, abderraouf.benali, frederic.kratz)@ensi-bourges.fr Abstract: This paper deals with a new method to compute the left inverse for a class of dynamical systems. This method is based on a projection which decompose the state space into two transverse foliations: one is tangent to the unknown inputs and the other is transverse. Contrary to the traditional methods, our method enables us to write the inverse dynamic system under consideration. Within this work we highlight our propositions by examples. Keywords: Inputs reconstruction, projector, foliation, nonlinear systems 1. INTRODUCTION The problem of unknown inputs and fault de- tection and isolation (FDI) is the design of a filter (or residual generator) which enables us to detect the occurrence of a component of the fault signal independently each of other and converges asymptotically to zero whenever the component in question is identically zero. The FDI problem was addressed by Beard and Jones. The well- known solution of this problem was addressed by (Massoumnia, 1986) for linear time invariant systems. His method is to reduce the dynamic via a quotient space obtained by canceling an un- observable subspace, by mean of output-injection and output-reduction. In (Frank and Ding, 1994), the author presents a frequency domain approach. LPV fault detec- tion and isolation can be found in (Balas and Bakor, 2000). Failure detection for Bilinear and non linear systems is investigated in (Hammouri et al., 1999) and (Persis et al., 2001) respectively. An other approach was introduced in (Barbot et al., 2005) based on the socalled fictive outputs to compute the unknown inputs. In this paper, we will use a projector to decompose the state space into two foliations: one is tangent to the unknown inputs the other is transverse. We construct after that the unknown inputs or the fault signal using the invertibility concept of a dynamical systems. This problem is the unknown input reconstruction by mean of a filter or a de- tector using the input and output measurements. In (Sira-Ramirez, 1999), in the case of mono- variable systems, this projector was introduced to design a suitable pre-feedback which make passive a dynamical system by stabilizing its zero dy- namic. This projector was generalized in (Boutat et al., 2000) for a multi outputs square regular dynamical systems with relative degree 1. The outline of the paper is as follows: in section 2, we give notations and some assumptions needed for our propose. In section 3, we define the pro- jector and we use it to compute the unknown inputs and to find the inverse dynamic. In section 4, we give a sufficient condition under which the projector can be extended to a neighborhood of the sub-manifold S . Some illustrative examples are given along the paper. Copyright © 2007 IFAC