Asian Journal of Control, Vol. 5, No. 4, pp. 513-527, December 2003 513 IMPLICIT TRIANGULAR OBSERVER FORM DEDICATED TO A SLIDING MODE OBSERVER FOR SYSTEMS WITH UNKNOWN INPUTS T. Boukhobza, M. Djemai, and J.P. Barbot ABSTRACT It has been presented in previous works that every uniformly observable sin- gle-output system can be put on a triangular observation form. For this structure a special kind of sliding mode observer has been designed by authors, which ensures a finite-time state reconstruction using a step by step observation algorithm. In this pa- per, we show that the multi-output case is more delicate to study especially when the system has some unknown inputs. Thus, in order to generalizes the triangular observer form, from single to multi-output case, we define an Implicit Triangular Observer (ITO) form. For such a form, two results are given. Firstly, we design a finite time converging observer for all values of the unknown inputs. Secondly, we give the nec- essary and sufficient condition, including a matching condition, for the existence of a coordinate change to put the system into this form. It is also shown that this class of systems is a subset of the uniform observable class of systems. KeyWords: Sliding mode observer, multi-output systems, unknown inputs, Implicit Triangular observer form. I. INTRODUCTION The techniques employed for the observation of the systems with unknown inputs are applicable to many problems ([33]) such as the fault detection and distur- bance rejection. We can distinguish two approaches: An approach which consists in observing the unknown in- puts or at least in determining the thresholds such as in the diagnosis as described in [8,9,13,15,17,18,24,29] for the linear and time variant case and [16,20,35] for the bilinear case. The other approach consists in observing the states of the system in the presence of unknown in- puts which can be regarded as disturbances. The present work deals with the problem of the observation of systems with unknown inputs in the multi-output case. The objective is to obtain an observer converging for all possible values of bounded unknown inputs. We treat on the case where the system can be put on the “ITO(Implicit Triangular Observer) form”. For such a form we construct an observer which carries out the objective of convergence defined above and we give the matching condition. To solve this problem, we use a step by step sliding mode observation algorithm which generalize the idea of the geometrical linearization of the observation error dynamics. In fact, the idea to linearize the observation error dynamics presented in [21,22], is developed by the authors for the output and output derivative injection form ([7]), and a step by step sliding mode observer with linear error dynamics was introduced. The works [31,32], gave the conditions under which a nonlinear system can be put with a coordinate change on the so-called ”output injection form”. Always for a single output, and using sliding mode method to obtain a finite time convergence instead of an exponential one as in [28], in [2] the authors gave a tri- angular observation form which is a generalization of the output injection form [21]. This work will be recalled in section 2 of the present paper. Thus, in this paper a more general form is introduced : the ITO form. The paper is organised as follows : In the 2 nd sec- tion we present the single output triangular form, the matching condition is given for this case, and the diffi- Manuscript received January 15, 2003; accepted April 1, 2003. T. Boukhobza is at IUT de Kourou, Universit´e des Antilles Guyane, Avenue Bois Chaudat, B.P. 725, 97 387 Kourou CEDEX, France. M. Djemai and J.P. Barbot are with Equipe Commande des Syst`emes (ECS), ENSEA, 6 Av. du Ponceau, 95014 Cergy Cedex, France.