High-order sliding modes observer for linear systems with unbounded unknown inputs ⋆ Leonid Fridman Arie Levant Jorge Davila Engineering Faculty, National Autonomous University of Mexico (UNAM), Ciudad Universitaria, 04510, Mexico, D.F. (e-mail: lfridman@unam.mx) Applied Mathematics Dept., School of Mathematical Sciences,Tel-Aviv University, Ramat-Aviv. Tel-Aviv 69978, Israel (e-mail: levant@post.tau.ac.il) General Direction of Academic Computing Services, National Autonomous University of Mexico (UNAM), Ciudad Universitaria, 04510, Mexico, D.F. (e-mail: jadavila@unam.mx) Abstract: In this article is provided a high order sliding mode observer for linear systems with unbounded unknown inputs. Linear time-invariant systems satisfying conditions of strong observability are studied and an observer basing on high-order sliding-modes di erentiator is designed. 1. INTRODUCTION The sliding-mode-based observation is an active area in the automatic control theory (see Utkin et al. [1999], Barbot et al. [2002], Edwards et al. [2002], Poznyak [2003], Frid- man et al. [2008a]). Sliding modes has such attractive fea- tures as insensitivity (more than robustness) with respect to unknown inputs, and possibility to use the equivalent output injection in order to obtain additional information (e.g., the reconstruction of the unknown inputs). The main restriction of these observers was that the estimation of the observable states were made by using the equivalent control. It generates their main limitation: the output of the system should have a relative degree one with respect to the unknown input. This condition is very restrictive even for velocity observers for mechanical systems Alvarez et al. [2000], Davila et al. [2005], Davila et al. [2006]. Step-by-step vector-state reconstruction by means of slid- ing modes is studied by Hashimoto et al. [1990], Ahmed- Ali and Lamnabhi-Lagarrigue [1999], Floquet and Barbot [2006]. These observers are based on a system transfor- mation to a triangular form and successive estimation of the state vector using the equivalent output injection. Some su cient conditions for observation of linear time- invariant systems with unknown inputs were obtained in Floquet and Barbot [2006]. Moreover the above-mentioned observers theoretically ensure nite-time convergence for all system states. Unfortunately, the realization of step- by-step sliding-mode observers is based on conventional sliding modes requiring ltration at each step due to imperfections of analog devices or discretization e ects. ⋆ Work supported in part by Mexican CONACyT (Consejo Nacional de Ciencia y Tecnologia), grants no. 56819, Programa de Apoyo a Proyectos de Investigacion e Innovacion Tecnolgica (PAPIIT) UNAM, grant no. 111206 In order to avoid the ltration, the hierarchical observers were recently developed in Bejarano et al. [2007] iteratively using the continuous super-twisting algorithm. The High-Order Sliding-Mode observers recently devel- oped in Fridman et al. [2007], Fridman et al. [2008b] provides for the global nite-time convergence to zero of the estimation error in strongly observable case and for the best possible accuracy. However, the application of that observer is con ned to the class of the systems having bounded unknown inputs and a well defined vector relative degree with respect to the unknown inputs. It turns that this is just the restriction of transformation method suggested in the above cited papers. A novel method to reconstruct the derivatives of a signals with unbounded higher derivatives had been proposed in Levant [2006]. In this article that di erentiator will be applied to linear systems with unknown inputs. As far as we known this is the rst article providing robust exact nite-time observation of linear time-invariant systems with unbounded unknown inputs satisfying the su cient and necessary conditions of strong observability basing on high-order sliding-modes di erentiation. 2. PROBLEM STATEMENT AND MAIN NOTIONS 2.1 System description Consider a Linear Time-Invariant System with Unknown Inputs (LTISUI) ˙ x = Ax + Bu(t)+ Eζ (t), (1) y = Cx + Du(t)+ Fζ (t), (2) where x ∈X R n are the system states, y ∈Y R p is the vector of the system outputs, u(t) ∈U R q0 is a vector control input, ζ (t) ∈W R m , m p, are the