Author's personal copy Automatica 43 (2007) 1464 – 1469 www.elsevier.com/locate/automatica Brief paper LMI-based sensor fault diagnosis for nonlinear Lipschitz systems  A.M. Pertew , 1 , H.J. Marquez ∗ , Q. Zhao Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada T6G 2V4 Received 5 December 2005; received in revised form 31 July 2006; accepted 29 January 2007 Available online 18 June 2007 Abstract The problem of sensor fault diagnosis in the class of nonlinear Lipschitz systems is considered. A dynamic observer structure is used with the objective to make the residual converge to the faults vector achieving detection and estimation at the same time. It is shown that, unlike the classical constant gain structure, this objective is achievable by minimizing the faults effect on the estimation error of the dynamic observer. The use of appropriate weightings to solve the design problem in a standard convex optimization framework is also demonstrated. An LMI design procedure solvable using commercially available software is presented.  2007 Elsevier Ltd. All rights reserved. Keywords: Dynamic observers; Lipschitz systems; Faults; Sensors; LMI 1. Introduction The fault diagnosis problem is gaining increasing considera- tion worldwide in both theory and application. This is due to the growing demand for higher reliability in control systems, and hence the importance of having a monitoring system to detect the existing faults and specify their locations and significance in the control loop. The observer-based approach is one of the most popular techniques used for fault diagnosis. Many stan- dard observer-based techniques exist in the literature providing different solutions to both the theoretical and practical aspects of the problem for the linear time-invariant (LTI) case (see Frank, 1990; Willsky, 1976 for good surveys). The basic idea behind this approach is to estimate the outputs of the system from the measurements by using either static gain observers in a deterministic framework (Zhong, Ding, Lam, & Wang, 2003) or Kalman filters in a stochastic framework (Chen, Mingori, & Speyer, 2003). The output estimation error is then used as  This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Ben M. Chen under the direction of Editor Ian Petersen. ∗ Corresponding author. Tel.: +1 780 492 3333; fax: +1 780 492 8506. E-mail addresses: pertew@ece.ualberta.ca (A.M. Pertew), marquez@ece.ualberta.ca (H.J. Marquez). 1 Also with the Computer and Systems Engineering Department, University of Alexandria, Alexandria, Egypt. 0005-1098/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2007.01.015 the residual. In contrast to the LTI case, however, the nonlinear problem lacks a universal approach and is currently an active area of research (see Adjallah, Maquin, & Ragot, 1994; Garcia & Frank, 1997; Hammouri, Kinnaret, & Elyaagoubi, 1999; Kabore & Wang, 2001;Vemuri, 2001; Wang, Huang, & Daley, 1997; Yu & Shields, 1996 for some important nonlinear re- sults). The main obstacle in the solution of the observer-based nonlinear fault detection problem is the lack of a universal ap- proach for nonlinear observer synthesis. In this paper, we focus on the class of Lipschitz systems of the form ˙ x(t) = Ax(t) + (u, t ) + (x,u,t), (1) y(t) = Cx(t) + f (t ), A ∈ R n×n , C ∈ R p×n , (2) where (A, C) is detectable, f(t) represent sensor faults, and (x,u,t) satisfies ‖(x 1 ,u,t) - (x 2 ,u,t)‖  ‖x 1 - x 2 ‖ (3) ∀u ∈ R m and t ∈ R and ∀x 1 and x 2 ∈ D, where D is a closed and bounded region containing the origin (see Pertew, Marquez, & Zhao, 2006 for more details about Lipschitz sys- tems, their importance, and previous works on the Lipschitz observer design problem). In this paper, we make use of the dy- namic observer structure introduced in Pertew et al. (2006) for the sensor fault estimation problem where the objective of esti- mating the fault magnitude is considered in addition to detection