IFAC PapersOnLine 51-24 (2018) 1239–1246 ScienceDirect ScienceDirect Available online at www.sciencedirect.com 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2018.09.693 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. 1. INTRODUCTION Components of a physical system are subject to deterioration and alterations caused by unexpected phenomena beyond the control of operators. This problem has motivated the design of automatic monitoring and diagnostic systems that indicate to operators where the faults are and what is their magnitude (Isermann, 2006). The basic principle for detecting faults is the existence of coherence between characteristics of the system, its statistics under nominal conditions and the loss of such coherence under faults (Chen and Patton, 1999). In general, models with dynamical variables x ∈ ℜ n and static variables x s ∈ ℜ s are described by a set of nonlinear equations of the type Σ    ˙ x(t )= f m (x(t ), u(t ), x s (t ), f (t ), ¯ f (t )) 0 s = g(x(t ), u(t ), y(t ), x s (t ), f (t ), ¯ f (t )) y(t )= h(x(t ), u(t ), x s (t ), f (t ), ¯ f (t )), known as descriptor systems (Guan-Ren, 2010), by assuming that the model is well determined where u ∈ ℜ m is a vector of known exogenous signals and the vector y ∈ ℜ p corresponds to measurable variables. In this model, f (t ) ∈ ℜ f and ¯ f (t ) ∈ ℜ ¯ F denote the faults of interest and non-interest, respectively. The basic element in fault detection and isolation (FDI) is the residual r(t ) that has a zero average value in the absence of faults, and this deviates from zero in fault conditions. For nonlinear regular systems, the tools most used to generate residuals are observers. In this case, the solution is to design dynamic systems sensitive to faults of interest f (t ) and un- responsive to non-interest ¯ f (t ) with the structure of an ob- server. Even for additive faults and by assuming the absence of static variables, detection conditions reported by Hammouri et al. (1999) and De-Persis and Isidori (2001) are difficult to verify and guarantee the existence of a solution. For large- ⋆ Founding: Proyecto 280170 conv 2016-3, Fondo Sectorial CONACyT- Secretar´ ıa de Energ´ ıa-Hidrocarburos: DGAPA IT100716. scale systems or models with generic parameters, the design of unknown inputs observers is not trivial. Edwards et al. (2000) proposes the use of observers based on sliding modes for un- known entries and reconstructs f (t ) directly, without generating residuals. The conditions of such observers are difficult to fulfill for large-scale systems. Staroswiecki and Varga (2001) address the FDI problem of nonlinear systems in the framework of the space parity, with the disadvantage of requiring differentiators and imposing the condition of stability on the system. There are several investigations that cope with the FDI problem by using descriptor systems theory (Yeu and Kawaji (2001), Yeu et al. (2005), Hou (2000)). In general, most of them consider that the systems analyzed can be transformed into its regular form and solve the problem using traditional tools (Zheng et al., 2017). Several alternatives to designing FDI systems through tools of graph theory (Verde et al., 2013), (Dion et al., 2003) has been developed, because structural analysis does not differentiate between descriptor or regular systems. Krysander et al. (2008) have simplified the task, making use of equivalence relations, and his research group has developed a toolbox for the gener- ation of redundant relations based on the concept of minimal structurally overdetermined sets (MSO) (Frisk et al., 2017). Unlike the algorithm proposed in Krysander et al. (2008), the present work formulates the problem of residuals insensitive to a fault set and also uses the concept of equivalence rela- tion. This algorithm reduces the size of subsystems obtained although, MSO is not used. As a case study, a combined cycle power plant described by 37 equations with faults in its valves is considered. This work is structured in the following way. Section 2 de- scribes the basic concepts of structural analysis (SA). In Section 3, an algorithm is described for obtaining graphs with diagnos- tic properties, by considering equivalence relations. Section 4 presents the model of the case study, indicating the valves that are affected by the faults. In addition, the algorithm is applied to the case study. Section 5 presents the residual implementation Keywords: Fault diagnosis, structural analysis, large-scale systems, equivalence transformations, descriptor systems, observers. Abstract: This paper presents an algorithm to obtain redundant relations of a large-scale descriptor systems by using structural analysis tools and the concept of equivalence classes. As a case study, a combined cycle power plant (CCPP) is considered with a nonlinear model described by 37 algebraic and dynamic equations where faults are assumed in some process valves. The main advantage of this proposal is the reduction of possible constraints to generate residuals for decentralized descriptor subsystems by using observers and redundant relations; furthermore, insensitivity to a particular fault set is achieved. * Universidad Nacional Aut´ onoma de M´ exico, Coyoac´ an DF 04510, M´ exico, jvillanuevap@iingen.unam.mx ** verde@unam.mx A. Villanueva * C. Verde ** Fault diagnosis for descriptor systems by equivalent transformations ⋆