This article aims at solving the problem of robust H − /H ∞ fault detection (FD) over finite frequency domain for spatially interconnected systems (SISs) with polytopic uncertainties. A fault detection observer is designed to generate the residual signal, and in order for the observer to work, the well-posedness, robust stability, and finite frequency H − /H ∞ indexes are properly defined for the error systems. A sufficient condition for the well-posedness and robust stability of error systems is given based on the theory of continuous semigroup. Then, to obtain the H − /H ∞ performance indexes conditions, a reformed generalized Kalman-Yakubovich-Popov (GKYP) lemma for polytopic SISs is derived, which takes the noncasual structure of SISs into account. It turns out that the reformed GKYP lemma is an extension of some existing results in traditional multidimensional casual systems. By resorting to Finsler's lemma and a linearization technique, the existence conditions of finite frequency H − /H ∞ FD observer for polytopic SISs are given in terms of linear matrix inequalities. An illustrative example is given to demonstrate the effectiveness of the proposed method. K E Y W O R D S H − /H ∞ indexes, fault detection, finite frequency, generalized Kalman-Yakubovich-Popov lemma, polytopic uncertainties, spatially interconnected systems 1 INTRODUCTION Over the past several decades, spatially interconnected systems (SISs) have been intensively studied due to the extensive use of distributed sensor-actor arrays. 1 As the name implies, SISs are systems consisting of a large number of spatially interconnected identical units, and each unit is equipped with actuating and sensing capabilities. In D'Andre and Dullerud, 2 a noncausal multidimensional (MD) state space model was originally proposed for SISs, and the attractive feature of this representation lays in the fact that the noncasualities of spatial operations are taken into account, leading to more explicit physical interpretations compared with traditional two-dimensional (2D) Roesser model 3,4 and Fornasini-Marchesini model. 5,6 Based on this representation, a great deal of attention has been focused on the analysis and synthesis of SISs, such as distributed control, 7 stability analysis, 8,9 and model reduction. 10