## 1 Introduction
Based on the MAC, several other criteria for the correlation between modal ordinate vectors were proposed: the Partial MAC (PMAC) [19], in which only part of the modal ordinate vector is used in the calculations, focusing the analysis in a particular component or direction; the Extended MAC (MAC) [20], with an expanded assertiveness for complex modes, and its enhanced version (MACXP) [20], which, by a weighting with the poles of the corresponding mode shapes, improves the performance of the criterion under reduced spatial resolution; and the Weighted MAC (WMAC), also known as Normalized Cross Orthogonality (NCO), which incorporates the mass or stiffness matrices as a weighting matrix for the calculation of the MAC, remedying the shortcoming of the MAC not being a true orthogonality check [13]. However, due to the fact that usually not all degrees of freedom are instrumented, the application of the WMAC requires condensing the mass or stiffness matrices through model-reduction techniques [13; 17]. Another issue regarding the WMAC is that, for complex mode shapes, a different formulation based on the state formulation of the orthogonality condition is required. This is due to the orthogonality conditions with respect to mass and stiffness matrices not being met for complex modes [21]. The mathematical formulation of some of these criteria will be briefly presented in Section 2 of the manuscript. A more detailed description of these and other criteria can be consulted, for example, in the References section [17; 19; 21; 22]. A common issue associated with the application of these classic mode-pairing criteria arises when dealing with structures where local modes are associated to a particular structural component, as well as a set of components or parts of the structure that do not constitute the object of interest but also have global components, even if very small, with a shape similar to the true global modes. These modes, especially when they have close natural frequencies, can easily generate errors in the matching process and compromise the stability of the model-optimization algorithm [12; 23]. Seeking to solve this issue, [PERSON] et al. [12] proposed an innovative criterion, the EMAC, which is based on the weighting of the MAC by the relative modal strain energy associated with different parts of the structure denominated as clusters. The authorsdemonstrated that, through the appropriate choice of clusters, it is possible to isolate modes from different parts of the structure quite efficiently. Applications of the EMAC for mode pairing can be consulted in the works of [3; 5; 23; 24], in which its use proved to be essential in applications involving the model updating of FE models of railway bridges including the track. Since it relies on the orthogonality conditions between the vibration modes and the stiffness matrix to calculate the modal strain energy, the EMAC, according to its original formulation, cannot be applied to problems involving complex modes. In many cases, especially in civil engineering structures where the modes are real or almost real, this does not represent a problem, and often, the pairing and model updating are performed considering the undamped modal problem. However, in the case of vehicles and structures with localized dampers, modes with a significant degree of complexity are present. In these situations, the EMAC ends up being at a disadvantage compared to criteria such as the FDAC, the MSF and the MAC, which can be applied to complex modes, and even more in relation to the MACX, which was specifically developed for these applications. In this framework, the present work intends to share innovative contributions that, according to the authors' knowledge, are not sufficiently detailed in the existing literature, namely:  
* The development of a mode-pairing formulation dedicated to complex modes based on an energy-based criterion and relying on a state-space formulation.