## 1 Introduction
A measure of the impact on an eigenvalue (\(\lambda\)) when one changes the individual elements a from the matrix of system A is the elasticity of the eigenvalue. While the technique allows determination of the poles of the system, it focuses on the internal dynamics of the system, focusing on how changes in the parameters of the model affect the behavior modes of the system. This technique provides information on the sensitivity and stability of the system against variations in internal parameters. On the other hand, the proposed approach takes into account both the internal and external dynamics of the system. By relating the output equation to input through the acquisition of the transfer matrix, it is possible to analyze the system's behavior in the face of multiple inputs and perturbations. The state-space approach adopted in this research allows us to model the system in a more precise and detailed way, providing a mathematical representation that considers both the inputs and the outputs of the system. This allows a more complete analysis of the system's internal and external interactions, providing a deeper understanding of its global behavior. The technique (LEEA) implemented in [23] shows that it is possible to express the polynomial characteristic of the matrix of system A (that is, the polynomial whose zeros are the eigenvalues of A) in terms of the gains of the loops, in what the author called a set of independent loops. Furthermore, he proved that a fully connected system with N links and n variables has a total of \(n+1\) independent loops, and he provided a procedure to construct this set and to calculate the elasticities of the loops, which was a great contribution to the state of the art in the formality of system dynamics; however, the eigenvalues method only analyzes the internal dynamics, and its results do not have any certainty about how the system will behave with different inputs and disturbances that the system under study may experience. The eigenvalue technique is used to determine the poles of a given matrix. On the other hand, by having the space-state representation of a system, both the transfer matrix and the system poles can be calculated. In addition, state-space representation allows results for multiple inputs and the calculation of control using modern control techniques. Compared to the approach of calculating only poles using the eigenvalue technique, the state-space method offers several significant advantages. By obtaining the transfer matrix, a complete description of the system is obtained that includes both the poles and the zeros, which allows a more complete and precise analysis of the behavior of the system. In addition, state-space representation is more flexible and versatile, as it can handle systems with multiple inputs and outputs. By using the state-space method, a wide range of modern control techniques can be applied to design effective controllers and achieve desired objectives in terms of system performance and stability. This provides a greater degree of control and optimization compared to the system's pole-based approach. Another relevant contribution in the search for formality in system dynamics is presented in [24]. In that work, an analysis of the general trajectory of a state variable is performed instead of focusing only on a specific behavior mode. It is emphasized that this general trajectory is determined by a linear combination of the products of the components of the eigenvectors and the modes of behavior associated with said eigenvectors. The study notes that changes in eigenvectors are related to short-term transients in the response of the state variable, while eigenvectors are linked to long-term behaviors. This contribution represents a significant advance in the formality of mathematical models when considering the different dynamics affecting the state variable. However, a major limitation of the proposed mathematical model is the lack of control over the desired values of the system.