## 1 Introduction
The validation, analysis and explanation of dynamic systems in control engineering is based on determining the transfer function, in order to analyze the external stability of the system based on the canonical decomposition of controllable and observable characteristic modes at the poles or eigenvalues of the transfer function, and to predict the output behavior at various changes in the determined inputs in a linear combination of the coefficients of the zeros by weighting the exponential response of the poles or eigenvalues of the polynomials of the transfer function, which enter the system and do not depend on any internal system variable. The research proposal is based on presenting an alternative of analytical validation for more robust decision-making to systems dynamics models, based on the explanation of the system structure through a transfer function, and its analysis of stability and external controllability for the system dynamics model under study. The results are visually analyzed in a root diagram. The link between system structure and dynamic behavior is one of the elements defined in the system dynamics paradigm. Recently, rigorous mathematical and systematic methods have been explored in this link that has been initiated and has become available ([PERSON]). Although several methods have recently been proposed to understand the causes of the model's behavior, formal model analysis remains an important and challenging area in the SD [23]. A SD model can be reduced to a linearized system of time-invariant differential equations and analyzed with the state space technique through a state Equation (1) and an output Equation (2). \[\dot{\mathrm{X}}=\mathrm{Ax}+\mathrm{Bu} \tag{1}\]  
\[\mathrm{y}=\mathrm{Cx} \tag{2}\]  
where Equation (1) explains the internal dynamics and controllability of the system. The internal dynamics of the system is analyzed with the first term of the equation, based on explaining the stability of the model, and the relationship with the system inputs is analyzed with the second term to define the controllability of the system. Equation (2) determines the observability of the system by obtaining response measurements in relation to the internal dynamics of the system. Recent studies have proposed several analytical methods based on the theory of linear systems and control engineering, particularly the analysis of values and eigenvectors as a pragmatic support. This is intended to be a link to explain the dominant structure of the system based on the dynamic behavior of the loops present in the DS model, such as the studies carried out by [24; 25; 26; 27; 28; 29; 30; 31; 32; 33], which investigate the impact of system dominance and elasticity through the determination of vectors and eigenvalues, noting the importance of their influence on the different feedback loops present in the system dynamics model. In these investigations, they reduce the model of system dynamics to a linearized system of first-order time-invariant differential equations, establishing the matrix of partial derivatives A of order nXn in order to analyze the internal dynamics of the system. They determine the values and eigenvectors, which are used to analyze the dominance of existing loops, and try to explain their behavior through an analytical tool. It should be noted that the authors only consider the first term of Equation (1) in the reduction of the system dynamics model to a linearized system invariant in time, making incomplete analysis of the linearized model and reducing the analysis only to the internal dynamics of the \(\dot{\mathrm{X}}=\mathrm{Ax}\) system.