## 1 Introduction
In addition, the technique used to analyze the internal dynamics of the system, based on the matrix of partial derivatives in control engineering theory, has the objective of defining the internal stability of the system through the decomposition and interpretation of its characteristic canonical modes, which may belong to one of the following combinations: controllable and observable CO; not controllable and observable CO; controllable and not observable \(\overline{\mathrm{CO}}\); and not controllable and not observable \(\overline{\mathrm{C}}\,\overline{\mathrm{O}}\), which are not mentioned, or discussed, in the above researches. This presents an area of opportunity to perform a complete analysis of the systems dynamics models, and to strengthen the theory that supports the explanation of the system structure link-dynamic behavior for decision making. The reason for this research, is that by proposing a methodology based on conventional control engineering theory, we can determine a transfer function of the SD model based on the Laplace transform. This explains the setting state variable trajectories to impulse or system inputs, transforming the time domain differential equations for solution to a frequency domain, and analyzing the external stability and controllability of the system based on the decomposition of controllable characteristic modes (poles) and observable (zeros) in the polynomial of the transfer function. Additionally, it aims to control and manipulate the behaviors of the model's significant variables and more robust decision-making, in contrast to the research cited where only the dominance of feedback loops is analyzed. Further, the analysis of the internal and external stability of the system, based on the decomposition and interpretation of characteristic modes (eigenvalues) is not contemplated, or the determination of a function that explains the particular trajectories to predict the response of the system to changes in the input variables, which as mentioned above, is not concluded by not interpreting its result to determine whether the system is stable or not stable, and not to generate the particular trajectory of each state variable as a system response. One of the applications of great relevance for the proposed research is that generated by [PERSON] (2016). Its main contribution is the determination of a function in time that explains the internal dynamics of the particular trajectory of each state variable, instead of just analyzing the internal dynamics with corresponding eigenvalues. The trajectory of the internal dynamics of the state variable laying consists of a linear combination of the product of the components of the eigenvector as coefficients, and the modes of behavior generated by eigenvalues as a response to the internal dynamics of the system. The contribution concludes that the derivatives of the own vectors are associated with the impact of the transients in the short term, and the own values are associated with the impact in the long term. It should be noted that the author uses the same technique of reducing the model to an invariant in the time of spaces of states, and in the same way that the other authors mentioned only using the first term of Equation (1) to perform the analysis of the internal stability of the system, and do not conclude defining whether there is stability or instability based on behavior through the calculation of values and eigenvectors of the partial derivative matrix. The author finds a function to analyze the internal dynamics of the Ax system, making a transformation of coordinates of the original matrix to the corresponding products of the eigenvector with the corresponding eigenvalue. However, in the proposed function to analyze the internal dynamics of the system, it is not able to explain the behavior of the response to the internal dynamics and, likewise, it is not able to respond to the inputs or impulses of the system since they are not related in the equation found.