Development and application of a generalized method of determination the roots of the characteristic equation a system tuned to the technical optimum

The object of the conducted research is dynamically unstable objects and processes of the third order with a transfer function of the third order The problem that has been considered and solved is related to the provision of dynamically unstable systems with such a stable state in which transients are set to the technical optimum. Analytical formulas have been obtained to determine the values of the roots of a third-order characteristic equation tuned to a technical optimum. The only variable of the obtained mathematical expressions is the time constant of the system or object, which determines the time of establishment or termination of periodic processes. Since the obtained analytical expressions directly determine the constant coefficients of the polynomial of the characteristic equation based on just one variable, which in the physical sense is a normalized value of the transition time, the synthesis of any dynamically unstable system leads to a state in which the system is optimized, according to the condition of technical optimum. In other words, only the analytical expressions obtained for the coefficients of the characteristic equation determine the system tuned to the technical optimum. Before synthesizing the regulator and feedback, this time constant is set as a set requirement for the time of termination of the transient process. Thus, the formulation of an equation for the coefficients of the characteristic equation of the system is equated to the found equations and the unknown coefficients of the feedback and rcgulator are calculated from the resulting system equation. The practical applications of the obtained results can be limited with two conditions: it is assumed that the dynamically unstable systems under consideration are linear or linelized; the selected solutions and feedback elements are tactically feasible. The results obtained after applying the proposed technique allow to conclude that by changing the ratio between the values of the adjustable coefficients, the performance is increased by about two times. At the same time, the overshoot does not exceed 6 %, which is normalized by the conditions of the technical optimum