Algorithmical determination of the equilibrium/stead position of a dynamic system by the finite-step method

Abstract

The problem of algorithmic estimation of the equilibrium/stationary state of the dynamic system based on the minimum number of measurements of the output quantity at equidistant time points is considered. For this, a finite-step method was used, which consists in the formation, according to certain rules, of the sums and differences of the values of the output quantity, which are located symmetrically on the time axis relative to a certain moment that can be determined a priori. As a result, overdetermined systems of linear equations are formed with respect to the introduced fictitious unknowns, and based on the necessary condition for the existence of a solution to these systems, formulas are found to determine the equilibrium/stationary position of the dynamic system. Algorithms for determining the equilibrium /stationary state for the following mathematical models of the output value are given: in the form of a constant component and a damped exponent, in the form of a constant component and a damped sinusoid, in the form of a constant component and two undamped sinusoids, in the form of a constant component and a damped sinusoid. It is shown that in the absence of measurement errors of the output value, the error in estimating the equilibrium/stationary state of the transient process depends only on the error in solving the system of linear equations. Ways of using the redundant number of measurements are discussed.
 Keywords: equilibrium position, stationary state, dynamic system, mathematical model, finite-step method, numerical algorithm.