Parameter estimation for an overdamped dynamic second order system, a new approach

To design a control system it is necessary having it's most accurate mathematical model as posible, so that the results of its implementation be as expected. In the case of second order systems of the underdamped type, from the overshoot and the peak time, two of its three parameters can be determined, its angular frequency and its damping ratio if it is less than 0.8, from mathematical expressions that relate them. For second-order systems with a damping ratio greater than one, called overdamped, there are no mathematical expressions that allow the determination of their parameters, so graphic methods are used, the results of which depend on the precision with which the reading is made. Several methods have been proposed to determine their system parameters, some of them use the response curve to step function and based on the information of some points of the curve and with the help of graphics and standardized curves, the parameters are determined. Other methods make use of identification techniques such as minimum squares, or through genetic algorithms. This paper discusses how to determine the transfer function parameters for a second order overdamped system, $\zeta,\omega_{o}$ and $K_{DC}$, but unlike other approaches based on the aforementioned parameters, the problem is expressed in terms of $p_{1}, \alpha$ and $K_{DC}$, with $p_{1}$ being the dominant pole, $\alpha$ the ratio between $p_{1}$ and $p_{2}$, the fastest pole $(p_{2}/p_{1}=\alpha)$ and $K_{DC}$. It is shown that proposing the response to the step in terms of $p_{1}$ and $\alpha$ allows consider an approximation to the response and thus a way to estimate $p_{1}$ and $\alpha$ from two points in the reaction curve. The choice of these two points is very important for the success of the method. The algorithm for estimating and parameters is shown, in an analytical way, by using two points in the reaction curve and without the use of graphics as indicated by. Some examples are made to show the application of the method and to compare its results with other methods that have been proposed and that are references in the specialized literature. It is shown that for overdamped systems the proposed method has the best results with respect to the other methods considered. The results of the proposed method are also compared against those obtained by other methods for higher order systems indicated in one of the references. To evaluate the performance of the proposed method, the criterion of square of the error's integral is used so that its advantage can be observed.