New formulation for the step response to the overdamped dynamic second order systems

Linear and time-invariant dynamic systems (L.T.I.) of second order, called overdamped, are characterized because their poles are real and differents. Usually the step response has been proposed in terms of $\xi, w_{0}$, and $K_{DC}$ [1]–[3], which are called parameters of the transfer function, $G(s)$; others have set it in terms of time constants $\tau_{1}$ and $\tau_{2}$ [4]–[6]. This article discusses the use of the dominant pole $p1$ and the relationship between the fastest pole and that one, $\alpha:=p2/p1,\ p1 < p2$, to state the expression of the system response signal, $y(t)$. This approach leads to a formulation of the expression for the simplest system output signal, setting $\xi$ and $w_{0}$ in terms of $p_{1}$ and $\alpha$. It should be noted that this approach allows determining certain characteristics of the signal from output to input step, for example, the determination of the settling time $t_{s}$, rise time $t_{r}$, fitting it with a polynomial for an easy calculation, also the time in which the change of concavity occurs with which the output corresponding to that instant of time is determined, there it is shown that this value is not greater than 27% of $K_{DC}$. Finally, the general expression for the impulse response is presented, parameterized by $p_{1},\ K_{DC}$ and $\alpha$. It is important to mention that the previous parameters cannot be obtained from the other known representations, with which from this work will have mathematical expressions to be able to completely analyze the overdamped second order systems, as the case for the first order systems, without having to simulate the output signal and obtain them from measurements on it.