Settling Time

Abstract

The settling time is defined as the time at which the output is within some margin of the steady state value $x(\infty )$, we tend to take this value as $5\%$ , or $x(t_s)=0.95x(\infty )$ The time constant of a first order dynamic system determines the Settling Time $t_s$ of the system.

Extra

Rise Time Peak Time Maximum Overshoot

First Order

For a First Order Step Input System (definition) we get:

$$\begin{array}{rl}0.95K\tau a& =K\tau a(1-e^{\frac{t_s}{\tau }})\\ \therefore e^{-\frac{t_s}{\tau }}& =0.05\\ -\frac{t_s}{\tau }& =\ln (0.05)\\ \frac{t_s}{\tau }& \approx 3\\ t_s& =3\tau \end{array}$$

So as we can see, we will reach within 5% of the $x(\infty )$ at a time that is 3 times the time constant.

Underdamped Second Order

For a Second Order Step Input System (definition) we get:

$$\begin{array}{rl}x(t)& =\frac{Ka}{{\omega }_n^2}(1-e^{-\xi {\omega }_{n}t})\\ \text{Settling }& \text{Time:}\\ 0.95\frac{Ka}{{\omega }_n^2}& =\frac{Ka}{{\omega }_n^2}(1-e^{-\xi {\omega }_{n}t})\\ \Rightarrow 0.05& =e^{-\xi {\omega }_{n}t}\\ \Rightarrow \ln 0.05& =-\xi {\omega }_{n}t\\ 3& \approx \xi {\omega }_{n}t\\ t_s=\frac{3}{\xi {\omega }_n}& \end{array}$$