Instructions: Drag the poles (a complex conjugate pair or two real poles).
Second-order system dynamics are important to understand since the response of higher-order systems is composed of first- and second-order responses. As one would expect, second-order responses are more complex than first-order responses and such some extra time is needed to understand the issue thoroughly.
Assume a closed-loop system (or open-loop) system is described by the following differential equation:
Let's apply Laplace transform - with zero initial conditions. The resulting transfer function between the input and output is:
This is the simplest second-order system - there are no zeroes, just poles.
The poles of this second order system are located at:
The poles of the system give us information about how the system responds because the poles encode all of the information about the natural frequency and the damping ratio.
The decaying exponential has a time constant equal to:
And the damped natural frequency is equal to:
The damped natural frequency is typically close to the natural frequency - and is the frequency of thedecaying sinusoid (underdamped system).
ωn is the undamped natural frequency.
ζ is the damping ratio:
Pole Location Example
The plane below shows the damping frequency and damping coefficient "zeta" graphically.
Note the following:
The example below is a second-order transfer function:
The natural frequency ω is ~ 5.65 rad/s and the damping coefficient ζ is 0.707. The system is underdamped. See the simulation example above.
The pole locations are:
Based on the natural frequency and damping coefficient values we might conclude that the overshoot is quite small (about 5%) and the system is well damped. The time-domain response will not oscillate for more than period. See below (the pole locations are just slightly off).
The poles of the system can be written in a slightly different form as:
In time domain:
Or in Laplace domain:
Time domain solution can be easily obtained by using the Inverse Laplace Transform. Reference (1) - @ MIT contains the time-domain solution to underdamped, overdamped, and critically damped cases.
The next figure shows the time-domain response based on pole location in the Laplace domain.
Version of this article is 11/19/2017.