5 Transform Methods

Three common specifications of the step response are percent overshoot, rise time, and settling time. We will consider these in the time domain, getting corresponding s plane regions, then map these to the z plane for design purposes. These will all be with reference to a continuous second-order underdamped system with no zeros, which has transfer function

$$\frac{\omega_n^2}{s^2+2\zeta\omega_n+\omega_n^2}$$

An example step response plot for this kind of system is shown in Figure 1 (with $\zeta = 0.5$ and $\omega_n = 1$), where the percent overshoot, rise time, and settling time are marked on the plot

2nd order response
Figure 1: Transient response specifications for system with $\zeta$=0.50 and $\omega_n$=1.0, showing percent overshoot PO, rise time $t_r$, and settling time $t_s$.

 

Use the controls below to change $\zeta$ and $\omega_n$ in Figure 1. Observe how they affect the percent overshoot PO, rise time $t_r$, and settling time $t_s$.



  1. How does changing $\omega_n$ change the percent overshoot PO?
  2. How do you have to adjust $\zeta$ and $\omega_n$ to achieve the smallest settling time?
  3. Although the controls limit the possible values for $\zeta$, how might the graph change if $\zeta \ge 1$?