5 Transform Methods

5.2.2 Transient ResponseAll Activities

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 $\zeta \ge 1$?