Second-Order System Approximation

If the poles of a system are dominated by a pair of complex poles, then the system can be approximated by a second-order system.

Provided we are provided with specs of settling time and percent overshoot (PO), we can determine the system by working with the formulas below.

• Settling time:

$$t_s=\frac{4}{\zeta {\omega }_n}$$

• Percent overshoot:

$$\text{PO}=e^{-\zeta \pi /\sqrt{1-{\zeta }^2}}\times 100\%$$

Design inequalities derived from these:

$$\zeta {\omega }_n\ge \frac{4}{t_s},\zeta \ge \frac{-\ln (\text{PO})}{\sqrt{{\pi }^2+((\ln (\text{PO}){)}^2}}$$

These help place dominant poles in the $s$-plane to meet desired transient performance.

We can place the dominant poles in the desired regions, and place the less dominant poles far away.