General Second Order Systems

Sources

1. General Second Order EK2. (Lecture Slides)

General Second Order Transfer Function

$G(s)=\frac{{\omega }_n^2}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}$

Where

• $a=2\zeta {\omega }_n$
• ${\omega }_n=\sqrt{b}$
• $\zeta =\frac{a}{2\sqrt{b}}$

Type of Response

Determining the Poles and Step Response Given the Damping Ratio

Damping Ratio	Poles	Step Response
$\zeta =0$	$\pm j{\omega }_n$	Undamped
$0<\zeta <1$	$\pm j{\omega }_n\sqrt{1-{\zeta }^2}$	Underdamped
$\zeta =1$	$-\zeta {\omega }_n$ (repeated)	Critically Damped
$\zeta >1$	$-\zeta {\omega }_n\pm {\omega }_n\sqrt{{\zeta }^2-1}$	Overdamped

Natural Frequency

The natural frequency ${\omega }_n$ indicates how much the system oscillates given that there is no damping.

$$\begin{array}{r}{w}_n=\sqrt{b}\\ \pm j\sqrt{b}\end{array}$$

Where

• $b$ is a system pole element

Damping Ratio

$$\begin{array}{rl}\zeta & =\frac{\text{Exponential Decay Frequency}}{{\omega }_n\text{(in radians per seconds)}}\\ \zeta & =\frac{1}{2\pi }\left(\frac{\frac{1}{{\omega }_n}\text{(in seconds)}}{\text{Exponential time constant}}\right)\end{array}$$

Characteristics of a General Second-Order System Transient Response

Rise Time

The rise time $T_r$ refers to the time needed for the response to go from 10% to 90% of its steady state value.

Damping Ratio	Normalized Rise Time
0.1	1.104
0.2	1.203
0.3	1.321
0.4	1.463
0.5	1.638
0.6	1.854
0.7	2.126
0.8	2.467
0.9	2.883

$$T_r=\frac{\text{Normalized Rise Time}}{{\omega }_n}$$

Peak Time

The peak time $T_p$ refers to the length in time for the waveform to either reach the first or the maximum peak.

$$T_p=\frac{\pi }{{\omega }_n\sqrt{1-{\zeta }^2}}$$

Percent Overshoot

The percent overshoot $\%OS$ measures how much the waveform go beyond the steady state (expressed as a percentage of the steady state). It could also refer to the peak time’s final value.

$$\%OS=e^{-(\zeta \pi /\sqrt{1-{\zeta }^2})}\times 100\%$$

Settling Time

The settling time $T_s$ refers to the time it takes for the response to reach and settle to 2% of its steady state value.

$$T_s=\frac{-\ln (0.02\sqrt{1-{\zeta }^2})}{\zeta {\omega }_n}$$

It can be approximated using the formula

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