Natural frequency and damping ratio
damp
computes the natural frequency, time constant, and damping
ratio of the system poles as defined in the following table:
Continuous Time  Discrete Time with Sample Time Ts  

Pole Location 
$$s$$

$$z$$

Equivalent ContinuousTime Pole 
$$\text{Notapplicable}$$

$$s=\frac{ln(z)}{{T}_{s}}$$

Natural Frequency 
$${\omega}_{n}=\lefts\right$$

$${\omega}_{n}=\lefts\right=\left\frac{ln(z)}{{T}_{s}}\right$$

Damping Ratio 
$$\zeta =cos(\angle s)$$

$$\begin{array}{lll}\zeta \hfill & =cos(\angle s)\hfill & =cos(\angle ln(z))\hfill \end{array}$$

Time Constant 
$$\tau =\frac{1}{{\omega}_{n}\zeta}$$

$$\tau =\frac{1}{{\omega}_{n}\zeta}$$

If the sample time is not specified, then damp
assumes a sample
time value of 1 and calculates zeta
accordingly.