I understand that you want to derive damping ratios and natural frequencies from a linear model. This model may be a state-space model or a transfer function.
If you have access to the System Identification Toolbox, you can use the "damp" function. For example, the following command displays a list of damping ratios and natural frequencies:
In this example, input 'sys' is the linear model. Output 'Wn' is a vector of natural frequencies and 'zeta' is a vector of damping ratios.
For more details about the "damp" function, refer to the following link: