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1 - Do you have some general idea of what your transfer function looks like? Or at least a frequency response? Could you perform some kind of step response? You could then try to fit an order-2 model on your step response and then try to tune your controller. 2 - You mention that your system is underdamped. I assume that you have underdamped stable poles (...

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Try https://www.dsprelated.com/ They have members well versed in this area.

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There are different ways of expressing a control system. For example, Observable Canonical Form Controllable Canonical Form If you express your system in controllable canonical form, then $B$ take the form of a column matrix with all zeros except one in the top position. Link: https://en.wikibooks.org/wiki/Control_Systems/Standard_Forms

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