Why do we need state feedback in the H-infinity concept instead of output feedback?

I have faced control of a flexible manipulator which has a zero dynamic related to flexible parts dynamic. The zero dynamic states (generalized states) are not observable in the input-output dynamic and usually require the observer to estimate them in state feedback applications.

I see in the manuscripts that in H-infinity based controllers, in penalty/performance vector definition, the internal dynamic states are participated.

I have a question: is it possible to have output feedback containing just external dynamic states? Is there any controller based on H-infinity which leads to output feedback, assuming the C matrix in $$y=Cx$$ is not full rank?

This can be compared to how a LQR + Observer works, the observer estimates something based on previous inputs and outputs and the LQR uses that something (in this case the state estimation) to compute the new input. However, if this previous data would not be collected (which makes the controller a true output feedback controller), it is practically impossible to guarantee that the controller will achieve the control goal (or even stability). This assuming $$C$$ is not full rank.