This study proposes an exosystem-observer approach to an output feedback problem with an output regulation constraint. The authors analyse the error covariance of a recently proposed exosystem observer and newly derive the degenerate Riccati equation and linear matrix inequalities for designing the optimal and suboptimal exosystem observer gains. Furthermore, they show that the output feedback controller based on the exosystem observer is equivalently implemented in a Youla-parameter-based structure, where the Youla parameter is realised as an observer for the exosystem state. By utilising the Youla-parameter-based structure, they provide output feedback controller design methods for achieving the optimal and suboptimal performances.