This paper proposes a novel observer/controller identification method for identifying the minimally realised equivalent (reduced-order) mathematical models in the block observer/controller-canonical forms of the unknown (i) open-loop system, (ii) existing feedback/feedforward controllers and/or (iii) observer, based on available measurements of the operating closed-loop system. By skipping the singular value decomposition procedure and without involving the model conversion of the identified model from the general coordinate into the block observer/controller-canonical forms during the identification process, the proposed method is able to directly realise the identified parameters in the minimally realised block observer/controller-canonical forms. This simplifies the system identification process. The new procedures enable us to enhance the computational aspects of designing self-tuning controllers for online adaptive control of (a class of) multivariable systems and to improve the tracking performance considerably. As a result, the newly proposed compensation improvement approach is able to compensate the undesirable operating controller.