This paper is an extension of our previous work regarding realization of LQ optimal systems using output feedback in which we showed that the solutions existed when the plant satisfied certain conditions, In this paper, we remove the restrictions set forth in our previous work and present a design which is suitable for any LQ optimal system. This design involves feedback consisting of scalar output, even when a multivariable system is the subject of the design. Moreover, both the discrete-time and the continuous-time formulations of the proposed design are formulated. The present method also combines plant model estimation and control design into a single procedure by using a technique which achieves an output feedback LQ design directly from plant input and output data, thereby eliminating the need for explicit identification of a parametric plant model.