  Recent works have approached the data-driven design of output-feedback
controllers for discrete-time LTI systems by constructing non-minimal state
vectors composed of past inputs and outputs. Depending on the system's
complexity (order, lag and number of inputs), it was observed in several works
that such an approach presents certain limitations, but no methods were
proposed to overcome them. In this note, we clarify these limitations and solve
them by proposing the construction of (alternative) non-minimal state vectors
that facilitate output-feedback control of MIMO discrete-time LTI systems.
