This paper studies the non-linear observer design problem by observer canonical forms. We present necessary and sufficient conditions in terms of the ranks of matrices for the existence of a non-linear state coordinate change under which a time-variable system is transformed into an observer canonical form. These conditions give the multiple versions of Li and Tao (1986) for the single-output case. Then a comparison is made with the Lie algebraic conditions obtained by Krener and Respondek (1985), and modified by the authors in a recent paper (Xia and Gao 1987).