This paper presents theory for multivariable system identification using matrix fraction descriptions and the matrix continued fraction description approach which, in turn, yields a lattice-type order-recursive structure. Once the matrix continued-fraction expansion has been determined, it is straightforward to obtain solutions to both the left and right coprime factorizations of transfer function estimates and, in addition, a solution to problems of state estimation (observer design) and pole-assignment control. An important and attractive technical property is that calculation of transfer functions on the form of right and left coprime factorizations, calculation of state variable observers, and regulators all can be made using causal polynomial transfer functions defined by means of matrix sequences of the continued-fraction expansion applied in causal and stable forward-order and backward-order recursions.