In this paper a positive commuting expansion problem is presented and solved. The problem is set up in the framework of a minimal unitary dilation of a contraction acting on a Hilbert space and includes the Carathéodory and other classical interpolation problems. By combining the geometry of the minimal unitary dilation with state space techniques from system theory, a special solution is constructed. Next using the band method approach and spectral factorizations of this special solution a linear fractional parameterization of all solutions is obtained. Explicit state space formulas and applications to some classical interpolation problems are given.