State/Driving-Variable Representation of 2D Systems

We characterize the discrete 2D systems with kernel representation that admit a state/driving-variable (SDV) representation. This characterization is based on the possibility of decomposing a behaviour ß as the sum of its controllable part with a suitable autonomous part (controllable-autonomous decomposition). We show that ß has a SDV representation if and only if it allows for a controllable-autonomous decomposition where the autonomous part is SDV representable. This means that ß has a kernel representation matrix which can be decomposed as the product of two 2D L-polynomial matrices such that the left factor is factor left prime and the right factor is square and properly invertible.