Flatness of Discrete-time Systems, a simple Approach

Flatness of discrete-time systems can be characterized by two simple properties. There exists a map, a submersion, from the flat coordinates and their forward shifts to the state and the input of the discrete-time system, such that the system equations are fulfilled identically. Flat coordinates, together with their shifts, also describe simple shift systems. Therefore, this map transfers dynamic systems to dynamic systems. Based on these facts, necessary conditions for a system, to be flat, are derived. It is worth mentioning that methods from differential geometry are not required. But these methods are used to derive algorithms for the test, whether a system is flat or not, and they are used to derive a flat parametrization.