Differential–geometric decomposition of flat nonlinear discrete-time systems

We prove that every flat nonlinear discrete-time system can be decomposed by coordinate transformations into a smaller-dimensional subsystem and an endogenous dynamic feedback. For continuous-time systems, no comparable result is available. The advantage of such decomposition the complete if only flat. Thus, repeating at most n−1 times, where n dimension state space, flatness checked in algorithmic way. If flat, then algorithm yields output which depends on variables. Hence, has does not depend inputs their forward-shifts. Again, for systems requires each step construction state- input transformations, are obtained straightening out certain vector fields or distributions with flow-box theorem Frobenius theorem. from computational point view, calculation flows solution algebraic equations needed. illustrate our results two examples.