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 flat continuous-time systems, no comparable result is available. The advantage of such a decomposition is that the complete system is flat if and only if the subsystem is flat. Thus, by repeating the decomposition at most n - 1 times, where n is the dimension of the state space, the flatness of a discrete-time system can be checked in an algorithmic way. If the system is flat, then the algorithm yields a flat output which only depends on the state variables. Hence, every flat discrete-time system has a flat output which does not depend on the inputs and their forward-shifts. Again, no comparable result for flat continuous-time systems is available. The algorithm requires in each decomposition step the construction of state-and input transformations, which are obtained by straightening out certain vector fields or distributions with the flow-box theorem or the Frobenius theorem. Thus, from a computational point of view, only the calculation of flows and the solution of algebraic equations is needed. We illustrate our results by two examples. (C) 2021 Elsevier Ltd. All rights reserved.

Automated summary

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. No comparable result is available for flat continuous-time systems. The advantage of such a decomposition is that the complete system is flat if and only if the subsystem is flat. The algorithm for decomposition requires constructing state-and input transformations, obtained by straightening out certain vector fields or distributions, making the computation mainly involve the calculation of flows and the solution of algebraic equations.