Flatness of dynamically linearizable systems
Analysis and design of control laws for nonlinear systems is a difficult task in general. In the 1990s, Michel Fliess and colleagues realized however that some of the existing tools for linear systems control can be also applied to a particular class of nonlinear systems with smooth trajectories, called flat systems. Roughly speaking, a nonlinear system is flat if there exists a set of explicit functions, called flat outputs, such that all the system variables (state, inputs, outputs) can be expressed via these functions and their time derivatives. Interestingly, many physically relevant classes of nonlinear systems turn out to be flat. It has been known that flatness of a nonlinear system implies dynamic linearizability, a property which ensures that a control law can be applied so that the resulting closed-loop nonlinear system can be reduced to a linear system via a suitable change of variables. For this reason flat outputs are also sometimes called linearizing outputs. The main contribution of this paper is to show that the converse is also true, i.e. dynamical linearizability implies flatness. In order to prove this result, the author develops very technical proofs using an infinite-dimensional geometrical approach originally developed for partial differential equations.
scientific article; zbMATH DE number 2170253