# Why differential flatness is called “flat”?

A system is differentially flat if the state and control input can be written as functions of the flat outputs and their time derivatives.

In other words, if a system is differentially flat, we can represent the state of the system and control signal as: $$x = x(y, \dot y, \ddot y, ... , y^{(p)}$$ $$u = u(y, \dot y, \ddot y, ... , y^{(q)}$$

It's really a nice property of the system. But I don't quite get the name of this property. Why it is called differential "flatness". What does it have to do with "flat"?

• Not a complete answer, but (Fliess 1995) says: "the terminology flat is due to the fact that y plays a somehow analogous role to the flat coordinates in a differential geometric approach to the Frobenius theorem" (p. 3). Also, there is a geometric definition of "flat" which might be coming into play. – Parker Lusk Oct 3 '18 at 18:37