ABSTRACT

A nonlinear system with one input is said to be differentially flat if there exist a differential function of the state (i.e., it does not satisfy any differential equation by itself and, additionally, it is a function of the state and of a finite number of its time derivatives), called the flat output, such that all variables in the system: i.e., states, outputs and inputs, are, in turn, expressible as differential functions of the flat output. Flatness was introduced, by Prof. M. Fliess and his colleagues in a series of very interesting papers ([6]-[8]) some years ago. Contrary to unwarranted belief, flatness is not just another way to do feedback linearization for nonlinear systems. It is, in fact, a structural property of the system that allows one to establish all the salient features which are needed for the application of a particular feedback controller design technique (like,: back-stepping, passivity, and, of course, feedback linearization). It is a property that trivializes the exact linearization problem in a nonlinear system, whether or not the system is multivariable, and whether or not it is affine in the control inputs. Moreover, flatness and its consequences directly applies to any nonlinear system, regardless of the nonlinear, or affine, nature of the control inputs in the system equations. Flatness immediately yields the required open loop (nominal) behavior of the system for a particular desired trajectory tracking task. It is, therefore, most suitable for trajectory planning, controller saturation avoidance, and predictive control, specially for those output trajectory tracking cases involving non-minimum phase outputs (see [15] and [9]). We also emphasize a less recognized feature of flatness, which is related to the possibilities of nonlinear system analysis. While it provides a natural and useful parametrization of constant equilibria, it also helps in determining the minimum or non-minimum phase character of system outputs. Constant input state detectability can also be determined through flatness in a direct fashion, just to mention but a few.