THE DIFFERENTIAL ALGEBRAIC APPROACH TO NONLINEAR SYSTEMS
Michel Fliess (ENS de Cachan, France), Jean Lévine (Ecole
des Mines de Paris, France), Pierre Rouchon (Ecole des Mines de Paris,
France), and Joachim Rudolph (TU Dresden, Germany)
Differential algebraic methods are used in nonlinear control theory since 1985 and since then led to a deeper understanding of the underlying concepts and to definitions of most useful new concepts. Although flatness is principally not tied to differential algebra, probably the most important outcome of this approach is the notion of differential flatness. Broadly speaking, differentially flat systems are those admitting a complete, finite and free differential parametrisation. This means that they can be described by a finite set of variables the trajectories of which can be assigned independently. The class of differentially flat systems, hence, generalises the class of linear controllable systems. Its importance is due to two facts: On the one hand, many mathematical models of technological processes have been shown to be flat systems; on the other hand, for the flat systems powerful and simple systematic methods are available for the motion planning and the design of feedback laws for stable trajectory tracking. Finally, this notion can be most fruitfully generalised to linear and nonlinear infinite dimensional systems, in particular to boundary controlled distributed parameter systems and linear or nonlinear systems with delays.
The lectures, presented by leading experts of the field, are thought to give an introduction to the differential algebraic approach to nonlinear systems, with an emphasis on differential flatness, and to show the bridges to linear and nonlinear infinite dimensional systems. Herein, the linear systems of finite or infinite dimension are treated in a module theoretic framework, the "linear analogue'' of differential algebra. Concepts and methods are illustrated on an important number of technological applications.