ABSTRACT
Some mathematically challenging physical nonlinear systems are known to be non-differentially flat, like the popular “ball and beam”, the “double inverted pendulum”, the famous “Caltech ducted fan”, the “Kapitsa” pendulum, the “Furuta rotational pendulum” and some interesting chemical reactors (a collection of such systems has been reported in [22]). The lack of flatness is addressed as the defect and it is usually represented by the set of state variables of the system which are not expressible as differential functions of the flat outputs associated with the largest flat subsystem. A special class of non-flat systems, introduced by Chelouah [10] and Chelouah and Petitot [11], exist in which the defect variables can be expressed in terms of quadratures of differential functions of the flat outputs. These systems, addressed as “Liouvillian” systems, allow for systematic controller design procedures which exploit flatness at least at the off-line planning and analysis stages (see [33], [34]). The variable length pendulum, the underactuated ship, a simplified helicopter system, studied in this section, happen to be Liouvillian.
