Controlling Nonlinear Systems by Flatness

A Lie-Backlund approach to equivalence and flatness of nonlinear systems

“…Consider the dynamics (16) where is smooth on an open subset , is the state, and is the control input. The mapping is in fact an infinite collection of vector fields parameterized by .…”

“…This infinite-dimensional dependence on the input is not well-adapted when considering dynamic feedback. According to the examples of the preceding section, we develop a slightly different standpoint where the integral curves of (16) (17) with . Therefore, (17) defines a vector field, in the classic sense, on the infinite-dimensional manifold .…”

“…The differential-algebraic framework is now replaced by a differential geometric one (see [16], [20], [21], and [24]). A related and independent approach has also been developed 0018-9286/99$10.00 © 1999 IEEE by Pomet [61]; see also an algebraic alternative approach by Jakubczyk [38], [39].…”

“…Precisely, according to the flat output properties, system trajectories exactly joining a collection of points with given velocities, accelerations, jerks, etc., are easily generated (see [22]), thus replacing difficult dynamical computations by static interpolation techniques. Moreover, a tracking architecture for which the controller is fixed once and for all, independently of the particular trajectory to follow, may be obtained with stability (see [16]). Finally, let us mention the major difficulties to obtain computable criteria for checking flatness.…”

“…Nevertheless, a large number of engineering examples from many fields stemming from mechanics to chemical or biological processes, enhance the authors' belief that, since flat outputs contain all the required dynamical informations to run the system, they may often be found by inspection among the key physical variables. This feature is particularly evident on the VTOL example (see [31], [45], and [49]), whose flat output is made up with the coordinates of the Huygens oscillation center and in the numerous applications to food engineering processes [5], car equipment design [6], car parking [16], [22], [67], crane control [22], [27], [44], tracking observers [28], magnetic bearings control [43], aircraft [48], [49], and helicopter control [55], induction motors [53], chemical reactors [64], [65], etc., recently treated by this approach.…”