## A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems (1999)

Venue: | 922–937. IN INDIRECT OPTIMAL CONTROL OF MULTIVARIABLE SYSTEMS 317 |

Citations: | 39 - 6 self |

### BibTeX

@INPROCEEDINGS{Fliess99alie-bäcklund,

author = {Michel Fliess and Jean Lévine and Pierre Rouchon},

title = {A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems},

booktitle = {922–937. IN INDIRECT OPTIMAL CONTROL OF MULTIVARIABLE SYSTEMS 317},

year = {1999}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract — In this paper, a new system equivalence relation, using the framework of differential geometry of jets and prolongations of infinite order, is studied. In this setting, two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of the other system and of a finite number of their time derivatives. This is a Lie–Bäcklund isomorphism. This quite natural, though unusual, equivalence is presented in an elementary way by the inverted pendulum and the vertical take-off and landing (VTOL) aircraft. The authors prove that, although the state dimension is not preserved, the number of input channels is kept fixed. They also prove that a Lie–Bäcklund isomorphism can be realized by an endogenous feedback, i.e., a special type of dynamic feedback. Differentially flat nonlinear systems, which were introduced by the authors in 1992 via differential algebraic techniques, are generalized here and the new notion of orbitally flat systems is defined. They correspond to systems which are equivalent to a trivial one, with time preservation or not. Trivial systems are, in turn, equivalent to any linear controllable system with the same number of inputs, and consequently flat systems are linearizable by endogenous feedback. The endogenous linearizing feedback is explicitly computed in the case of the VTOL aircraft to track given reference trajectories with stability; simulations are presented. Index Terms — Dynamic feedback, flatness, infinite-order prolongations, Lie–Bäcklund equivalence, nonlinear systems.

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Citation Context ... number of derivatives of , such that the resulting square system is left and right input–output invertible, with a trivial zero-dynamics. This proposition, which is an easy adaptation of a result of =-=[37]-=-, may be considered as an alternative definition of a flat system. However, with such a definition, the underlying feedback equivalence relation as well as the geometry of the problem would be less ap... |

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Citation Context ... 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 parki... |

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Citation Context ... 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 a... |

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Citation Context ...to be able to embrace, in particular, systems defined by a general vector differential equation of the form . 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 IEEEFLEISS et al.: APPROACH TO EQUIVALENCE AND FLATNESS 923 by Pomet [61]; see also an al... |

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Citation Context ...plications 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. This point of view is here enhanced by sketching both the motion plan... |

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Citation Context ...ori written in the above form, but as a mixture of differential and nondifferential, i.e., implicit equations (remember that such a situation often occurs in practice; see, e.g., [7], [22], [27], and =-=[54]-=-). The924 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 5, MAY 1999 Fig. 1. The VTOL aircraft in the vertical plane. equivalence we have in mind then becomes in this setting a change of (infin... |