Copyright © IFAC Nonlinear Control Systems, Stuttgart, Germany, 2004 ON TIMFrINVARlANT SYSTEMS POSSESSING TIMFrDEPENDENT FLAT OUTPUTS Paulo Sergio Pereira da Silva *,1 Pierre Rouchon .. ,1 * Universidade de Sao Paulo Escola Politecnica - PTC Dep. Eng. Telec. e Controle, Av. Prof. Luciano Gualberto, Trav. 03, no. 158, 05508-900, sAo PAULO - SP BRAZIL Email: paulo@lac.usp.br ** Ecole des Mines de Paris, Centre Automatique et Systemes, 60 Bd Saint-Michel, 75272 Paris cedex 06, FRANCE. Email: pierre.rouchon@ensmp.fr ELSEVIER IFAC PUBLICATIONS www.elsevier.comllocatdifac Abstract: The notion of flatness based on Cartan's absolute equivalence may admit a time-dependent flat output y( t, x) even for time-invariant systems. A natural question that arises is the following: does any flat time-invariant system always admit a time-invariant flat-output? In this paper, we show that this problem is equivalent to a (nonclassical) one-parameter group rectifiability. Copyright © 2004 IFAC Keywords: Nonlinear systems, time-invariant systems, flatness. 1. INTRODUCTION In (van Nieuwstadt et al., 1998), Murray and co- workers propose a definition of flat systems based on Cartan's absolute equivalence, that is slightly different than the original one due to Fliess and co-workers (Fliess et al., 1992; Fliess et al., 1995; Fliess et al., 1999). The difference between the two definitions mainly relies on the fact that a time-invariant system is flat, in the original setting, if it admits a time-invariant flat-output whereas in the second setting the flat output map could also depend on time.The following question becomes natural: does any time-invariant flat system always admit a time-invariant flat output? In this paper, we show that this open problem is equivalent to a rectifiability property of a one parameter group of transformations on an 1 The authors are partially supported by CAPES- COFECUB. The first author is also partially supported by Conselho Nacional de Desenvolvimento Cientffico e Tec- nol6gico-CNPq. infinite dimensional space, a trivial diffiety (Fliess et al., 1999). Take a control system :i; = f(x, u) with a time dependent flat-output y = het, x). This means that x and u depends on y and a finite number of its derivatives: x = A(t, y(O:)), u = BO(t, y(0:+1)) for some derivation order Q and where y(O:) stands for (y, iI, ... , y(O:)). Then a time translation t 1-+ T = t + a does not change the system. Thus y = het, x) and Y = het + a, x) are flat-outputs of the same system. This implies that for each a, the map (t, y, iI, .... ) 1 Ea (T = t + a, Y = het + a, A(t, y(O:))), Y, ... ) is invertible with inverse (T, Y, Y, .... ) 1 E- a (t = T - a, y = heT - a, A(T, y(O:))), iI, ... ). 20]