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

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.

Control via state estimations of some nonlinear systems

“…The presentation of all these system theoretic notions may also be given in a differential geometric framework by utilising diffieties, i.e., prolongations and jets of infinite order (see, e.g., Fliess et al, 1999, and the references therein).…”

“…Although the utilisation of (differentially) flat nonlinear systems (Fliess et al, 1995a(Fliess et al, , 1999 (see, also, Rudolph, 2003a, Sira-Ramírez & Agrawal, 2004 has become quite widespread in industry, a fundamental question like the estimation of state variables is still far from being fully understood. This communication is proposing a clear cut solution which is extending the state reconstructors for linear systems obtained in .…”

Flatness based trajectory generation of quantum systems

“…In (Vettori, 2002;Ferrante et al, 2002;Mirrahimi and Rouchon, 2004a;Mirrahimi and Rouchon, 2004b) Lyapounov based methods are proposed. This paper proposes flatness based methods (Fliess et al, 1995;Fliess et al, 1999) for steering from a pure state to another pure state. The main advantage of such method is its computational simplicity and its simple interpretation in physical terms.…”

Control of an industrial polymerization reactor using flatness

“…The flatness property of a (nonlinear) dynamical system x : =f(x, u) with x 2 R n , u 2 R m , (n, m) 2 N is described as follows [3,4] …”