We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman's controllability. The distance to flatness is measured by a non-negative integer, the defect. We utilize differential algebra which suits well to the fact that, in accordance with Willems' standpoint, flatness and defect are best defined without distinguishing between input, state, output and other variables. Many realistic classes of examples are flat. We treat two popular ones: the crane and the car with n trailers, the motion planning of which is obtained via elementary properties of planar curves. The three non-flat examples, the simple, double and variable length pendulums, are borrowed from nonlinear physics. A high frequency control strategy is proposed such that the averaged systems become flat.

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.

Abstract.A closed loop parametrical identification procedure for continuous-time constant linear systems is introduced. This approach which exhibits good robustness properties with respect to a large variety of additive perturbations is based on the following mathematical tools:(1) module theory; (2) differential algebra; (3) operational calculus. Several concrete case-studies with computer simulations demonstrate the efficiency of our on-line identification scheme.1991 Mathematics Subject Classification. 93B30, 93B35, 93C05, 93C73, 93C95, 12H05, 13C05, 44A40.

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