Towards a computer algebraic algorithm for flat output determination

Abstract

This contribution deals with nonlinear control systems. More precisely, we are interested in the formal computation of a so-called flat output, a particular generalized output whose property is, roughly speaking, that all the integral curves of the system may be expressed as smooth functions of the components of this flat output and their successive time derivatives up to a finite order (to be determined). Recently, a characterization of such flat output has been obtained in [14,15], in the framework of manifolds of jets of infinite order (see e.g. [18,9]), that yields an abstract algorithm for its computation. In this paper it is discussed how these conditions can be checked using computer algebra. All steps of the algorithm are discussed for the simple (but rich enough) example of a non holonomic car.

On necessary and sufficient conditions for differential flatness

“…. , n − m, the system is in the implicit form (2), and since ∂F ∂ẋ = diag{I n−m , G}, G being the matrix made of the entries − m k=1 ∂f i ∂u k ∂µ k ∂ẋ j for i, j = n − m + 1, . .…”

“…Cartan's notion of absolute equivalence [7] (already noted by W. Shadwick [60]) and via D. Hilbert's concept of invertible and without integral transformations (umkehrbar, integrallos transformationen in German) [25]. The second one is related to the notion of parameterization: using the definition presented in this paper, with, in place of (1), the set of n − m implicit equations (2) introduced in the next section, where the control variables u are eliminated, flatness may be seen as a generalization in the framework of manifolds of jets of infinite order of the uniformization of analytic functions of Hilbert 's 22nd problem [24], solved by Poincaré [45] in 1907 (see [5] for a modern presentation of this subject and recent extensions and results). This problem consists, roughly speaking, given a set of complex polynomial equations in one complex variable, in finding an open dense subset D of the complex plane C and a holomorphic function s from D to C such that s is surjective and s(p) identically satisfies the given equations for all values of the "parameter" p ∈ D. In our setting, C is replaced by a (real) manifold of jets of infinite order, a flat output y 1 , .…”

On Symbolic Computation of Flat Outputs for Differentially Flat Systems

“…Constructive algorithms, relying on standard computer algebra environments, may be found e.g. in [23,24] for nonlinear finite-dimensional systems, or [25,26] for linear systems over Ore algebras.…”

An efficient algorithm for checking hyper-regularity of matrices

“…By adaption of the base field it is also possible to apply the algorithm for the flatness analysis of nonlinear control systems as proposed in [7,8,1].…”