On Necessary and Sufficient Conditions for Differential Flatness

This paper is devoted to the characterization of differentially flat nonlinear systems in implicit representation, after elimination of the input variables, in the differential geometric framework of manifolds of jets of infinite order. We extend the notion of Lie-B\"acklund equivalence, introduced in Fliess et al. (1999), to this implicit context and focus attention on Lie-B\"acklund isomorphisms associated to flat systems, called trivializations. They can be locally characterized in terms of polynomial matrices of the indeterminate \ddt, whose range is equal to the kernel of the polynomial matrix associated to the implicit variational system. Such polynomial matrices are useful to compute the ideal of differential forms generated by the differentials of all possible trivializations. We introduce the notion of a strongly closed ideal of differential forms, and prove that flatness is equivalent to the strong closedness of the latter ideal, which, in turn, is equivalent to the existence of solutions of the so-called generalized moving frame structure equations. Two sequential procedures to effectively compute flat outputs are deduced and various examples and consequences are presented.Comment: Version 3 is the published versio

On Barriers in State and Input Constrained Nonlinear Systems

In this paper, the problem of state and input constrained control is addressed, with multidimensional constraints. We obtain a local description of the boundary of the admissible subset of the state space where the state and input constraints can be satisfied \emph{for all times}. This boundary is made of two disjoint parts: the subset of the state constraint boundary on which there are trajectories pointing towards the interior of the admissible set or tangentially to it; and a barrier, namely a semipermeable surface which is constructed via a minimum-like principle.Comment: 36 pages, 8 figures, submitte

On the computation of π\pi-flat outputs for differential-delay systems

We introduce a new definition of $\pi$-flatness for linear differential delay systems with time-varying coefficients. We characterize $\pi$- and $\pi$-0-flat outputs and provide an algorithm to efficiently compute such outputs. We present an academic example of motion planning to discuss the pertinence of the approach.Comment: Minor corrections to fit with the journal versio

Differential Flatness by Pure Prolongation: Necessary and Sufficient Conditions

In this article, we introduce the notion of differential flatness by pure prolongation: loosely speaking, a system admits this property if, and only if, there exists a pure prolongation of finite order such that the prolonged system is feedback linearizable. We obtain Lie-algebraic necessary and sufficient conditions for a general nonlinear multi-input system to satisfy this property. These conditions are comprised of the involutivity and relative invariance of a pair of filtrations of distributions of vector fields. An algorithm computing the minimal prolongation lengths of the input channels that achieve the system linearization, yielding the associated flat outputs, is deduced. Examples that show the efficiency and computational tractability of the approach are then presented

Flatness Characterization: Two Approaches

We survey two approaches to flatness necessary and sufficient conditions and compare them on examples

Towards a Computer Algebraic Algorithm for Flat Output Determination

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

A flatness-based nonlinear predictive approach for crane control

We study in this paper a flatness-based nonlinear predictive control law for a reduced size model of a crane studied in (Kiss, 2001; Kiss et al., 1999; Kiss et al., 2000a; Kiss et al., 2000b). The controller is composed of two parts: the first one is a traditional PD output feedback to track the reference trajectory and reject small perturbations, the second one consists of updating the reference trajectory from the current estimated state of the crane to the desired equilibrium point on a receding horizon each time the pursuit error exceeds a given threshold. Simulations are presented to illustrate its performances

Active estimation of the initial phase for brushless synchronous motors

International audienceThis paper addresses the initial phase estimation problem for brushless synchronous motors. Only displacement measurements are used (no current) and friction, load and motor parameters are supposed to be unknown. Because of friction, the system is modelled by a differential equation with discontinuous right-hand side. Specific open-loop inputs are designed (active method) to get the initial phase as a function of the magnitude of the displacements along the corresponding trajectories. The estimation relies on a complete classification of the possible dynamical behaviours of the considered discontinuous right-hand side system with periodic input, whatever values the unknown parameters may take. We propose an approximated formula of the initial phase. Some experimental results are given, together with a comparison of our method to a classical procedur