Flat outputs of two-input driftless control systems

Abstract

Abstract.We study the problem of flatness of two-input driftless control systems. Although a characterization of flat systems of that class is known, the problems of describing all flat outputs and of calculating them is open and we solve it in the paper. We show that all x-flat outputs are parameterized by an arbitrary function of three canonically defined variables. We also construct a system of 1st order PDE's whose solutions give all x-flat outputs of two-input driftless systems. We illustrate our results by describing all x-flat outputs of models of a nonholonomic car and the n-trailer system.Mathematics Subject Classification. 93B27, 93C10, 93C95.

Multi-input control-affine systems static feedback equivalent to a triangular form and their flatness

“…This implies that there is more flat outputs for Σ af f than for the associated Σ lin . Actually, the condition (FO1) applied to Σ lin implies that (L g ϕ 0 , L g ϕ 1 )(x * ) = (0, 0) (thus obtaining the same necessary and sufficient conditions as those given in (Li and Respondek, 2012) for two-input control-linear systems ), whereas (FO1) applied to Σ af f still admits systems for which (L g ϕ 0 , L g ϕ 1 )(x * ) = (0, 0) as the following example shows.…”

“…As an immediate corollary of Theorem 3, we obtain a system of first order PDE's, described by Proposition 2 below, whose solutions give all x-flat outputs. Like for systems equivalent to the chained form (see (Li and Respondek, 2012)), x-flat outputs for the systems feedback equivalent to the triangular form T Ch k 1 are far from being unique: since the distribution C k−2 is involutive and of corank three, there are as many functions ϕ 0 satisfying L c ϕ 0 = 0, for any c ∈ C k−2 , as functions of three variables. Indeed, according to the following proposition, ϕ 0 can be chosen as any function of the three independent functions, whose differentials annihilate C k−2 , and if moreover, < dϕ 0 , G 0 > (x * ) = 0, then there exists a unique ϕ 1 (up to a diffeomorphism) completing it to an x-flat output.…”

“…Proof of (F2). It was shown in (Li and Respondek, 2012) that conditions (F O2) and (F O2) ′ are equivalent (for control-linear systems Σ lin but notice that (F O2) and (F O2) ′ do not involve the drift f ). We deduce immediately that (ii) ⇔ (iii).…”

Cascade-Like Modular Tracking Controller for non-Standard N-Trailers

“…One may predict that the recent trend for automation of guidance systems in commercial vehicles will also find applications in the domain of articulated vehicles which are especially difficult to control. Due to specific properties of N-trailer kinematics (investigated, e.g., in [1], [8], [10], [12], [13], and [26]), feedback control design for these systems is generally nontrivial. Most control solutions proposed in the literature so far for truly N-trailers (i.e., admitting arbitrary number of trailers) concern the time-noncritical tasks like the set-point stabilization and path following (see, for instance, [2]- [4], [9], [14], [15], [18], [21], [24], [27]), or address the control problems for the differentially flat so-called Standard N-Trailers (SNT) equipped solely with on-axle hitches (see [7], [19], [22], [24], [27]).…”

On computing flat outputs through Goursat normal form

“…Unfortunately, necessary and sufficient conditions to check flatness for a general nonlinear system do not exist. Since mid nineties, extensive work has been done in this direction, but only some particular cases have been solved ( [6], [8], [9], [11]). Control systems are usually presented in state space form.…”