12 research outputs found
A Flat System Possessing no (x,u)-Flat Output
In general, flat outputs of a nonlinear system may depend on the system's
state and input as well as on an arbitrary number of time derivatives of the
latter. If a flat output which also depends on time derivatives of the input is
known, one may pose the question whether there also exists a flat output which
is independent of these time derivatives, i.e., an (x,u)-flat output. Until
now, the question whether every flat system also possesses an (x,u)-flat output
has been open. In this contribution, this conjecture is disproved by means of a
counterexample. We present a two-input system which is differentially flat with
a flat output depending on the state, the input and first-order time
derivatives of the input, but which does not possess any (x,u)-flat output. The
proof relies on the fact that every (x,u)-flat two-input system can be exactly
linearized after an at most dim(x)-fold prolongation of one of its (new) inputs
after a suitable input transformation has been applied
Flatness of two-input control-affine systems linearizable via a two-fold prolongation
International audienceWe study flatness of two-input control-affine systems, defined on an n-dimensional state-space. We give a geometric characterization of systems that become static feedback linearizable after a two-fold prolongation of a suitably chosen control. They form a particular class of flat systems: they are of differential weight n + 4. We present a normal form compatible with the minimal flat outputs
A Structurally Flat Triangular Form Based on the Extended Chained Form
In this paper, we present a structurally flat triangular form which is based
on the extended chained form. We provide a complete geometric characterization
of the proposed triangular form in terms of necessary and sufficient conditions
for an affine input system with two inputs to be static feedback equivalent to
this triangular form. This yields a sufficient condition for an affine input
system to be flat.Comment: arXiv admin note: substantial text overlap with arXiv:2002.0120
Dynamic Feedback Linearization of Control Systems with Symmetry
Control systems of interest are often invariant under Lie groups of
transformations. Given such a control system, assumed to not be static feedback
linearizable, a verifiable geometric condition is described and proven to
guarantee its dynamic feedback linearizability. Additionally, a systematic
procedure for obtaining all the system trajectories is shown to follow from
this condition. Besides smoothness and the existence of symmetry, no further
assumption is made on the local form of a control system, which is therefore
permitted to be fully nonlinear and time varying. Likewise, no constraints are
imposed on the local form of the dynamic compensator. Particular attention is
given to those systems requiring non-trivial dynamic extensions; that is,
beyond augmentation by chains of integrators. Nevertheless, the results are
illustrated by an example of each type. Firstly, a control system that can be
dynamically linearized by a chain of integrators, and secondly, one which does
not possess any linearizing chains of integrators and for which a dynamic
feedback linearization is nevertheless derived. These systems are discussed in
some detail. The constructions have been automated in the Maple package
DifferentialGeometry.Comment: 41 pages, minor revisions and error correction
Algorithmic Transverse Feedback Linearization
The feedback equivalence problem, that there exists a state and feedback transformation between two control systems, has been used to solve a wide range of problems both in linear and nonlinear control theory. Its significance is in asking whether a particular system can be made equivalent to a possibly simpler system for which the control problem is easier to solve. The equivalence can then be utilized to transform a solution to the simpler control problem into one for the original control system.
Transverse feedback linearization is one such feedback equivalence problem. It is a feedback equivalence problem first introduced by Banaszuk and Hauser for feedback linearizing the dynamics transverse to an orbit in the state-space. In particular, it asks to find an equivalence between the original nonlinear control-affine system and two subsystems: one that is nonlinear but acts tangent to the orbit, and another that is a controllable, linear system and acts transverse to the orbit. If this controllable, linear subsystem is stabilized, the original system converges upon the orbit.
Nielsen and Maggiore generalized this problem to arbitrary smooth manifolds of the state-space, and produced conditions upon which the problem was solvable. Those conditions do not help in finding the specific transformation required to implement the control design, but they did suggest one method to find the required transformation. It relies on the construction of a mathematical object that is difficult to do without system-specific insight.
This thesis proposes an algorithm for transverse feedback linearization that computes the required transformation. Inspired by literature that looked at the feedback linearization and dynamic feedback linearization problems, this work suggests turning to the "dual space" and using a tool known as the derived flag. The algorithm proposed is geometric in nature, and gives a different perspective on, not just transverse feedback linearization, but feedback linearization problems more broadly
New Directions for Contact Integrators
Contact integrators are a family of geometric numerical schemes which
guarantee the conservation of the contact structure. In this work we review the
construction of both the variational and Hamiltonian versions of these methods.
We illustrate some of the advantages of geometric integration in the
dissipative setting by focusing on models inspired by recent studies in
celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282