18 research outputs found
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
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
Necessary and Sufficient Conditions for Difference Flatness
We show that the flatness of a nonlinear discrete-time system can be checked
by computing a unique sequence of involutive distributions. The well-known test
for static feedback linearizability is included as a special case. Since the
computation of the sequence of distributions requires only the solution of
algebraic equations, it allows an efficient implementation in a computer
algebra program. In case of a positive result, a flat output can be obtained by
straightening out the involutive distributions with the Frobenius theorem. The
resulting coordinate transformation can be used to transform the system into a
structurally flat implicit triangular form. We illustrate our results by an
example.Comment: arXiv admin note: text overlap with arXiv:1907.0059