Mathematical Theory of Networks and Systems 2006
The following articles were presented at
MTNS 2006, Kyoto (Japan):
Computing Invariants of Multidimensional Linear Systems on an Abstract Homological Level
Proceedings of MTNS 2006, Kyoto (Japan)
[.pdf]
Methods from homological algebra Rotman (1979)
play a more and more important role
in the study of multidimensional linear systems
Quadrat (1999),
Pommaret (2001),
Chyzak et al (2005).
The use of modules allows an algebraic treatment of linear systems
which is independent of their presentations by systems of equations.
The type of linear system (ordinary/partial differential equations,
time-delay systems, discrete systems...) is encoded in the
(non-commutative) ring of (differential, shift, ...) operators
over which the modules are defined. In this framework, homological
algebra gives very general information about the structural
properties of linear systems.
Homological algebra is a natural extension of the theory of modules over
rings. The category of modules and their homomorphisms is replaced by the
category of chain complexes and their chain maps. A module is represented
by any of its resolutions. The module is then recovered as the only
non-trivial homology of the resolution.
The notions of derived functors and their homologies,
connecting homomorphism and the resulting long exact homology sequences play
a central role in homological algebra.
The Maple package homalg
Barakat, Robertz (2006),
Barakat, Robertz
provides a way to deal with these
powerful notions. The package is abstract in the sense that it is
independent of any specific ring arithmetic. If one specifies a ring in
which one can solve the ideal membership problem and compute syzygies,
the above homological algebra constructions over that ring become
accessible using homalg.
In this paper we introduce the package
homalg and present several
applications of
homalg
to the study of multidimensional linear systems
using available Maple packages which provide the
ring arithmetics, e.g.
OreModules
Chyzak et al (2004),
Chyzak et al
and Janet Blinkov et al (2003),
Plesken, Robertz (2005).
Maple worksheets:
See also the Library of Examples.
Using morphism computations for factoring and decomposing general linear functional systems
INRIA Report 5942 and Proceedings of MTNS 2006, Kyoto (Japan)
[.pdf]
see also
OreModules subpackage OreMorphisms
Within a constructive homological algebra approach, we study the factorization and
decomposition problems for general linear functional systems and, in particular, for
multidimensional linear systems appearing in control theory. Using the concept of Ore
algebras of functional operators (e.g., ordinary/partial differential operators, shift
operators, time-delay operators), we first concentrate on the computation of morphisms
from a finitely presented left module M over an Ore algebra to another one
M', where M (resp., M') is a module intrinsically associated with
the linear functional system R y=0 (resp., R' z=0). These morphisms
define applications sending solutions of the system R' z=0 to the ones of R
y=0. We explicitly
characterize the kernel, image, cokernel and coimage of a general morphism. We then
show that the existence of a non-injective endomorphism of the module M is
equivalent to the existence of a non-trivial factorization
R=R2 R1 of the
system matrix R. The corresponding system can then be integrated in cascade.
Under certain conditions, we also show that
the system R y=0 is equivalent to a system R' z=0, where R' is a
block-triangular matrix.
We show that the existence of
projectors of the ring of endomorphisms of the module M allows us to reduce the
integration of
the system R y=0 to the integration of
two independent systems R1 y1=0 and
R2 y2=0. Furthermore, we prove
that, under certain
conditions, idempotents provide decompositions of the
system R y=0, i.e., they allow us to compute an equivalent system
R' z=0, where R' is a block-diagonal matrix.
Many applications of these results in mathematical physics and control
theory are given. Finally, the different algorithms of the paper are implemented in a
package
OreMorphisms
based on the library
OreModules.
Flat multidimensional linear systems with constant coefficients are equivalent to controllable 1-D linear systems
Proceedings of MTNS 2006, Kyoto (Japan)
[.pdf]
Based on constructive proofs of the Quillen-Suslin theorem, the purpose of
this paper is to show that every flat multidimensional linear
system with constant coefficients is equivalent to a controllable 1-D linear system.
This
result looks like the classical result in non-linear control theory stating that every
flat ordinary
differential non-linear system is equivalent to a controllable ordinary
differential linear system. In particular, we prove that every flat
differential time-delay linear system is equivalent to the ordinary
differential controllable linear system obtained by setting all
the delay amplitudes to 0. This result allows to transfer synthesis problems onto the
equivalent ordinary differential linear system without delays, which sometimes
simplifies the construction of stabilizing controllers. Finally, using algorithmic
versions of the Quillen-Suslin theorem, we give a constructive proof of Pommaret's proof
Pommaret (2001)
of the Lin-Bose conjecture
Lin, Bose (2001)
and we show how to compute
(weakly) doubly coprime factorizations of rational transfer matrices. All the results
are illustrated on explicit examples and the different algorithms have been implemented
in
OreModules.
See also: A package for computing bases of free modules over commutative polynomial rings.
Constructive computation of flat outputs of a class of multidimensional linear systems with variable coefficients
Proceedings of MTNS 2006, Kyoto (Japan)
[.pdf]
The purpose of this paper is to give a constructive algorithm for the computation of
bases of finitely presented free modules over the Weyl algebras of differential
operators with polynomial or rational coefficients.
In particular, we show how to use these results in order to recognize when a
multidimensional linear system defined by partial differential equations with polynomial
or rational coefficients is flat and, if so, to compute flat outputs and the injective
image representations of the system. These new results are based on recent constructive
proofs of a famous result in non-commutative algebra due to J. T. Stafford
Stafford (1978).
The different algorithms have been
implemented in the package
Stafford
based on
OreModules.
These results allow us to achieve the general solution of the so-called Monge problem
for multidimensional linear systems defined by partial differential equations with
polynomial or rational coefficients.
Finally, we constructively answer an open question posed by Datta
(Datta (2001))
on the possibility to generalize the results of
Malrait et al (2001)
to multi-input
multi-output polynomial time-varying controllable linear systems. We show that every
controllable ordinary differential linear system with at least two inputs and polynomial
coefficients is flat.
Maple worksheets:
See also the Library of Examples.
On the Monge problem and multidimensional optimal control
Proceedings of MTNS 2006, Kyoto (Japan)
[.pdf]
Using new results on the general Monge parametrization (see
Zervos (1932),
and the references therein) recently
obtained in
Quadrat, Robertz (2005),
i.e., on the possibility to extend the concept of
image representation to non-controllable
multidimensional linear systems, we show that we can
transform some quadratic variational problems (e.g., optimal control problems) with
differential
constraints into free variational ones directly solvable by means of the standard
Euler-Lagrange equations. This result generalizes
for non-controllable multidimensional linear systems the results obtained in
Pillai, Willems (2002),
Pommaret, Quadrat (2004)
for controllable ones. In particular, in the 1-D
case, this result allows us to avoid the controllability condition commonly
used in the behavioural approach literature for the study of
optimal control problems with a finite horizon and replace it by the
stabilizability condition for the ones with an infinite horizon.