Scientific Context:
Algebraic methods have been successfully applied to problems arising in systems theory for a very long time now (see Kalman, Rosenbrock, Kucera, Kailath).
The most important reasons are the following. Algebra provides structural information about the systems under consideration, which could not be obtained by
using numerical methods alone. An algebraic approach to linear control systems leads to a classification of structural properties of the system which can be
characterized in algebraic terms and checked by symbolic computation, e.g., the fundamental notions of controllability,observability, and flatness.
Using this algebraic language, multidimensional linear systems, also called n-dimensional systems, defined by partial differential equations, differential time-delay systems,
discrete systems, repetitive systems, hybrid systems etc. can be treated simultaneously; depending on the type of the linear system under consideration one deals with modules
defined over a (possibly non-commutative) polynomial ring which contains all functional operators that are present in the governing equations (e.g., differential/time-delay/shift operators).
Therefore algebra provides a unified approach to system theoretic phenomena which are common to many types of multidimensional linear systems.
More precisely, the general approach that we are using and developing is the following one:
- Translate the structural properties of multidimensional systems theory (e.g., controllability, observability, flatness) into algebraic terms by using classifications appearing in
module theory over non-commutative polynomial rings of functional operators (e.g., differential/time-delay/shift operators).
- Characterize the corresponding module properties in the classification (e.g., torsion, torsion-free, reflexive, projective, stably free, free, injective, cogenerator modules) in
terms of homological algebra.
- Develop computational methods (e.g., Gröbner and Janet bases over some classes of non-commutative polynomial rings) to turn the tools from homological algebra
constructive (e.g., free resolution, extension and torsion functors, projective dimensions).
In recent years such a framework has been developed (Oberst, Pommaret, Quadrat, Zerz, ChyzakQuadratRobertz), and new insight about the classification of structural properties of multidimensional linear control systems and the synthesis of control laws has already been obtained in a collaboration between Lehrstuhl B für Mathematik, RWTH Aachen University (Germany) and INRIA Sophia Antipolis (France). All results have been constructive and led to algorithms that were implemented in the Maple package OreModules, which implements
some methods from homological algebra (Cartan, MacLane, Rotman) and module theory with regard to control theoretic applications.
Teams:
France:
- Moulay Barkatou, Professor, University of Limoges, CNRS ; XLIM UMR 6172, DMI, 123 avenue Albert Thomas, 87060 Limoges cedex, France
- Thomas Cluzeau, Assistant Professor, University of Limoges, CNRS ; XLIM UMR 6172, DMI, 123 avenue Albert Thomas, 87060 Limoges cedex, France (leader of the french team)
- Alban Quadrat,
Research Scientist, INRIA SACLAY - ÎLE-DE-FRANCE, DISCO
project, L2S, Supélec, 3 rue Joliot Curie, 91192
Gif-sur-Yvette cedex, France
- Georg Regensburger, Research Scientist (PostDoc), INRIA
SACLAY - ÎLE-DE-FRANCE, DISCO project, L2S,
Supélec, 3 rue Joliot Curie, 91192 Gif-sur-Yvette cedex, France
- Jacques-Arthur Weil, Assistant Professor, University of Limoges, CNRS ; XLIM UMR 6172, DMI, 123 avenue Albert Thomas, 87060 Limoges cedex, France
- Carole El Bacha, PhD student, University of Limoges, CNRS ; XLIM UMR 6172, DMI, 123 avenue Albert Thomas, 87060 Limoges cedex, France
- Abdelkarim Chakhar, PhD student, University of Limoges, CNRS ; XLIM UMR 6172, DMI, 123 avenue Albert Thomas, 87060 Limoges cedex, France
Germany: Lehrstuhl B für Mathematik, RWTH Aachen University, http://wwwb.math.rwth-aachen.de and http://www.math.rwth-aachen.de
- Wolf Daniel Andres, Diploma student
- Martina Esser, Master student
- Sebastian Gutsche, Bachelor student
- Matthias Wisotzky, Bachelor student
- Sebastian Posur, Bachelor student
Objective of the collaboration:
See Section 2 in Proposal
References:
H. Cartan, S. Eilenberg, Homological Algebra, Princeton University Press, 1956.
F. Chyzak, A. Quadrat, D. Robertz, Effective algorithms for parametrizing linear control systems over Ore algebras, Appl. Algebra Engrg. Comm. Comput., 16 (2005), 319-376.
T. Kailath, Linear Systems, Prentice-Hall, 1980.
R. E. Kalman, P. L. Falb, M. A. Arbib, Topics in Mathematical System Theory, McGraw-Hill, 1969.
V. Kucera, Discrete Linear Control: The Polynomial Equation Approach, Wiley, Chichester 1979.
S. MacLane, Homology, Springer Verlag, 1995.
U. Oberst, Multidimensional constant linear systems, Acta Appl. Math., 20 (1990), 1-175.
J.-F. Pommaret, Partial Differential Control Theory, Kluwer Academic Publishers, Mathematics and Its Applications, 2001.
A. Quadrat, The fractional
representation approach to synthesis problems: an algebraic analysis
viewpoint. Part~I: (Weakly) doubly coprime factorizations, SIAM
J. Control Optim., 42 (2003), 266-299.
H. H. Rosenbrock, State Space and Multivariable Theory, Wiley, 1970.
J. J. Rotman, An Introduction to Homological Algebra, 2nd edition, Springer, 2009.
E. Zerz, Topics in Multidimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences 256, Springer, 2000.