The algebraic structure of DD systems : a behavioral perspective

This paper presents a survey on the recent contributions to linear time-invariant delaydifferential systems in the behavioral approach. In this survey both systems with commensurate and with incommensurate delays will be considered. The emphasis lies on the investigation of the relationship between various systems descriptions. While this can be understood in a completely algebraic setting for systems with commensurate delays, this is not the case for systems with incommensurate delays. In the study of this class of systems functional analitic methods need to be introduced and general convolutional equations have to be incorporated. Whenever it is possible, the results will be linked to the relevant control theoretic notions.

[1]  R. Tennant.  Algebra , 1941, Nature.

[2]  Olaf Helmer,et al.  The elementary divisor theorem for certain rings without chain condition , 1943 .

[3]  L. Schwartz.  Theorie Generale des Fonctions Moyenne-Periodiques , 1947 .

[4]  Irving Kaplansky,et al.  Elementary divisors and modules , 1949 .

[5]  Ivan Niven.  Irrational Numbers: THE GENERALIZED LINDEMANN THEOREM , 1956 .

[6]  L. Ehrenpreis,et al.  Solutions of Some Problems of Division: Part III. Division in the Spaces, D , H, Q A , O , 1956 .

[7]  B. Malgrange.  Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution , 1956 .

[8]  L. Ehrenpreis,et al.  SOLUTIONS OF SOME PROBLEMS OF DIVISION.* Part V. Hyperbolic Operators. , 1962 .

[9]  A. Zemanian,et al.  Distribution theory and transform analysis , 1966 .

[10]  F. Trèves.  Topological vector spaces, distributions and kernels , 1967 .

[11]  P. Cohn.  Free rings and their relations , 1973 .

[12]  Carlos A. Berenstein,et al.  The Ritt theorem in several variables , 1974 .

[13]  N. Jacobson,et al.  Basic Algebra I , 1976 .

[14]  Ahmad Diab.  Quotient et zéros communs de deux polynômes exponentiels. , 1975 .

[15]  G. Meisters.  Periodic distributions and non-Liouville numbers , 1977 .

[16]  H. Brezis.  Analyse fonctionnelle : théorie et applications , 1983 .

[17]  E. Kamen,et al.  Proper stable Bezout factorizations and feedback control of linear time-delay systems† , 1986 .

[18]  C. Berenstein,et al.  Ideals generated by exponential-polynomials , 1986 .

[19]  N. Bourbaki,et al.  Topological Vector Spaces: Chapters 1–5 , 1987 .

[20]  Luciano Pandolfi,et al.  Null controllability of a class of functional differential systems , 1988 .

[21]  François Parreau,et al.  Schwartz's theorem on mean periodic vector-valued functions , 1989 .

[22]  Ulrich Oberst,et al.  Multidimensional constant linear systems , 1990, EUROCAST.

[23]  G. Folland.  Fourier analysis and its applications , 1992 .

[24]  C. Berenstein,et al.  Complex Analysis and Convolution Equations , 1993 .

[25]  Heinz Kredel,et al.  Gröbner Bases: A Computational Approach to Commutative Algebra , 1993 .

[26]  J. Willems.  On interconnections, control, and feedback , 1997, IEEE Trans. Autom. Control..

[27]  Xi Ciclo.  Delay{Dierential Systems in the Behavioral Approach , 1998 .

[28]  Hugues Mounier,et al.  Algebraic interpretations of the spectral controllability of a linear delay system , 1998 .

[29]  O. Lezama,et al.  On the Simultaneous Basis Property in Prüfer Domains , 1998 .

[30]  Arjeh M. Cohen,et al.  Some tapas of computer algebra , 1999, Algorithms and computation in mathematics.

[31]  Luc C. G. J. M. Habets,et al.  System Equivalence for AR-Systems over Rings—with an Application to Delay-Differential Systems , 1997, 1997 European Control Conference (ECC).

[32]  Sandro Zampieri,et al.  Controllability of Systems Described by Convolutional or Delay-Differential Equations , 2000, SIAM J. Control. Optim..

[33]  Heide Gluesing-Luerssen,et al.  A Convolution Algebra of Delay-Differential Operators and a Related Problem of Finite Spectrum Assignability , 2000, Math. Control. Signals Syst..

[34]  L. Habets,et al.  Behavioral controllability of time-delay systems with incommensurable delays , 2000 .

[35]  Paula Rocha,et al.  Trajectory Control and Interconnection of 1D and nD Systems , 2001, SIAM J. Control. Optim..

[36]  Heide Gluesing-Luerssen,et al.  Linear Delay-Differential Systems with Commensurate Delays: An Algebraic Approach , 2001 .

[37]  Sandro Zampieri,et al.  Some results on systems described by convolutional equations , 2001, IEEE Trans. Autom. Control..

[38]  Jeffrey S. Rosenthal,et al.  Introduction to mathematical systems theory. a behavioral approach [Book Review] , 2002, IEEE Transactions on Automatic Control.