Modules and behaviours in nD systems theory

Multidimensional Systems and Signal Processing

Volumn 11, Issue 1, 2000, Pages 11-48

  Wood, Jeffrey ^a  

^a UNIVERSITY OF SOUTHAMPTON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL COMPLEXITY; LINEAR CONTROL SYSTEMS; MATRIX ALGEBRA; OBSERVABILITY; STATE SPACE METHODS; SYSTEMS ANALYSIS; THEOREM PROVING;

BEHAVIOURAL THEORY; MODULE THEORY; ROSENBROCK SYSTEM MATRICES; SYSTEM COMPLEXITY; SYSTEM OBSERVABLES; TRANSFER MATRICES;

SYSTEM THEORY;

EID: 0033717468     PISSN: 09236082     EISSN: None     Source Type: Journal    
DOI: 10.1023/a:1008430528456     Document Type: Article

References (67)

1. T. Becker and V. Weispfenning, Gröbner Bases: A Computational Approach to Commutative Algebra, volume 141 of Graduate Texts in Mathematics, Springer-Verlag, 1993.
2. J.-E. Björk, Rings of Differential Operators, North-Holland Publishing Company, 1979.
3. H. Bourlès and M. Fliess, "Finite Poles and Zeros of Linear Systems: An Intrinsic Approach," Int. J. of Control, vol. 68, no. 4, 1997, pp. 897-922.
4. B. Buchberger, "Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory," In N. K. Bose, editor, Multidimensional systems theory, D. Reidel, 1985, pp. 184-232.
5. P. M. Cohn, Algebra, volume 2. Chinchester, UK, John Wiley and Sons, 2nd edition, 1989.
6. D. Cox, J. Little, and D. O'Shea, Ideals, Varieties and Algorithms, Springer-Verlag, 2nd edition, 1996.
7. E. Delaleau and M. Fliess, "Algorithme de Structure, Filtrations et Découplage," Computes Rendus Hebdomadaires des Séances de l'Academie des Sciences, Serie I, vol. 315, 1992, pp. 101-106.
8. D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, volume 150 of Graduate Texts in Mathematics, Springer-Verlag, 1995.
9. M. Fliess, "Generalized Controller Canonical Forms for Linear and Nonlinear Dynamics," IEEE Trans. on Auto. Contr., vol. 35, 1990, pp. 994-1001.
10. M. Fliess, "Some Basic Structural Properties of Generalized Linear Systems," Systems and Control Letters, vol. 15, 1990, pp. 391-396.
11. M. Fliess, "Controllability Revisited," In A. C. Antoulas, editor, Mathematical System Theory (the influence of R. E. Kalman), pp. 463-474, Springer-Verlag, 1991.
12. M. Fliess, "A Remark on Willems' Trajectory Characterization of Linear Controllability," Syst. Contr. Lett., vol. 19, 1992, pp. 43-45.
13. M. Fliess, "Une Interprétation Algébrique de la Transformation de Laplace et des Matrices de Transfert," Linear Algebra and Its Applications, vol. 203-204, 1994, pp. 429-442.
14. M. Fliess, J. Lévine, P. Martin, and P. Rouchon, "Flatness and Defect of Non-Linear Systems: Introductory Theory and Examples," International Journal of Control, vol. 61, no. 6, 1995, pp. 1327-1361.
15. E. Fornasini, P. Rocha, and S. Zampieri, "State Space Realization of 2-D Finte-Dimensional Behaviours," SIAM J. on Contr. and Opt., vol. 31, no. 6, 1993, pp. 1502-1517.
16. E. Fornasini and M. E. Valcher, "Algebraic Aspects of 2D Convolutional Codes," IEEE Trans. in Inf. Tech., vol. 33, 1993, pp. 1210-1225.
17. E. Fornasini and M. E. Valcher. "nD Polynomial Matrices with Applications to Multidimensional Signal Analysis," Multidimensional Systems and Signal Processing, vol. 8, no. 4, 1997, pp. 387-408.
18. S. Fröhler and U. Oberst, "Continuous Time-Varying Linear Systems, Syst. Contr. Lett., vol. 35, no. 2, 1998, pp. 97-110.
19. H. Glüsing-Lüerßen, "A Behavioural Approach to Delay-Differential Systems," SIAM J. Control Optim., vol. 35, no. 2, 1997, pp. 480-499.
20. H. Goldschmidt, "Integrability Criterion for Systems of Nonlinear Partial Differential Equations," J. Differential Geom., vol. 1, 1969, pp. 269-307.
21. L. C. G. J. M. Habets. "System Equivalence for Delay-Differential Systems with Incommensurable Delays," Presented at the 13th International Symposium on the Mathematical Theory of Networks and Systems, Padua, Italy, 1998.
22. M. Janet, "Les Systèms d'Équations aux Dérivées Partielles," J. Math. Pure Appl., vol. 3, 1920, p. 65.
23. R. E. Kalman, P. L. Falb, and M. A. Arbib, Topics in Mathematical Systems Theory, International Series in Pure and Applied Mathematics, McGraw-Hill, 1969.
24. S. Kleon and U. Oberst, "Transfer Operators and State Spaces for Discrete Multidimensional Linear Systems," Acta Applicandae Mathematicae, vol. 57, 1999, pp. 1-82.
25. J. Komorník, P. Rocha, and J. C. Willems. "Closed Subspaces, Polynomial Operators in the Shift, and ARMA Representations," Applied Mathematics Letters, vol. 4, no. 3, 1991, pp. 15-19.
26. V. Lomadze, "Finite-Dimensional Time-Invariant Linear Dynamical Systems: Algebraic Theory," Acta Appliacandae Mathematicae, vol. 19, 1990, pp. 149-201.
27. H. Mounier, Propriétés Structurelles des Systèmes Linéaires A Retards: Aspects Théoriques et Pratiques, PhD thesis, Université Paris-Sud, 1995.
28. M. Napoli and S. Zampieri, "2D Proper Rational Matrices and Causal Input/Output Representations of 2D Behavioural Systems," SIAM J. on Control and Optimization, vol. 37, 1999, pp. 1538-1552.
29. U. Oberst, "Multidimensional Constant Linear Systems," Acta Applicandae Mathematicae, vol. 20, 1990, pp. 1-175.
30. U. Oberst, "On the Minimal Number of Trajectories Determining a Multidimensional System," Mathematics of Control, Signals and Systems, vol. 6, 1993, pp. 264-288.
31. U. Oberst, "Variations on the Fundamental Principle for Linear Systems of Partial Differential and Difference Equations with Constant Coefficients," Applicable Algebra in Engineering, Communication and Computing, vol. 6, no. 4/5, 1995.
32. U. Oberst, "Finite Dimensional Systems of Partial Differential or Difference Equations," Advances in Applied Mathematics, vol. 17, 1996, pp. 337-356.
33. V. P. Palamodov. Linear Differntial Operators with Constant Coefficients, Springer-Verlag, 1970.
34. F. Pauer and A. Unterkircher, "Gröbner Bases for Ideas in Laurent Polynomial Rings and Their Application to Systems of Difference Equations," Applicable Algebra in Engineering, Communication and Computing, vol. 9, 1999, pp. 271-291.
35. F. Pauer and S. Zampieri, "Gröbner Bases with respect to Generalized Term Orders and Their Application to the Modelling Problem," J. Symbolic Computation, vol. 21, 1996, pp. 155-168.
36. H. Pillai and S. Shankar, "A Behavioural Approach to Control of Distributed Systems," SIAM Journal on Control and Optimization, vol. 37, 1999, pp. 388-408.
37. J. W. Polderman and J. C. Willems, Introduction to Mathematical Systems Theory: A Behavioural Approach, volume 26 of Texts in Applied Mathematics, Springer-Verlag, 1997.
38. J.-F. Pommaret, "New Perspectives in Control Theory for Partial Differential Equations," IMA Journal on Mathematical Control and Information, vol. 9, 1992, pp. 305-330.
39. J.-F. Pommaret, Partial Differential Equations and Group Theory: New Perspectives for Applications, volume 293 of Mathematics and Its Applications, Dordrecht, 1994.
40. J.-F. Pommaret and A. Quadrat, "Formal Elimination for Multidimensional Systems and Applications to Control Theory," Technical Report 97-108, CERMICS/INRIA ENPC, France, 1997. Submitted for publication.
41. J.-F. Pommaret and A. Quadrat, "Algebraic Analysis of Linear Multidimensional Control Systems," IMA J. on Math. Control and Information, vol. 16, 1999, pp. 275-297.
42. J.-F. Pommaret and A. Quadrat, "Generalized Bezout Identity," Applicable Algebra in Engineering, Communication and Computing, vol. 9, no. 2, 1998, pp. 91-116.
43. Ch. Riquier, La Méthode des Fonctions Majorantes et les Systèmes d'Équations aux Dérivées Partielles, volume XXXII of Mémorial Sci. Math., Gauthier-Villars, Paris, 1910.
44. P. Rocha, Structure and Representation of 2D Systems, PhD thesis, University of Groningen, The Netherlands, 1990.
45. 2 2D Systems," In Proc. of the European Control Conference 1997, 1997.
46. P. Rocha and J. C. Willems, "State for 2-D Systems," J. Linear Algebra Appl., vol. 121-123, 1989, pp. 1003-1038.
47. P. Rocha and J. Wood, "A Foundation for the Control Theory of nD Behaviours," Presented at the 13th International Symposium on the Mathematical Theory of Networks and Systems, Padua, Italy, 1997.
48. P. Rocha and J. Wood, "A New Perspective on Controllability Properties for Dynamical Systems," Applied Mathematics and Computer Science, vol. 7, no. 4, 1997, pp. 869-879.
49. J. Rosenthal, "Some Interesting Problems in Systems Theory which are of Fundamental Importance in Coding Theory," In Proc. of the 36th IEEE Conference on Decision and Control, IEEE, 1997, pp. 4574-4579.
50. J. Rudolph, "Duality in Time-Varying Linear Systems - A Module Theoretic Approach," Linear Algebra and Its Applications, vol. 245, 1996, pp. 83-106.
51. D. C. Spencer, "Overdetermined Systems of Partial Differential Equations," Bull. Am. Math. Soc., vol. 75, 1965, pp. 1-114.
52. M. E. Valcher, "Characteristic Cones and Stability Properties of Two-Dimensional Autonomous Behaviours," To appear in IEEE Trans. Circ. and Sys. Part I, 2000.
53. M. E. Valcher, "On the Decomposition of Two-Dimensional Behaviours," Submitted for publication, 1998.
54. J. C. Willems, "From Time Series to Linear System - Part I: Finte-Dimensional Linear Time Invariant Systems," Automatica, vol. 22, no. 5, 1986, pp. 561-580.
55. J. C. Willems, "Paradigms and Puzzles in the Theory of Dynamical Systems," IEEE Trans, on Auto. Contr., vol. 36, no. 3, 1991, pp. 259-294.
56. J. C. Willems, "On Interconnections, Control, and Feedback," IEEE Trans. on Auto. Contr., vol. 42, no. 4, 1997, pp. 458-472.
57. J. Wood, U. Oberst, E. Rogers, and D. H. Owens, "A Behavioural Approach to the Pole Structure of ID and nD Linear Systems," To appear in SIAM J. on Contr. and Opt., 1998.
58. J. Wood, P. Rocha, E. Rogers, and D. H. Owens, "Structure Indices for Multidimensional Systems," To appear in the IMA J. of Mathematical Control and Information, vol. 10, 1999, pp. 33-69.
59. J. Wood, E. Rogers, and D. H. Owens, "A Formal Theory of Matrix Primeness," Mathematics of Control, Signals and Systems, vol. 11, no. 1, 1998, pp. 40-78.
60. J. Wood, E. Rogers, and D. H. Owens, "Controllable and Autonomous nD Systems," Mult. Systems and Signal Processing, vol. 10, 1999, pp. 33-69.
61. J. Wood and E. Zerz, "Notes on the Definition of Behavioural Controllability," Sys. and Contr. Lett., vol. 37, 1999, pp. 31-37.
62. S. Zampieri, "A Solution of the Cauchy Problem for Multidimensional Discrete Linear Shift-Invariant Systems," Linear Algebra and Its Applications, vol. 202, 1994, pp. 143-162.
63. S. Zampieri, "A Behavioural Approach to Identifiability to 2D Scalar Systems," Automatica, vol. 33, no. 1, 1997, pp. 49-61.
64. S. Zampieri, "Causal Input/Output Representation of 2D Systems in the Behavioural Approach," the SIAM Journal on Mathematical Control and Optimization, vol. 36, 1998, pp. 1133-1146.
65. E. Zerz, "Primeness of Multivariate Polynomial Matrices," Systems and Control Letters, vol. 29, no. 3, 1996, pp. 139-146.
66. r, Acta Applicandae Mathematicae, vol. 31, no. 3, 1993, pp. 249-273.
67. Y. F. Zheng, J. C. Willems, and C. S. Zhang, "Common Factors and Controllability of Nonlinear Systems," In Proceedings of the 36th IEEE Conference on Decision and Control, 1997, pp. 2584-2589.