| Title:
|
The algebraic structure of delay-differential systems: a behavioral perspective (English) |
| Author:
|
Gluesing-Luerssen, Heide |
| Author:
|
Vettori, Paolo |
| Author:
|
Zampieri, Sandro |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
37 |
| Issue:
|
4 |
| Year:
|
2001 |
| Pages:
|
[397]-426 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper presents a survey on the recent contributions to linear time- invariant delay-differential systems in the behavioral approach. In this survey both systems with commensurate and with noncommensurate 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 noncommensurate delays. In the study of this class of systems functional analytic 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. (English) |
| Keyword:
|
delay-differential system |
| Keyword:
|
algebraic methods |
| Keyword:
|
general convolution equations |
| Keyword:
|
noncommensurate delays |
| Keyword:
|
delay-differential systems |
| Keyword:
|
behavioral approach |
| MSC:
|
34K35 |
| MSC:
|
47N70 |
| MSC:
|
93B25 |
| MSC:
|
93C05 |
| MSC:
|
93C23 |
| idZBL:
|
Zbl 1265.93064 |
| idMR:
|
MR1859094 |
| . |
| Date available:
|
2009-09-24T19:40:45Z |
| Last updated:
|
2015-03-26 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/135419 |
| . |
| Reference:
|
[1] Becker T., Weispfennig V.: Gröbner Bases: A Computational Approach to Commutative Algebra.Springer, New York 1993 MR 1213453 |
| Reference:
|
[2] Berenstein C. A., Dostal M. A.: The Ritt theorem in several variables.Ark. Mat. 12 (1974), 267–280 Zbl 0293.33001, MR 0377111, 10.1007/BF02384763 |
| Reference:
|
[3] Berenstein C. A., Struppa D. C.: Complex analysis and convolution equations.Several complex variables. Encyclopedia Math. Sci. 54 (1993), 1–108 |
| Reference:
|
[4] Berenstein C. A., Yger A.: Ideals generated by exponential-polynomials.Adv. in Math. 60 (1986), 1–80 Zbl 0586.32019, MR 0839482, 10.1016/0001-8708(86)90002-2 |
| Reference:
|
[5] Brezis H.: Analyse fonctionnelle: theorie et applications.Masson, Paris 1983 Zbl 1147.46300, MR 0697382 |
| Reference:
|
[6] Cohen A. M., Cuypers, H., (eds.) H. Sterk: Some Tapas of Computer Algebra.Springer, Berlin 1999 Zbl 0924.13021, MR 1679917 |
| Reference:
|
[7] Cohn P. M.: Free Rings and Their Relations.Academic Press, London 1985. Second edition Zbl 0659.16001, MR 0800091 |
| Reference:
|
[8] Diab A.: Sur les zéros communs des polynômes exponentiels.C. R. Acad. Sci. Paris Sér. A 281 (1975), 757–758 Zbl 0323.30003, MR 0390186 |
| Reference:
|
[9] Ehrenpreis L.: Solutions of some problems of division.Part III. Division in the spaces ${\mathcal D}^{\prime },\,{\mathcal H},\,{\mathcal Q}_A,\,{\mathcal O}$. Amer. J. Math. 78 (1956), 685–715 Zbl 0072.32801, MR 0083690 |
| Reference:
|
[10] Folland G. B.: Fourier Analysis and its Applications.Wadsworth & Brooks, Pacific Grove 1992 Zbl 1222.42001, MR 1145236 |
| Reference:
|
[11] Gluesing–Luerssen H.: A convolution algebra of delay-differential operators and a related problem of finite spectrum assignability.Math. Control Signal Systems 13 (2000), 22–40 Zbl 0954.93007, MR 1742138, 10.1007/PL00009859 |
| Reference:
|
[12] Gluesing–Luerssen H.: A behavioral approach to delay differential equations.SIAM J. Control Optim. 35 (1997), 480–499 MR 1436634, 10.1137/S0363012995281869 |
| Reference:
|
[13] Gluesing–Luerssen H.: Linear delay-differential systems with commensurate delays: An algebraic approach.Habilitationsschrift at the University of Oldenburg 2000. Accepted for publication as Lecture Notes in Mathematics, Springer Zbl 0989.34001, MR 1874340 |
| Reference:
|
[14] Habets L. C. G. J. M.: System equivalence for AR-systems over rings – With an application to delay-differential systems.Math. Control Signal Systems 12 (1999), 219–244 Zbl 0951.93015, MR 1707899, 10.1007/PL00009851 |
| Reference:
|
[15] Habets L. C. G. J. M., Eijndhoven S. J. L.: Behavioral controllability of time-delay systems with incommensurate delays.In: Proc. IFAC Workshop on Linear Time Delay Systems (A. M. Perdon, ed.), Ancona 2000, pp. 195–201 |
| Reference:
|
[16] Helmer O.: The elementary divisor theorem for certain rings without chain condition.Bull. Amer. Math. Soc. 49 (1943), 225–236 Zbl 0060.07606, MR 0007744, 10.1090/S0002-9904-1943-07886-X |
| Reference:
|
[17] Jacobson N.: Basic Algebra I.Second edition. W. H. Freeman, New York 1985 Zbl 0557.16001, MR 0780184 |
| Reference:
|
[18] Kamen E. W.: On an algebraic theory of systems defined by convolution operators.Math. Systems Theory 9 (1975), 57–74 Zbl 0318.93003, MR 0395953, 10.1007/BF01698126 |
| Reference:
|
[19] Kamen E. W., Khargonekar P. P., Tannenbaum A.: Proper stable Bezout factorizations and feedback control of linear time-delay systems.Internat. J. Control 43 (1986), 837–857 Zbl 0599.93047, MR 0828360, 10.1080/00207178608933506 |
| Reference:
|
[20] Kaplansky I.: Elementary divisors and modules.Trans. Amer. Math. Soc. 66 (1949), 464–491 Zbl 0036.01903, MR 0031470, 10.1090/S0002-9947-1949-0031470-3 |
| Reference:
|
[21] Kelley J. L., Namioka I.: Topological Vector Spaces.Van Nostrand, 1963 MR 0166578 |
| Reference:
|
[22] Lang S.: Algebra.Second edition. Addison–Wesley, Reading, N.J. 1984 Zbl 1063.00002, MR 0783636 |
| Reference:
|
[23] Lezama O., Vasquez O.: On the simultaneous basis property in Prüfer domains.Acta Math. Hungar. 80 (1998), 169–176 MR 1624566, 10.1023/A:1006537212456 |
| Reference:
|
[24] Malgrange B.: Existence et approximations des solutions des équations aux dérivées partielles et des équations de convolution.Ann. Inst. Fourier 6 (1955/1956), 271–355 MR 0086990, 10.5802/aif.65 |
| Reference:
|
[25] Meisters G. H.: Periodic distributions and non-Liouville numbers.J. Funct. Anal. 26 (1977), 68–88 Zbl 0359.46027, MR 0448068, 10.1016/0022-1236(77)90016-7 |
| Reference:
|
[26] Mounier H.: Algebraic interpretations of the spectral controllability of a linear delay system.Forum Math. 10 (1998), 39–58 Zbl 0891.93014, MR 1490137, 10.1515/form.10.1.39 |
| Reference:
|
[27] Niven I.: Irrational Numbers.Wiley, New York 1956 Zbl 0146.27703, MR 0080123 |
| Reference:
|
[28] Oberst U.: Multidimensional constant linear systems.Acta Appl. Math. 20 (1990), 1–175 Zbl 0715.93014, MR 1078671, 10.1007/BF00046908 |
| Reference:
|
[29] Olbrot A. W., Pandolfi L.: Null controllability of a class of functional differential systems.Internat. J. Control 47 (1988), 193–208 Zbl 0662.93008, MR 0929735, 10.1080/00207178808906006 |
| Reference:
|
[30] Parreau F., Weit Y.: Schwartz’s theorem on mean periodic vector-valued functions.Bull. Soc. Math. France 117 (1989), 3, 319–325 Zbl 0704.46011, MR 1020109 |
| Reference:
|
[31] Polderman J. W., Willems J. C.: Introduction to Mathematical Systems Theory.A behavioral approach. Springer, Boston 1998 Zbl 0940.93002, MR 1480665 |
| Reference:
|
[32] Rocha P., Wood J.: Trajectory control and interconnection of 1D and $n$D systems.SIAM J. Control Optim. 40 (2001), 107–134 MR 1855308, 10.1137/S0363012999362797 |
| Reference:
|
[33] Schwartz L.: Théorie génerale des fonctions moyennes-périodiques.Ann. of Math. (2) 48 (1947), 857–929 10.2307/1969386 |
| Reference:
|
[34] Treves F.: Topological Vector Spaces, Distributions and Kernels.Academic Press, New York 1967 Zbl 1111.46001, MR 0225131 |
| Reference:
|
[35] Eijndhoven S. J. L. van, Habets L. C. G. J. M.: Equivalence of Convolution Systems in a Behavioral Framework.Report RANA 99-25. Eindhoven University of Technology 1999 |
| Reference:
|
[36] Poorten A. J. van der, Tijdeman R.: On common zeros of exponential polynomials.Enseign. Math. (2) 21 (1975), 57–67 MR 0379387 |
| Reference:
|
[37] Vettori P.: Delay Differential Systems in the Behavioral Approach.Ph. D. Thesis, Università di Padova 1999 |
| Reference:
|
[38] Vettori P., Zampieri S.: Controllability of systems described by convolutional or delay-differential equations.SIAM J. Control Optim. 39 (2000), 728–756 Zbl 0976.34070, MR 1786327, 10.1137/S0363012999359718 |
| Reference:
|
[39] Vettori P., Zampieri S.: Some results on systems described by convolutional equations.IEEE Trans. Automat Control. AC–46 (2001), 793–797 Zbl 1009.93013, MR 1833038, 10.1109/9.920803 |
| Reference:
|
[40] Willems J. C.: On interconnection, control, and feedback.IEEE Trans. Automat. Control AC-42 (1997), 326–339 MR 1435822, 10.1109/9.557576 |
| Reference:
|
[41] Zemanian A. H.: Distribution Theory and Transform Analysis.McGraw–Hill, New York 1965 Zbl 0643.46028, MR 0177293 |
| . |
