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Publications about 'systems over rings'
Articles in journal or book chapters
  1. E.D. Sontag. Constant McMillan degree and the continuous stabilization of families of transfer matrices. In Control of uncertain systems (Bremen, 1989), volume 6 of Progr. Systems Control Theory, pages 289-295. Birkhäuser Boston, Boston, MA, 1990. [PDF] Keyword(s): systems over rings.


  2. E.D. Sontag and Y. Wang. Pole shifting for families of linear systems depending on at most three parameters. Linear Algebra Appl., 137/138:3-38, 1990. [PDF] Keyword(s): systems over rings, systems over rings.
    Abstract:
    We prove that for any family of n-dimensional controllable linear systems, continuously parameterized by up to three parameters, and for any continuous selection of n eigenvalues (in complex conjugate pairs), there is some dynamic controller of dimension 3n which is itself continuously parameterized and for which the closed-loop eigenvalues are these same eigenvalues, each counted 4 times. An analogous result holds also for smooth parameterizations.


  3. M. L. J. Hautus and E.D. Sontag. New results on pole-shifting for parametrized families of systems. J. Pure Appl. Algebra, 40(3):229-244, 1986. [PDF] Keyword(s): systems over rings.
    Abstract:
    New results are given on the pole-shifting problem for commutative rings, and these are then applied to conclude that rings of continuous, smooth, or real-analytic functions on a manifold X are PA rings if and only if X is one-dimensional.


  4. E.D. Sontag. Comments on: ``Some results on pole-placement and reachability'' [Systems Control Lett. 6 (1986), no. 5, 325--328; MR0821927 (87c:93032)] by P. K. Sharma. Systems Control Lett., 8(1):79-83, 1986. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(86)90034-4]
    Abstract:
    We present various comments on a question about systems over rings posed in a recent note by Sharma, proving that a ring R is pole-assignable if and only if, for every reachable system (F,G), G contains a rank-one summand of the state space. We also provide a generalization to deal with dynamic feedback.


  5. E.D. Sontag. Continuous stabilizers and high-gain feedback. IMA Journal of Mathematical Control and Information, 3:237-253, 1986. [PDF] Keyword(s): adaptive control, systems over rings.
    Abstract:
    A controller is shown to exist, universal for the family of all systems of fixed dimension n, and m controls, which stabilizes those systems that are stabilizable, if certain gains are large enough. The controller parameters are continuous, in fact polynomial, functions of the entries of the plant. As a consequence, a result is proved on polynomial stabilization of families of systems.


  6. E.D. Sontag. An introduction to the stabilization problem for parametrized families of linear systems. In Linear algebra and its role in systems theory (Brunswick, Maine, 1984), volume 47 of Contemp. Math., pages 369-400. Amer. Math. Soc., Providence, RI, 1985. [PDF] Keyword(s): systems over rings.
    Abstract:
    This paper provides an introduction to definitions and known facts relating to the stabilization of parametrized families of linear systems using static and dynamic controllers. New results are given in the rational and polynomial cases.


  7. E.D. Sontag. Parametric stabilization is easy. Systems Control Lett., 4(4):181-188, 1984. [PDF] Keyword(s): systems over rings.
    Abstract:
    A polynomially parametrized family of continuous-time controllable linear systems is always stabilizable by polynomially parametrized feedback. (Note: appendix had a MACSYMA computation. I cannot find the source file for that. Please look at journal if interested, but this is not very important. Also, two figures involving root loci are not in the web version.)


  8. R.T. Bumby and E.D. Sontag. Stabilization of polynomially parametrized families of linear systems. The single-input case. Systems Control Lett., 3(5):251-254, 1983. [PDF] Keyword(s): systems over rings.
    Abstract:
    Given a continuous-time family of finite dimensional single input linear systems, parametrized polynomially, such that each of the systems in the family is controllable, there exists a polynomially parametrized control law making each of the systems in the family stable.


  9. E.D. Sontag. Linear systems over commutative rings: a (partial) updated survey. In Control science and technology for the progress of society, Vol. 1 (Kyoto, 1981), pages 325-330. IFAC, Laxenburg, 1982. Keyword(s): systems over rings.


  10. P.P. Khargonekar and E.D. Sontag. On the relation between stable matrix fraction factorizations and regulable realizations of linear systems over rings. IEEE Trans. Automat. Control, 27(3):627-638, 1982. [PDF] Keyword(s): systems over rings.
    Abstract:
    Various types of transfer matrix factorizations are of interest when designing regulators for generalized types of linear systems (delay differential. 2-D, and families of systems). This paper studies the existence of stable and of stable proper factorizations, in the context of the thery of systems over rings. Factorability is related to stabilizability and detectability properties of realizations of the transfer matrix. The original formulas for coprime factorizations (which are valid, in particular, over the field of reals) were given in this paper.


  11. R.T. Bumby, E.D. Sontag, H.J. Sussmann, and W. Vasconcelos. Remarks on the pole-shifting problem over rings. J. Pure Appl. Algebra, 20(2):113-127, 1981. [PDF] Keyword(s): systems over rings, systems over rings.
    Abstract:
    Problems that appear in trying to extend linear control results to systems over rings R have attracted considerable attention lately. This interest has been due mainly to applications-oriented motivations (in particular, dealing with delay-differential equations), and partly to a purely algebraic interest. Given a square n-matrix F and an n-row matrix G. pole-shifting problems consist in obtaining more or less arbitrary characteristic polynomials for F+GK, for suitable ("feedback") matrices K. A review of known facts is given, various partial results are proved, and the case n=2 is studied in some detail.


  12. Y. Rouchaleau and E.D. Sontag. On the existence of minimal realizations of linear dynamical systems over Noetherian integral domains. J. Comput. System Sci., 18(1):65-75, 1979. [PDF] Keyword(s): systems over rings.
    Abstract:
    This paper studies the problem of obtaining minimal realizations of linear input/output maps defined over rings. In particular, it is shown that, contrary to the case of systems over fields, it is in general impossible to obtain realizations whose dimiension equals the rank of the Hankel matrix. A characterization is given of those (Noetherian) rings over which realizations of such dimensions can he always obtained, and the result is applied to delay-differential systems.


  13. E.D. Sontag. On finitary linear systems. Kybernetika (Prague), 15(5):349-358, 1979. [PDF] Keyword(s): systems over rings.
    Abstract:
    An abstract operator approach is introduced, permitting a unified study of discrete- and continuous-time linear control systems. As an application, an algorithm is given for deciding if a linear system can be built from any fixed set of linear components. Finally, a criterion is given for reachability of the abstract systems introduced, giving thus a unified proof of known reachability results for discrete-time, continuous-time, and delay-differential systems.


  14. E.D. Sontag. On split realizations of response maps over rings. Information and Control, 37(1):23-33, 1978. [PDF] Keyword(s): systems over rings.
    Abstract:
    This paper deals with observability properties of realizations of linear response maps defined over commutative rings. A characterization is given for those maps which admit realizations which are simultaneously reachable and observable in a strong sense. Applications are given to delay-differential systems.


  15. E.D. Sontag. The lattice of minimal realizations of response maps over rings. Math. Systems Theory, 11(2):169-175, 1977. [PDF] Keyword(s): systems over rings.
    Abstract:
    A lattice characterization is given for the class of minimal-rank realizations of a linear response map defined over a (commutative) Noetherian integral domain. As a corollary, it is proved that there are only finitely many nonisomorphic minimal-rank realizations of a response map over the integers, while for delay -differential systems these are classified by a lattice of subspaces of a finite-dimensional real vector space.


  16. E.D. Sontag and Y. Rouchaleau. Sur les anneaux de Fatou forts. C. R. Acad. Sci. Paris Sér. A-B, 284(5):A331-A333, 1977. [PDF] Keyword(s): systems over rings.
    Abstract:
    It is well known that principal rings are strong Fatou rings. We construct here a more general type of strong Fatou rings. We also prove that the monoid of divisor classes of a noetherian strong Fatou ring contains only the zero element, and that the dimension of such a ring is at most two.


  17. E.D. Sontag. Linear systems over commutative rings: A survey. Ricerche di Automatica, 7:1-34, 1976. [PDF] Keyword(s): systems over rings.
    Abstract:
    An elementary presentation is given of some of the main motivations and known results on linear systems over rings, including questions of realization and control. The analogies and differences with the more standard case of systems over fields are emphasized throughout.


  18. E.D. Sontag. On finitely accessible and finitely observable rings. J. Pure Appl. Algebra, 8(1):97-104, 1976. [PDF] Keyword(s): systems over rings, observability, noncommutative rings.
    Abstract:
    Two classes of rings which occur in linear system theory are introduced and compared. Characterizations of one of them are given in terms, of integral extensions (every finite extension of R is integral) and Cayley--Hamilton type matrix condition. A comparison is made in the case of no zero-divisors with Ore domains.


  19. E.D. Sontag. On linear systems and noncommutative rings. Math. Systems Theory, 9(4):327-344, 1975. [PDF] Keyword(s): systems over rings.
    Abstract:
    This paper studies some problems appearing in the extension of the theory of linear dynamical systems to the case in which parameters are taken from noncommutative rings. Purely algebraic statements of some of the problems are also obtained. Through systems defined by operator rings, the theory of linear systems over rings may be applied to other areas of automata and control theory; several such applications are outlined.


Conference articles
  1. P.P. Khargonekar and E.D. Sontag. On the relation between stable matrix fraction decompositions and regulable realizations of systems over rings. In Proc. IEEE Conf.Dec. and Control, San Diego, Dec. 1981, pages 1006-1011, 1981. Keyword(s): systems over rings.



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