1. E.D. Sontag. Polynomial Response Maps, volume 13 of Lecture Notes in Control and Information Sciences. Springer-Verlag, Berlin, 1979. [PDF] Keyword(s): realization theory, discrete-time, real algebraic geometry. Abstract:

   (This is a monograph based upon Eduardo Sontag's Ph.D. thesis. The contents are basically the same as the thesis, except for a very few revisions and extensions.) This work deals the realization theory of discrete-time systems (with inputs and outputs, in the sense of control theory) defined by polynomial update equations. It is based upon the premise that the natural tools for the study of the structural-algebraic properties (in particular, realization theory) of polynomial input/output maps are provided by algebraic geometry and commutative algebra, perhaps as much as linear algebra provides the natural tools for studying linear systems. Basic ideas from algebraic geometry are used throughout in system-theoretic applications (Hilbert's basis theorem to finite-time observability, dimension theory to minimal realizations, Zariski's Main Theorem to uniqueness of canonical realizations, etc). In order to keep the level elementary (in particular, not utilizing sheaf-theoretic concepts), certain ideas like nonaffine varieties are used only implicitly (eg., quasi-affine as open sets in affine varieties) or in technical parts of a few proofs, and the terminology is similarly simplified (e.g., "polynomial map" instead of "scheme morphism restricted to k-points", or "k-space" instead of "k-points of an affine k-scheme").

1. 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, parametric classes of systems. 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.




2. E.D. Sontag. On finitary linear systems. Kybernetika (Prague), 15(5):349-358, 1979. [PDF] Keyword(s): systems over rings, parametric classes of systems. 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.




3. E.D. Sontag. On the observability of polynomial systems. I. Finite-time problems. SIAM J. Control Optim., 17(1):139-151, 1979. [PDF] Keyword(s): observability, observability, polynomial systems. Abstract:

   Different notions of observability are compared for systems defined by polynomial difference equations. The main result states that, for systems having the standard property of (multiple-experiment initial-state) observability, the response to a generic input sequence is sufficient for final-state determination. Some remarks are made on results for nonpolynomial and/or continuous-time systems. An identifiability result is derived from the above.




4. E.D. Sontag. Realization theory of discrete-time nonlinear systems. I. The bounded case. IEEE Trans. Circuits and Systems, 26(5):342-356, 1979. [PDF] Keyword(s): discrete-time systems, nonlinear systems, realization theory, bilinear systems, state-affine systems. Abstract:

   A state-space realization theory is presented for a wide class of discrete time input/output behaviors. Although In many ways restricted, this class does include as particular cases those treated in the literature (linear, multilinear, internally bilinear, homogeneous), as well as certain nonanalytic nonlinearities. The theory is conceptually simple, and matrix-theoretic algorithms are straightforward. Finite-realizability of these behaviors by state-affine systems is shown to be equivalent both to the existence of high-order input/output equations and to realizability by more general types of systems.

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