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 Books and proceedings
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").

 Articles in journal or book chapters
1. D. Nesic, A.R. Teel, and E.D. Sontag. Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems. Systems Control Lett., 38(1):49-60, 1999. [PDF] Keyword(s): input to state stability, sampled-data systems, discrete-time systems, sampling.
Abstract:
 We provide an explicit KL stability or input-to-state stability (ISS) estimate for a sampled-data nonlinear system in terms of the KL estimate for the corresponding discrete-time system and a K function describing inter-sample growth. It is quite obvious that a uniform inter-sample growth condition, plus an ISS property for the exact discrete-time model of a closed-loop system, implies uniform ISS of the sampled-data nonlinear system; our results serve to quantify these facts by means of comparison functions. Our results can be used as an alternative to prove and extend results of Aeyels et al and extend some results by Chen et al to a class of nonlinear systems. Finally, the formulas we establish can be used as a tool for some other problems which we indicate.

2. E.D. Sontag and F.R. Wirth. Remarks on universal nonsingular controls for discrete-time systems. Systems Control Lett., 33(2):81-88, 1998. [PDF] [doi:http://dx.doi.org/10.1016/S0167-6911(97)00117-5] Keyword(s): discrete time, controllability.
Abstract:
 For analytic discrete-time systems, it is shown that uniform forward accessibility implies the generic existence of universal nonsingular control sequences. A particular application is given by considering forward accessible systems on compact manifolds. For general systems, it is proved that the complement of the set of universal sequences of infinite length is of the first category. For classes of systems satisfying a descending chain condition, and in particular for systems defined by polynomial dynamics, forward accessibility implies uniform forward accessibility.

3. Y. Yang, E.D. Sontag, and H.J. Sussmann. Global stabilization of linear discrete-time systems with bounded feedback. Systems Control Lett., 30(5):273-281, 1997. [PDF] [doi:http://dx.doi.org/10.1016/S0167-6911(97)00021-2] Keyword(s): discrete-time, saturation.
Abstract:
 This paper deals with the problem of global stabilization of linear discrete time systems by means of bounded feedback laws. The main result proved is an analog of one proved for the continuous time case by the authors, and shows that such stabilization is possible if and only if the system is stabilizable with arbitrary controls and the transition matrix has spectral radius less or equal to one. The proof provides in principle an algorithm for the construction of such feedback laws, which can be implemented either as cascades or as parallel connections (single hidden layer neural networks'') of simple saturation functions.

4. B. Jakubczyk and E.D. Sontag. Controllability of nonlinear discrete-time systems: a Lie-algebraic approach. SIAM J. Control Optim., 28(1):1-33, 1990. [PDF] [doi:http://dx.doi.org/10.1137/0328001] Keyword(s): discrete-time.
Abstract:
 This paper presents a geometric study of controllability for discrete-time nonlinear systems. Various accessibility properties are characterized in terms of Lie algebras of vector fields. Some of the results obtained are parallel to analogous ones in continuous-time, but in many respects the theory is substantially different and many new phenomena appear.

5. B. Jakubczyk and E.D. Sontag. Nonlinear discrete-time systems. Accessibility conditions. In Modern optimal control, volume 119 of Lecture Notes in Pure and Appl. Math., pages 173-185. Dekker, New York, 1989. [PDF]

6. A. Arapostathis, B. Jakubczyk, H.-G. Lee, S. I. Marcus, and E.D. Sontag. The effect of sampling on linear equivalence and feedback linearization. Systems Control Lett., 13(5):373-381, 1989. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(89)90103-5] Keyword(s): discrete-time, sampled-data systems, discrete-time systems, sampling.
Abstract:
 We investigate the effect of sampling on linearization for continuous time systems. It is shown that the discretized system is linearizable by state coordinate change for an open set of sampling times if and only if the continuous time system is linearizable by state coordinate change. Also, it is shown that linearizability via digital feedback imposes highly nongeneric constraints on the structure of the plant, even if this is known to be linearizable with continuous-time feedback.

7. E.D. Sontag and Y. Rouchaleau. On discrete-time polynomial systems. Nonlinear Anal., 1(1):55-64, 1976. [PDF] Keyword(s): identifiability, observability, polynomial systems, realization theory, discrete-time.
Abstract:
 Considered here are a type of discrete-time systems which have algebraic constraints on their state set and for which the state transitions are given by (arbitrary) polynomial functions of the inputs and state variables. The paper studies reachability in bounded time, the problem of deciding whether two systems have the same external behavior by applying finitely many inputs, the fact that finitely many inputs (which can be chosen quite arbitrarily) are sufficient to separate those states of a system which are distinguishable, and introduces the subject of realization theory for this class of systems.

 Conference articles
1. D. Nesic, A.R. Teel, and E.D. Sontag. On stability and input-to-state stability ${\cal K}{\cal L}$ estimates of discrete-time and sampled-data nonlinear systems. In Proc. American Control Conf., San Diego, June 1999, pages 3990-3994, 1999. Keyword(s): input to state stability, sampled-data systems, discrete-time systems, sampling.

2. E.D. Sontag. Algebraic-geometric methods in the realization of discrete-time systems. In Proc. Conf. Inform. Sci. and Systems, John Hopkins Univ. (1978), pages 158-162, 1978.

 Internal reports
1. E.D. Sontag and F.R. Wirth. Remarks on universal nonsingular controls for discrete-time systems. Technical report 381, Institute for Dynamical Systems, University of Bremen, 1996.

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