Publications about 'continuous time'
Articles in journal or book chapters
  1. J. Hanson, M. Raginsky, and E.D. Sontag. Learning recurrent neural net models of nonlinear systems. Proc. of Machine Learning Research, 144:1-11, 2021. [PDF] Keyword(s): machine learning, empirical risk minimization, recurrent neural networks, dynamical systems, continuous time, system identification, statistical learning theory, generalization bounds.
    This paper considers the following learning problem: given sample pairs of input and output signals generated by an unknown nonlinear system (which is not assumed to be causal or time-invariant), one wishes to find a continuous-time recurrent neural net, with activation function tanh, that approximately reproduces the underlying i/o behavior with high confidence. Leveraging earlier work concerned with matching derivatives up to a finite order of the input and output signals the problem is reformulated in familiar system-theoretic language and quantitative guarantees on the sup-norm risk of the learned model are derived, in terms of the number of neurons, the sample size, the number of derivatives being matched, and the regularity properties of the inputs, the outputs, and the unknown i/o map.

  2. 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:] Keyword(s): discrete-time, saturation, bounded inputs.
    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.

  3. F. Albertini and E.D. Sontag. State observability in recurrent neural networks. Systems Control Lett., 22(4):235-244, 1994. [PDF] [doi:] Keyword(s): machine learning, neural networks, recurrent neural networks, observability, identifiability.
    This paper concerns recurrent networks x'=s(Ax+Bu), y=Cx, where s is a sigmoid, in both discrete time and continuous time. Our main result is that observability can be characterized, if one assumes certain conditions on the nonlinearity and on the system, in a manner very analogous to that of the linear case. Recall that for the latter, observability is equivalent to the requirement that there not be any nontrivial A-invariant subspace included in the kernel of C. We show that the result generalizes in a natural manner, except that one now needs to restrict attention to certain special "coordinate" subspaces.

  4. E.D. Sontag. Kalman's controllability rank condition: from linear to nonlinear. In Mathematical system theory, pages 453-462. Springer, Berlin, 1991. [PDF] Keyword(s): controllability.
    The notion of controllability was identified by Kalman as one of the central properties determining system behavior. His simple rank condition is ubiquitous in linear systems analysis. This article presents an elementary and expository overview of the generalizations of this test to a condition for testing accessibility of discrete and continuous time nonlinear systems.

  5. E.D. Sontag. Integrability of certain distributions associated with actions on manifolds and applications to control problems. In Nonlinear controllability and optimal control, volume 133 of Monogr. Textbooks Pure Appl. Math., pages 81-131. Dekker, New York, 1990. [PDF] Keyword(s): controllability.
    Results are given on the integrability of certain distributions which arise from smoothly parametrized families of diffeomorphisms acting on manifolds. Applications to control problems and in particular to the problem of sampling are discussed. Pages 42-50 apply the results to the control of continuous time systems; this is an exposition of some of the basic results of the Lie algebraic accessibility theory.

  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:] Keyword(s): discrete-time, sampled-data systems, discrete-time systems, sampling.
    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. Smooth stabilization implies coprime factorization. IEEE Trans. Automat. Control, 34(4):435-443, 1989. [PDF] Keyword(s): input to state stability, ISS, input to state stability.
    This paper shows that coprime right factorizations exist for the input to state mapping of a continuous time nonlinear system provided that the smooth feedback stabilization problem be solvable for this system. In particular, it follows that feedback linearizable systems admit such factorizations. In order to establish the result a Lyapunov-theoretic definition is proposed for bounded input bounded output stability. The main technical fact proved relates the notion of stabilizability studied in the state space nonlinear control literature to a notion of stability under bounded control perturbations analogous to those studied in operator theoretic approaches to systems; it states that smooth stabilization implies smooth input-to-state stabilization. (Note: This is the original ISS paper, but the ISS results have been much improved in later papers. The material on coprime factorizations is still of interest, but the 89 CDC paper has some improvements and should be read too.)

  8. E.D. Sontag. Bilinear realizability is equivalent to existence of a singular affine differential I/O equation. Systems Control Lett., 11(3):181-187, 1988. [PDF] [doi:] Keyword(s): identification, identifiability, observability, observation space, real-analytic functions.
    For continuous time analytic input/output maps, the existence of a singular differential equation relating derivatives of controls and outputs is shown to be equivalent to bilinear realizability. A similar result holds for the problem of immersion into bilinear systems. The proof is very analogous to that of the corresponding, and previously known, result for discrete time.

  9. E.D. Sontag. An algebraic approach to bounded controllability of linear systems. Internat. J. Control, 39(1):181-188, 1984. [PDF] Keyword(s): saturation, bounded inputs.
    In this note we present an algebraic approach to the proof that a linear system with matrices (A,B) is null-controllable using bounded inputs iff it is null-controllable (with unbounded inputs) and all eigenvalues of A have nonpositive real parts (continuous time) or magnitude not greater than one (discrete time). We also give the analogous results for the asymptotic case. Finally, we give an interpretation of these results in the context of local nonlinear controllability.

Conference articles
  1. F. Albertini and E.D. Sontag. Uniqueness of weights for recurrent nets. In Systems and Networks: Mathematical Theory and Applications, Proc. MTNS '93, Vol. 2, Akad. Verlag, Regensburg, pages 599-602, 1993. Note: Full version, never submitted for publication, is here: [PDF] Keyword(s): machine learning, neural networks, identifiability, recurrent neural networks.
    This paper concerns recurrent networks x'=s(Ax+Bu), y=Cx, where s is a sigmoid, in both discrete time and continuous time. The paper establishes parameter identifiability under stronger assumptions on the activation than in "For neural networks, function determines form", but on the other hand deals with arbitrary (nonzero) initial states.

  2. E.D. Sontag and Y. Wang. I/O equations in discrete and continuous time. In Proc. IEEE Conf. Decision and Control, Tucson, Dec. 1992, IEEE Publications, 1992, pages 3661-3662, 1992. Keyword(s): identifiability, observability, realization theory.

  3. F. Albertini and E.D. Sontag. Some connections between chaotic dynamical systems and control systems. In Proc. European Control Conf. , Vol 1, Grenoble, July 1991, pages 58-163, 1991. [PDF] Keyword(s): chaotic systems, controllability.
    This paper shows how to extend recent results of Colonius and Kliemann, regarding connections between chaos and controllability, from continuous to discrete time. The extension is nontrivial because the results all rely on basic properties of the accessibility Lie algebra which fail to hold in discrete time. Thus, this paper first develops further results in nonlinear accessibility, and then shows how a theorem can be proved, which while analogous to the one given in the work by Colonius and Klieman, also exhibits some important differences. A counterexample is used to show that the theorem given in continuous time cannot be generalized in a straightforward manner.

  4. E.D. Sontag and Y. Wang. I/O equations for nonlinear systems and observation spaces. In Proc. IEEE Conf. Decision and Control, Brighton, UK, Dec. 1991, IEEE Publications, 1991, pages 720-725, 1991. [PDF] Keyword(s): identifiability, observability, realization theory, real-analytic functions.
    This paper studies various types of input/output representations for nonlinear continuous time systems. The algebraic and analytic i/o equations studied in previous papers by the authors are generalized to integral and integro-differential equations, and an abstract notion is also considered. New results are given on generic observability, and these results are then applied to give conditions under which that the minimal order of an equation equals the minimal possible dimension of a realization, just as with linear systems but in contrast to the discrete time nonlinear theory.

  5. E.D. Sontag. Some complexity questions regarding controllability. In Proc. IEEE Conf. Decision and Control, Austin, Dec. 1988, pages 1326-1329, 1988. [PDF] Keyword(s): theory of computing and complexity, computational complexity, controllability, computational complexity.
    It has been known for a long time that certain controllability properties are more difficult to verify than others. This article makes this fact precise, comparing controllability with accessibility, for a wide class of nonlinear continuous time systems. The original contribution is in formalizing this comparison in the context of computational complexity. (This paper placed here by special request.)



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