BACK TO INDEX

Publications about 'real-analytic functions'
Articles in journal or book chapters
  1. E.D. Sontag and Y. Wang. Uniformly Universal Inputs. In Alessandro Astolfi, editor, Analysis and Design of Nonlinear Control Systems, volume 224, pages 9-24. Springer-Verlag, London, 2007. [PDF] Keyword(s): observability, identification, real-analytic functions.
    Abstract:
    A result is presented showing the existence of inputs universal for observability, uniformly with respect to the class of all continuous-time analytic systems. This represents an ultimate generalization of a 1977 theorem, for bilinear systems, due to Alberto Isidori and Osvaldo Grasselli.


  2. E.D. Sontag. For differential equations with r parameters, 2r+1 experiments are enough for identification. J. Nonlinear Sci., 12(6):553-583, 2002. [PDF] Keyword(s): identifiability, observability, systems biology, biochemical networks, parameter identification, real-analytic functions.
    Abstract:
    Given a set of differential equations whose description involves unknown parameters, such as reaction constants in chemical kinetics, and supposing that one may at any time measure the values of some of the variables and possibly apply external inputs to help excite the system, how many experiments are sufficient in order to obtain all the information that is potentially available about the parameters? This paper shows that the best possible answer (assuming exact measurements and real analiticity) is 2r+1 experiments, where r is the number of parameters.


  3. E.D. Sontag. A general approach to path planning for systems without drift. In J. Baillieul, S. S. Sastry, and H.J. Sussmann, editors, Essays on mathematical robotics (Minneapolis, MN, 1993), volume 104 of IMA Vol. Math. Appl., pages 151-168. Springer, New York, 1998. [PDF] Keyword(s): path-planning, systems without drift, nonlinear control, controllability, real-analytic functions.
    Abstract:
    This paper proposes a generally applicable technique for the control of analytic systems with no drift. The method is based on the generation of "nonsingular loops" that allow linearized controllability. One can then implement Newton and/or gradient searches in the search for a control. A general convergence theorem is proved.


  4. 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, real-analytic functions.
    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.


  5. E.D. Sontag. Shattering all sets of k points in `general position' requires (k-1)/2 parameters. Neural Comput., 9(2):337-348, 1997. [PDF] Keyword(s): machine learning, neural networks, VC dimension, real-analytic functions.
    Abstract:
    For classes of concepts defined by certain classes of analytic functions depending on k parameters, there are nonempty open sets of samples of length 2k+2 which cannot be shattered. A slighly weaker result is also proved for piecewise-analytic functions. The special case of neural networks is discussed.


  6. E.D. Sontag. Critical points for least-squares problems involving certain analytic functions, with applications to sigmoidal nets. Adv. Comput. Math., 5(2-3):245-268, 1996. [PDF] Keyword(s): machine learning, subanalytic sets, semianalytic sets, critical points, approximation theory, neural networks, real-analytic functions.
    Abstract:
    This paper deals with nonlinear least-squares problems involving the fitting to data of parameterized analytic functions. For generic regression data, a general result establishes the countability, and under stronger assumptions finiteness, of the set of functions giving rise to critical points of the quadratic loss function. In the special case of what are usually called "single-hidden layer neural networks", which are built upon the standard sigmoidal activation tanh(x) or equivalently 1/(1+exp(-x)), a rough upper bound for this cardinality is provided as well.


  7. E.D. Sontag. Control of systems without drift via generic loops. IEEE Trans. Automat. Control, 40(7):1210-1219, 1995. [PDF] Keyword(s): stabilization, non-holonomic systems, path-planning, systems without drift, nonlinear control, controllability, real-analytic functions.
    Abstract:
    This paper proposes a simple numerical technique for the steering of arbitrary analytic systems with no drift. It is based on the generation of "nonsingular loops" which allow linearized controllability along suitable trajetories. Once such loops are available, it is possible to employ standard Newton or steepest descent methods, as classically done in numerical control. The theoretical justification of the approach relies on recent results establishing the genericity of nonsingular controls, as well as a simple convergence lemma.


  8. Y. Wang and E.D. Sontag. Orders of input/output differential equations and state-space dimensions. SIAM J. Control Optim., 33(4):1102-1126, 1995. [PDF] [doi:http://dx.doi.org/10.1137/S0363012993246828] Keyword(s): identifiability, observability, realization theory, real-analytic functions.
    Abstract:
    This paper deals with the orders of input/output equations satisfied by nonlinear systems. Such equations represent differential (or difference, in the discrete-time case) relations between high-order derivatives (or shifts, respectively) of input and output signals. It is shown that, under analyticity assumptions, there cannot exist equations of order less than the minimal dimension of any observable realization; this generalizes the known situation in the classical linear case. The results depend on new facts, themselves of considerable interest in control theory, regarding universal inputs for observability in the discrete case, and observation spaces in both the discrete and continuous cases. Included in the paper is also a new and simple self-contained proof of Sussmann's universal input theorem for continuous-time analytic systems.


  9. F. Albertini and E.D. Sontag. Discrete-time transitivity and accessibility: analytic systems. SIAM J. Control Optim., 31(6):1599-1622, 1993. [PDF] [doi:http://dx.doi.org/10.1137/0331075] Keyword(s): controllability, discrete-time systems, accessibility, real-analytic functions.
    Abstract:
    A basic open question for discrete-time nonlinear systems is that of determining when, in analogy with the classical continuous-time "positive form of Chow's Lemma", accessibility follows from transitivity of a natural group action. This paper studies the problem, and establishes the desired implication for analytic systems in several cases: (i) compact state space, (ii) under a Poisson stability condition, and (iii) in a generic sense. In addition, the paper studies accessibility properties of the "control sets" recently introduced in the context of dynamical systems studies. Finally, various examples and counterexamples are provided relating the various Lie algebras introduced in past work.


  10. E.D. Sontag. Universal nonsingular controls. Systems Control Lett., 19(3):221-224, 1992. Note: Erratum appeared in SCL 20(1993), p. 77, can be found in same file.[PDF] [doi:http://dx.doi.org/10.1016/0167-6911(92)90116-A] Keyword(s): controllability, real-analytic functions.
    Abstract:
    For analytic systems satisfying the strong accessibility rank condition, generic inputs produce trajectories along which the linearized system is controllable. Applications to the steering of systems without drift are briefly mentioned.


  11. Y. Wang and E.D. Sontag. Generating series and nonlinear systems: analytic aspects, local realizability, and i/o representations. Forum Math., 4(3):299-322, 1992. [PDF] Keyword(s): identifiability, observability, realization theory, input/output system representations, real-analytic functions.
    Abstract:
    This paper studies fundamental analytic properties of generating series for nonlinear control systems, and of the operators they define. It then applies the results obtained to the extension of facts, which relate realizability and algebraic input/output equations, to local realizability and analytic equations.


  12. Y. Lin and E.D. Sontag. A universal formula for stabilization with bounded controls. Systems Control Lett., 16(6):393-397, 1991. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(91)90111-Q] Keyword(s): stabilization, nonlinear systems, saturation, bounded inputs, control-Lyapunov functions, real-analytic functions.
    Abstract:
    We provide a formula for a stabilizing feedback law using a bounded control, under the assumption that an appropriate control-Lyapunov function is known. Such a feedback, smooth away from the origin and continuous everywhere, is known to exist via Artstein's Theorem. As in the unbounded-control case treated in a previous note, we provide an explicit and ``universal'' formula given by an algebraic function of Lie derivatives. In particular, we extend to the bounded case the result that the feedback can be chosen analytic if the Lyapunov function and the vector fields defining the system are analytic.


  13. E.D. Sontag. A ``universal'' construction of Artstein's theorem on nonlinear stabilization. Systems Control Lett., 13(2):117-123, 1989. [PDF] Keyword(s): control-Lyapunov functions, stabilization, real-analytic functions.
    Abstract:
    This note presents an explicit proof of the theorem - due to Artstein - which states that the existence of a smooth control-Lyapunov function implies smooth stabilizability. Moreover, the result is extended to the real-analytic and rational cases as well. The proof uses a "universal" formula given by an algebraic function of Lie derivatives; this formula originates in the solution of a simple Riccati equation.


  14. 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:http://dx.doi.org/10.1016/0167-6911(88)90057-6] Keyword(s): identification, identifiability, observability, observation space, real-analytic functions.
    Abstract:
    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.


  15. 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, pole-shifting, parametric classes of systems, real-analytic functions.
    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.


  16. E.D. Sontag. Remarks on the preservation of various controllability properties under sampling. In Mathematical tools and models for control, systems analysis and signal processing, Vol. 3 (Toulouse/Paris, 1981/1982), Travaux Rech. Coop. Programme 567, pages 623-637. CNRS, Paris, 1983. [PDF] Keyword(s): controllability, sampling, nonlinear systems, real-analytic functions.
    Abstract:
    This note studies the preservation of controllability (and other properties) under sampling of a nonlinear system. More detailed results are obtained in the cases of analytic systems and of systems with finite dimensional Lie algebras.


  17. E.D. Sontag. Conditions for abstract nonlinear regulation. Inform. and Control, 51(2):105-127, 1981. [PDF] Keyword(s): feedback stabilization, nonlinear systems, real-analytic functions.
    Abstract:
    A paper that introduces a separation principle for general finite dimensional analytic continuous-time systems, proving the equivalence between existence of an output regulator (which is an abstract dynamical system) and certain "0-detectability" and asymptotic controllability assumptions.


Conference articles
  1. A. Macintyre and E.D. Sontag. Finiteness results for sigmoidal neural networks. In STOC '93: Proceedings of the twenty-fifth annual ACM symposium on Theory of computing, New York, NY, USA, pages 325-334, 1993. ACM Press. [PDF] [doi:http://doi.acm.org/10.1145/167088.167192] Keyword(s): machine learning, neural networks, theory of computing and complexity, real-analytic functions.
    Abstract:
    This paper deals with analog circuits. It establishes the finiteness of VC dimension, teaching dimension, and several other measures of sample complexity which arise in learning theory. It also shows that the equivalence of behaviors, and the loading problem, are effectively decidable, modulo a widely believed conjecture in number theory. The results, the first ones that are independent of weight size, apply when the gate function is the "standard sigmoid" commonly used in neural networks research. The proofs rely on very recent developments in the elementary theory of real numbers with exponentiation. (Some weaker conclusions are also given for more general analytic gate functions.) Applications to learnability of sparse polynomials are also mentioned.


  2. E.D. Sontag. Gradient techniques for systems with no drift: A classical idea revisited. In Proc. IEEE Conf. Decision and Control, San Antonio, Dec. 1993, IEEE Publications, 1993, pages 2706-2711, 1993. [PDF] Keyword(s): path-planning, systems without drift, nonlinear control, controllability, real-analytic functions.
    Abstract:
    This paper proposes a technique for the control of analytic systems with no drift. It is based on the generation of "nonsingular loops" which allow linearized controllability. Once such loops are available, it is possible to employ standard Newton or steepest descent methods. The theoretical justification of the approach relies on results on genericity of nonsingular controls as well as a simple convergence lemma.


  3. 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.
    Abstract:
    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.


  4. Y. Wang and E.D. Sontag. Realization and input/output relations: the analytic case. In Proceedings of the 28th IEEE Conference on Decision and Control, Vol. 1--3 (Tampa, FL, 1989), New York, pages 1975-1980, 1989. IEEE. Keyword(s): identifiability, observability, realization theory, real-analytic functions.



BACK TO INDEX




Disclaimer:

This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders.




Last modified: Thu Oct 5 13:19:22 2023
Author: sontag.


This document was translated from BibTEX by bibtex2html