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Publications by Eduardo D. Sontag in year 1995
Articles in journal or book chapters
  1. E.D. Sontag. State-space and i/o stability for nonlinear systems. In Feedback control, nonlinear systems, and complexity (Montreal, PQ, 1994), volume 202 of Lecture Notes in Control and Inform. Sci., pages 215-235. Springer, London, 1995. Note: (Expository paper, placed online per request. The paper ``Input to state stability: Basic concepts and results'' is far more up to date and should be downloaded instead of this one!). [PDF] Keyword(s): input to state stability.


  2. A.R. Teel, T.T. Georgiou, L. Praly, and E.D. Sontag. Input-Output Stability. In W. S. Levine, editor, The Control Handbook, pages 895-908. CRC Press, Boca Raton, 1995. [PDF]
    Abstract:
    An encyclopedia-type article on foundations of input/output stability.


  3. Y. Chitour, W. Liu, and E.D. Sontag. On the continuity and incremental-gain properties of certain saturated linear feedback loops. Internat. J. Robust Nonlinear Control, 5(5):413-440, 1995. [PDF] Keyword(s): saturation.
    Abstract:
    This paper discusses various continuity and incremental-gain properties for neutrally stable linear systems under linear feedback subject to actuator saturation. The results complement our previous ones, which applied to the same class of problems and provided finite-gain stability.


  4. M. A. Dahleh, E.D. Sontag, D. N. C. Tse, and J. N. Tsitsiklis. Worst-case identification of nonlinear fading memory systems. Automatica, 31(3):503-508, 1995. [PDF] [doi:http://dx.doi.org/10.1016/0005-1098(94)00131-2]
    Abstract:
    We consider the problem of characterizing possible supply functions for a given dissipative nonlinear system, and provide a result that allows some freedom in the modification of such functions.


  5. B. DasGupta, H.T. Siegelmann, and E.D. Sontag. On the complexity of training neural networks with continuous activation functions. IEEE Trans. Neural Networks, 6:1490-1504, 1995. [PDF] Keyword(s): neural networks, analog computing, theory of computing, neural networks, computational complexity, machine learning.
    Abstract:
    Blum and Rivest showed that any possible neural net learning algorithm based on fixed architectures faces severe computational barriers. This paper extends their NP-completeness result, which applied only to nets based on hard threshold activations, to nets that employ a particular continuous activation. In view of neural network practice, this is a more relevant result to understanding the limitations of backpropagation and related techniques.


  6. Y. Lin and E.D. Sontag. Control-Lyapunov universal formulas for restricted inputs. Control Theory Adv. Tech., 10(4, part 5):1981-2004, 1995. [PDF] Keyword(s): control-Lyapunov functions, saturation.
    Abstract:
    We deal with the question of obtaining explicit feedback control laws that stabilize a nonlinear system, under the assumption that a "control Lyapunov function" is known. In previous work, the case of unbounded controls was considered. Here we obtain results for bounded and/or positive controls. We also provide some simple preliminary remarks regarding a set stability version of the problem and a version for systems subject to disturbances.


  7. Y. Lin, E.D. Sontag, and Y. Wang. Input to state stabilizability for parametrized families of systems. Internat. J. Robust Nonlinear Control, 5(3):187-205, 1995. [PDF]
    Abstract:
    This paper studies various stability issues for parameterized families of systems, including problems of stabilization with respect to sets. The study of such families is motivated by robust control applications. A Lyapunov-theoretic necessary and sufficient characterization is obtained for a natural notion of robust uniform set stability; this characterization allows replacing ad hoc conditions found in the literature by more conceptual stability notions. We then use these techniques to establish a result linking state space stability to ``input to state'' (bounded-input bounded-state) stability. In addition, the preservation of stabilizability under certain types of cascade interconnections is analyzed.


  8. H. T. Siegelmann and E.D. Sontag. On the computational power of neural nets. J. Comput. System Sci., 50(1):132-150, 1995. [PDF] [doi:http://dx.doi.org/10.1006/jcss.1995.1013] Keyword(s): neural networks, recurrent neural networks, analog computing, theory of computing, neural networks, computational complexity, super-Turing computation.
    Abstract:
    This paper deals with finite size networks which consist of interconnections of synchronously evolving processors. Each processor updates its state by applying a "sigmoidal" function to a rational-coefficient linear combination of the previous states of all units. We prove that one may simulate all Turing Machines by such nets. In particular, one can simulate any multi-stack Turing Machine in real time, and there is a net made up of 886 processors which computes a universal partial-recursive function. Products (high order nets) are not required, contrary to what had been stated in the literature. Non-deterministic Turing Machines can be simulated by non-deterministic rational nets, also in real time. The simulation result has many consequences regarding the decidability, or more generally the complexity, of questions about recursive nets.


  9. E.D. Sontag. Control of systems without drift via generic loops. IEEE Trans. Automat. Control, 40(7):1210-1219, 1995. [PDF] Keyword(s): controllability.
    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.


  10. E.D. Sontag. On the input-to-state stability property. European J. Control, 1:24-36, 1995. [PDF] Keyword(s): input to state stability.
    Abstract:
    The "input to state stability" (ISS) property provides a natural framework in which to formulate notions of stability with respect to input perturbations. In this expository paper, we review various equivalent definitions expressed in stability, Lyapunov-theoretic, and dissipation terms. We sketch some applications to the stabilization of cascades of systems and of linear systems subject to control saturation.


  11. E.D. Sontag and A.R. Teel. Changing supply functions in input/state stable systems. IEEE Trans. Automat. Control, 40(8):1476-1478, 1995. [PDF] Keyword(s): input to state stability, Lyapunov functions.
    Abstract:
    We consider the problem of characterizing possible supply functions for a given dissipative nonlinear system, and provide a result that allows some freedom in the modification of such functions.


  12. E.D. Sontag and Y. Wang. On characterizations of the input-to-state stability property. Systems Control Lett., 24(5):351-359, 1995. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(94)00050-6] Keyword(s): input to state stability.
    Abstract:
    We show that the well-known Lyapunov sufficient condition for input-to-state stability is also necessary, settling positively an open question raised by several authors during the past few years. Additional characterizations of the ISS property, including one in terms of nonlinear stability margins, are also provided.


  13. 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.
    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.


Conference articles
  1. E.D. Sontag. An abstract approach to dissipation. In Proc. IEEE Conf. Decision and Control, New Orleans, Dec. 1995, IEEE Publications, 1995, pages 2702-2703, 1995. Note: Full version, never submitted, is here: http://www.math.rutgers.edu/(tilde)sontag/FTPDIR/dissipation.pdf. [PDF]
    Abstract:
    We suggest that a very natural mathematical framework for the study of dissipation -in the sense of Willems, Moylan and Hill, and others- is that of indefinite quasimetric spaces. Several basic facts about dissipative systems are seen to be simple consequences of the properties of such spaces. Quasimetric spaces provide also one natural context for optimal control problems, and even for "gap" formulations of robustness.


  2. E.D. Sontag. Critical points for neural net least-squares problems. In Proc. 1995 IEEE Internat. Conf. Neural Networks, IEEE Publications, 1995, pages 2949-2954, 1995. Keyword(s): neural networks.


  3. E.D. Sontag. From linear to nonlinear: some complexity comparisons. In Proc. IEEE Conf. Decision and Control, New Orleans, Dec. 1995, IEEE Publications, 1995, pages 2916-2920, 1995. [PDF] Keyword(s): theory of computing and complexity, computational complexity, controllability, observability.
    Abstract:
    This paper deals with the computational complexity, and in some cases undecidability, of several problems in nonlinear control. The objective is to compare the theoretical difficulty of solving such problems to the corresponding problems for linear systems. In particular, the problem of null-controllability for systems with saturations (of a "neural network" type) is mentioned, as well as problems regarding piecewise linear (hybrid) systems. A comparison of accessibility, which can be checked fairly simply by Lie-algebraic methods, and controllability, which is at least NP-hard for bilinear systems, is carried out. Finally, some remarks are given on analog computation in this context.


  4. E.D. Sontag. Spaces of observables in nonlinear control. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Basel, pages 1532-1545, 1995. Birkhäuser. [PDF] Keyword(s): observability, dynamical systems.
    Abstract:
    Invited talk at the 1994 ICM. Paper deals with the notion of observables for nonlinear systems, and their role in realization theory, minimality, and several control and path planning questions.


  5. E.D. Sontag and H.J. Sussmann. Nonsmooth control-Lyapunov functions. In Proc. IEEE Conf. Decision and Control, New Orleans, Dec. 1995, IEEE Publications, 1995, pages 2799-2805, 1995. [PDF] Keyword(s): control-Lyapunov functions.
    Abstract:
    It is shown that the existence of a continuous control-Lyapunov function (CLF) is necessary and sufficient for null asymptotic controllability of nonlinear finite-dimensional control systems. The CLF condition is expressed in terms of a concept of generalized derivative (upper contingent derivative). This result generalizes to the non-smooth case the theorem of Artstein relating closed-loop feedback stabilization to smooth CLF's. It relies on viability theory as well as optimal control techniques. A "non-strict" version of the results, analogous to the LaSalle Invariance Principle, is also provided.


  6. E.D. Sontag and Y. Wang. On characterizations of input-to-state stability with respect to compact sets. In Proceedings of IFAC Non-Linear Control Systems Design Symposium, (NOLCOS '95), Tahoe City, CA, June 1995, pages 226-231, 1995. [PDF] Keyword(s): input to state stability.
    Abstract:
    Previous characterizations of ISS-stability are shown to generalize without change to the case of stability with respect to sets. Some results on ISS-stabilizability are mentioned as well.


  7. E.D. Sontag and Y. Wang. Various results concerning set input-to-state stability. In Proc. IEEE Conf. Decision and Control, New Orleans, Dec. 1995, IEEE Publications, 1995, pages 1330-1335, 1995. Keyword(s): input to state stability.



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