1. 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]



2. 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.




3. 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.




4. E.D. Sontag. Sigmoids distinguish more efficiently than Heavisides. Neural Computation, 1:470-472, 1989. [PDF] Keyword(s): machine learning, neural networks, boolean systems. Abstract:

   Every dichotomy on a 2k-point set in Rn can be implemented by a neural net with a single hidden layer containing k sigmoidal neurons. If the neurons were of a hardlimiter (Heaviside) type, 2k-1 would be in general needed.




5. 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. Abstract:

   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.)




6. E.D. Sontag and H.J. Sussmann. Backpropagation can give rise to spurious local minima even for networks without hidden layers. Complex Systems, 3(1):91-106, 1989. [PDF] Keyword(s): machine learning, neural networks. Abstract:

   We give an example of a neural net without hidden layers and with a sigmoid transfer function, together with a training set of binary vectors, for which the sum of the squared errors, regarded as a function of the weights, has a local minimum which is not a global minimum. The example consists of a set of 125 training instances, with four weights and a threshold to be learnt. We do not know if substantially smaller binary examples exist.




7. E.D. Sontag and H.J. Sussmann. Further comments on the stabilizability of the angular velocity of a rigid body. Systems Control Lett., 12(3):213-217, 1989. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(89)90052-2] Keyword(s): satellite control, feedback stabilization. Abstract:

   We prove that the angular velocity equations can be smoothly stabilized with a single torque controller for bodies having an axis of symmetry. This complements a recent result of Aeyels and Szafranski.




8. E.D. Sontag and Y. Yamamoto. On the existence of approximately coprime factorizations for retarded systems. Systems Control Lett., 13(1):53-58, 1989. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(89)90020-0] Keyword(s): delay-differential systems. Abstract:

   This note establishes a result linking algebraically coprime factorizations of transfer matrices of delay systems to approximately coprime factorizations in the sense of distributions. The latter have been employed by the second author in the study of function-space controllability for such systems.




9. Y. Wang and E.D. Sontag. On two definitions of observation spaces. Systems Control Lett., 13(4):279-289, 1989. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(89)90116-3] Keyword(s): observability, identifiability, observables, observation space, realization theory. Abstract:

   This paper establishes the equality of the observation spaces defined by means of piecewise constant controls with those defined in terms of differentiable controls.

1. E.D. Sontag. Remarks on stabilization and input-to-state stability. In Proceedings of the 28th IEEE Conference on Decision and Control, Vol. 1--3 (Tampa, FL, 1989), New York, pages 1376-1378, 1989. IEEE. [PDF] Keyword(s): input to state stability, ISS, stabilization. Abstract:

   This paper describes how notions of input-to-state stabilization are useful when stabilizing cascades of systems. The simplest result along these lines is local, and it states that a cascade of two locally asymptotically stable systems is again asystable. A global result is obtained if both systems have the origin as a globally asymptotically stable state and the "converging input bounded state" property holds for the second system. Relations to input to state stability and the "bounded input bounded state" property as mentioned as well.




2. E.D. Sontag. Remarks on the time-optimal control of a class of Hamiltonian systems. In Proceedings of the 28th IEEE Conference on Decision and Control, Vol. 1--3 (Tampa, FL, 1989), New York, pages 217-221, 1989. IEEE. [PDF] Keyword(s): robotics, optimal control. Abstract:

   This paper introduces a subclass of Hamiltonian control systems motivated by mechanical models. It deals with time-optimal control problems. The main results characterize regions of the state space where singular trajectories cannot exist, and provide high-order conditions for optimality.




3. E.D. Sontag. Some connections between stabilization and factorization. In Proceedings of the 28th IEEE Conference on Decision and Control, Vol. 1--3 (Tampa, FL, 1989), New York, pages 990-995, 1989. IEEE. [PDF] Abstract:

   Coprime right fraction representations are obtained for nonlinear systems defined by differential equations, under assumptions of stabilizability and detectability. A result is also given on left (not necessarily coprime) factorizations.




4. E.D. Sontag. Some recent results on nonlinear feedback. In Proc. Conf. Info. Sciences and Systems, Johns Hopkins University Press, 1989, pages 151-156, 1989.



5. E.D. Sontag and H.J. Sussmann. Backpropagation Separates when Perceptrons Do. In Proc. IEEE Int. Conf. Neural Networks, Washington, DC, June 1989, pages 639-642, 1989. [PDF]



6. E.D. Sontag and H.J. Sussmann. Remarks on local minima in backpropagation. In Proc. Conf. Info. Sciences and Systems, Johns Hopkins University Press, 1989, pages 432-435, 1989. Keyword(s): machine learning, neural networks.



7. Y. Wang and E.D. Sontag. A new result on the relation between differential-algebraic realizability and state space realizations. In Proc. Conf. Info. Sciences and Systems, Johns Hopkins University Press, 1989, pages 143-147, 1989. Keyword(s): observables, observation space, identifiability, observability, realization theory.



8. 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.

1. E.D. Sontag. Sigmoids distinguish more efficiently than Heavisides. Technical report SYCON-89-12, Rutgers Center for Systems and Control, 1989. Keyword(s): machine learning, neural networks.

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