1. E.D. Sontag. An introduction to the stabilization problem for parametrized families of linear systems. In Linear algebra and its role in systems theory (Brunswick, Maine, 1984), volume 47 of Contemp. Math., pages 369-400. Amer. Math. Soc., Providence, RI, 1985. [PDF] Keyword(s): systems over rings, systems over rings, parametric classes of systems. Abstract:

   This paper provides an introduction to definitions and known facts relating to the stabilization of parametrized families of linear systems using static and dynamic controllers. New results are given in the rational and polynomial cases.




2. B.W. Dickinson and E.D. Sontag. Dynamic realizations of sufficient sequences. IEEE Trans. Inform. Theory, 31(5):670-676, 1985. [PDF] Keyword(s): realization theory, statistics, innovations, sufficient statistics. Abstract:

   Let Ul, U2, ... be a sequence of observed random variables and (T1(U1),T2(Ul,U2),...) be a corresponding sequence of sufficient statistics (a sufficient sequence). Under certain regularity conditions, the sufficient sequence defines the input/output map of a time-varying, discrete-time nonlinear system. This system provides a recursive way of updating the sufficient statistic as new observations are made. Conditions are provided assuring that such a system evolves in a state space of minimal dimension. Several examples are provided to illustrate how this notion of dimensional minimality is related to other properties of sufficient sequences. The results can be used to verify the form of the minimum dimension (discrete-time) nonlinear filter associated with the autoregressive parameter estimation problem.




3. E.D. Sontag. Real addition and the polynomial hierarchy. Inform. Process. Lett., 20(3):115-120, 1985. [PDF] Abstract:

   The k-th alternation level of the theory of real numbers under addition and order is log-complete for the k-th level of the polynomial hierarchy.

1. E.D. Sontag. Further results on accessibility under sampling. In Proc.Conf. Info. Sci. and Systems, Johns Hopkins University, March 1985, 1985.



2. E.D. Sontag and H.J. Sussmann. Image restoration and segmentation using the annealing algorithm. In Proc. IEEE Conf. Dec. and Control, 1985, pages 768-773, 1985. [PDF] Keyword(s): image processing, optimization. Abstract:

   We consider the problem of estimating a signal, which is known -- or assumed -- to be constant on each of the members of a partition of a square lattice into m unknown regions, from the observation of the signal plus Gaussian noise. This is a nonlinear estimation problem, for which it is not appropriate to use the conditional expectation as the estimate. We show that, at least in principle, the "maximum iikelihood estimator" (MLE) proposed by Geman and Geman lends itself to numerical computation using the annealing algorithm. We argue that the MLE by itself can be, under certain conditions (low signal to noise ratio), a very unsatisfactory estimator, in that it does worse than just deciding that the signal was zero. However, if combined with a rule which we propose, for deciding when to use and when to ignore it, the MLE can provide a reasonable suboptimal estimator. We then discuss preliminary numerical data obtained using the annealing method. These results indicate that: (a) the annealing algorithm performs remarkably well, and (b) a criterion can be formulated in terms of quantities computed from the observed image (without using a priori knowledge of the signal-to-noise ratio) for deciding when to keep the MLE.




3. E.D. Sontag and H.J. Sussmann. Remarks on the time-optimal control of two-link manipulators. In Proc. IEEE Conf. Dec. and Control, 1985, pages 1646-1652, 1985. [PDF] Keyword(s): optimal control, robotics.

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