Elsevier

Information Sciences

Volume 1, Issue 1, December 1968, Pages 55-85
Information Sciences

On optimal stochastic control with smoothed information

https://doi.org/10.1016/0020-0255(68)90007-8Get rights and content

Abstract

This paper presents a generalization of the Separation Theorem of stochastic control. The generalization consists in assuming observations ahead of running time. We consider the following problem of optimal pursuit: Given noisy incomplete observations of a linear stochastic system, control another linear stochastic system so that a quadratic functional of the difference (in some generalized sense) between the two processes is minimized. Provided that we have access to observations only up to the time of control, the Separation Theorem states that the solution to this problem is given by a linear combination of the Kalman filtering estimates. Now suppose instead that we have observations ahead of running time. (Consider for instance an airplane with terrain following radar.) Then the basic result of this paper is that a weighted integral of the minimum variance smoothing estimate should be included in the optimal control. This result is obtained by use of Fejér kernels and enlargement of the state space. Finally we provide proof of optimality.

References (15)

  • N.I. Achieser

    Vorlesungen über Approximationstheorie

    (1953)
  • A.E. Bryson et al.

    Smoothing for Linear and Nonlinear Dynamic Systems

  • J.L. Doob

    Stochastic Processes

    (1953)
  • R.E. Kalman et al.

    New results in linear filtering and prediction theory

    J. Basic Eng.

    (March 1961)
  • A.N. Kolmogorov et al.

    Measure, Lebesgue Integrals, and Hilbert Space

    (1961)
  • M. Loève

    Probability Theory

    (1963)
  • J.S. Meditch

    Orthogonal projection and discrete optimal linear smoothing

    SIAM J. on Control

    (Feb. 1967)
There are more references available in the full text version of this article.

Cited by (10)

  • Kriging filters for space-time interpolation

    2002, Advances in Imaging and Electron Physics
    Citation Excerpt :

    This algorithm is best for off-line use after all observations have been collected. We refer to this as temporal smoothing, which is the terminology adopted by the Kalman filtering community (e.g., Lindquist, 1968). Then, minimizing the error variance can be accomplished by individually mini-mizing both terms in this equation.

  • The separation principle in stochastic control, redux

    2013, IEEE Transactions on Automatic Control
  • Revisiting the separation principle in stochastic control

    2012, Proceedings of the IEEE Conference on Decision and Control
  • A framework for discrete-time H<inf>2</inf> preview control

    2010, Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME
View all citing articles on Scopus
View full text