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Filtering and Stochastic Control

Contributed by Ljubo Vlacic.
A comprehensive survey of linear filtering theory can be found in [75].
  • H. W. Bode and C. E. Shannon proposed the solution to the problem of prediction and smoothing ([76]). A modern account of the solution can be found in [77] and more detailed treatment of the ideas are presented in [78] [79].

  • 1960
  • R. E. Kalman ([81],[82],[83]) made explicit that an effective solution to the Wiener-Hopf equation using method of spectral factorization ([80]) could be obtained when the continuous process had a rational spectral density.
  • Stratanovich derived the conditional density equation using the so-called Stratanovich calculus ([111]).

  • 1960-63
  • The theory of optimal stochastic control in the fully observable case is quite similar to that of non-linear filtering in connection with the linear quadratic stochastic control problem ([79]). Early works in this area are due to Howard ([121]), Florentin ([91]), and Fleming ([122]); See also [123].

  • 1960-64
  • Inspired by the development of Dynamic Programming by Bellman ([85]) and the ideas of Caratheodory ([86]) related to Hamilton-Jacobi Theory, the development of optimal control of nonlinear dynamical systems took place ([87], [88]), see [89], [84], [90] for further details of the ideas.

  • 1961-1973
  • The solution to quadratic cost optimal control for linear stochastic dynamical systems was provided by Florentin ([91], [92]), by Joseph in discrete-time ([93]), and by Kushner ([94]). The definitive treatment of the problem was proposed by Wonham ([95]), see also [96].

  • 1962
  • The partially observable stochastic control problem treated by Florentin ([92]), Davis and Varayia ([125]) and Fleming and Pardouz ([126]). Detailed discussions can be found in [127] and the references therein.

  • 1964
  • For a good discussion on the distinction between open-loop stochastic control and feedback control see [99].

  • 1965
  • Non-linear filters are almost always infinite dimensional and there are only a few known examples where the filter is known to be a finite dimension. The Kalman filter is an example and the other finite-state cases are first discussed in [104] [114] and also [112] [113].

  • 1967-79
  • A difficulty is that one of the fundamental equations of non-linear filtering turns out to be a non-linear stochastic partial differential equation ([79]). Zakai ([105]), Duncan ([106]), and Mortensen ([107]) proposed alternative solutions to the above difficulty which involves a linear stochastic differential equation.

  • 1971
  • Giransov introduced the idea of measure transformation in stochastic differential equation, see [79], [110], [115] and the references therein for details.

  • 1971-72
  • The earlier ideas of nonlinear filtering were developed and introduced by Forest and Kailath ([100]), and in definitive form by Fujisaki, Kallianpur, Kunita ([101]).

  • 1976
  • Bobrovsky and Zakai proposed a method for obtaining lower bounds on the mean-squared error ([119]).

  • 1978
  • As an attempt to address some of the issues with non-linear filtering, pathwise non-linear filtering was considered where the filter depends continuously on the output ([117], [118]).

  • 1996
  • The Linear Quadratic Gaussian methodology and optimal non-linear stochastic control have found a wide variety of applications in aerospace, multi-variable control design systems, finance, etc. ([115], [128]).

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