A probabilistic likelihood approach to trajectory sensitivity

  • Authors:

  • Y. Hontoir;J. B. Cruz, Jr.

  • Affiliations:

  • Traction et Electricite, rue de la Science 31, 1040 Brussels, Belgium;Coordinated Science Laboratory and Department of Electrical Engineering, University of Illinois, Urbana, Illinois 61801, U.S.A.

  • Venue:
  • Automatica (Journal of IFAC)
  • Year:
  • 1974

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Abstract

Comparative trajectory sensitivity is investigated with respect to small probabilistic perturbations in initial state and plant parameters when a neighboring feedback control rather than the nominally equivalent open-loop control is used. The uncertainty in initial state and plant parameters is modeled by jointly normally distributed random variables with known mean and known covariance matrix. The nominal control is assumed to yield a satisfactory nominal trajectory and it is desired to preserve this shape in non-nominal situations. The likelihood of the state provides a measure of insensitivity at time t. Using the system equations linearized around the nominal trajectory, the joint distributions of the incremental state are computed in both open-loop and closed-loop cases. It is established that the augmented nominal state when a neighboring optimal feedback control is used is at least as likely as that when the nominally equivalent open-loop control is used. The domain where the augmented incremental state for the optimal closed-loop control is more densely distributed than that corresponding to open-loop control is shown to be a hyper-hyperboloid. This domain is unbounded in certain directions, thus pointing out a potential disadvantage in using optimal feedback control.