Linear estimation in Krein Spaces Applications

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Linear estimation in Krein Spaces/Applications/

• Janusz Klejsa,
• Optimal Filtering, Special Topic Presentation
• 2008-11-06

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Outline

• Background
• H8 estimation
• Example Suboptimal solution to H8 filtering
  problem
• Solution to H8 estimation problem in Krein Space
• Formulating the problem in Krein Space
• Solution by projections in the Krein Space a
  Kalman filter
• Comparison to the conventional Kalman filter
• Risk sensitive estimation
• Optimal risk sensitive estimator
• Connection to the H8 estimation
• A brief summary

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H8 estimation (1)

• State space model transfer operator

• Context
• Estimate
• i.e. find
• Definition
• The H8 of a transfer operator is defined
  as
• where is the h2-norm of the causal
  sequence

• Interpretation
• H8 norm corresponds to the maximum energy gain
  from the input to the output

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H8 estimation (2)

• Optimal H8 problem (filtering prediction)
• Find H8 optimal estimation strategies
• and
  that respectively minimize
• and and obtain the
  resulting
  ,
• .
• Example filtering
• Sub-obtimal H8 problem
• Given scalars find H8
  sub-optimal estimation
• strategies (a posteriori
  filter)
• and (a priori filter), that
  respectively
• achieve and
  .

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Solution to the suboptimal H8 filtering problem
(1)

• First step associate an indefinite quadratic
  form with the filtering problem (a posteriori or
  a priori)
• Second step construct a Krein Space state-space
  model, derive a Kalman filter in the Krein Space
• Indefinite quadratic form (example a posteriori
  filter)
• Krein-Space state-space model

indefinite matrix
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Solution to the suboptimal H8 filtering problem
(2)

• H8 a posteriori filtering problem
• For a given if have full rank,
  then the estimator that achieves
  exists iff
• where and satisfies
  the Riccati recursion
• with
• If this is the case one possible level-
  filter is given by

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Existence condition

• Goal to check whether minimum exists for
• for
• We assume that have full rank
• The minimum exists iff

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Comparison to the conventional Kalman filter

• H2-Kalman the estimate of any linear combination
  of the state is given by that linear combination
  of the state estimates
• H8 estimators depend on the linear combinations
  of the states (via Riccati recursion)
• H2-Kalman covariance mat. of the estimation
  error is positive definite
• H8 additional condition must be satisfied for
  the filter to exist
• H2-Kalman positive definite covariance matrices
• H8 indefinite covariance matrices
• As the Riccati recursion reduces
  to the Kalman filter recursion
• gt H2-Kalman filter performs poorly w.r.t. H8
  criterion
• H2-smoother is one of the H8 smoothers (no
  dependence on )
• Quadratic forms in H8 need not always have
  minima or maxma

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Risk sensitive estimation (1)

• State space model
• Context estimate from the
  observations
• Conventional estimators (risk-neutral)
• where
• Risk-sensitive estimator
• risk-seeking estimator,
  risk-averse estimator

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Risk sensitive estimation (2)

• Solution introduce an auxiliary Krein space
  model corresponding to the (possibly indefinite)
  quadratic form
• The derivation of the filters follows the same
  derivation as in the case of H8 filters

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Summary

• Conventional H2 Kalman algorithms can be
  extended to the H8 setting
• Projections in the Krein space allows to derive
  H8 estimators
• Risk-sensitive estimation can be considered in a
  similar framework to the H8 estimation

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References

• 1 B. Hassibi, A.H. Sayed and T. Kailath,
  Linear estimation in Krein spaces - part I
  Theory, IEEE Transactions on Automatic Control,
  vol. 41, no. 1, pp. 18-33, Jan. 1996.
• 2 B. Hassibi, A.H. Sayed and T. Kailath, Linear
  estimation in Krein spaces - part II
  Applications, IEEE Transactions on Automatic
  Control, vol. 41, no. 1, pp. 34-49, Jan. 1996.

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Appendix Krein Spaces (1)

• Definition
• An abstract vector space
  that satisfies the following requirements is
  called a Krein Space
• is a linear space over , the complex
  numbers
• there exist a bilinear form
  on such as
• (i)
• (ii)
• for any
• the vector space admits a direct orthogonal
  sum decomposition
• such that and
  are Hilbert spaces
• and for any
  and .
• A Hilbert Space satisfies additionally

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Appendix Krein Spaces (2)

• Some properties and a relation to Hilbert Spaces
• Krein Space is an indefinite metric space
• Projections
• In Hilbert Space projections always exist and
  always are unique
• In Krein Space projections exist iff a certain
  Gramm matrix is nonsingular
• Behavior of quadratic forms
• In Hilbert Space maxima (or minima) always exist
• In Krein Space one can assert that the quadratic
  forms have stationary points
• Nonsingularity and strict positive definiteness
• In Hilbert Space the above are equivalent
• In Krein Space the above are not necessarily
  equivalent.