Robust Extended Kalman Filtering

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Robust Extended Kalman Filtering

• Soheil Salehpour

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Topics

• The Extended Kalman Filter (EKF)
• solution for a linear filtering problem
• Extended filter
• Application
• Conclusion and viewpoint

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1. The Extended Kalman Filter (EKF)

• The nonlinear terms are approximated by first
  order a Taylor series
• 
  
• 
  (1)
• It can be expanded in a Taylor series about
  and
• 
  
  (2)
• 
  
  (3)

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• The higher order terms
  set to zero
• We use a correlated noise argument of the
  Kalman filter
• 
  
• 
  
• Fundamental assumptions in the derivation of the
  Kalman filter
• The system is described by a linear
  state-space
• The noise terms are white and Gaussian with
  zero-mean
• Obs! When the system is nonlinear, the first
  condition is violated.

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2. solution for a linear filtering problem

• The linear problem
• (9)
• Minimize the maximum
  overall input noise
• A solution to the filtering problem is as usual
  Kalman,exept the following equations for
  covariance
• A minimum value for needs to be found by
  searching over such that

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3. Extended filter

• An extended filter defined by
• Satisfies and
• An auxiliary problem defined
• Where
• Proof
• implies

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5. Application

• Figure 1 The histogram of the frequency
  estimation for .(-. ) Extended
  filter with ?1,)
  .(__) Extended filter with ?1,
  , .(---)EKT
• (Example 1)

• Example 1 The FM model
• Example 2 A nonlinear channel as follows

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• Figure 2 The histogram of the frequency
  estimation for .(___) Extended filter
  with ?1.4.(---)EKT
• (Example 1)

• Figure 3 The result of simulations with ?20,
  
  .
  And for

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6. Conclusion and viewpoint

• 1. The benefit result is obtained when the
  problem is sufficiently nonlinear.
• 2. When the state dynamics are linear the
  conservation of yields degraded performance
• 3. Some parameters should be tuned
• 4. The computational over heads are not
  significant affected Since the order of the
  problem remains changed
• Criticism
• The can consider zero in trial and
  error.
• The Result of Example 2 in the paper demonstrates
  for , ,implies
  which is like the solution
  for a linear filtering problem