Mark L. Psiaki,

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Kalman Filtering Smoothing to Estimate
Real-Valued States Integer Constants

• Mark L. Psiaki,
• Sibley School of Mechanical Aerospace
  EngineeringCornell University

2
Goal

• Improve estimation algorithms for systems that
  have integer measurement ambiguities
• CDGPS with double-differenced integer ambiguities
• Systems using carrier-phase measurements of TDMA
  signals

Strategies

• Use SRIF/LAMBDA-type formulation to deal with
  mixed real/integer problem
• Develop optimal suboptimal Kalman filter
  smoother algorithms
• Optimal keep all ambiguities treat as integers
• Suboptimal retain integers in a finite time
  window

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Outline of Talk

• Related research
• Problem definition
• Mixed real/integer Kalman filter
• Optimal, retains all past integers
• Suboptimal, retains finite window of past
  integers
• Mixed real/integer fixed-interval smoother
• Optimal, retains all integers of fixed interval
• Suboptimal, retains finite window of past
  future integers relative to each time point
• Truth-model simulation results
• Conclusions

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Related Research

• Batch estimation w/integer ambiguities
• The LAMBDA method, Teunissen (1995) follow-ons
• Other methods, e.g., Chen Lachapelle (1995)
• SRIF LAMBDA-like method, Psiaki Mohiuddin
  (2007)
• Kalman filtering w/integer ambiguities
• Standard Covariance EKF, Kroes et al. (2005)
• SRIF-based EKF, Mohiuddin Psiaki (2008)
• Sub-optimal dropping of each integer ambiguity
  immediately after its last occurrence in a
  measurement
• Smoothing w/integer ambiguities
• Nothing

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Dynamics Model
Real-state dynamics
Growth of integer state with sample number
Partitioning of integer states by affected
measurement sample times (past, past present,
past, present future)
Or dynamic re-partitioning
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Measurement Model
using integer vector partitions
using full integer vector
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Example Sensitivities of Different Measurement
Types to Different Integers
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Kalman Filtering/Smoothing Problem

• find x0, , xk1, w0, , wk, nk1 dn0
  dnk
• to minimize
• subject to xj1 Fjxj Gjwj hj for j 0,
  1, 2, ..., k nk1 is an integer-valued vector

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Optimal SRIF Kalman Filter

• Stage-k a posterior info
• Combined information eqs. w/dynamics substitution
  for xk
• New stage-(k1) a posterior info after QR
  factorization

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Measurement Update via Integer Linear
Least-Squares Solution

• Solve integer linear least-squares problem to
  determine integer a posteriori estimate
• Back-substitute to compute real-valued states

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Suboptimal KF Retention of Exact Integers within
a Window of Samples
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Suboptimal SRIF Kalman Filter

• Stage-k a posterior info
• Combined information eqs. w/dynamics substitution
  for xk mk
• New stage-(k1) a posterior info after QR
  factorization

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Optimal RTS Smoother in SRIF Form

• Terminal sample K initialization
• 1-sample backwards recursion starts w/filtered wk
  smoothed xk1 info. eqs. uses dynamics to get
• QR factorize to isolate smoothed xk info. eq.

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Suboptimal RTS Smoother Retention of Exact
Integers within a Window of Samples
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Suboptimal RTS Smoother (1 of 2)

• Terminal sample K initialization
• 1-sample backwards recursion starts w/filtered wk
  Dnk-i smoothed xk1 lk1 info. eqs. uses
  dynamics integer permutation/partitions to get

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Suboptimal RTS Smoother (2 of 2)

• New stage-k smoothed xk lk square-root
  information equations after QR factorization
• is the integer vector that minimizes
• The real part of the state is determined by back
  substitution

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Example 1-Dimensional CDGPS-Type Problem with
3rd-Order Dynamics

• Dynamics
• Measurements

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x1 Errors for Three Kalman Filters
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x1 Errors for Three Smoothers
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Integer-Part Computational Cost of Optimal
Suboptimal Algorithms
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Summary Conclusions

• Developed optimal suboptimal Kalman filters
  fixed-interval smoothers for mixed real/integer
  estimation problems
• Constant integer ambiguities enter only
  measurements
• Optimal algorithms consider all integers in data
  batch
• Suboptimal algorithms drop integers that affect
  measurements only in remote past or future
• Tested using data from truth-model simulation
• Optimal suboptimal filter achieve modest
  accuracy gains vs. all-reals approximate filter
• Filter accuracy gains may be greater for
  different problem
• Optimal suboptimal smoother significantly more
  accurate than all-reals smoother
• Suboptimal smoother nearly as accurate as optimal
  smoother
• Suboptimal algorithms reduce required processing
  by at least 65 through reductions in dimensions
  of measurement update integer linear
  least-squares problems