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We developed a novel state estimation framework for noisy systems, offering continuous time estimates beyond discrete measurements. This maximum likelihood approach generalizes Kalman filtering, improving data accuracy and performance.

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Area of Science:

  • Control Theory
  • Signal Processing
  • Applied Mathematics

Background:

  • State estimation is crucial for systems with continuous dynamics and discrete measurements.
  • Existing methods like Kalman filtering have limitations due to assumptions on linearity and Gaussian noise.

Purpose of the Study:

  • To develop a general state estimation framework applicable to systems with noise-polluted continuous time dynamics and discrete time noisy measurements.
  • To overcome limitations of traditional methods by not assuming linear mappings or Gaussian noise distributions.

Main Methods:

  • Utilizes maximum likelihood estimation.
  • Employs the calculus of variations to derive optimality conditions for continuous time functions.
  • Interprets the optimal solution as a continuous time spline.

Main Results:

  • The framework provides a generalized approach beyond Kalman filtering/smoothing.
  • The optimal solution is a continuous time spline whose properties depend on system dynamics and noise distributions.
  • Achieves increased data accuracy and provides continuous estimates between measurements.
  • Demonstrates significant performance improvement over existing methods in simulations.

Conclusions:

  • The proposed framework offers a more general and powerful tool for state estimation.
  • The spline-based solution enhances accuracy and provides continuous temporal estimation.
  • The approach shows broad applicability to both linear and nonlinear systems.